Integrand size = 35, antiderivative size = 748 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {a^2 \sqrt {c+d x} (A b c+a c C-a B d+(b B c-A b d-a C d) x)}{3 b^3 \left (b c^2-a d^2\right ) \left (a-b x^2\right )^{3/2}}-\frac {a \sqrt {c+d x} \left (4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )+\left (15 a^2 C d^3+b^2 c^2 (14 B c-13 A d)-a b d \left (19 c^2 C+10 B c d-9 A d^2\right )\right ) x\right )}{6 b^3 \left (b c^2-a d^2\right )^2 \sqrt {a-b x^2}}+\frac {2 (7 c C-5 B d) \sqrt {c+d x} \sqrt {a-b x^2}}{15 b^3 d^2}-\frac {2 C (c+d x)^{3/2} \sqrt {a-b x^2}}{5 b^3 d^2}-\frac {\sqrt {a} \left (231 a^3 C d^6+a b^2 c^2 d^2 \left (92 c^2 C+150 B c d-185 A d^2\right )-15 a^2 b d^4 \left (25 c^2 C+6 B c d-7 A d^2\right )+4 b^3 c^4 \left (8 c^2 C-10 B c d+15 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{30 b^{7/2} d^3 \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\sqrt {a} \left (3 a^2 d^4 (51 c C-25 B d)-a b c d^2 \left (116 c^2 C-30 B c d-65 A d^2\right )-4 b^2 c^3 \left (8 c^2 C-10 B c d+15 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{30 b^{7/2} d^3 \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
1/3*a^2*(d*x+c)^(1/2)*(A*b*c+C*a*c-B*a*d+(-A*b*d+B*b*c-C*a*d)*x)/b^3/(-a*d ^2+b*c^2)/(-b*x^2+a)^(3/2)-1/6*a*(d*x+c)^(1/2)*(4*A*b*c*(-2*a*d^2+3*b*c^2) +a*(b*c^2*(-17*B*d+18*C*c)-a*d^2*(-13*B*d+14*C*c))+(15*a^2*C*d^3+b^2*c^2*( -13*A*d+14*B*c)-a*b*d*(-9*A*d^2+10*B*c*d+19*C*c^2))*x)/b^3/(-a*d^2+b*c^2)^ 2/(-b*x^2+a)^(1/2)+2/15*(-5*B*d+7*C*c)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^3/ d^2-2/5*C*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b^3/d^2-1/30*a^(1/2)*(231*a^3*C*d ^6+a*b^2*c^2*d^2*(-185*A*d^2+150*B*c*d+92*C*c^2)-15*a^2*b*d^4*(-7*A*d^2+6* B*c*d+25*C*c^2)+4*b^3*c^4*(15*A*d^2-10*B*c*d+8*C*c^2))*(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^(7/2)/d^3/(-a*d^2+b*c^2)^2/((d* x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-1/30*a^(1/2)*(3*a^2*d^4 *(-25*B*d+51*C*c)-a*b*c*d^2*(-65*A*d^2-30*B*c*d+116*C*c^2)-4*b^2*c^3*(15*A *d^2-10*B*c*d+8*C*c^2))*((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^(7/2)/d^3/(-a*d^2+b*c^2)/(d*x+c)^(1/2) /(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 33.28 (sec) , antiderivative size = 1155, normalized size of antiderivative = 1.54 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
Integrate[(x^5*(A + B*x + C*x^2))/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
Output:
Sqrt[c + d*x]*Sqrt[a - b*x^2]*((-2*(-4*c*C + 5*B*d))/(15*b^3*d^2) - (2*C*x )/(5*b^3*d) + (a^2*A*b*c + a^3*c*C - a^3*B*d + a^2*b*B*c*x - a^2*A*b*d*x - a^3*C*d*x)/(3*b^3*(b*c^2 - a*d^2)*(-a + b*x^2)^2) + (12*a*A*b^2*c^3 + 18* a^2*b*c^3*C - 17*a^2*b*B*c^2*d - 8*a^2*A*b*c*d^2 - 14*a^3*c*C*d^2 + 13*a^3 *B*d^3 + 14*a*b^2*B*c^3*x - 13*a*A*b^2*c^2*d*x - 19*a^2*b*c^2*C*d*x - 10*a ^2*b*B*c*d^2*x + 9*a^2*A*b*d^3*x + 15*a^3*C*d^3*x)/(6*b^3*(b*c^2 - a*d^2)^ 2*(-a + b*x^2))) + (Sqrt[a - (b*(c + d*x)^2*(-1 + c/(c + d*x))^2)/d^2]*(-( Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(231*a^3*C*d^6 + a*b^2*c^2*d^2*(92*c^2*C + 150*B*c*d - 185*A*d^2) + 15*a^2*b*d^4*(-25*c^2*C - 6*B*c*d + 7*A*d^2) + 4* b^3*c^4*(8*c^2*C - 10*B*c*d + 15*A*d^2))*(-((a*d^2)/(c + d*x)^2) + b*(-1 + c/(c + d*x))^2)) + (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(231*a^3*C*d^6 + a* b^2*c^2*d^2*(92*c^2*C + 150*B*c*d - 185*A*d^2) + 15*a^2*b*d^4*(-25*c^2*C - 6*B*c*d + 7*A*d^2) + 4*b^3*c^4*(8*c^2*C - 10*B*c*d + 15*A*d^2))*Sqrt[1 - c/(c + d*x) - (Sqrt[a]*d)/(Sqrt[b]*(c + d*x))]*Sqrt[1 - c/(c + d*x) + (Sqr t[a]*d)/(Sqrt[b]*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqr t[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/Sq rt[c + d*x] + (I*Sqrt[a]*Sqrt[b]*d*(Sqrt[b]*c - Sqrt[a]*d)*(-231*a^(5/2)*C *d^5 + 3*a^2*Sqrt[b]*d^4*(-51*c*C + 25*B*d) + a*b^(3/2)*c*d^2*(116*c^2*C - 30*B*c*d - 65*A*d^2) + 3*a^(3/2)*b*d^3*(74*c^2*C + 55*B*c*d - 35*A*d^2) + 12*Sqrt[a]*b^2*c^2*d*(2*c^2*C - 15*B*c*d + 10*A*d^2) + 4*b^(5/2)*c^3*(...
Time = 4.95 (sec) , antiderivative size = 819, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2180, 27, 2180, 27, 2185, 27, 2185, 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.
| \(\displaystyle \int \frac {x^5 \left (A+B x+C x^2\right )}{\left (a-b x^2\right )^{5/2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2180 |
| \(\displaystyle \frac {\int -\frac {6 a C \left (c^2-\frac {a d^2}{b}\right ) x^5+6 a B \left (c^2-\frac {a d^2}{b}\right ) x^4+\frac {6 a (A b+a C) \left (b c^2-a d^2\right ) x^3}{b^2}+\frac {6 a^2 B \left (b c^2-a d^2\right ) x^2}{b^2}+\frac {3 a^2 \left (A b \left (2 b c^2-a d^2\right )-a \left (a C d^2-b c (2 c C-B d)\right )\right ) x}{b^3}+\frac {a^3 (b c (2 B c-A d)-a d (c C+B d))}{b^3}}{2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{3 a \left (b c^2-a d^2\right )}+\frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\int \frac {6 a C \left (c^2-\frac {a d^2}{b}\right ) x^5+6 a B \left (c^2-\frac {a d^2}{b}\right ) x^4+\frac {6 a (A b+a C) \left (b c^2-a d^2\right ) x^3}{b^2}+\frac {6 a^2 B \left (b c^2-a d^2\right ) x^2}{b^2}+\frac {3 a^2 \left (A b \left (2 b c^2-a d^2\right )-a \left (a C d^2-b c (2 c C-B d)\right )\right ) x}{b^3}+\frac {a^3 (b c (2 B c-A d)-a d (c C+B d))}{b^3}}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2180 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\int -\frac {\frac {\left (12 b^2 (2 B c-A d) c^3-a b d \left (18 C c^2+31 B d c-8 A d^2\right ) c+a^2 d^3 (14 c C+11 B d)\right ) a^3}{b^3}+\frac {12 C \left (b c^2-a d^2\right )^2 x^3 a^2}{b^2}+\frac {12 B \left (b c^2-a d^2\right )^2 x^2 a^2}{b^2}+\frac {\left (A b \left (12 b^2 c^4-37 a b d^2 c^2+21 a^2 d^4\right )+a \left (39 a^2 C d^4-a b c (67 c C+10 B d) d^2+2 b^2 c^3 (12 c C+7 B d)\right )\right ) x a^2}{b^3}}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}+\frac {a^2 \sqrt {c+d x} \left (x \left (15 a^2 C d^3-a b d \left (-9 A d^2+10 B c d+19 c^2 C\right )+b^2 c^2 (14 B c-13 A d)\right )+4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \sqrt {c+d x} \left (x \left (15 a^2 C d^3-a b d \left (-9 A d^2+10 B c d+19 c^2 C\right )+b^2 c^2 (14 B c-13 A d)\right )+4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\int \frac {\frac {\left (12 b^2 (2 B c-A d) c^3-a b d \left (18 C c^2+31 B d c-8 A d^2\right ) c+a^2 d^3 (14 c C+11 B d)\right ) a^3}{b^3}+\frac {12 C \left (b c^2-a d^2\right )^2 x^3 a^2}{b^2}+\frac {12 B \left (b c^2-a d^2\right )^2 x^2 a^2}{b^2}+\frac {\left (A b \left (12 b^2 c^4-37 a b d^2 c^2+21 a^2 d^4\right )+a \left (39 a^2 C d^4-a b c (67 c C+10 B d) d^2+2 b^2 c^3 (12 c C+7 B d)\right )\right ) x a^2}{b^3}}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \sqrt {c+d x} \left (x \left (15 a^2 C d^3-a b d \left (-9 A d^2+10 B c d+19 c^2 C\right )+b^2 c^2 (14 B c-13 A d)\right )+4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {-\frac {2 \int -\frac {\frac {d^2 \left (a^2 (106 c C+55 B d) d^4-a b c \left (162 C c^2+155 B d c-40 A d^2\right ) d^2+12 b^2 c^3 \left (3 C c^2+10 B d c-5 A d^2\right )\right ) a^3}{b^2}-\frac {12 d^2 (7 c C-5 B d) \left (b c^2-a d^2\right )^2 x^2 a^2}{b}+\frac {d \left (231 a^3 C d^6-a^2 b \left (431 C c^2+50 B d c-105 A d^2\right ) d^4+a b^2 c^2 \left (204 C c^2+70 B d c-185 A d^2\right ) d^2-b^3 \left (24 c^6 C-60 A c^4 d^2\right )\right ) x a^2}{b^2}}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {24 a^2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \sqrt {c+d x} \left (x \left (15 a^2 C d^3-a b d \left (-9 A d^2+10 B c d+19 c^2 C\right )+b^2 c^2 (14 B c-13 A d)\right )+4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {\int \frac {\frac {d^2 \left (a^2 (106 c C+55 B d) d^4-a b c \left (162 C c^2+155 B d c-40 A d^2\right ) d^2+12 b^2 c^3 \left (3 C c^2+10 B d c-5 A d^2\right )\right ) a^3}{b^2}-\frac {12 d^2 (7 c C-5 B d) \left (b c^2-a d^2\right )^2 x^2 a^2}{b}+\frac {d \left (231 a^3 C d^6-a^2 b \left (431 C c^2+50 B d c-105 A d^2\right ) d^4+a b^2 c^2 \left (204 C c^2+70 B d c-185 A d^2\right ) d^2-b^3 \left (24 c^6 C-60 A c^4 d^2\right )\right ) x a^2}{b^2}}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {24 a^2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \sqrt {c+d x} \left (x \left (15 a^2 C d^3-a b d \left (-9 A d^2+10 B c d+19 c^2 C\right )+b^2 c^2 (14 B c-13 A d)\right )+4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {8 a^2 d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2 (7 c C-5 B d)}{b^2}-\frac {2 \int -\frac {3 a^2 d^3 \left (a d \left (3 a^2 (26 c C+25 B d) d^4-a b c \left (106 C c^2+195 B d c-40 A d^2\right ) d^2+4 b^2 c^3 \left (2 C c^2+35 B d c-15 A d^2\right )\right )+\left (231 a^3 C d^6-15 a^2 b \left (25 C c^2+6 B d c-7 A d^2\right ) d^4+a b^2 c^2 \left (92 C c^2+150 B d c-185 A d^2\right ) d^2+4 b^3 c^4 \left (8 C c^2-10 B d c+15 A d^2\right )\right ) x\right )}{2 b \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}}{5 b d^3}-\frac {24 a^2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \sqrt {c+d x} \left (x \left (15 a^2 C d^3-a b d \left (-9 A d^2+10 B c d+19 c^2 C\right )+b^2 c^2 (14 B c-13 A d)\right )+4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {a^2 d \int \frac {a d \left (3 a^2 (26 c C+25 B d) d^4-a b c \left (106 C c^2+195 B d c-40 A d^2\right ) d^2+4 b^2 c^3 \left (2 C c^2+35 B d c-15 A d^2\right )\right )+\left (231 a^3 C d^6-15 a^2 b \left (25 C c^2+6 B d c-7 A d^2\right ) d^4+a b^2 c^2 \left (92 C c^2+150 B d c-185 A d^2\right ) d^2+4 b^3 c^4 \left (8 C c^2-10 B d c+15 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b^2}+\frac {8 a^2 d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2 (7 c C-5 B d)}{b^2}}{5 b d^3}-\frac {24 a^2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \sqrt {c+d x} \left (x \left (15 a^2 C d^3-a b d \left (-9 A d^2+10 B c d+19 c^2 C\right )+b^2 c^2 (14 B c-13 A d)\right )+4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {a^2 d \left (\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (51 c C-25 B d)-a b c d^2 \left (-65 A d^2-30 B c d+116 c^2 C\right )-4 b^2 c^3 \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {\left (231 a^3 C d^6-15 a^2 b d^4 \left (-7 A d^2+6 B c d+25 c^2 C\right )+a b^2 c^2 d^2 \left (-185 A d^2+150 B c d+92 c^2 C\right )+4 b^3 c^4 \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{b^2}+\frac {8 a^2 d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2 (7 c C-5 B d)}{b^2}}{5 b d^3}-\frac {24 a^2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \sqrt {c+d x} \left (x \left (15 a^2 C d^3-a b d \left (-9 A d^2+10 B c d+19 c^2 C\right )+b^2 c^2 (14 B c-13 A d)\right )+4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {a^2 d \left (\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (51 c C-25 B d)-a b c d^2 \left (-65 A d^2-30 B c d+116 c^2 C\right )-4 b^2 c^3 \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {\sqrt {1-\frac {b x^2}{a}} \left (231 a^3 C d^6-15 a^2 b d^4 \left (-7 A d^2+6 B c d+25 c^2 C\right )+a b^2 c^2 d^2 \left (-185 A d^2+150 B c d+92 c^2 C\right )+4 b^3 c^4 \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{b^2}+\frac {8 a^2 d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2 (7 c C-5 B d)}{b^2}}{5 b d^3}-\frac {24 a^2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \sqrt {c+d x} \left (x \left (15 a^2 C d^3-a b d \left (-9 A d^2+10 B c d+19 c^2 C\right )+b^2 c^2 (14 B c-13 A d)\right )+4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {a^2 d \left (\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (51 c C-25 B d)-a b c d^2 \left (-65 A d^2-30 B c d+116 c^2 C\right )-4 b^2 c^3 \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (231 a^3 C d^6-15 a^2 b d^4 \left (-7 A d^2+6 B c d+25 c^2 C\right )+a b^2 c^2 d^2 \left (-185 A d^2+150 B c d+92 c^2 C\right )+4 b^3 c^4 \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b^2}+\frac {8 a^2 d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2 (7 c C-5 B d)}{b^2}}{5 b d^3}-\frac {24 a^2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \sqrt {c+d x} \left (x \left (15 a^2 C d^3-a b d \left (-9 A d^2+10 B c d+19 c^2 C\right )+b^2 c^2 (14 B c-13 A d)\right )+4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {a^2 d \left (\frac {\left (b c^2-a d^2\right ) \left (3 a^2 d^4 (51 c C-25 B d)-a b c d^2 \left (-65 A d^2-30 B c d+116 c^2 C\right )-4 b^2 c^3 \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (231 a^3 C d^6-15 a^2 b d^4 \left (-7 A d^2+6 B c d+25 c^2 C\right )+a b^2 c^2 d^2 \left (-185 A d^2+150 B c d+92 c^2 C\right )+4 b^3 c^4 \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b^2}+\frac {8 a^2 d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2 (7 c C-5 B d)}{b^2}}{5 b d^3}-\frac {24 a^2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^3 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a^2 \sqrt {c+d x} \left (x \left (15 a^2 C d^3-a b d \left (-9 A d^2+10 B c d+19 c^2 C\right )+b^2 c^2 (14 B c-13 A d)\right )+4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )\right )}{b^3 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {\frac {\frac {a^2 d \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a^2 d^4 (51 c C-25 B d)-a b c d^2 \left (-65 A d^2-30 B c d+116 c^2 C\right )-4 b^2 c^3 \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (231 a^3 C d^6-15 a^2 b d^4 \left (-7 A d^2+6 B c d+25 c^2 C\right )+a b^2 c^2 d^2 \left (-185 A d^2+150 B c d+92 c^2 C\right )+4 b^3 c^4 \left (15 A d^2-10 B c d+8 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b^2}+\frac {8 a^2 d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )^2 (7 c C-5 B d)}{b^2}}{5 b d^3}-\frac {24 a^2 C \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )^2}{5 b^3 d^2}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (A b c+a C c-a B d+(b B c-A b d-a C d) x)}{3 b^3 \left (b c^2-a d^2\right ) \left (a-b x^2\right )^{3/2}}-\frac {\frac {a^2 \sqrt {c+d x} \left (4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )+\left (15 a^2 C d^3-a b \left (19 C c^2+10 B d c-9 A d^2\right ) d+b^2 c^2 (14 B c-13 A d)\right ) x\right )}{b^3 \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}-\frac {\frac {\frac {d \left (-\frac {2 \sqrt {a} \left (231 a^3 C d^6-15 a^2 b \left (25 C c^2+6 B d c-7 A d^2\right ) d^4+a b^2 c^2 \left (92 C c^2+150 B d c-185 A d^2\right ) d^2+4 b^3 c^4 \left (8 C c^2-10 B d c+15 A d^2\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (3 a^2 (51 c C-25 B d) d^4-a b c \left (116 C c^2-30 B d c-65 A d^2\right ) d^2-4 b^2 c^3 \left (8 C c^2-10 B d c+15 A d^2\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right ) a^2}{b^2}+\frac {8 d (7 c C-5 B d) \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2} a^2}{b^2}}{5 b d^3}-\frac {24 a^2 C \left (b c^2-a d^2\right )^2 (c+d x)^{3/2} \sqrt {a-b x^2}}{5 b^3 d^2}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {a^2 \sqrt {c+d x} (A b c+a C c-a B d+(b B c-A b d-a C d) x)}{3 b^3 \left (b c^2-a d^2\right ) \left (a-b x^2\right )^{3/2}}-\frac {\frac {a^2 \sqrt {c+d x} \left (4 A b c \left (3 b c^2-2 a d^2\right )+a \left (b c^2 (18 c C-17 B d)-a d^2 (14 c C-13 B d)\right )+\left (15 a^2 C d^3-a b \left (19 C c^2+10 B d c-9 A d^2\right ) d+b^2 c^2 (14 B c-13 A d)\right ) x\right )}{b^3 \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}-\frac {\frac {\frac {d \left (-\frac {2 \sqrt {a} \left (231 a^3 C d^6-15 a^2 b \left (25 C c^2+6 B d c-7 A d^2\right ) d^4+a b^2 c^2 \left (92 C c^2+150 B d c-185 A d^2\right ) d^2+4 b^3 c^4 \left (8 C c^2-10 B d c+15 A d^2\right )\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (b c^2-a d^2\right ) \left (3 a^2 (51 c C-25 B d) d^4-a b c \left (116 C c^2-30 B d c-65 A d^2\right ) d^2-4 b^2 c^3 \left (8 C c^2-10 B d c+15 A d^2\right )\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {c+d x} \sqrt {a-b x^2}}\right ) a^2}{b^2}+\frac {8 d (7 c C-5 B d) \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2} a^2}{b^2}}{5 b d^3}-\frac {24 a^2 C \left (b c^2-a d^2\right )^2 (c+d x)^{3/2} \sqrt {a-b x^2}}{5 b^3 d^2}}{2 a \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\) |
Input:
Int[(x^5*(A + B*x + C*x^2))/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
Output:
(a^2*Sqrt[c + d*x]*(A*b*c + a*c*C - a*B*d + (b*B*c - A*b*d - a*C*d)*x))/(3 *b^3*(b*c^2 - a*d^2)*(a - b*x^2)^(3/2)) - ((a^2*Sqrt[c + d*x]*(4*A*b*c*(3* b*c^2 - 2*a*d^2) + a*(b*c^2*(18*c*C - 17*B*d) - a*d^2*(14*c*C - 13*B*d)) + (15*a^2*C*d^3 + b^2*c^2*(14*B*c - 13*A*d) - a*b*d*(19*c^2*C + 10*B*c*d - 9*A*d^2))*x))/(b^3*(b*c^2 - a*d^2)*Sqrt[a - b*x^2]) - ((-24*a^2*C*(b*c^2 - a*d^2)^2*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/(5*b^3*d^2) + ((8*a^2*d*(7*c*C - 5*B*d)*(b*c^2 - a*d^2)^2*Sqrt[c + d*x]*Sqrt[a - b*x^2])/b^2 + (a^2*d*((- 2*Sqrt[a]*(231*a^3*C*d^6 + a*b^2*c^2*d^2*(92*c^2*C + 150*B*c*d - 185*A*d^2 ) - 15*a^2*b*d^4*(25*c^2*C + 6*B*c*d - 7*A*d^2) + 4*b^3*c^4*(8*c^2*C - 10* B*c*d + 15*A*d^2))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt [1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqr t[b]*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)*(3*a^2*d^4*(51*c*C - 25*B*d) - a*b*c*d^2*(116 *c^2*C - 30*B*c*d - 65*A*d^2) - 4*b^2*c^3*(8*c^2*C - 10*B*c*d + 15*A*d^2)) *Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*Ell ipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/S qrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/b^2)/(5*b*d^3))/ (2*a*(b*c^2 - a*d^2)))/(6*a*(b*c^2 - a*d^2))
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[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 , x], x, 1]}, Simp[(-(d + e*x)^(m + 1))*(a + b*x^2)^(p + 1)*((a*(e*R - d*S) + (b*d*R + a*e*S)*x)/(2*a*(p + 1)*(b*d^2 + a*e^2))), x] + Simp[1/(2*a*(p + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a* (p + 1)*(b*d^2 + a*e^2)*Qx + b*d^2*R*(2*p + 3) - a*e*(d*S*m - e*R*(m + 2*p + 3)) + e*(b*d*R + a*e*S)*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, d, e , m}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && !(IGtQ[ m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
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]))
Time = 15.64 (sec) , antiderivative size = 1273, normalized size of antiderivative = 1.70
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1273\) |
| risch | \(\text {Expression too large to display}\) | \(2218\) |
| default | \(\text {Expression too large to display}\) | \(10909\) |
Input:
int(x^5*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x,method=_RETURNVERBO SE)
Output:
((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*((1/3*a^2*(A*b*d -B*b*c+C*a*d)/(a*d^2-b*c^2)/b^5*x-1/3*a^2*(A*b*c-B*a*d+C*a*c)/(a*d^2-b*c^2 )/b^5)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x^2-a/b)^2-2*(-b*d*x-b*c)*(-1/1 2*a*(9*A*a*b*d^3-13*A*b^2*c^2*d-10*B*a*b*c*d^2+14*B*b^2*c^3+15*C*a^2*d^3-1 9*C*a*b*c^2*d)/b^4/(a*d^2-b*c^2)^2*x+1/12*a*(8*A*a*b*c*d^2-12*A*b^2*c^3-13 *B*a^2*d^3+17*B*a*b*c^2*d+14*C*a^2*c*d^2-18*C*a*b*c^3)/b^4/(a*d^2-b*c^2)^2 )/((x^2-a/b)*(-b*d*x-b*c))^(1/2)-2/5*C/b^3/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c )^(1/2)-2/3*(B/b^2-4/5*C/b^2/d*c)/b/d*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2 *(2*B*a/b^3-1/6*a*(A*b*c*d+13*B*a*d^2-14*B*b*c^2+C*a*c*d)/b^3/(a*d^2-b*c^2 )-1/12/b^3*d*a*(8*A*a*b*c*d^2-12*A*b^2*c^3-13*B*a^2*d^3+17*B*a*b*c^2*d+14* C*a^2*c*d^2-18*C*a*b*c^3)/(a*d^2-b*c^2)^2+1/6/b^3*c*a*(9*A*a*b*d^3-13*A*b^ 2*c^2*d-10*B*a*b*c*d^2+14*B*b^2*c^3+15*C*a^2*d^3-19*C*a*b*c^2*d)/(a*d^2-b* c^2)^2+2/5*C/b^3/d*a*c+1/3*(B/b^2-4/5*C/b^2/d*c)/b*a)*(c/d-1/b*(a*b)^(1/2) )*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a* b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d* x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/ 2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*((A*b+2*C*a)/b ^3+1/12*d*a*(9*A*a*b*d^3-13*A*b^2*c^2*d-10*B*a*b*c*d^2+14*B*b^2*c^3+15*C*a ^2*d^3-19*C*a*b*c^2*d)/b^3/(a*d^2-b*c^2)^2+3/5*C/b^3*a-2/3*(B/b^2-4/5*C/b^ 2/d*c)/d*c)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)...
Leaf count of result is larger than twice the leaf count of optimal. 1621 vs. \(2 (674) = 1348\).
Time = 0.15 (sec) , antiderivative size = 1621, normalized size of antiderivative = 2.17 \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(x^5*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="f ricas")
Output:
1/90*((32*C*a^2*b^3*c^7 - 40*B*a^2*b^3*c^6*d - 270*B*a^3*b^2*c^4*d^3 + 495 *B*a^4*b*c^2*d^5 - 225*B*a^5*d^7 + 4*(17*C*a^3*b^2 + 15*A*a^2*b^3)*c^5*d^2 - (57*C*a^4*b + 5*A*a^3*b^2)*c^3*d^4 - 3*(C*a^5 + 5*A*a^4*b)*c*d^6 + (32* C*b^5*c^7 - 40*B*b^5*c^6*d - 270*B*a*b^4*c^4*d^3 + 495*B*a^2*b^3*c^2*d^5 - 225*B*a^3*b^2*d^7 + 4*(17*C*a*b^4 + 15*A*b^5)*c^5*d^2 - (57*C*a^2*b^3 + 5 *A*a*b^4)*c^3*d^4 - 3*(C*a^3*b^2 + 5*A*a^2*b^3)*c*d^6)*x^4 - 2*(32*C*a*b^4 *c^7 - 40*B*a*b^4*c^6*d - 270*B*a^2*b^3*c^4*d^3 + 495*B*a^3*b^2*c^2*d^5 - 225*B*a^4*b*d^7 + 4*(17*C*a^2*b^3 + 15*A*a*b^4)*c^5*d^2 - (57*C*a^3*b^2 + 5*A*a^2*b^3)*c^3*d^4 - 3*(C*a^4*b + 5*A*a^3*b^2)*c*d^6)*x^2)*sqrt(-b*d)*we ierstrassPInverse(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*(32*C*a^2*b^3*c^6*d - 40*B*a^2*b^3*c^5*d^ 2 + 150*B*a^3*b^2*c^3*d^4 - 90*B*a^4*b*c*d^6 + 4*(23*C*a^3*b^2 + 15*A*a^2* b^3)*c^4*d^3 - 5*(75*C*a^4*b + 37*A*a^3*b^2)*c^2*d^5 + 21*(11*C*a^5 + 5*A* a^4*b)*d^7 + (32*C*b^5*c^6*d - 40*B*b^5*c^5*d^2 + 150*B*a*b^4*c^3*d^4 - 90 *B*a^2*b^3*c*d^6 + 4*(23*C*a*b^4 + 15*A*b^5)*c^4*d^3 - 5*(75*C*a^2*b^3 + 3 7*A*a*b^4)*c^2*d^5 + 21*(11*C*a^3*b^2 + 5*A*a^2*b^3)*d^7)*x^4 - 2*(32*C*a* b^4*c^6*d - 40*B*a*b^4*c^5*d^2 + 150*B*a^2*b^3*c^3*d^4 - 90*B*a^3*b^2*c*d^ 6 + 4*(23*C*a^2*b^3 + 15*A*a*b^4)*c^4*d^3 - 5*(75*C*a^3*b^2 + 37*A*a^2*b^3 )*c^2*d^5 + 21*(11*C*a^4*b + 5*A*a^3*b^2)*d^7)*x^2)*sqrt(-b*d)*weierstrass Zeta(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), ...
Timed out. \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(x**5*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(-b*x**2+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{5}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^5*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="m axima")
Output:
integrate((C*x^2 + B*x + A)*x^5/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
\[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{5}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^5*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="g iac")
Output:
integrate((C*x^2 + B*x + A)*x^5/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^5\,\left (C\,x^2+B\,x+A\right )}{{\left (a-b\,x^2\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int((x^5*(A + B*x + C*x^2))/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)),x)
Output:
int((x^5*(A + B*x + C*x^2))/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {x^5 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \left (C \,x^{2}+B x +A \right )}{\sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:
int(x^5*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)
Output:
int(x^5*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)