\(\int \frac {\sqrt {c+d x} (A+B x+C x^2+D x^3)}{(a-b x^2)^{5/2}} \, dx\) [163]

Optimal result

Integrand size = 37, antiderivative size = 532 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\frac {\left (a \left (B+\frac {a D}{b}\right )+(A b+a C) x\right ) \sqrt {c+d x}}{3 a b \left (a-b x^2\right )^{3/2}}-\frac {\sqrt {c+d x} \left (a \left (A b^2 c d-a \left (7 a d^2 D-b \left (c C d-B d^2+6 c^2 D\right )\right )\right )-b \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (2 c C+B d)-a d (3 C d-c D))\right ) x\right )}{6 a^2 b^2 \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}+\frac {\left (A b \left (4 b c^2-3 a d^2\right )-a (b c (2 c C+B d)-a d (3 C d-c D))\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 a^{3/2} b^{3/2} \left (b c^2-a d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\left (4 A b^2 c-a (b (2 c C+B d)-5 a d D)\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 a^{3/2} b^{5/2} \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

1/3*(a*(B+a*D/b)+(A*b+C*a)*x)*(d*x+c)^(1/2)/a/b/(-b*x^2+a)^(3/2)-1/6*(d*x+ 
c)^(1/2)*(a*(A*b^2*c*d-a*(7*a*d^2*D-b*(-B*d^2+C*c*d+6*D*c^2)))-b*(A*b*(-3* 
a*d^2+4*b*c^2)-a*(b*c*(B*d+2*C*c)-a*d*(3*C*d-D*c)))*x)/a^2/b^2/(-a*d^2+b*c 
^2)/(-b*x^2+a)^(1/2)+1/6*(A*b*(-3*a*d^2+4*b*c^2)-a*(b*c*(B*d+2*C*c)-a*d*(3 
*C*d-D*c)))*(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)/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2 
+a)^(1/2)-1/6*(4*A*b^2*c-a*(b*(B*d+2*C*c)-5*D*a*d))*((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 = 29.43 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (2 a \left (b c^2-a d^2\right ) \left (a^2 D+A b^2 x+a b (B+C x)\right )-\left (a-b x^2\right ) \left (-7 a^3 d^2 D-4 A b^3 c^2 x+a b^2 (c (2 c C+B d) x+A d (c+3 d x))+a^2 b \left (6 c^2 D-d^2 (B+3 C x)+c d (C+D x)\right )\right )\right )}{\left (a-b x^2\right )^2}+\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (2 c C+B d)+a d (-3 C d+c D))\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (2 c C+B d)+a d (-3 C d+c D))\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (A \left (4 b^2 c+3 \sqrt {a} b^{3/2} d\right )+a \left (-b (2 c C+B d)+5 a d D-3 \sqrt {a} \sqrt {b} (C d-2 c D)\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{6 a^2 b^2 \left (b c^2-a d^2\right ) \sqrt {c+d x}} \] Input:

Integrate[(Sqrt[c + d*x]*(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)*(2*a*(b*c^2 - a*d^2)*(a^2*D + A*b^2*x + a*b*( 
B + C*x)) - (a - b*x^2)*(-7*a^3*d^2*D - 4*A*b^3*c^2*x + a*b^2*(c*(2*c*C + 
B*d)*x + A*d*(c + 3*d*x)) + a^2*b*(6*c^2*D - d^2*(B + 3*C*x) + c*d*(C + D* 
x)))))/(a - b*x^2)^2 + (d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(A*b*(4*b*c^2 - 
 3*a*d^2) - a*(b*c*(2*c*C + B*d) + a*d*(-3*C*d + c*D)))*(a - b*x^2) + I*Sq 
rt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(A*b*(4*b*c^2 - 3*a*d^2) - a*(b*c*(2*c*C + B 
*d) + a*d*(-3*C*d + 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)*EllipticE[I*ArcS 
inh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d) 
/(Sqrt[b]*c - Sqrt[a]*d)] + I*Sqrt[a]*d*(Sqrt[b]*c - Sqrt[a]*d)*(A*(4*b^2* 
c + 3*Sqrt[a]*b^(3/2)*d) + a*(-(b*(2*c*C + B*d)) + 5*a*d*D - 3*Sqrt[a]*Sqr 
t[b]*(C*d - 2*c*D)))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sq 
rt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sq 
rt[-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)*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 556, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {2176, 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.

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

\(\Big \downarrow \) 2176

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2180

\(\displaystyle \frac {\frac {\int -\frac {d \left (a \left (A c d b^2+a \left (5 a D d^2+b \left (-6 D c^2+C d c-B d^2\right )\right )\right )+b \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (2 c C+B d)-a d (3 C d-c D))\right ) x\right )}{2 b \sqrt {c+d x} \sqrt {a-b x^2}}dx}{a \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (a \left (-\frac {d^2 (7 a D+b B)}{b}+6 c^2 D+c C d\right )+A b c d\right )-x \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (B d+2 c C)-a d (3 C d-c D))\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b}+\frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {d \int \frac {a \left (A c d b^2+a \left (5 a D d^2+b \left (-6 D c^2+C d c-B d^2\right )\right )\right )+b \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (2 c C+B d)-a d (3 C d-c D))\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (a \left (-\frac {d^2 (7 a D+b B)}{b}+6 c^2 D+c C d\right )+A b c d\right )-x \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (B d+2 c C)-a d (3 C d-c D))\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b}+\frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {-\frac {d \left (\frac {b \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (B d+2 c C)-a d (3 C d-c D))\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c-a (-5 a d D+b B d+2 b c C)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{2 a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (a \left (-\frac {d^2 (7 a D+b B)}{b}+6 c^2 D+c C d\right )+A b c d\right )-x \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (B d+2 c C)-a d (3 C d-c D))\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b}+\frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {-\frac {d \left (\frac {b \sqrt {1-\frac {b x^2}{a}} \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (B d+2 c C)-a d (3 C d-c D))\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c-a (-5 a d D+b B d+2 b c C)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{2 a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (a \left (-\frac {d^2 (7 a D+b B)}{b}+6 c^2 D+c C d\right )+A b c d\right )-x \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (B d+2 c C)-a d (3 C d-c D))\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b}+\frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {-\frac {d \left (-\frac {\left (b c^2-a d^2\right ) \left (4 A b^2 c-a (-5 a d D+b B d+2 b c C)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (B d+2 c C)-a d (3 C d-c D))\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 a b \left (b c^2-a d^2\right )}-\frac {\sqrt {c+d x} \left (a \left (a \left (-\frac {d^2 (7 a D+b B)}{b}+6 c^2 D+c C d\right )+A b c d\right )-x \left (A b \left (4 b c^2-3 a d^2\right )-a (b c (B d+2 c C)-a d (3 C d-c D))\right )\right )}{a \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a b}+\frac {\sqrt {c+d x} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{3 a b \left (a-b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

Int[(Sqrt[c + d*x]*(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)*Sqrt[c + d*x])/(3*a*b*(a - b*x^2)^(3/2) 
) + (-((Sqrt[c + d*x]*(a*(A*b*c*d + a*(c*C*d + 6*c^2*D - (d^2*(b*B + 7*a*D 
))/b)) - (A*b*(4*b*c^2 - 3*a*d^2) - a*(b*c*(2*c*C + B*d) - a*d*(3*C*d - c* 
D)))*x))/(a*(b*c^2 - a*d^2)*Sqrt[a - b*x^2])) - (d*((-2*Sqrt[a]*Sqrt[b]*(A 
*b*(4*b*c^2 - 3*a*d^2) - a*(b*c*(2*c*C + B*d) - a*d*(3*C*d - c*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)])/(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)*(4*A 
*b^2*c - a*(2*b*c*C + b*B*d - 5*a*d*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*(b*c^2 - a*d^2)))/(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]))
 

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. \(1014\) vs. \(2(466)=932\).

Time = 5.64 (sec) , antiderivative size = 1015, normalized size of antiderivative = 1.91

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1015\)
default	\(\text {Expression too large to display}\)	\(7594\)

Input:

int((d*x+c)^(1/2)*(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+C*a 
)/b^3/a*x+1/3*(B*b+D*a)/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/b^2*(3*A*a*b*d^2-4*A*b^2*c^2+B*a*b*c*d-3*C*a^2*d^2+ 
2*C*a*b*c^2+D*a^2*c*d)/a^2/(a*d^2-b*c^2)*x+1/12*(A*b^2*c*d-B*a*b*d^2+C*a*b 
*c*d-7*D*a^2*d^2+6*D*a*b*c^2)/(a*d^2-b*c^2)/a/b^3)/((x^2-a/b)*(-b*d*x-b*c) 
)^(1/2)+2*(D*d/b^2+1/6/b^2*(4*A*b^2*c-B*a*b*d-2*C*a*b*c-7*D*a^2*d)/a^2-1/1 
2/b^2*d*(A*b^2*c*d-B*a*b*d^2+C*a*b*c*d-7*D*a^2*d^2+6*D*a*b*c^2)/(a*d^2-b*c 
^2)/a-1/6/b*c*(3*A*a*b*d^2-4*A*b^2*c^2+B*a*b*c*d-3*C*a^2*d^2+2*C*a*b*c^2+D 
*a^2*c*d)/a^2/(a*d^2-b*c^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*(3*A*a*b*d^2-4*A*b^2*c^2+B*a*b*c*d- 
3*C*a^2*d^2+2*C*a*b*c^2+D*a^2*c*d)/a^2/b/(a*d^2-b*c^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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 974 vs. \(2 (466) = 932\).

Time = 0.13 (sec) , antiderivative size = 974, normalized size of antiderivative = 1.83 \[ \int \frac {\sqrt {c+d x} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/18*((6*A*a^3*b^2*c*d^2 + (6*A*a*b^4*c*d^2 + 2*(C*a*b^4 - 2*A*b^5)*c^3 - 
(17*D*a^2*b^3 - B*a*b^4)*c^2*d + 3*(5*D*a^3*b^2 - B*a^2*b^3)*d^3)*x^4 + 2* 
(C*a^3*b^2 - 2*A*a^2*b^3)*c^3 - (17*D*a^4*b - B*a^3*b^2)*c^2*d + 3*(5*D*a^ 
5 - B*a^4*b)*d^3 - 2*(6*A*a^2*b^3*c*d^2 + 2*(C*a^2*b^3 - 2*A*a*b^4)*c^3 - 
(17*D*a^3*b^2 - B*a^2*b^3)*c^2*d + 3*(5*D*a^4*b - B*a^3*b^2)*d^3)*x^2)*sqr 
t(-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^4 - 2*A*b^5)*c^2*d + 
 (D*a^2*b^3 + B*a*b^4)*c*d^2 - 3*(C*a^2*b^3 - A*a*b^4)*d^3)*x^4 + 2*(C*a^3 
*b^2 - 2*A*a^2*b^3)*c^2*d + (D*a^4*b + B*a^3*b^2)*c*d^2 - 3*(C*a^4*b - A*a 
^3*b^2)*d^3 - 2*(2*(C*a^2*b^3 - 2*A*a*b^4)*c^2*d + (D*a^3*b^2 + B*a^2*b^3) 
*c*d^2 - 3*(C*a^3*b^2 - A*a^2*b^3)*d^3)*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), weierstras 
sPInverse(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*(2*D*a^3*b^2 - B*a^2*b^3)*c^2*d + (C*a^3*b^2 
+ A*a^2*b^3)*c*d^2 - (5*D*a^4*b - B*a^3*b^2)*d^3 - (2*(C*a*b^4 - 2*A*b^5)* 
c^2*d + (D*a^2*b^3 + B*a*b^4)*c*d^2 - 3*(C*a^2*b^3 - A*a*b^4)*d^3)*x^3 - ( 
6*D*a^2*b^3*c^2*d + (C*a^2*b^3 + A*a*b^4)*c*d^2 - (7*D*a^3*b^2 + B*a^2*b^3 
)*d^3)*x^2 - (6*A*a*b^4*c^2*d - (D*a^3*b^2 + B*a^2*b^3)*c*d^2 + (C*a^3*b^2 
 - 5*A*a^2*b^3)*d^3)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a^4*b^4*c^2*d - a 
^5*b^3*d^3 + (a^2*b^6*c^2*d - a^3*b^5*d^3)*x^4 - 2*(a^3*b^5*c^2*d - a^4...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((d*x+c)^(1/2)*(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)*sqrt(d*x + c)/(-b*x^2 + a)^(5/2), x)
 

Giac [F]

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

integrate((d*x+c)^(1/2)*(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)*sqrt(d*x + c)/(-b*x^2 + a)^(5/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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