\(\int \frac {(c+d x)^{3/2} (A+B x+C x^2+D x^3)}{(a-b x^2)^{5/2}} \, dx\) [162]

Optimal result

Integrand size = 37, antiderivative size = 461 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\frac {\left (a \left (B+\frac {a D}{b}\right )+(A b+a C) x\right ) (c+d x)^{3/2}}{3 a b \left (a-b x^2\right )^{3/2}}+\frac {\sqrt {c+d x} \left (a (A b d-5 a C d-6 a c D)+\left (4 A b^2 c-a (2 b c C+3 b B d+9 a d D)\right ) x\right )}{6 a^2 b^2 \sqrt {a-b x^2}}+\frac {\left (4 A b^2 c-a (2 b c C+3 b B d+21 a d D)\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 a^{3/2} b^{5/2} \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\left (A b \left (4 b c^2-a d^2\right )-a (b c (2 c C+3 B d)-a d (5 C d-3 c D))\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 a^{3/2} b^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

1/3*(a*(B+a*D/b)+(A*b+C*a)*x)*(d*x+c)^(3/2)/a/b/(-b*x^2+a)^(3/2)+1/6*(d*x+ 
c)^(1/2)*(a*(A*b*d-5*C*a*d-6*D*a*c)+(4*A*b^2*c-a*(3*B*b*d+2*C*b*c+9*D*a*d) 
)*x)/a^2/b^2/(-b*x^2+a)^(1/2)+1/6*(4*A*b^2*c-a*(3*B*b*d+2*C*b*c+21*D*a*d)) 
*(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^(5/ 
2)/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-1/6*(A*b*(-a*d^2 
+4*b*c^2)-a*(b*c*(3*B*d+2*C*c)-a*d*(5*C*d-3*D*c)))*((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))/a^(3/2)/b^(5/ 
2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.02 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.40 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (-4 A b^3 c x^3-a^3 (5 C d+4 c D+7 d D x)+a b^2 x \left ((2 c C+3 B d) x^2+A (6 c+d x)\right )+a^2 b \left (A d+B (2 c-d x)+x^2 (7 C d+6 c D+9 d D x)\right )\right )}{a^2 b^2 \left (a-b x^2\right )^2}+\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (4 A b^2 c-a (2 b c C+3 b B d+21 a d D)\right ) \left (-a+b x^2\right )-i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (4 A b^2 c-a (2 b c C+3 b B d+21 a d D)\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} \sqrt {b} d \left (A \left (4 b^2 c-\sqrt {a} b^{3/2} d\right )+a \left (-b (2 c C+3 B d)-21 a d D+\sqrt {a} \sqrt {b} (5 C d+18 c D)\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{a^2 b^3 d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{6 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*(((c + d*x)*(-4*A*b^3*c*x^3 - a^3*(5*C*d + 4*c*D + 7*d*D* 
x) + a*b^2*x*((2*c*C + 3*B*d)*x^2 + A*(6*c + d*x)) + a^2*b*(A*d + B*(2*c - 
 d*x) + x^2*(7*C*d + 6*c*D + 9*d*D*x))))/(a^2*b^2*(a - b*x^2)^2) + (d^2*Sq 
rt[-c + (Sqrt[a]*d)/Sqrt[b]]*(4*A*b^2*c - a*(2*b*c*C + 3*b*B*d + 21*a*d*D) 
)*(-a + b*x^2) - I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(4*A*b^2*c - a*(2*b*c*C 
 + 3*b*B*d + 21*a*d*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)/(S 
qrt[b]*c - Sqrt[a]*d)] - I*Sqrt[a]*Sqrt[b]*d*(A*(4*b^2*c - Sqrt[a]*b^(3/2) 
*d) + a*(-(b*(2*c*C + 3*B*d)) - 21*a*d*D + Sqrt[a]*Sqrt[b]*(5*C*d + 18*c*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 
)])/(a^2*b^3*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2))))/(6*Sqrt[c + 
d*x])
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 481, 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 = {2176, 27, 2176, 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.

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

\(\Big \downarrow \) 2176

\(\displaystyle \frac {\int \frac {\sqrt {c+d x} \left (-6 a d D x^2+(A b d-5 a C d-6 a c D) x+4 A b c-\frac {a (2 b c C+3 b B d+3 a d D)}{b}\right )}{2 \left (a-b x^2\right )^{3/2}}dx}{3 a b}+\frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c+d x} \left (-6 a d D x^2+(A b d-5 a C d-6 a c D) x+4 A b c-\frac {a (2 b c C+3 b B d+3 a d D)}{b}\right )}{\left (a-b x^2\right )^{3/2}}dx}{6 a b}+\frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2176

\(\displaystyle \frac {\frac {\int -\frac {d \left (a (A b d-5 a C d-18 a c D)+\left (4 A b^2 c-a (2 b c C+3 b B d+21 a d D)\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a b}+\frac {\sqrt {c+d x} \left (x \left (4 A b^2 c-a (9 a d D+3 b B d+2 b c C)\right )+a (-6 a c D-5 a C d+A b d)\right )}{a b \sqrt {a-b x^2}}}{6 a b}+\frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {c+d x} \left (x \left (4 A b^2 c-a (9 a d D+3 b B d+2 b c C)\right )+a (-6 a c D-5 a C d+A b d)\right )}{a b \sqrt {a-b x^2}}-\frac {d \int \frac {a (A b d-5 a C d-18 a c D)+\left (4 A b^2 c-a (2 b c C+3 b B d+21 a d D)\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a b}}{6 a b}+\frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

((a*(B + (a*D)/b) + (A*b + a*C)*x)*(c + d*x)^(3/2))/(3*a*b*(a - b*x^2)^(3/ 
2)) + ((Sqrt[c + d*x]*(a*(A*b*d - 5*a*C*d - 6*a*c*D) + (4*A*b^2*c - a*(2*b 
*c*C + 3*b*B*d + 9*a*d*D))*x))/(a*b*Sqrt[a - b*x^2]) - (d*((-2*Sqrt[a]*(4* 
A*b^2*c - a*(2*b*c*C + 3*b*B*d + 21*a*d*D))*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]*(A*b*(4*b*c^2 - a*d^2) - a*(b*c*(2*c 
*C + 3*B*d) - a*d*(5*C*d - 3*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))/(6*a*b)
 

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 2176

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*(a + b*x^2)^(p + 1)*((a*S - b*R*x)/(2*a*b*(p 
 + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^(p 
 + 1)*ExpandToSum[2*a*b*(p + 1)*(d + e*x)*Qx - a*e*S*m + b*d*R*(2*p + 3) + 
b*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] &&  !(IGtQ[m, 0] && R 
ationalQ[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. \(899\) vs. \(2(395)=790\).

Time = 6.80 (sec) , antiderivative size = 900, normalized size of antiderivative = 1.95

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {\left (\frac {\left (A \,b^{2} c +B a b d +C a b c +a^{2} d D\right ) x}{3 a \,b^{4}}+\frac {A b d +B b c +C a d +D a c}{3 b^{4}}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{\left (x^{2}-\frac {a}{b}\right )^{2}}-\frac {2 \left (-b d x -b c \right ) \left (\frac {\left (4 A \,b^{2} c -3 B a b d -2 C a b c -9 a^{2} d D\right ) x}{12 a^{2} b^{3}}-\frac {A b d +7 C a d +6 D a c}{12 a \,b^{3}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 \left (\frac {d \left (C d +2 D c \right )}{b^{2}}-\frac {A a b \,d^{2}-4 A \,b^{2} c^{2}+3 a b B c d +7 a^{2} C \,d^{2}+2 C a b \,c^{2}+15 a^{2} c d D}{6 b^{2} a^{2}}+\frac {d \left (A b d +7 C a d +6 D a c \right )}{12 b^{2} a}-\frac {c \left (4 A \,b^{2} c -3 B a b d -2 C a b c -9 a^{2} d D\right )}{6 b^{2} a^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (\frac {D d^{2}}{b^{2}}-\frac {\left (4 A \,b^{2} c -3 B a b d -2 C a b c -9 a^{2} d D\right ) d}{12 a^{2} b^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(900\)
default	\(\text {Expression too large to display}\)	\(5898\)

Input:

int((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(-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^4*x+1/3*(A*b*d+B*b*c+C*a*d+D*a*c)/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*(4*A*b^2*c- 
3*B*a*b*d-2*C*a*b*c-9*D*a^2*d)/a^2/b^3*x-1/12*(A*b*d+7*C*a*d+6*D*a*c)/a/b^ 
3)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2*(d*(C*d+2*D*c)/b^2-1/6/b^2*(A*a*b*d^2- 
4*A*b^2*c^2+3*B*a*b*c*d+7*C*a^2*d^2+2*C*a*b*c^2+15*D*a^2*c*d)/a^2+1/12/b^2 
*d*(A*b*d+7*C*a*d+6*D*a*c)/a-1/6/b^2*c*(4*A*b^2*c-3*B*a*b*d-2*C*a*b*c-9*D* 
a^2*d)/a^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*(D*d^2/b^2-1/12*(4*A*b^2*c-3*B*a*b*d-2*C*a*b*c-9*D*a^2*d 
)*d/a^2/b^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)^(1/2)))^(1/2))+1/b*(a*b)^(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))))
 

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.52 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/18*(((2*(C*a*b^3 - 2*A*b^4)*c^2 - 3*(11*D*a^2*b^2 - B*a*b^3)*c*d - 3*(5* 
C*a^2*b^2 - A*a*b^3)*d^2)*x^4 + 2*(C*a^3*b - 2*A*a^2*b^2)*c^2 - 3*(11*D*a^ 
4 - B*a^3*b)*c*d - 3*(5*C*a^4 - A*a^3*b)*d^2 - 2*(2*(C*a^2*b^2 - 2*A*a*b^3 
)*c^2 - 3*(11*D*a^3*b - B*a^2*b^2)*c*d - 3*(5*C*a^3*b - A*a^2*b^2)*d^2)*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*b^3 - 2*A*b^4)*c 
*d + 3*(7*D*a^2*b^2 + B*a*b^3)*d^2)*x^4 + 2*(C*a^3*b - 2*A*a^2*b^2)*c*d + 
3*(7*D*a^4 + B*a^3*b)*d^2 - 2*(2*(C*a^2*b^2 - 2*A*a*b^3)*c*d + 3*(7*D*a^3* 
b + B*a^2*b^2)*d^2)*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*((2*(C*a*b^3 - 2*A*b^4)*c*d + 3*(3*D*a^2*b^2 + B*a*b^3)*d^2)*x^3 - 2 
*(2*D*a^3*b - B*a^2*b^2)*c*d - (5*C*a^3*b - A*a^2*b^2)*d^2 + (6*D*a^2*b^2* 
c*d + (7*C*a^2*b^2 + A*a*b^3)*d^2)*x^2 + (6*A*a*b^3*c*d - (7*D*a^3*b + B*a 
^2*b^2)*d^2)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a^2*b^5*d*x^4 - 2*a^3*b^4 
*d*x^2 + a^4*b^3*d)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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