\(\int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} (a-b x^2)^{3/2}} \, dx\) [158]

Optimal result

Integrand size = 37, antiderivative size = 564 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {a (b B c-A b d-a C d+a c D)+b \left (c (A b+a C)-a d \left (B+\frac {a D}{b}\right )\right ) x}{a b \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\left (A b d \left (b c^2+3 a d^2\right )+a \left (a d^2 (C d-2 c D)+b c \left (3 c C d-4 B d^2-2 c^2 D\right )\right )\right ) \sqrt {a-b x^2}}{a b \left (b c^2-a d^2\right )^2 \sqrt {c+d x}}+\frac {\left (A b d \left (b c^2+3 a d^2\right )+a \left (a d^2 (C d-2 c D)+b c \left (3 c C d-4 B d^2-2 c^2 D\right )\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} \sqrt {b} d \left (b c^2-a d^2\right )^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\left (A b^2 c d+a \left (a d^2 D+b \left (c C d-B d^2-2 c^2 D\right )\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a} b^{3/2} d \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

(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)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-(A*b*d*(3*a*d^2+b*c^2)+a*(a*d^2*(C* 
d-2*D*c)+b*c*(-4*B*d^2+3*C*c*d-2*D*c^2)))*(-b*x^2+a)^(1/2)/a/b/(-a*d^2+b*c 
^2)^2/(d*x+c)^(1/2)+(A*b*d*(3*a*d^2+b*c^2)+a*(a*d^2*(C*d-2*D*c)+b*c*(-4*B* 
d^2+3*C*c*d-2*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^(1/2)/b^(1/2)/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)-(A*b^2*c*d+a*(a*d^2*D+b*(-B*d^2+C*c*d-2*D*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))/a^(1/2)/b^(3/2)/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 = 29.20 (sec) , antiderivative size = 749, normalized size of antiderivative = 1.33 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (A b d \left (b c^2+3 a d^2\right )+2 a b \left (-c^2 C d+B c d^2-A d^3+c^3 D\right )+a \left (a d^2 (C d-2 c D)+b c \left (3 c C d-4 B d^2-2 c^2 D\right )\right )-\frac {(c+d x) \left (a^3 d^2 D+A b^3 c^2 x+a b^2 \left (c^2 C x+B c (c-2 d x)+A d (-2 c+d x)\right )+a^2 b \left (c^2 D+d^2 (B+C x)-2 c d (C+D x)\right )\right )}{-a+b x^2}-\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (A b d \left (b c^2+3 a d^2\right )+a \left (a d^2 (C d-2 c D)+b c \left (3 c C d-4 B d^2-2 c^2 D\right )\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}-\frac {i \sqrt {a} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (A b^2 c d+\sqrt {a} b^{3/2} \left (-2 c^2 C+3 B c d-3 A d^2\right )+a^2 d^2 D+a^{3/2} \sqrt {b} d (-C d+3 c D)+a b \left (c C d-B d^2-2 c^2 D\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{a b \left (b c^2-a d^2\right )^2 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*(A*b*d*(b*c^2 + 3*a*d^2) + 2*a*b*(-(c^2*C*d) + B*c*d^2 - 
A*d^3 + c^3*D) + a*(a*d^2*(C*d - 2*c*D) + b*c*(3*c*C*d - 4*B*d^2 - 2*c^2*D 
)) - ((c + d*x)*(a^3*d^2*D + A*b^3*c^2*x + a*b^2*(c^2*C*x + B*c*(c - 2*d*x 
) + A*d*(-2*c + d*x)) + a^2*b*(c^2*D + d^2*(B + C*x) - 2*c*d*(C + D*x))))/ 
(-a + b*x^2) - (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(A*b*d*(b*c^2 + 3*a*d^2) 
 + a*(a*d^2*(C*d - 2*c*D) + b*c*(3*c*C*d - 4*B*d^2 - 2*c^2*D)))*Sqrt[(d*(S 
qrt[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]]/Sq 
rt[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)) - (I*Sqrt[a]*(Sqrt[b]*c - Sqrt[a]* 
d)*(A*b^2*c*d + Sqrt[a]*b^(3/2)*(-2*c^2*C + 3*B*c*d - 3*A*d^2) + a^2*d^2*D 
 + a^(3/2)*Sqrt[b]*d*(-(C*d) + 3*c*D) + a*b*(c*C*d - B*d^2 - 2*c^2*D))*Sqr 
t[(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))))/(a*b*(b*c^2 - a*d^2)^2*Sqrt 
[c + d*x])
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {2180, 27, 688, 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.

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

\(\Big \downarrow \) 2180

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

(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)/(a*b*(b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2]) - ((2*(A*b*d*(b*c^ 
2 + 3*a*d^2) + a*(a*d^2*(C*d - 2*c*D) + b*c*(3*c*C*d - 4*B*d^2 - 2*c^2*D)) 
)*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c + d*x]) + ((-2*Sqrt[a]*Sqrt[b]* 
(A*b*d*(b*c^2 + 3*a*d^2) + a*(a*d^2*(C*d - 2*c*D) + b*c*(3*c*C*d - 4*B*d^2 
 - 2*c^2*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)*(A*b^2*c*d + a*(a*d^2*D + b*(c*C*d - B*d^2 - 2*c^2*D)))*Sq 
rt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*Ellipt 
icF[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]))/(b*c^2 - a*d^2))/(2* 
a*b*(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 688

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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. \(1173\) vs. \(2(506)=1012\).

Time = 7.64 (sec) , antiderivative size = 1174, normalized size of antiderivative = 2.08

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1174\)
default	\(\text {Expression too large to display}\)	\(4431\)

Input:

int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2)/(-b*x^2+a)^(3/2),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)*(2*b*d*(1/2*(3 
*A*a*b*d^3+A*b^2*c^2*d-4*B*a*b*c*d^2+C*a^2*d^3+3*C*a*b*c^2*d-2*D*a^2*c*d^2 
-2*D*a*b*c^3)/b/d/(a*d^2-b*c^2)^2/a*x^2-1/2*(A*b^2*c-B*a*b*d+C*a*b*c-D*a^2 
*d)/d/a/b^2/(a*d^2-b*c^2)*x-1/2*(2*A*a*b*d^3+2*A*b^2*c^2*d-3*B*a*b*c*d^2-B 
*b^2*c^3+4*C*a*b*c^2*d-D*a^2*c*d^2-3*D*a*b*c^3)/b^2/d/(a^2*d^4-2*a*b*c^2*d 
^2+b^2*c^4))/(-(x^3+c/d*x^2-a*x/b-a/b*c/d)*b*d)^(1/2)+2*(-D/b/d-1/2*(6*A*a 
*b^2*c*d^3-2*A*b^3*c^3*d-3*B*a^2*b*d^4-B*a*b^2*c^2*d^2+4*C*a^2*b*c*d^3-3*D 
*a^3*d^4+D*a^2*b*c^2*d^2-2*D*a*b^2*c^4)/d/a/b/(a^2*d^4-2*a*b*c^2*d^2+b^2*c 
^4)+1/b*(A*b^2*c-B*a*b*d+C*a*b*c-D*a^2*d)/a/(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))+2*(3/2*(3*A 
*a*b*d^3+A*b^2*c^2*d-4*B*a*b*c*d^2+C*a^2*d^3+3*C*a*b*c^2*d-2*D*a^2*c*d^2-2 
*D*a*b*c^3)/a/(a^2*d^4-2*a*b*c^2*d^2+b^2*c^4)-2*(3*A*a*b*d^3+A*b^2*c^2*d-4 
*B*a*b*c*d^2+C*a^2*d^3+3*C*a*b*c^2*d-2*D*a^2*c*d^2-2*D*a*b*c^3)/(a*d^2-b*c 
^2)^2/a)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1 
/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1 
/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1356 vs. \(2 (517) = 1034\).

Time = 0.11 (sec) , antiderivative size = 1356, normalized size of antiderivative = 2.40 \[ \int \frac {A+B x+C x^2+D x^3}{(c+d x)^{3/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*((2*D*a^2*b^2*c^5 + (3*C*a^2*b^2 - A*a*b^3)*c^4*d - (13*D*a^3*b + 5*B* 
a^2*b^2)*c^3*d^2 + (5*C*a^3*b + 9*A*a^2*b^2)*c^2*d^3 + 3*(D*a^4 - B*a^3*b) 
*c*d^4 - (2*D*a*b^3*c^4*d + (3*C*a*b^3 - A*b^4)*c^3*d^2 - (13*D*a^2*b^2 + 
5*B*a*b^3)*c^2*d^3 + (5*C*a^2*b^2 + 9*A*a*b^3)*c*d^4 + 3*(D*a^3*b - B*a^2* 
b^2)*d^5)*x^3 - (2*D*a*b^3*c^5 + (3*C*a*b^3 - A*b^4)*c^4*d - (13*D*a^2*b^2 
 + 5*B*a*b^3)*c^3*d^2 + (5*C*a^2*b^2 + 9*A*a*b^3)*c^2*d^3 + 3*(D*a^3*b - B 
*a^2*b^2)*c*d^4)*x^2 + (2*D*a^2*b^2*c^4*d + (3*C*a^2*b^2 - A*a*b^3)*c^3*d^ 
2 - (13*D*a^3*b + 5*B*a^2*b^2)*c^2*d^3 + (5*C*a^3*b + 9*A*a^2*b^2)*c*d^4 + 
 3*(D*a^4 - B*a^3*b)*d^5)*x)*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*D*a^2*b^2*c^4*d - (3*C*a^2*b^2 + A*a*b^3)*c^3*d^2 + 2*(D*a^3*b + 2*B*a 
^2*b^2)*c^2*d^3 - (C*a^3*b + 3*A*a^2*b^2)*c*d^4 - (2*D*a*b^3*c^3*d^2 - (3* 
C*a*b^3 + A*b^4)*c^2*d^3 + 2*(D*a^2*b^2 + 2*B*a*b^3)*c*d^4 - (C*a^2*b^2 + 
3*A*a*b^3)*d^5)*x^3 - (2*D*a*b^3*c^4*d - (3*C*a*b^3 + A*b^4)*c^3*d^2 + 2*( 
D*a^2*b^2 + 2*B*a*b^3)*c^2*d^3 - (C*a^2*b^2 + 3*A*a*b^3)*c*d^4)*x^2 + (2*D 
*a^2*b^2*c^3*d^2 - (3*C*a^2*b^2 + A*a*b^3)*c^2*d^3 + 2*(D*a^3*b + 2*B*a^2* 
b^2)*c*d^4 - (C*a^3*b + 3*A*a^2*b^2)*d^5)*x)*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*A*a^2*b^2*d^5 - (3*D*a^2*b^2 + B*a*b^3)*c^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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