\(\int \frac {x^4}{\sqrt {c+d x} (a-b x^2)^{5/2}} \, dx\) [1581]

Optimal result

Integrand size = 25, antiderivative size = 454 \[ \int \frac {x^4}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=-\frac {a (a d-b c x) \sqrt {c+d x}}{3 b^2 \left (b c^2-a d^2\right ) \left (a-b x^2\right )^{3/2}}+\frac {\sqrt {c+d x} \left (a d \left (11 c^2-\frac {7 a d^2}{b}\right )-4 c \left (2 b c^2-a d^2\right ) x\right )}{6 b \left (b c^2-a d^2\right )^2 \sqrt {a-b x^2}}-\frac {2 \sqrt {a} c \left (2 b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {a} \left (4 b c^2-5 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 b^{5/2} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-1/3*a*(-b*c*x+a*d)*(d*x+c)^(1/2)/b^2/(-a*d^2+b*c^2)/(-b*x^2+a)^(3/2)+1/6* 
(d*x+c)^(1/2)*(a*d*(11*c^2-7*a*d^2/b)-4*c*(-a*d^2+2*b*c^2)*x)/b/(-a*d^2+b* 
c^2)^2/(-b*x^2+a)^(1/2)-2/3*a^(1/2)*c*(-a*d^2+2*b*c^2)*(d*x+c)^(1/2)*(1-b* 
x^2/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))/b^(3/2)/(-a*d^2+b*c^2)^2/(b^(1/2)*(d 
*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-1/6*a^(1/2)*(-5*a*d^2+ 
4*b*c^2)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1/2)*E 
llipticF(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)/(-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 = 23.86 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.32 \[ \int \frac {x^4}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (-5 a^3 d^3+8 b^3 c^3 x^3-a b^2 c x \left (6 c^2+11 c d x+4 d^2 x^2\right )+a^2 b d \left (9 c^2+2 c d x+7 d^2 x^2\right )\right )}{\left (a-b x^2\right )^2}-\frac {4 c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (2 b c^2-a d^2\right ) \left (a-b x^2\right )+4 i \sqrt {b} c \left (2 b^{3/2} c^3-2 \sqrt {a} b c^2 d-a \sqrt {b} c d^2+a^{3/2} d^3\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \left (12 b^2 c^4-8 \sqrt {a} b^{3/2} c^3 d-13 a b c^2 d^2+4 a^{3/2} \sqrt {b} c d^3+5 a^2 d^4\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (a-b x^2\right )}\right )}{6 b^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x}} \] Input:

Integrate[x^4/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
 

Output:

(Sqrt[a - b*x^2]*(((c + d*x)*(-5*a^3*d^3 + 8*b^3*c^3*x^3 - a*b^2*c*x*(6*c^ 
2 + 11*c*d*x + 4*d^2*x^2) + a^2*b*d*(9*c^2 + 2*c*d*x + 7*d^2*x^2)))/(a - b 
*x^2)^2 - (4*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(2*b*c^2 - a*d^2)*(a - b 
*x^2) + (4*I)*Sqrt[b]*c*(2*b^(3/2)*c^3 - 2*Sqrt[a]*b*c^2*d - a*Sqrt[b]*c*d 
^2 + a^(3/2)*d^3)*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)/(Sqrt[b] 
*c - Sqrt[a]*d)] - I*(12*b^2*c^4 - 8*Sqrt[a]*b^(3/2)*c^3*d - 13*a*b*c^2*d^ 
2 + 4*a^(3/2)*Sqrt[b]*c*d^3 + 5*a^2*d^4)*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)*El 
lipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]* 
c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] 
*(a - b*x^2))))/(6*b^2*(b*c^2 - a*d^2)^2*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {602, 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.

\(\displaystyle \int \frac {x^4}{\left (a-b x^2\right )^{5/2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 602

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2180

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

\(\displaystyle -\frac {-\frac {a \left (\left (4 b c^2-5 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+4 b c \left (2 b c^2-a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx\right )}{2 b^2 \left (b c^2-a d^2\right )}-\frac {a \sqrt {c+d x} \left (a d \left (11 b c^2-7 a d^2\right )-4 b^2 c x \left (2 c^2-\frac {a d^2}{b}\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}-\frac {a \sqrt {c+d x} (a d-b c x)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle -\frac {-\frac {a \left (\left (4 b c^2-5 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {4 b c \sqrt {1-\frac {b x^2}{a}} \left (2 b c^2-a d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}\right )}{2 b^2 \left (b c^2-a d^2\right )}-\frac {a \sqrt {c+d x} \left (a d \left (11 b c^2-7 a d^2\right )-4 b^2 c x \left (2 c^2-\frac {a d^2}{b}\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}-\frac {a \sqrt {c+d x} (a d-b c x)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle -\frac {-\frac {a \left (\left (4 b c^2-5 a d^2\right ) \left (b c^2-a d^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {8 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (2 b c^2-a d^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{2 b^2 \left (b c^2-a d^2\right )}-\frac {a \sqrt {c+d x} \left (a d \left (11 b c^2-7 a d^2\right )-4 b^2 c x \left (2 c^2-\frac {a d^2}{b}\right )\right )}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}}{6 a \left (b c^2-a d^2\right )}-\frac {a \sqrt {c+d x} (a d-b c x)}{3 b^2 \left (a-b x^2\right )^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

Int[x^4/(Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
 

Output:

-1/3*(a*(a*d - b*c*x)*Sqrt[c + d*x])/(b^2*(b*c^2 - a*d^2)*(a - b*x^2)^(3/2 
)) - (-((a*Sqrt[c + d*x]*(a*d*(11*b*c^2 - 7*a*d^2) - 4*b^2*c*(2*c^2 - (a*d 
^2)/b)*x))/(b^2*(b*c^2 - a*d^2)*Sqrt[a - b*x^2])) - (a*((-8*Sqrt[a]*Sqrt[b 
]*c*(2*b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[S 
qrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/( 
Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (2*Sq 
rt[a]*(4*b*c^2 - 5*a*d^2)*(b*c^2 - a*d^2)*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]*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 602

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m, a + b*x^2, x], e = Coeff[Polynomia 
lRemainder[x^m, a + b*x^2, x], x, 0], f = Coeff[PolynomialRemainder[x^m, a 
+ b*x^2, x], x, 1]}, Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^(p + 1)*((a*(d*e 
 - c*f) + (b*c*e + a*d*f)*x)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2 
*a*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToS 
um[2*a*(p + 1)*(b*c^2 + a*d^2)*Qx + e*(b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3 
)) - a*c*d*f*n + d*(b*c*e + a*d*f)*(n + 2*p + 4)*x, x], x], x]] /; FreeQ[{a 
, b, c, d, n}, x] && IGtQ[m, 1] && LtQ[p, -1] && NeQ[b*c^2 + a*d^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. \(859\) vs. \(2(384)=768\).

Time = 7.36 (sec) , antiderivative size = 860, normalized size of antiderivative = 1.89

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (\frac {\left (-\frac {c a x}{3 \left (a \,d^{2}-b \,c^{2}\right ) b^{3}}+\frac {d \,a^{2}}{3 \left (a \,d^{2}-b \,c^{2}\right ) 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 {c \left (a \,d^{2}-2 b \,c^{2}\right ) x}{3 b^{2} \left (a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {a d \left (7 a \,d^{2}-11 b \,c^{2}\right )}{12 \left (a \,d^{2}-b \,c^{2}\right )^{2} b^{3}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 \left (\frac {1}{b^{2}}-\frac {7 a \,d^{2}-8 b \,c^{2}}{6 \left (a \,d^{2}-b \,c^{2}\right ) b^{2}}+\frac {d^{2} a \left (7 a \,d^{2}-11 b \,c^{2}\right )}{12 b^{2} \left (a \,d^{2}-b \,c^{2}\right )^{2}}-\frac {2 c^{2} \left (a \,d^{2}-2 b \,c^{2}\right )}{3 b \left (a \,d^{2}-b \,c^{2}\right )^{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 c d \left (a \,d^{2}-2 b \,c^{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 )}{3 b \left (a \,d^{2}-b \,c^{2}\right )^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\)	\(860\)
default	\(\text {Expression too large to display}\)	\(2822\)

Input:

int(x^4/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((-1/3/(a*d^2-b* 
c^2)/b^3*c*a*x+1/3*d/(a*d^2-b*c^2)*a^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)*(1/3/b^2*c*(a*d^2-2*b*c^2)/(a*d^2-b*c^2)^2 
*x-1/12*a*d*(7*a*d^2-11*b*c^2)/(a*d^2-b*c^2)^2/b^3)/((x^2-a/b)*(-b*d*x-b*c 
))^(1/2)+2*(1/b^2-1/6/(a*d^2-b*c^2)/b^2*(7*a*d^2-8*b*c^2)+1/12/b^2*d^2*a*( 
7*a*d^2-11*b*c^2)/(a*d^2-b*c^2)^2-2/3/b*c^2*(a*d^2-2*b*c^2)/(a*d^2-b*c^2)^ 
2)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a* 
b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a* 
b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/ 
d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^ 
(1/2))-2/3*c*d*(a*d^2-2*b*c^2)/b/(a*d^2-b*c^2)^2*(c/d-1/b*(a*b)^(1/2))*((x 
+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1 
/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*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 = 0.09 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.28 \[ \int \frac {x^4}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=-\frac {{\left (28 \, a^{2} b^{2} c^{4} - 35 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} d^{4} + {\left (28 \, b^{4} c^{4} - 35 \, a b^{3} c^{2} d^{2} + 15 \, a^{2} b^{2} d^{4}\right )} x^{4} - 2 \, {\left (28 \, a b^{3} c^{4} - 35 \, a^{2} b^{2} c^{2} d^{2} + 15 \, a^{3} b d^{4}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) - 12 \, {\left (2 \, a^{2} b^{2} c^{3} d - a^{3} b c d^{3} + {\left (2 \, b^{4} c^{3} d - a b^{3} c d^{3}\right )} x^{4} - 2 \, {\left (2 \, a b^{3} c^{3} d - a^{2} b^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) - 3 \, {\left (9 \, a^{2} b^{2} c^{2} d^{2} - 5 \, a^{3} b d^{4} + 4 \, {\left (2 \, b^{4} c^{3} d - a b^{3} c d^{3}\right )} x^{3} - {\left (11 \, a b^{3} c^{2} d^{2} - 7 \, a^{2} b^{2} d^{4}\right )} x^{2} - 2 \, {\left (3 \, a b^{3} c^{3} d - a^{2} b^{2} c d^{3}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{18 \, {\left (a^{2} b^{5} c^{4} d - 2 \, a^{3} b^{4} c^{2} d^{3} + a^{4} b^{3} d^{5} + {\left (b^{7} c^{4} d - 2 \, a b^{6} c^{2} d^{3} + a^{2} b^{5} d^{5}\right )} x^{4} - 2 \, {\left (a b^{6} c^{4} d - 2 \, a^{2} b^{5} c^{2} d^{3} + a^{3} b^{4} d^{5}\right )} x^{2}\right )}} \] Input:

integrate(x^4/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

-1/18*((28*a^2*b^2*c^4 - 35*a^3*b*c^2*d^2 + 15*a^4*d^4 + (28*b^4*c^4 - 35* 
a*b^3*c^2*d^2 + 15*a^2*b^2*d^4)*x^4 - 2*(28*a*b^3*c^4 - 35*a^2*b^2*c^2*d^2 
 + 15*a^3*b*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) - 12*(2*a^ 
2*b^2*c^3*d - a^3*b*c*d^3 + (2*b^4*c^3*d - a*b^3*c*d^3)*x^4 - 2*(2*a*b^3*c 
^3*d - a^2*b^2*c*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), 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*a^2*b^2*c^2*d^2 - 5*a^3*b*d^4 + 4*(2*b^4*c^3*d - a*b^3*c*d^3)*x 
^3 - (11*a*b^3*c^2*d^2 - 7*a^2*b^2*d^4)*x^2 - 2*(3*a*b^3*c^3*d - a^2*b^2*c 
*d^3)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a^2*b^5*c^4*d - 2*a^3*b^4*c^2*d^ 
3 + a^4*b^3*d^5 + (b^7*c^4*d - 2*a*b^6*c^2*d^3 + a^2*b^5*d^5)*x^4 - 2*(a*b 
^6*c^4*d - 2*a^2*b^5*c^2*d^3 + a^3*b^4*d^5)*x^2)
 

Sympy [F]

\[ \int \frac {x^4}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^{4}}{\left (a - b x^{2}\right )^{\frac {5}{2}} \sqrt {c + d x}}\, dx \] Input:

integrate(x**4/(d*x+c)**(1/2)/(-b*x**2+a)**(5/2),x)
 

Output:

Integral(x**4/((a - b*x**2)**(5/2)*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {x^4}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:

integrate(x^4/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate(x^4/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
 

Giac [F]

\[ \int \frac {x^4}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c}} \,d x } \] Input:

integrate(x^4/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

integrate(x^4/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {x^4}{{\left (a-b\,x^2\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int(x^4/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)),x)
 

Output:

int(x^4/((a - b*x^2)^(5/2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^4}{\sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:

int(x^4/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)
 

Output:

( - 9*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqr 
t(a - b*x**2)*a**2*d**2*x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a 
 - b*x**2)*a*b*d**2*x**4 + sqrt(a - b*x**2)*b**2*c**2*x**4 - sqrt(a - b*x* 
*2)*b**2*d**2*x**6),x)*a**4*c*d**3 + 12*sqrt(a - b*x**2)*int(sqrt(c + d*x) 
/(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d**2*x**2 - 2*sqrt(a 
- b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)*a*b*d**2*x**4 + sqrt(a - b*x* 
*2)*b**2*c**2*x**4 - sqrt(a - b*x**2)*b**2*d**2*x**6),x)*a**3*b*c**3*d + 1 
8*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a 
- b*x**2)*a**2*d**2*x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b 
*x**2)*a*b*d**2*x**4 + sqrt(a - b*x**2)*b**2*c**2*x**4 - sqrt(a - b*x**2)* 
b**2*d**2*x**6),x)*a**3*b*c*d**3*x**2 - 24*sqrt(a - b*x**2)*int(sqrt(c + d 
*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d**2*x**2 - 2*sqrt 
(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)*a*b*d**2*x**4 + sqrt(a - b 
*x**2)*b**2*c**2*x**4 - sqrt(a - b*x**2)*b**2*d**2*x**6),x)*a**2*b**2*c**3 
*d*x**2 - 9*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 
 - sqrt(a - b*x**2)*a**2*d**2*x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2* 
sqrt(a - b*x**2)*a*b*d**2*x**4 + sqrt(a - b*x**2)*b**2*c**2*x**4 - sqrt(a 
- b*x**2)*b**2*d**2*x**6),x)*a**2*b**2*c*d**3*x**4 + 12*sqrt(a - b*x**2)*i 
nt(sqrt(c + d*x)/(sqrt(a - b*x**2)*a**2*c**2 - sqrt(a - b*x**2)*a**2*d**2* 
x**2 - 2*sqrt(a - b*x**2)*a*b*c**2*x**2 + 2*sqrt(a - b*x**2)*a*b*d**2*x...