Integrand size = 37, antiderivative size = 791 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \left (b c^2-a d^2\right ) \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a-b x^2}}{d^6 \sqrt {c+d x}}-\frac {2 \left (180 a^2 d^4 D+3 a b d^2 \left (1166 c C d-495 B d^2-1929 c^2 D\right )-b^2 c \left (6710 c^2 C d-4653 B c d^2+2772 A d^3-8895 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{3465 b d^6}+\frac {2 \left (a d^2 (847 C d-2733 c D)-b \left (4840 c^2 C d-2277 B c d^2+693 A d^3-8430 c^3 D\right )\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}{3465 d^6}+\frac {2 \left (117 a d^2 D+b \left (418 c C d-99 B d^2-1086 c^2 D\right )\right ) (c+d x)^{5/2} \sqrt {a-b x^2}}{693 d^6}-\frac {2 b (11 C d-57 c D) (c+d x)^{7/2} \sqrt {a-b x^2}}{99 d^6}-\frac {2 b D (c+d x)^{9/2} \sqrt {a-b x^2}}{11 d^6}-\frac {8 \sqrt {a} \left (3 a^2 d^4 (77 C d-123 c D)-3 a b d^2 \left (1166 c^2 C d-957 B c d^2+693 A d^3-1344 c^3 D\right )+4 b^2 c^2 \left (880 c^2 C d-792 B c d^2+693 A d^3-960 c^3 D\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3465 \sqrt {b} d^7 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}+\frac {8 \sqrt {a} \left (b c^2-a d^2\right ) \left (45 a^2 d^4 D-3 a b d^2 \left (286 c C d-165 B d^2-384 c^2 D\right )+4 b^2 c \left (880 c^2 C d-792 B c d^2+693 A d^3-960 c^3 D\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3465 b^{3/2} d^7 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
2*(-a*d^2+b*c^2)*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(-b*x^2+a)^(1/2)/d^6/(d*x+c )^(1/2)-2/3465*(180*a^2*d^4*D+3*a*b*d^2*(-495*B*d^2+1166*C*c*d-1929*D*c^2) -b^2*c*(2772*A*d^3-4653*B*c*d^2+6710*C*c^2*d-8895*D*c^3))*(d*x+c)^(1/2)*(- b*x^2+a)^(1/2)/b/d^6+2/3465*(a*d^2*(847*C*d-2733*D*c)-b*(693*A*d^3-2277*B* c*d^2+4840*C*c^2*d-8430*D*c^3))*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/d^6+2/693*( 117*a*d^2*D+b*(-99*B*d^2+418*C*c*d-1086*D*c^2))*(d*x+c)^(5/2)*(-b*x^2+a)^( 1/2)/d^6-2/99*b*(11*C*d-57*D*c)*(d*x+c)^(7/2)*(-b*x^2+a)^(1/2)/d^6-2/11*b* D*(d*x+c)^(9/2)*(-b*x^2+a)^(1/2)/d^6-8/3465*a^(1/2)*(3*a^2*d^4*(77*C*d-123 *D*c)-3*a*b*d^2*(693*A*d^3-957*B*c*d^2+1166*C*c^2*d-1344*D*c^3)+4*b^2*c^2* (693*A*d^3-792*B*c*d^2+880*C*c^2*d-960*D*c^3))*(d*x+c)^(1/2)*((-b*x^2+a)/a )^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2) *d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(1/2)/d^7/((d*x+c)/(c+a^(1/2)*d/b^(1/2) ))^(1/2)/(-b*x^2+a)^(1/2)+8/3465*a^(1/2)*(-a*d^2+b*c^2)*(45*a^2*d^4*D-3*a* b*d^2*(-165*B*d^2+286*C*c*d-384*D*c^2)+4*b^2*c*(693*A*d^3-792*B*c*d^2+880* C*c^2*d-960*D*c^3))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^( 1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/ (b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(3/2)/d^7/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 34.26 (sec) , antiderivative size = 978, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:
Integrate[((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]
Output:
(2*Sqrt[a - b*x^2]*(3465*b*(b*c^2 - a*d^2)*(c^2*C*d - B*c*d^2 + A*d^3 - c^ 3*D) + (-180*a^2*d^4*D + a*b*d^2*(-2651*c*C*d + 1485*B*d^2 + 3639*c^2*D) + b^2*c*(3575*c^2*C*d - 2871*B*c*d^2 + 2079*A*d^3 - 4215*c^3*D))*(c + d*x) - b*d*(a*d^2*(-847*C*d + 1563*c*D) - 3*b*(-605*c^2*C*d + 429*B*c*d^2 - 231 *A*d^3 + 765*c^3*D))*x*(c + d*x) + 5*b*d^2*(117*a*d^2*D + b*(187*c*C*d - 9 9*B*d^2 - 267*c^2*D))*x^2*(c + d*x) - 35*b^2*d^3*(11*C*d - 21*c*D)*x^3*(c + d*x) - 315*b^2*d^4*D*x^4*(c + d*x) + (4*(d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[ b]]*(3*a^2*d^4*(-77*C*d + 123*c*D) + 3*a*b*d^2*(1166*c^2*C*d - 957*B*c*d^2 + 693*A*d^3 - 1344*c^3*D) + 4*b^2*c^2*(-880*c^2*C*d + 792*B*c*d^2 - 693*A *d^3 + 960*c^3*D))*(a - b*x^2) + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(3*a^2* d^4*(-77*C*d + 123*c*D) + 3*a*b*d^2*(1166*c^2*C*d - 957*B*c*d^2 + 693*A*d^ 3 - 1344*c^3*D) + 4*b^2*c^2*(-880*c^2*C*d + 792*B*c*d^2 - 693*A*d^3 + 960* c^3*D))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt [b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt [a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[ a]*d)] - I*Sqrt[a]*d*(Sqrt[b]*c - Sqrt[a]*d)*(45*a^2*d^4*D + 3*a^(3/2)*Sqr t[b]*d^3*(-77*C*d + 138*c*D) + 3*a*b*d^2*(-286*c*C*d + 165*B*d^2 + 384*c^2 *D) + 3*Sqrt[a]*b^(3/2)*d*(880*c^2*C*d - 792*B*c*d^2 + 693*A*d^3 - 960*c^3 *D) - 4*b^2*c*(-880*c^2*C*d + 792*B*c*d^2 - 693*A*d^3 + 960*c^3*D))*Sqrt[( d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/...
Time = 1.91 (sec) , antiderivative size = 889, normalized size of antiderivative = 1.12, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.432, Rules used = {2182, 27, 2185, 27, 682, 27, 682, 27, 27, 600, 509, 508, 327, 512, 511, 321}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {2 \int \frac {\left (a-b x^2\right )^{3/2} \left (\left (\frac {b c^2}{d}-a d\right ) D x^2-\left (a (C d-c D)+b \left (\frac {10 D c^3}{d^2}-\frac {10 C c^2}{d}+9 B c-9 A d\right )\right ) x+\frac {-a D c^2+A b d c+a C d c-a B d^2}{d}\right )}{2 \sqrt {c+d x}}dx}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (a-b x^2\right )^{3/2} \left (\left (\frac {b c^2}{d}-a d\right ) D x^2-\left (a (C d-c D)+b \left (\frac {10 D c^3}{d^2}-\frac {10 C c^2}{d}+9 B c-9 A d\right )\right ) x+A b c+a \left (-\frac {D c^2}{d}+C c-B d\right )\right )}{\sqrt {c+d x}}dx}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{11} D \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )-\frac {2 \int -\frac {\left (d \left (11 A b^2 c d-a \left (a d^2 D-b \left (-10 D c^2+11 C d c-11 B d^2\right )\right )\right )-b \left (a d^2 (11 C d-21 c D)-b \left (-120 D c^3+110 C d c^2-99 B d^2 c+99 A d^3\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{2 \sqrt {c+d x}}dx}{11 b d^2}}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (d \left (11 A b^2 c d-a \left (a d^2 D-b \left (-10 D c^2+11 C d c-11 B d^2\right )\right )\right )-b \left (a d^2 (11 C d-21 c D)-b \left (-120 D c^3+110 C d c^2-99 B d^2 c+99 A d^3\right )\right ) x\right ) \left (a-b x^2\right )^{3/2}}{\sqrt {c+d x}}dx}{11 b d^2}+\frac {2}{11} D \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {-\frac {4 \int -\frac {b \left (a d \left (b c^2-a d^2\right ) \left (9 a d^2 D-b \left (-120 D c^2+110 C d c-99 B d^2\right )\right )+b \left (b c^2-a d^2\right ) \left (a d^2 (77 C d-138 c D)-b \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{2 \sqrt {c+d x}}dx}{21 b d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a^2 d^4 D+7 b d x \left (a d^2 (11 C d-21 c D)-b \left (99 A d^3-99 B c d^2-120 c^3 D+110 c^2 C d\right )\right )-a b d^2 \left (-99 B d^2-258 c^2 D+187 c C d\right )+b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{63 d^2}}{11 b d^2}+\frac {2}{11} D \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {\left (a d \left (b c^2-a d^2\right ) \left (9 a d^2 D-b \left (-120 D c^2+110 C d c-99 B d^2\right )\right )+b \left (b c^2-a d^2\right ) \left (a d^2 (77 C d-138 c D)-b \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) x\right ) \sqrt {a-b x^2}}{\sqrt {c+d x}}dx}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a^2 d^4 D+7 b d x \left (a d^2 (11 C d-21 c D)-b \left (99 A d^3-99 B c d^2-120 c^3 D+110 c^2 C d\right )\right )-a b d^2 \left (-99 B d^2-258 c^2 D+187 c C d\right )+b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{63 d^2}}{11 b d^2}+\frac {2}{11} D \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) \left (45 a^2 d^4 D+3 b d x \left (a d^2 (77 C d-138 c D)-b \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )-3 a b d^2 \left (-165 B d^2-384 c^2 D+286 c C d\right )+4 b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{15 d^2}-\frac {4 \int -\frac {b \left (a d \left (b c^2-a d^2\right ) \left (45 a^2 D d^4-3 a b \left (-246 D c^2+209 C d c-165 B d^2\right ) d^2+b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right )+b \left (b c^2-a d^2\right ) \left (3 a^2 (77 C d-123 c D) d^4-3 a b \left (-1344 D c^3+1166 C d c^2-957 B d^2 c+693 A d^3\right ) d^2+4 b^2 c^2 \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 b d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a^2 d^4 D+7 b d x \left (a d^2 (11 C d-21 c D)-b \left (99 A d^3-99 B c d^2-120 c^3 D+110 c^2 C d\right )\right )-a b d^2 \left (-99 B d^2-258 c^2 D+187 c C d\right )+b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{63 d^2}}{11 b d^2}+\frac {2}{11} D \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {2 \int \frac {\left (b c^2-a d^2\right ) \left (a d \left (45 a^2 D d^4-3 a b \left (-246 D c^2+209 C d c-165 B d^2\right ) d^2+b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right )+b \left (3 a^2 (77 C d-123 c D) d^4-3 a b \left (-1344 D c^3+1166 C d c^2-957 B d^2 c+693 A d^3\right ) d^2+4 b^2 c^2 \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) x\right )}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) \left (45 a^2 d^4 D+3 b d x \left (a d^2 (77 C d-138 c D)-b \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )-3 a b d^2 \left (-165 B d^2-384 c^2 D+286 c C d\right )+4 b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a^2 d^4 D+7 b d x \left (a d^2 (11 C d-21 c D)-b \left (99 A d^3-99 B c d^2-120 c^3 D+110 c^2 C d\right )\right )-a b d^2 \left (-99 B d^2-258 c^2 D+187 c C d\right )+b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{63 d^2}}{11 b d^2}+\frac {2}{11} D \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {2 \left (b c^2-a d^2\right ) \int \frac {a d \left (45 a^2 D d^4-3 a b \left (-246 D c^2+209 C d c-165 B d^2\right ) d^2+b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right )+b \left (3 a^2 (77 C d-123 c D) d^4-3 a b \left (-1344 D c^3+1166 C d c^2-957 B d^2 c+693 A d^3\right ) d^2+4 b^2 c^2 \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) \left (45 a^2 d^4 D+3 b d x \left (a d^2 (77 C d-138 c D)-b \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )-3 a b d^2 \left (-165 B d^2-384 c^2 D+286 c C d\right )+4 b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a^2 d^4 D+7 b d x \left (a d^2 (11 C d-21 c D)-b \left (99 A d^3-99 B c d^2-120 c^3 D+110 c^2 C d\right )\right )-a b d^2 \left (-99 B d^2-258 c^2 D+187 c C d\right )+b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{63 d^2}}{11 b d^2}+\frac {2}{11} D \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {2 \left (b c^2-a d^2\right ) \left (\frac {b \left (3 a^2 d^4 (77 C d-123 c D)-3 a b d^2 \left (693 A d^3-957 B c d^2-1344 c^3 D+1166 c^2 C d\right )+4 b^2 c^2 \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (45 a^2 d^4 D-3 a b d^2 \left (-165 B d^2-384 c^2 D+286 c C d\right )+4 b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) \left (45 a^2 d^4 D+3 b d x \left (a d^2 (77 C d-138 c D)-b \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )-3 a b d^2 \left (-165 B d^2-384 c^2 D+286 c C d\right )+4 b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a^2 d^4 D+7 b d x \left (a d^2 (11 C d-21 c D)-b \left (99 A d^3-99 B c d^2-120 c^3 D+110 c^2 C d\right )\right )-a b d^2 \left (-99 B d^2-258 c^2 D+187 c C d\right )+b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{63 d^2}}{11 b d^2}+\frac {2}{11} D \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {2 \left (b c^2-a d^2\right ) \left (\frac {b \sqrt {1-\frac {b x^2}{a}} \left (3 a^2 d^4 (77 C d-123 c D)-3 a b d^2 \left (693 A d^3-957 B c d^2-1344 c^3 D+1166 c^2 C d\right )+4 b^2 c^2 \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (45 a^2 d^4 D-3 a b d^2 \left (-165 B d^2-384 c^2 D+286 c C d\right )+4 b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) \left (45 a^2 d^4 D+3 b d x \left (a d^2 (77 C d-138 c D)-b \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )-3 a b d^2 \left (-165 B d^2-384 c^2 D+286 c C d\right )+4 b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a^2 d^4 D+7 b d x \left (a d^2 (11 C d-21 c D)-b \left (99 A d^3-99 B c d^2-120 c^3 D+110 c^2 C d\right )\right )-a b d^2 \left (-99 B d^2-258 c^2 D+187 c C d\right )+b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{63 d^2}}{11 b d^2}+\frac {2}{11} D \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {\frac {2}{11} \left (\frac {a}{b}-\frac {c^2}{d^2}\right ) D \sqrt {c+d x} \left (a-b x^2\right )^{5/2}+\frac {\frac {2 \left (\frac {2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2} \left (45 a^2 D d^4-3 a b \left (-384 D c^2+286 C d c-165 B d^2\right ) d^2+3 b \left (a d^2 (77 C d-138 c D)-b \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) x d+4 b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right )}{15 d^2}+\frac {2 \left (b c^2-a d^2\right ) \left (-\frac {\left (b c^2-a d^2\right ) \left (45 a^2 D d^4-3 a b \left (-384 D c^2+286 C d c-165 B d^2\right ) d^2+4 b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^2 (77 C d-123 c D) d^4-3 a b \left (-1344 D c^3+1166 C d c^2-957 B d^2 c+693 A d^3\right ) d^2+4 b^2 c^2 \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \sqrt {c+d x} \left (9 a^2 D d^4-a b \left (-258 D c^2+187 C d c-99 B d^2\right ) d^2+7 b \left (a d^2 (11 C d-21 c D)-b \left (-120 D c^3+110 C d c^2-99 B d^2 c+99 A d^3\right )\right ) x d+b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{63 d^2}}{11 b d^2}}{b c^2-a d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {2 \left (b c^2-a d^2\right ) \left (-\frac {\left (b c^2-a d^2\right ) \left (45 a^2 d^4 D-3 a b d^2 \left (-165 B d^2-384 c^2 D+286 c C d\right )+4 b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (3 a^2 d^4 (77 C d-123 c D)-3 a b d^2 \left (693 A d^3-957 B c d^2-1344 c^3 D+1166 c^2 C d\right )+4 b^2 c^2 \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{15 d^2}+\frac {2 \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right ) \left (45 a^2 d^4 D+3 b d x \left (a d^2 (77 C d-138 c D)-b \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )-3 a b d^2 \left (-165 B d^2-384 c^2 D+286 c C d\right )+4 b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x} \left (9 a^2 d^4 D+7 b d x \left (a d^2 (11 C d-21 c D)-b \left (99 A d^3-99 B c d^2-120 c^3 D+110 c^2 C d\right )\right )-a b d^2 \left (-99 B d^2-258 c^2 D+187 c C d\right )+b^2 c \left (693 A d^3-792 B c d^2-960 c^3 D+880 c^2 C d\right )\right )}{63 d^2}}{11 b d^2}+\frac {2}{11} D \left (a-b x^2\right )^{5/2} \sqrt {c+d x} \left (\frac {a}{b}-\frac {c^2}{d^2}\right )}{b c^2-a d^2}+\frac {2 \left (a-b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 \sqrt {c+d x} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {\frac {2}{11} \left (\frac {a}{b}-\frac {c^2}{d^2}\right ) D \sqrt {c+d x} \left (a-b x^2\right )^{5/2}+\frac {\frac {2 \left (\frac {2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2} \left (45 a^2 D d^4-3 a b \left (-384 D c^2+286 C d c-165 B d^2\right ) d^2+3 b \left (a d^2 (77 C d-138 c D)-b \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) x d+4 b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right )}{15 d^2}+\frac {2 \left (b c^2-a d^2\right ) \left (-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^2 (77 C d-123 c D) d^4-3 a b \left (-1344 D c^3+1166 C d c^2-957 B d^2 c+693 A d^3\right ) d^2+4 b^2 c^2 \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (45 a^2 D d^4-3 a b \left (-384 D c^2+286 C d c-165 B d^2\right ) d^2+4 b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \sqrt {c+d x} \left (9 a^2 D d^4-a b \left (-258 D c^2+187 C d c-99 B d^2\right ) d^2+7 b \left (a d^2 (11 C d-21 c D)-b \left (-120 D c^3+110 C d c^2-99 B d^2 c+99 A d^3\right )\right ) x d+b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{63 d^2}}{11 b d^2}}{b c^2-a d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {\frac {2}{11} \left (\frac {a}{b}-\frac {c^2}{d^2}\right ) D \sqrt {c+d x} \left (a-b x^2\right )^{5/2}+\frac {\frac {2 \left (\frac {2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2} \left (45 a^2 D d^4-3 a b \left (-384 D c^2+286 C d c-165 B d^2\right ) d^2+3 b \left (a d^2 (77 C d-138 c D)-b \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) x d+4 b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right )}{15 d^2}+\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (45 a^2 D d^4-3 a b \left (-384 D c^2+286 C d c-165 B d^2\right ) d^2+4 b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^2 (77 C d-123 c D) d^4-3 a b \left (-1344 D c^3+1166 C d c^2-957 B d^2 c+693 A d^3\right ) d^2+4 b^2 c^2 \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \sqrt {c+d x} \left (9 a^2 D d^4-a b \left (-258 D c^2+187 C d c-99 B d^2\right ) d^2+7 b \left (a d^2 (11 C d-21 c D)-b \left (-120 D c^3+110 C d c^2-99 B d^2 c+99 A d^3\right )\right ) x d+b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{63 d^2}}{11 b d^2}}{b c^2-a d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 \left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (a-b x^2\right )^{5/2}}{d^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}}+\frac {\frac {2}{11} \left (\frac {a}{b}-\frac {c^2}{d^2}\right ) D \sqrt {c+d x} \left (a-b x^2\right )^{5/2}+\frac {\frac {2 \left (\frac {2 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2} \left (45 a^2 D d^4-3 a b \left (-384 D c^2+286 C d c-165 B d^2\right ) d^2+3 b \left (a d^2 (77 C d-138 c D)-b \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) x d+4 b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right )}{15 d^2}+\frac {2 \left (b c^2-a d^2\right ) \left (\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (45 a^2 D d^4-3 a b \left (-384 D c^2+286 C d c-165 B d^2\right ) d^2+4 b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \left (3 a^2 (77 C d-123 c D) d^4-3 a b \left (-1344 D c^3+1166 C d c^2-957 B d^2 c+693 A d^3\right ) d^2+4 b^2 c^2 \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}\right )}{15 d^2}\right )}{21 d^2}-\frac {2 \sqrt {c+d x} \left (9 a^2 D d^4-a b \left (-258 D c^2+187 C d c-99 B d^2\right ) d^2+7 b \left (a d^2 (11 C d-21 c D)-b \left (-120 D c^3+110 C d c^2-99 B d^2 c+99 A d^3\right )\right ) x d+b^2 c \left (-960 D c^3+880 C d c^2-792 B d^2 c+693 A d^3\right )\right ) \left (a-b x^2\right )^{3/2}}{63 d^2}}{11 b d^2}}{b c^2-a d^2}\) |
Input:
Int[((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(3/2),x]
Output:
(2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a - b*x^2)^(5/2))/(d^2*(b*c^2 - a* d^2)*Sqrt[c + d*x]) + ((2*(a/b - c^2/d^2)*D*Sqrt[c + d*x]*(a - b*x^2)^(5/2 ))/11 + ((-2*Sqrt[c + d*x]*(9*a^2*d^4*D - a*b*d^2*(187*c*C*d - 99*B*d^2 - 258*c^2*D) + b^2*c*(880*c^2*C*d - 792*B*c*d^2 + 693*A*d^3 - 960*c^3*D) + 7 *b*d*(a*d^2*(11*C*d - 21*c*D) - b*(110*c^2*C*d - 99*B*c*d^2 + 99*A*d^3 - 1 20*c^3*D))*x)*(a - b*x^2)^(3/2))/(63*d^2) + (2*((2*(b*c^2 - a*d^2)*Sqrt[c + d*x]*(45*a^2*d^4*D - 3*a*b*d^2*(286*c*C*d - 165*B*d^2 - 384*c^2*D) + 4*b ^2*c*(880*c^2*C*d - 792*B*c*d^2 + 693*A*d^3 - 960*c^3*D) + 3*b*d*(a*d^2*(7 7*C*d - 138*c*D) - b*(880*c^2*C*d - 792*B*c*d^2 + 693*A*d^3 - 960*c^3*D))* x)*Sqrt[a - b*x^2])/(15*d^2) + (2*(b*c^2 - a*d^2)*((-2*Sqrt[a]*Sqrt[b]*(3* a^2*d^4*(77*C*d - 123*c*D) - 3*a*b*d^2*(1166*c^2*C*d - 957*B*c*d^2 + 693*A *d^3 - 1344*c^3*D) + 4*b^2*c^2*(880*c^2*C*d - 792*B*c*d^2 + 693*A*d^3 - 96 0*c^3*D))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqr t[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(d*Sqrt[(Sqrt [b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c ^2 - a*d^2)*(45*a^2*d^4*D - 3*a*b*d^2*(286*c*C*d - 165*B*d^2 - 384*c^2*D) + 4*b^2*c*(880*c^2*C*d - 792*B*c*d^2 + 693*A*d^3 - 960*c^3*D))*Sqrt[(Sqrt[ b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSi n[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)] )/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(15*d^2)))/(21*d^2))/(11*...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(3010\) vs. \(2(703)=1406\).
Time = 5.06 (sec) , antiderivative size = 3011, normalized size of antiderivative = 3.81
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(3011\) |
| default | \(\text {Expression too large to display}\) | \(5965\) |
Input:
int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x,method=_RETURNVER BOSE)
Output:
1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((d*x+c)*(-b*x^2+a))^(1/2)*(-2*(-b*d*x^2+ a*d)*(A*a*d^5-A*b*c^2*d^3-B*a*c*d^4+B*b*c^3*d^2+C*a*c^2*d^3-C*b*c^4*d-D*a* c^3*d^2+D*b*c^5)/d^7/((x+c/d)*(-b*d*x^2+a*d))^(1/2)-2/11*b/d^2*D*x^4*(-b*d *x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/9*(b^2/d^2*(C*d-D*c)-10/11*b^2/d^2*D*c)/b/ d*x^3*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/7*(b/d^3*(B*b*d^2-C*b*c*d-2*D*a *d^2+D*b*c^2)+9/11*b/d*D*a-8/9*(b^2/d^2*(C*d-D*c)-10/11*b^2/d^2*D*c)/d*c)/ b/d*x^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(b/d^4*(A*b*d^3-B*b*c*d^2-2 *C*a*d^3+C*b*c^2*d+2*D*a*c*d^2-D*b*c^3)+8/11*b/d^2*D*a*c+7/9*(b^2/d^2*(C*d -D*c)-10/11*b^2/d^2*D*c)/b*a-6/7*(b/d^3*(B*b*d^2-C*b*c*d-2*D*a*d^2+D*b*c^2 )+9/11*b/d*D*a-8/9*(b^2/d^2*(C*d-D*c)-10/11*b^2/d^2*D*c)/d*c)/d*c)/b/d*x*( -b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(-1/d^5*(A*b^2*c*d^3+2*B*a*b*d^4-B*b ^2*c^2*d^2-2*C*a*b*c*d^3+C*b^2*c^3*d-D*a^2*d^4+2*D*a*b*c^2*d^2-D*b^2*c^4)+ 2/3*(b^2/d^2*(C*d-D*c)-10/11*b^2/d^2*D*c)/b/d*a*c+5/7*(b/d^3*(B*b*d^2-C*b* c*d-2*D*a*d^2+D*b*c^2)+9/11*b/d*D*a-8/9*(b^2/d^2*(C*d-D*c)-10/11*b^2/d^2*D *c)/d*c)/b*a-4/5*(b/d^4*(A*b*d^3-B*b*c*d^2-2*C*a*d^3+C*b*c^2*d+2*D*a*c*d^2 -D*b*c^3)+8/11*b/d^2*D*a*c+7/9*(b^2/d^2*(C*d-D*c)-10/11*b^2/d^2*D*c)/b*a-6 /7*(b/d^3*(B*b*d^2-C*b*c*d-2*D*a*d^2+D*b*c^2)+9/11*b/d*D*a-8/9*(b^2/d^2*(C *d-D*c)-10/11*b^2/d^2*D*c)/d*c)/d*c)/d*c)/b/d*(-b*d*x^3-b*c*x^2+a*d*x+a*c) ^(1/2)+2*((2*A*a*b*c*d^5-A*b^2*c^3*d^3+B*a^2*d^6-2*B*a*b*c^2*d^4+B*b^2*c^4 *d^2-C*a^2*c*d^5+2*C*a*b*c^3*d^3-C*b^2*c^5*d+D*a^2*c^2*d^4-2*D*a*b*c^4*...
Time = 0.13 (sec) , antiderivative size = 1028, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm= "fricas")
Output:
-2/10395*(4*(3840*D*b^3*c^7 - 3520*C*b^3*c^6*d - 288*(24*D*a*b^2 - 11*B*b^ 3)*c^5*d^2 + 198*(31*C*a*b^2 - 14*A*b^3)*c^4*d^3 + 9*(287*D*a^2*b - 583*B* a*b^2)*c^3*d^4 - 66*(32*C*a^2*b - 63*A*a*b^2)*c^2*d^5 + 135*(D*a^3 + 11*B* a^2*b)*c*d^6 + (3840*D*b^3*c^6*d - 3520*C*b^3*c^5*d^2 - 288*(24*D*a*b^2 - 11*B*b^3)*c^4*d^3 + 198*(31*C*a*b^2 - 14*A*b^3)*c^3*d^4 + 9*(287*D*a^2*b - 583*B*a*b^2)*c^2*d^5 - 66*(32*C*a^2*b - 63*A*a*b^2)*c*d^6 + 135*(D*a^3 + 11*B*a^2*b)*d^7)*x)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/( b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 12*(3840*D *b^3*c^6*d - 3520*C*b^3*c^5*d^2 - 288*(14*D*a*b^2 - 11*B*b^3)*c^4*d^3 + 66 *(53*C*a*b^2 - 42*A*b^3)*c^3*d^4 + 9*(41*D*a^2*b - 319*B*a*b^2)*c^2*d^5 - 231*(C*a^2*b - 9*A*a*b^2)*c*d^6 + (3840*D*b^3*c^5*d^2 - 3520*C*b^3*c^4*d^3 - 288*(14*D*a*b^2 - 11*B*b^3)*c^3*d^4 + 66*(53*C*a*b^2 - 42*A*b^3)*c^2*d^ 5 + 9*(41*D*a^2*b - 319*B*a*b^2)*c*d^6 - 231*(C*a^2*b - 9*A*a*b^2)*d^7)*x) *sqrt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/ 27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(315*D*b^3*d^7*x^5 + 7680*D*b^3*c^5*d^2 - 7040*C*b^3*c^4*d^3 + 3465*A*a*b^2*d^7 - 192*(37*D* a*b^2 - 33*B*b^3)*c^3*d^4 + 44*(139*C*a*b^2 - 126*A*b^3)*c^2*d^5 + 90*(2*D *a^2*b - 55*B*a*b^2)*c*d^6 - 35*(12*D*b^3*c*d^6 - 11*C*b^3*d^7)*x^4 + 5*(1 20*D*b^3*c^2*d^5 - 110*C*b^3*c*d^6 - 9*(13*D*a*b^2 - 11*B*b^3)*d^7)*x^3...
\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((-b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(3/2),x)
Output:
Integral((a - b*x**2)**(3/2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**(3/2), x)
\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm= "maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)/(d*x + c)^(3/2), x)
\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x, algorithm= "giac")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*(-b*x^2 + a)^(3/2)/(d*x + c)^(3/2), x)
Timed out. \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(3/2),x)
Output:
int(((a - b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(3/2), x)
\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{3/2}} \, dx=\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{\frac {3}{2}}}d x \] Input:
int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x)
Output:
int((-b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x)