\(\int \frac {x^3 (A+B x+C x^2)}{\sqrt {c+d x} (a-b x^2)^{5/2}} \, dx\) [233]

Optimal result

Integrand size = 35, antiderivative size = 608 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {a \sqrt {c+d x} (A b c+a c C-a B d+(b B c-A b d-a C d) x)}{3 b^2 \left (b c^2-a d^2\right ) \left (a-b x^2\right )^{3/2}}-\frac {\sqrt {c+d x} \left (2 A b c \left (3 b c^2-a d^2\right )+a \left (b c^2 (12 c C-11 B d)-a d^2 (8 c C-7 B d)\right )+\left (9 a^2 C d^3+b^2 c^2 (8 B c-7 A d)-a b d \left (13 c^2 C+4 B c d-3 A d^2\right )\right ) x\right )}{6 b^2 \left (b c^2-a d^2\right )^2 \sqrt {a-b x^2}}-\frac {\sqrt {a} \left (21 a^2 C d^4+b^2 c^2 \left (12 c^2 C+8 B c d-7 A d^2\right )-a b d^2 \left (37 c^2 C+4 B c d-3 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 b^{5/2} d \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\sqrt {a} \left (a d^2 (13 c C-5 B d)-b c \left (12 c^2 C-4 B c d-A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 b^{5/2} d \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

1/3*a*(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^2/(-a*d^2 
+b*c^2)/(-b*x^2+a)^(3/2)-1/6*(d*x+c)^(1/2)*(2*A*b*c*(-a*d^2+3*b*c^2)+a*(b* 
c^2*(-11*B*d+12*C*c)-a*d^2*(-7*B*d+8*C*c))+(9*a^2*C*d^3+b^2*c^2*(-7*A*d+8* 
B*c)-a*b*d*(-3*A*d^2+4*B*c*d+13*C*c^2))*x)/b^2/(-a*d^2+b*c^2)^2/(-b*x^2+a) 
^(1/2)-1/6*a^(1/2)*(21*a^2*C*d^4+b^2*c^2*(-7*A*d^2+8*B*c*d+12*C*c^2)-a*b*d 
^2*(-3*A*d^2+4*B*c*d+37*C*c^2))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*Ellipti 
cE(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^(5/2)/d/(-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^(1/2)*(a*d^2*(-5*B*d+13*C*c)-b*c*(-A*d^2-4 
*B*c*d+12*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^(5/2)/d/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2 
+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 30.59 (sec) , antiderivative size = 813, normalized size of antiderivative = 1.34 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {b (c+d x) \left (2 a \left (b c^2-a d^2\right ) (a c C+b B c x-a d (B+C x)+A b (c-d x))+\left (-a+b x^2\right ) \left (8 b^2 B c^3 x+a^2 d^2 (-8 c C+7 B d+9 C d x)+a b c \left (12 c^2 C-11 B c d-13 c C d x-4 B d^2 x\right )+A b \left (b c^2 (6 c-7 d x)+a d^2 (-2 c+3 d x)\right )\right )\right )}{\left (a-b x^2\right )^2}-\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (21 a^2 C d^4+b^2 c^2 \left (12 c^2 C+8 B c d-7 A d^2\right )+a b d^2 \left (-37 c^2 C-4 B c d+3 A d^2\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (21 a^2 C d^4+b^2 c^2 \left (12 c^2 C+8 B c d-7 A d^2\right )+a b d^2 \left (-37 c^2 C-4 B c d+3 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {b} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (21 a^2 C d^3+6 b^2 c^2 (2 B c-A d)+a^{3/2} \sqrt {b} d^2 (13 c C-5 B d)+3 a b d \left (-8 c^2 C-3 B c d+A d^2\right )+\sqrt {a} b^{3/2} c \left (-12 c^2 C+4 B c d+A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{6 b^3 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}} \] Input:

Integrate[(x^3*(A + B*x + C*x^2))/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
 

Output:

(Sqrt[a - b*x^2]*((b*(c + d*x)*(2*a*(b*c^2 - a*d^2)*(a*c*C + b*B*c*x - a*d 
*(B + C*x) + A*b*(c - d*x)) + (-a + b*x^2)*(8*b^2*B*c^3*x + a^2*d^2*(-8*c* 
C + 7*B*d + 9*C*d*x) + a*b*c*(12*c^2*C - 11*B*c*d - 13*c*C*d*x - 4*B*d^2*x 
) + A*b*(b*c^2*(6*c - 7*d*x) + a*d^2*(-2*c + 3*d*x)))))/(a - b*x^2)^2 - (d 
^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(21*a^2*C*d^4 + b^2*c^2*(12*c^2*C + 8*B* 
c*d - 7*A*d^2) + a*b*d^2*(-37*c^2*C - 4*B*c*d + 3*A*d^2))*(a - b*x^2) + I* 
Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(21*a^2*C*d^4 + b^2*c^2*(12*c^2*C + 8*B*c* 
d - 7*A*d^2) + a*b*d^2*(-37*c^2*C - 4*B*c*d + 3*A*d^2))*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[b]*d*(Sqrt 
[b]*c - Sqrt[a]*d)*(21*a^2*C*d^3 + 6*b^2*c^2*(2*B*c - A*d) + a^(3/2)*Sqrt[ 
b]*d^2*(13*c*C - 5*B*d) + 3*a*b*d*(-8*c^2*C - 3*B*c*d + A*d^2) + Sqrt[a]*b 
^(3/2)*c*(-12*c^2*C + 4*B*c*d + A*d^2))*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)*Ell 
ipticF[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^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b] 
]*(a - b*x^2))))/(6*b^3*(b*c^2 - a*d^2)^2*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 2.50 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2180, 27, 2180, 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.

\(\displaystyle \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a-b x^2\right )^{5/2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 2180

\(\displaystyle \frac {\int -\frac {6 a C \left (c^2-\frac {a d^2}{b}\right ) x^3+6 a B \left (c^2-\frac {a d^2}{b}\right ) x^2+\frac {3 a \left (A b \left (2 b c^2-a d^2\right )-a \left (a C d^2-b c (2 c C-B d)\right )\right ) x}{b^2}+\frac {a^2 (b c (2 B c-A d)-a d (c C+B d))}{b^2}}{2 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{3 a \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\int \frac {6 a C \left (c^2-\frac {a d^2}{b}\right ) x^3+6 a B \left (c^2-\frac {a d^2}{b}\right ) x^2+\frac {3 a \left (A b \left (2 b c^2-a d^2\right )-a \left (a C d^2-b c (2 c C-B d)\right )\right ) x}{b^2}+\frac {a^2 (b c (2 B c-A d)-a d (c C+B d))}{b^2}}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}}dx}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 2180

\(\displaystyle \frac {a \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {\int -\frac {a^2 \left (6 b^2 (2 B c-A d) c^3-a b d \left (12 C c^2+13 B d c-2 A d^2\right ) c+a^2 d^3 (8 c C+5 B d)+\left (21 a^2 C d^4-a b \left (37 C c^2+4 B d c-3 A d^2\right ) d^2+b^2 c^2 \left (12 C c^2+8 B d c-7 A d^2\right )\right ) x\right )}{2 b^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}+\frac {a \sqrt {c+d x} \left (x \left (9 a^2 C d^3-a b d \left (-3 A d^2+4 B c d+13 c^2 C\right )+b^2 c^2 (8 B c-7 A d)\right )+2 A b c \left (3 b c^2-a d^2\right )+a \left (b c^2 (12 c C-11 B d)-a d^2 (8 c C-7 B d)\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (x \left (9 a^2 C d^3-a b d \left (-3 A d^2+4 B c d+13 c^2 C\right )+b^2 c^2 (8 B c-7 A d)\right )+2 A b c \left (3 b c^2-a d^2\right )+a \left (b c^2 (12 c C-11 B d)-a d^2 (8 c C-7 B d)\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {a \int \frac {6 b^2 (2 B c-A d) c^3-a b d \left (12 C c^2+13 B d c-2 A d^2\right ) c+a^2 d^3 (8 c C+5 B d)+\left (21 a^2 C d^4-a b \left (37 C c^2+4 B d c-3 A d^2\right ) d^2+b^2 c^2 \left (12 C c^2+8 B d c-7 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {a \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (x \left (9 a^2 C d^3-a b d \left (-3 A d^2+4 B c d+13 c^2 C\right )+b^2 c^2 (8 B c-7 A d)\right )+2 A b c \left (3 b c^2-a d^2\right )+a \left (b c^2 (12 c C-11 B d)-a d^2 (8 c C-7 B d)\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {a \left (\frac {\left (21 a^2 C d^4-a b d^2 \left (-3 A d^2+4 B c d+37 c^2 C\right )+b^2 c^2 \left (-7 A d^2+8 B c d+12 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}+\frac {\left (b c^2-a d^2\right ) \left (a d^2 (13 c C-5 B d)-b c \left (-A d^2-4 B c d+12 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {a \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (x \left (9 a^2 C d^3-a b d \left (-3 A d^2+4 B c d+13 c^2 C\right )+b^2 c^2 (8 B c-7 A d)\right )+2 A b c \left (3 b c^2-a d^2\right )+a \left (b c^2 (12 c C-11 B d)-a d^2 (8 c C-7 B d)\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {a \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (21 a^2 C d^4-a b d^2 \left (-3 A d^2+4 B c d+37 c^2 C\right )+b^2 c^2 \left (-7 A d^2+8 B c d+12 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}+\frac {\left (b c^2-a d^2\right ) \left (a d^2 (13 c C-5 B d)-b c \left (-A d^2-4 B c d+12 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {a \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (x \left (9 a^2 C d^3-a b d \left (-3 A d^2+4 B c d+13 c^2 C\right )+b^2 c^2 (8 B c-7 A d)\right )+2 A b c \left (3 b c^2-a d^2\right )+a \left (b c^2 (12 c C-11 B d)-a d^2 (8 c C-7 B d)\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (a d^2 (13 c C-5 B d)-b c \left (-A d^2-4 B c d+12 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (21 a^2 C d^4-a b d^2 \left (-3 A d^2+4 B c d+37 c^2 C\right )+b^2 c^2 \left (-7 A d^2+8 B c d+12 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {a \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (x \left (9 a^2 C d^3-a b d \left (-3 A d^2+4 B c d+13 c^2 C\right )+b^2 c^2 (8 B c-7 A d)\right )+2 A b c \left (3 b c^2-a d^2\right )+a \left (b c^2 (12 c C-11 B d)-a d^2 (8 c C-7 B d)\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {a \left (\frac {\left (b c^2-a d^2\right ) \left (a d^2 (13 c C-5 B d)-b c \left (-A d^2-4 B c d+12 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (21 a^2 C d^4-a b d^2 \left (-3 A d^2+4 B c d+37 c^2 C\right )+b^2 c^2 \left (-7 A d^2+8 B c d+12 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {a \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (x \left (9 a^2 C d^3-a b d \left (-3 A d^2+4 B c d+13 c^2 C\right )+b^2 c^2 (8 B c-7 A d)\right )+2 A b c \left (3 b c^2-a d^2\right )+a \left (b c^2 (12 c C-11 B d)-a d^2 (8 c C-7 B d)\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {a \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (a d^2 (13 c C-5 B d)-b c \left (-A d^2-4 B c d+12 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (21 a^2 C d^4-a b d^2 \left (-3 A d^2+4 B c d+37 c^2 C\right )+b^2 c^2 \left (-7 A d^2+8 B c d+12 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {a \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (x \left (9 a^2 C d^3-a b d \left (-3 A d^2+4 B c d+13 c^2 C\right )+b^2 c^2 (8 B c-7 A d)\right )+2 A b c \left (3 b c^2-a d^2\right )+a \left (b c^2 (12 c C-11 B d)-a d^2 (8 c C-7 B d)\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {a \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (a d^2 (13 c C-5 B d)-b c \left (-A d^2-4 B c d+12 c^2 C\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (21 a^2 C d^4-a b d^2 \left (-3 A d^2+4 B c d+37 c^2 C\right )+b^2 c^2 \left (-7 A d^2+8 B c d+12 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {a \sqrt {c+d x} (x (-a C d-A b d+b B c)-a B d+a c C+A b c)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}-\frac {\frac {a \sqrt {c+d x} \left (x \left (9 a^2 C d^3-a b d \left (-3 A d^2+4 B c d+13 c^2 C\right )+b^2 c^2 (8 B c-7 A d)\right )+2 A b c \left (3 b c^2-a d^2\right )+a \left (b c^2 (12 c C-11 B d)-a d^2 (8 c C-7 B d)\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {a \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (21 a^2 C d^4-a b d^2 \left (-3 A d^2+4 B c d+37 c^2 C\right )+b^2 c^2 \left (-7 A d^2+8 B c d+12 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (a d^2 (13 c C-5 B d)-b c \left (-A d^2-4 B c d+12 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )}{2 b^2 \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}\)

Input:

Int[(x^3*(A + B*x + C*x^2))/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
 

Output:

(a*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 
^2*(b*c^2 - a*d^2)*(a - b*x^2)^(3/2)) - ((a*Sqrt[c + d*x]*(2*A*b*c*(3*b*c^ 
2 - a*d^2) + a*(b*c^2*(12*c*C - 11*B*d) - a*d^2*(8*c*C - 7*B*d)) + (9*a^2* 
C*d^3 + b^2*c^2*(8*B*c - 7*A*d) - a*b*d*(13*c^2*C + 4*B*c*d - 3*A*d^2))*x) 
)/(b^2*(b*c^2 - a*d^2)*Sqrt[a - b*x^2]) - (a*((-2*Sqrt[a]*(21*a^2*C*d^4 + 
b^2*c^2*(12*c^2*C + 8*B*c*d - 7*A*d^2) - a*b*d^2*(37*c^2*C + 4*B*c*d - 3*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)])/(Sqrt[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)*(a*d^2*(13*c*C - 5*B*d) - b*c*(12*c^2*C - 4*B*c*d - A*d^2 
))*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*E 
llipticF[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*b^2*(b*c^2 
- a*d^2)))/(6*a*(b*c^2 - a*d^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

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])
 

rule 327

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]
 

rule 508

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]
 

rule 509

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]
 

rule 511

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]
 

rule 512

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]
 

rule 600

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]
 

rule 2180

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]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1109\) vs. \(2(542)=1084\).

Time = 7.51 (sec) , antiderivative size = 1110, normalized size of antiderivative = 1.83

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1110\)
default	\(\text {Expression too large to display}\)	\(8457\)

Input:

int(x^3*(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*(A*b*d-B 
*b*c+C*a*d)/(a*d^2-b*c^2)/b^4*x-1/3*a*(A*b*c-B*a*d+C*a*c)/(a*d^2-b*c^2)/b^ 
4)*(-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*(3 
*A*a*b*d^3-7*A*b^2*c^2*d-4*B*a*b*c*d^2+8*B*b^2*c^3+9*C*a^2*d^3-13*C*a*b*c^ 
2*d)/(a*d^2-b*c^2)^2/b^3*x+1/12*(2*A*a*b*c*d^2-6*A*b^2*c^3-7*B*a^2*d^3+11* 
B*a*b*c^2*d+8*C*a^2*c*d^2-12*C*a*b*c^3)/b^3/(a*d^2-b*c^2)^2)/((x^2-a/b)*(- 
b*d*x-b*c))^(1/2)+2*(B/b^2-1/6*(A*b*c*d+7*B*a*d^2-8*B*b*c^2+C*a*c*d)/b^2/( 
a*d^2-b*c^2)-1/12/b^2*d*(2*A*a*b*c*d^2-6*A*b^2*c^3-7*B*a^2*d^3+11*B*a*b*c^ 
2*d+8*C*a^2*c*d^2-12*C*a*b*c^3)/(a*d^2-b*c^2)^2+1/6/b^2*c*(3*A*a*b*d^3-7*A 
*b^2*c^2*d-4*B*a*b*c*d^2+8*B*b^2*c^3+9*C*a^2*d^3-13*C*a*b*c^2*d)/(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))+2*(C/b^2+1/12*d*(3*A*a*b*d^3-7*A*b^2*c^2*d-4*B*a*b*c*d^2+8*B*b 
^2*c^3+9*C*a^2*d^3-13*C*a*b*c^2*d)/b^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)*((-c/d-1/b*(a*b)^(1/2))*EllipticE(((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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1116 vs. \(2 (546) = 1092\).

Time = 0.12 (sec) , antiderivative size = 1116, normalized size of antiderivative = 1.84 \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(x^3*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="f 
ricas")
 

Output:

1/18*((12*C*a^2*b^2*c^5 - 28*B*a^2*b^2*c^4*d + 35*B*a^3*b*c^2*d^3 - 15*B*a 
^4*d^5 - (C*a^3*b - 11*A*a^2*b^2)*c^3*d^2 - 3*(C*a^4 + A*a^3*b)*c*d^4 + (1 
2*C*b^4*c^5 - 28*B*b^4*c^4*d + 35*B*a*b^3*c^2*d^3 - 15*B*a^2*b^2*d^5 - (C* 
a*b^3 - 11*A*b^4)*c^3*d^2 - 3*(C*a^2*b^2 + A*a*b^3)*c*d^4)*x^4 - 2*(12*C*a 
*b^3*c^5 - 28*B*a*b^3*c^4*d + 35*B*a^2*b^2*c^2*d^3 - 15*B*a^3*b*d^5 - (C*a 
^2*b^2 - 11*A*a*b^3)*c^3*d^2 - 3*(C*a^3*b + A*a^2*b^2)*c*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*(12*C*a^2*b^2*c^4*d + 8*B*a^2*b^2*c 
^3*d^2 - 4*B*a^3*b*c*d^4 - (37*C*a^3*b + 7*A*a^2*b^2)*c^2*d^3 + 3*(7*C*a^4 
 + A*a^3*b)*d^5 + (12*C*b^4*c^4*d + 8*B*b^4*c^3*d^2 - 4*B*a*b^3*c*d^4 - (3 
7*C*a*b^3 + 7*A*b^4)*c^2*d^3 + 3*(7*C*a^2*b^2 + A*a*b^3)*d^5)*x^4 - 2*(12* 
C*a*b^3*c^4*d + 8*B*a*b^3*c^3*d^2 - 4*B*a^2*b^2*c*d^4 - (37*C*a^2*b^2 + 7* 
A*a*b^3)*c^2*d^3 + 3*(7*C*a^3*b + A*a^2*b^2)*d^5)*x^2)*sqrt(-b*d)*weierstr 
assZeta(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*(9*B*a^2*b^2*c^2*d^3 + 6*C*a^3*b*c*d^4 
 - 5*B*a^3*b*d^5 - 2*(5*C*a^2*b^2 + 2*A*a*b^3)*c^3*d^2 + (8*B*b^4*c^3*d^2 
- 4*B*a*b^3*c*d^4 - (13*C*a*b^3 + 7*A*b^4)*c^2*d^3 + 3*(3*C*a^2*b^2 + A*a* 
b^3)*d^5)*x^3 - (11*B*a*b^3*c^2*d^3 - 7*B*a^2*b^2*d^5 - 6*(2*C*a*b^3 + A*b 
^4)*c^3*d^2 + 2*(4*C*a^2*b^2 + A*a*b^3)*c*d^4)*x^2 - (6*B*a*b^3*c^3*d^2...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**3*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(-b*x**2+a)**(5/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{3}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:

integrate(x^3*(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^3/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
 

Giac [F]

\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{3}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:

integrate(x^3*(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^3/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^3\,\left (C\,x^2+B\,x+A\right )}{{\left (a-b\,x^2\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int((x^3*(A + B*x + C*x^2))/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)),x)
 

Output:

int((x^3*(A + B*x + C*x^2))/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^{3} \left (C \,x^{2}+B x +A \right )}{\sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:

int(x^3*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)
 

Output:

int(x^3*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)