Integrand size = 37, antiderivative size = 671 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {c+d x} \left (a (b B c-A b d-a C d+a c D)+b \left (c (A b+a C)-a d \left (B+\frac {a D}{b}\right )\right ) x\right )}{3 a b \left (b c^2-a d^2\right ) \left (a-b x^2\right )^{3/2}}-\frac {\sqrt {c+d x} \left (a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (5 c C d-4 B d^2-6 c^2 D\right )\right )\right )-\left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (b^2 c^2 (2 c C-B d)+3 a^2 d^3 D+a b d \left (2 c C d-3 B d^2-7 c^2 D\right )\right )\right ) x\right )}{6 a^2 b \left (b c^2-a d^2\right )^2 \sqrt {a-b x^2}}+\frac {\left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (b^2 c^2 (2 c C-B d)+3 a^2 d^3 D+a b d \left (2 c C d-3 B d^2-7 c^2 D\right )\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 a^{3/2} b^{3/2} \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\left (A b \left (4 b c^2-5 a d^2\right )-a (b c (2 c C-B d)-a d (C d+c D))\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 a^{3/2} b^{3/2} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
1/3*(d*x+c)^(1/2)*(a*(-A*b*d+B*b*c-C*a*d+D*a*c)+b*(c*(A*b+C*a)-a*d*(B+a*D/ b))*x)/a/b/(-a*d^2+b*c^2)/(-b*x^2+a)^(3/2)-1/6*(d*x+c)^(1/2)*(a*(A*b*d*(-5 *a*d^2+b*c^2)+a*(a*d^2*(C*d-2*D*c)-b*c*(-4*B*d^2+5*C*c*d-6*D*c^2)))-(4*A*b ^2*c*(-2*a*d^2+b*c^2)-a*(b^2*c^2*(-B*d+2*C*c)+3*a^2*d^3*D+a*b*d*(-3*B*d^2+ 2*C*c*d-7*D*c^2)))*x)/a^2/b/(-a*d^2+b*c^2)^2/(-b*x^2+a)^(1/2)+1/6*(4*A*b^2 *c*(-2*a*d^2+b*c^2)-a*(b^2*c^2*(-B*d+2*C*c)+3*a^2*d^3*D+a*b*d*(-3*B*d^2+2* C*c*d-7*D*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 ))/a^(3/2)/b^(3/2)/(-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)-1/6*(A*b*(-5*a*d^2+4*b*c^2)-a*(b*c*(-B*d+2*C*c)-a*d*(C*d+ D*c)))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*Elliptic F(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))/a^(3/2)/b^(3/2)/(-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 = 31.91 (sec) , antiderivative size = 928, normalized size of antiderivative = 1.38 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (-\frac {b (c+d x) \left (4 A b^4 c^3 x^3+a^4 d^3 (-C+D x)-a b^3 c x \left (c (2 c C-B d) x^2+A \left (6 c^2+c d x+8 d^2 x^2\right )\right )+a^2 b^2 \left (c x^2 \left (5 c C d-6 c^2 D-2 C d^2 x+7 c d D x\right )+A d \left (3 c^2+10 c d x+5 d^2 x^2\right )+B \left (-2 c^3+c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )\right )+a^3 b \left (4 c^3 D-c^2 d (3 C+5 D x)+2 c d^2 (3 B+x (2 C+D x))-d^3 (7 A+x (5 B+x (C+3 D x)))\right )\right )}{\left (a-b x^2\right )^2}+\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (4 A b^2 c \left (b c^2-2 a d^2\right )+a \left (b^2 c^2 (-2 c C+B d)-3 a^2 d^3 D+a b d \left (-2 c C d+3 B d^2+7 c^2 D\right )\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (4 A b^2 c \left (b c^2-2 a d^2\right )+a \left (b^2 c^2 (-2 c C+B d)-3 a^2 d^3 D+a b d \left (-2 c C d+3 B d^2+7 c^2 D\right )\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \sqrt {b} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (A b^{3/2} \left (4 b c^2+3 \sqrt {a} \sqrt {b} c d-5 a d^2\right )+a \left (b^{3/2} c (-2 c C+B d)+3 a^{3/2} d^2 D+a \sqrt {b} d (C d+c D)-3 \sqrt {a} b \left (-c C d+B d^2+2 c^2 D\right )\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{6 a^2 b^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
Output:
(Sqrt[a - b*x^2]*(-((b*(c + d*x)*(4*A*b^4*c^3*x^3 + a^4*d^3*(-C + D*x) - a *b^3*c*x*(c*(2*c*C - B*d)*x^2 + A*(6*c^2 + c*d*x + 8*d^2*x^2)) + a^2*b^2*( c*x^2*(5*c*C*d - 6*c^2*D - 2*C*d^2*x + 7*c*d*D*x) + A*d*(3*c^2 + 10*c*d*x + 5*d^2*x^2) + B*(-2*c^3 + c^2*d*x - 4*c*d^2*x^2 + 3*d^3*x^3)) + a^3*b*(4* c^3*D - c^2*d*(3*C + 5*D*x) + 2*c*d^2*(3*B + x*(2*C + D*x)) - d^3*(7*A + x *(5*B + x*(C + 3*D*x))))))/(a - b*x^2)^2) + (d^2*Sqrt[-c + (Sqrt[a]*d)/Sqr t[b]]*(4*A*b^2*c*(b*c^2 - 2*a*d^2) + a*(b^2*c^2*(-2*c*C + B*d) - 3*a^2*d^3 *D + a*b*d*(-2*c*C*d + 3*B*d^2 + 7*c^2*D)))*(a - b*x^2) + I*Sqrt[b]*(Sqrt[ b]*c - Sqrt[a]*d)*(4*A*b^2*c*(b*c^2 - 2*a*d^2) + a*(b^2*c^2*(-2*c*C + B*d) - 3*a^2*d^3*D + a*b*d*(-2*c*C*d + 3*B*d^2 + 7*c^2*D)))*Sqrt[(d*(Sqrt[a]/S qrt[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]*Sqrt[b] *d*(Sqrt[b]*c - Sqrt[a]*d)*(A*b^(3/2)*(4*b*c^2 + 3*Sqrt[a]*Sqrt[b]*c*d - 5 *a*d^2) + a*(b^(3/2)*c*(-2*c*C + B*d) + 3*a^(3/2)*d^2*D + a*Sqrt[b]*d*(C*d + c*D) - 3*Sqrt[a]*b*(-(c*C*d) + B*d^2 + 2*c^2*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)*EllipticF[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)])/(d*Sqrt[-c + (Sqrt[a] *d)/Sqrt[b]]*(a - b*x^2))))/(6*a^2*b^2*(b*c^2 - a*d^2)^2*Sqrt[c + d*x])
Time = 1.28 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {2180, 27, 686, 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.
| \(\displaystyle \int \frac {A+B x+C x^2+D x^3}{\left (a-b x^2\right )^{5/2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2180 |
| \(\displaystyle \frac {\int \frac {A b \left (4 b c^2-5 a d^2\right )-a (b c (2 c C-B d)-a d (C d+c D))+3 b \left (A b c d+a \left (-2 D c^2+C d c-\frac {d^2 (b B-a D)}{b}\right )\right ) x}{2 b \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{3 a \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{3 a b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {A b \left (4 b c^2-5 a d^2\right )-a (b c (2 c C-B d)-a d (C d+c D))+3 \left (A c d b^2+a \left (a D d^2+b \left (-2 D c^2+C d c-B d^2\right )\right )\right ) x}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{6 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{3 a b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {-\frac {\int \frac {b d \left (a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (-6 D c^2+5 C d c-4 B d^2\right )\right )\right )+\left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 D d^3+a b \left (-7 D c^2+2 C d c-3 B d^2\right ) d+b^2 c^2 (2 c C-B d)\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (-4 B d^2-6 c^2 D+5 c C d\right )\right )\right )-x \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{3 a b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {d \int \frac {a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (-6 D c^2+5 C d c-4 B d^2\right )\right )\right )+\left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 D d^3+a b \left (-7 D c^2+2 C d c-3 B d^2\right ) d+b^2 c^2 (2 c C-B d)\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (-4 B d^2-6 c^2 D+5 c C d\right )\right )\right )-x \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{3 a b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {-\frac {d \left (\frac {\left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (A b \left (4 b c^2-5 a d^2\right )-a (b c (2 c C-B d)-a d (c D+C d))\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (-4 B d^2-6 c^2 D+5 c C d\right )\right )\right )-x \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{3 a b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {-\frac {d \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (A b \left (4 b c^2-5 a d^2\right )-a (b c (2 c C-B d)-a d (c D+C d))\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (-4 B d^2-6 c^2 D+5 c C d\right )\right )\right )-x \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{3 a b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {-\frac {d \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) \left (A b \left (4 b c^2-5 a d^2\right )-a (b c (2 c C-B d)-a d (c D+C d))\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (-4 B d^2-6 c^2 D+5 c C d\right )\right )\right )-x \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{3 a b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {-\frac {d \left (-\frac {\left (b c^2-a d^2\right ) \left (A b \left (4 b c^2-5 a d^2\right )-a (b c (2 c C-B d)-a d (c D+C d))\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (-4 B d^2-6 c^2 D+5 c C d\right )\right )\right )-x \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{3 a b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {-\frac {d \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (A b \left (4 b c^2-5 a d^2\right )-a (b c (2 c C-B d)-a d (c D+C d))\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (-4 B d^2-6 c^2 D+5 c C d\right )\right )\right )-x \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{3 a b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {-\frac {d \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (A b \left (4 b c^2-5 a d^2\right )-a (b c (2 c C-B d)-a d (c D+C d))\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (-4 B d^2-6 c^2 D+5 c C d\right )\right )\right )-x \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{3 a b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {-\frac {d \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (A b \left (4 b c^2-5 a d^2\right )-a (b c (2 c C-B d)-a d (c D+C d))\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (A b d \left (b c^2-5 a d^2\right )+a \left (a d^2 (C d-2 c D)-b c \left (-4 B d^2-6 c^2 D+5 c C d\right )\right )\right )-x \left (4 A b^2 c \left (b c^2-2 a d^2\right )-a \left (3 a^2 d^3 D+a b d \left (-3 B d^2-7 c^2 D+2 c C d\right )+b^2 c^2 (2 c C-B d)\right )\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b \left (b c^2-a d^2\right )}+\frac {\sqrt {c+d x} \left (b x \left (c (a C+A b)-a d \left (\frac {a D}{b}+B\right )\right )+a (a c D-a C d-A b d+b B c)\right )}{3 a b \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
Output:
(Sqrt[c + d*x]*(a*(b*B*c - A*b*d - a*C*d + a*c*D) + b*(c*(A*b + a*C) - a*d *(B + (a*D)/b))*x))/(3*a*b*(b*c^2 - a*d^2)*(a - b*x^2)^(3/2)) + (-((Sqrt[c + d*x]*(a*(A*b*d*(b*c^2 - 5*a*d^2) + a*(a*d^2*(C*d - 2*c*D) - b*c*(5*c*C* d - 4*B*d^2 - 6*c^2*D))) - (4*A*b^2*c*(b*c^2 - 2*a*d^2) - a*(b^2*c^2*(2*c* C - B*d) + 3*a^2*d^3*D + a*b*d*(2*c*C*d - 3*B*d^2 - 7*c^2*D)))*x))/(a*(b*c ^2 - a*d^2)*Sqrt[a - b*x^2])) - (d*((-2*Sqrt[a]*(4*A*b^2*c*(b*c^2 - 2*a*d^ 2) - a*(b^2*c^2*(2*c*C - B*d) + 3*a^2*d^3*D + a*b*d*(2*c*C*d - 3*B*d^2 - 7 *c^2*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)])/(Sqrt[b]*d*Sq rt[(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)*(A*b*(4*b*c^2 - 5*a*d^2) - a*(b*c*(2*c*C - B*d) - a*d* (C*d + c*D)))*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - ( b*x^2)/a]*EllipticF[ArcSin[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])))/(2* a*(b*c^2 - a*d^2)))/(6*a*b*(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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[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, 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]))
Leaf count of result is larger than twice the leaf count of optimal. \(1258\) vs. \(2(605)=1210\).
Time = 8.26 (sec) , antiderivative size = 1259, normalized size of antiderivative = 1.88
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1259\) |
| default | \(\text {Expression too large to display}\) | \(9534\) |
Input:
int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),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)*((-1/3*(A*b^2* c-B*a*b*d+C*a*b*c-D*a^2*d)/a/b^3/(a*d^2-b*c^2)*x+1/3*(A*b*d-B*b*c+C*a*d-D* a*c)/(a*d^2-b*c^2)/b^3)*(-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)*(-1/12*(8*A*a*b^2*c*d^2-4*A*b^3*c^3-3*B*a^2*b*d^3-B*a*b^2*c^2* d+2*C*a^2*b*c*d^2+2*C*a*b^2*c^3+3*D*a^3*d^3-7*D*a^2*b*c^2*d)/a^2/(a*d^2-b* c^2)^2/b^2*x+1/12*(5*A*a*b*d^3-A*b^2*c^2*d-4*B*a*b*c*d^2-C*a^2*d^3+5*C*a*b *c^2*d+2*D*a^2*c*d^2-6*D*a*b*c^3)/a/b^2/(a*d^2-b*c^2)^2)/((x^2-a/b)*(-b*d* x-b*c))^(1/2)+2*(1/6/(a*d^2-b*c^2)/b*(5*A*a*b*d^2-4*A*b^2*c^2-B*a*b*c*d-C* a^2*d^2+2*C*a*b*c^2-D*a^2*c*d)/a^2-1/12/b*d*(5*A*a*b*d^3-A*b^2*c^2*d-4*B*a *b*c*d^2-C*a^2*d^3+5*C*a*b*c^2*d+2*D*a^2*c*d^2-6*D*a*b*c^3)/a/(a*d^2-b*c^2 )^2+1/6/b*c*(8*A*a*b^2*c*d^2-4*A*b^3*c^3-3*B*a^2*b*d^3-B*a*b^2*c^2*d+2*C*a ^2*b*c*d^2+2*C*a*b^2*c^3+3*D*a^3*d^3-7*D*a^2*b*c^2*d)/a^2/(a*d^2-b*c^2)^2) *(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))+1/6*d*(8*A*a*b^2*c*d^2-4*A*b^3*c^3-3*B*a^2*b*d^3-B*a*b^2*c^2*d+2*C*a^ 2*b*c*d^2+2*C*a*b^2*c^3+3*D*a^3*d^3-7*D*a^2*b*c^2*d)/a^2/b/(a*d^2-b*c^2)^2 *(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...
Leaf count of result is larger than twice the leaf count of optimal. 1288 vs. \(2 (614) = 1228\).
Time = 0.13 (sec) , antiderivative size = 1288, normalized size of antiderivative = 1.92 \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm= "fricas")
Output:
1/18*((2*(C*a^3*b^2 - 2*A*a^2*b^3)*c^4 + (11*D*a^4*b - B*a^3*b^2)*c^3*d - (13*C*a^4*b - 11*A*a^3*b^2)*c^2*d^2 - 3*(D*a^5 - 3*B*a^4*b)*c*d^3 + 3*(C*a ^5 - 5*A*a^4*b)*d^4 + (2*(C*a*b^4 - 2*A*b^5)*c^4 + (11*D*a^2*b^3 - B*a*b^4 )*c^3*d - (13*C*a^2*b^3 - 11*A*a*b^4)*c^2*d^2 - 3*(D*a^3*b^2 - 3*B*a^2*b^3 )*c*d^3 + 3*(C*a^3*b^2 - 5*A*a^2*b^3)*d^4)*x^4 - 2*(2*(C*a^2*b^3 - 2*A*a*b ^4)*c^4 + (11*D*a^3*b^2 - B*a^2*b^3)*c^3*d - (13*C*a^3*b^2 - 11*A*a^2*b^3) *c^2*d^2 - 3*(D*a^4*b - 3*B*a^3*b^2)*c*d^3 + 3*(C*a^4*b - 5*A*a^3*b^2)*d^4 )*x^2)*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) + 3*(2*(C*a^3*b^2 - 2*A*a ^2*b^3)*c^3*d - (7*D*a^4*b + B*a^3*b^2)*c^2*d^2 + 2*(C*a^4*b + 4*A*a^3*b^2 )*c*d^3 + 3*(D*a^5 - B*a^4*b)*d^4 + (2*(C*a*b^4 - 2*A*b^5)*c^3*d - (7*D*a^ 2*b^3 + B*a*b^4)*c^2*d^2 + 2*(C*a^2*b^3 + 4*A*a*b^4)*c*d^3 + 3*(D*a^3*b^2 - B*a^2*b^3)*d^4)*x^4 - 2*(2*(C*a^2*b^3 - 2*A*a*b^4)*c^3*d - (7*D*a^3*b^2 + B*a^2*b^3)*c^2*d^2 + 2*(C*a^3*b^2 + 4*A*a^2*b^3)*c*d^3 + 3*(D*a^4*b - B* a^3*b^2)*d^4)*x^2)*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* (6*B*a^3*b^2*c*d^3 + 2*(2*D*a^3*b^2 - B*a^2*b^3)*c^3*d - 3*(C*a^3*b^2 - A* a^2*b^3)*c^2*d^2 - (C*a^4*b + 7*A*a^3*b^2)*d^4 - (2*(C*a*b^4 - 2*A*b^5)*c^ 3*d - (7*D*a^2*b^3 + B*a*b^4)*c^2*d^2 + 2*(C*a^2*b^3 + 4*A*a*b^4)*c*d^3...
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2)/(-b*x**2+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm= "maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm= "giac")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (a-b\,x^2\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int((A + B*x + C*x^2 + x^3*D)/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)),x)
Output:
int((A + B*x + C*x^2 + x^3*D)/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {A+B x+C x^2+D x^3}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {D x^{3}+C \,x^{2}+B x +A}{\sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:
int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)
Output:
int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)