\(\int \frac {x^5}{\sqrt {c+d x} (a-b x^2)^{3/2}} \, dx\) [1553]

Optimal result

Integrand size = 25, antiderivative size = 441 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}-\frac {14 c \sqrt {c+d x} \sqrt {a-b x^2}}{15 b^2 d^2}+\frac {2 (c+d x)^{3/2} \sqrt {a-b x^2}}{5 b^2 d^2}+\frac {\sqrt {a} \left (16 b^2 c^4+32 a b c^2 d^2-63 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 b^{5/2} d^3 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {4 \sqrt {a} c \left (4 b c^2+11 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 b^{5/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

a^2*(-d*x+c)*(d*x+c)^(1/2)/b^2/(-a*d^2+b*c^2)/(-b*x^2+a)^(1/2)-14/15*c*(d* 
x+c)^(1/2)*(-b*x^2+a)^(1/2)/b^2/d^2+2/5*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b^2 
/d^2+1/15*a^(1/2)*(-63*a^2*d^4+32*a*b*c^2*d^2+16*b^2*c^4)*(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^(5/2)/d^3/(-a*d^2+b*c^2)/(b^(1/ 
2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-4/15*a^(1/2)*c*(1 
1*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)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)* 
d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(5/2)/d^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.01 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.42 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (-\frac {8 c}{d^2}+\frac {6 x}{d}+\frac {15 a^2 (-c+d x)}{\left (-b c^2+a d^2\right ) \left (a-b x^2\right )}\right )}{b^2}-\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (16 b^2 c^4+32 a b c^2 d^2-63 a^2 d^4\right ) \left (-a+b x^2\right )-i \sqrt {b} \left (16 b^{5/2} c^5-16 \sqrt {a} b^2 c^4 d+32 a b^{3/2} c^3 d^2-32 a^{3/2} b c^2 d^3-63 a^2 \sqrt {b} c d^4+63 a^{5/2} d^5\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-i \sqrt {a} \sqrt {b} d \left (16 b^2 c^4-4 \sqrt {a} b^{3/2} c^3 d+32 a b c^2 d^2+19 a^{3/2} \sqrt {b} c d^3-63 a^2 d^4\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{b^3 d^4 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-b c^2+a d^2\right ) \left (-a+b x^2\right )}\right )}{15 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*(((c + d*x)*((-8*c)/d^2 + (6*x)/d + (15*a^2*(-c + d*x))/( 
(-(b*c^2) + a*d^2)*(a - b*x^2))))/b^2 - (d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b] 
]*(16*b^2*c^4 + 32*a*b*c^2*d^2 - 63*a^2*d^4)*(-a + b*x^2) - I*Sqrt[b]*(16* 
b^(5/2)*c^5 - 16*Sqrt[a]*b^2*c^4*d + 32*a*b^(3/2)*c^3*d^2 - 32*a^(3/2)*b*c 
^2*d^3 - 63*a^2*Sqrt[b]*c*d^4 + 63*a^(5/2)*d^5)*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]], (S 
qrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - I*Sqrt[a]*Sqrt[b]*d*(16*b 
^2*c^4 - 4*Sqrt[a]*b^(3/2)*c^3*d + 32*a*b*c^2*d^2 + 19*a^(3/2)*Sqrt[b]*c*d 
^3 - 63*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)*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)])/(b^3*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-(b*c^2) + a*d^2 
)*(-a + b*x^2))))/(15*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {602, 27, 2185, 27, 2185, 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.

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

\(\Big \downarrow \) 602

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {4 a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b^2 d^2}-\frac {2 \int \frac {-14 a c \left (b c^2-a d^2\right ) x^2 d^2+a^2 c \left (6 c^2-\frac {11 a d^2}{b}\right ) d^2-a \left (4 b c^4-20 a d^2 c^2+\frac {21 a^2 d^4}{b}\right ) x d}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {4 a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b^2 d^2}-\frac {\int \frac {-14 a c \left (b c^2-a d^2\right ) x^2 d^2+a^2 c \left (6 c^2-\frac {11 a d^2}{b}\right ) d^2-a \left (4 b c^4-20 a d^2 c^2+\frac {21 a^2 d^4}{b}\right ) x d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {4 a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b^2 d^2}-\frac {\frac {28 a c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b}-\frac {2 \int -\frac {a d^3 \left (a c d \left (4 b c^2-19 a d^2\right )+\left (16 b^2 c^4+32 a b d^2 c^2-63 a^2 d^4\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {4 a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b^2 d^2}-\frac {\frac {a d \int \frac {a c d \left (4 b c^2-19 a d^2\right )+\left (16 b^2 c^4+32 a b d^2 c^2-63 a^2 d^4\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}+\frac {28 a c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {4 a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b^2 d^2}-\frac {\frac {a d \left (\frac {\left (-63 a^2 d^4+32 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {4 c \left (b c^2-a d^2\right ) \left (11 a d^2+4 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{3 b}+\frac {28 a c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {4 a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b^2 d^2}-\frac {\frac {a d \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (-63 a^2 d^4+32 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {4 c \left (b c^2-a d^2\right ) \left (11 a d^2+4 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{3 b}+\frac {28 a c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {4 a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b^2 d^2}-\frac {\frac {a d \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-63 a^2 d^4+32 a b c^2 d^2+16 b^2 c^4\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {4 c \left (b c^2-a d^2\right ) \left (11 a d^2+4 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )}{3 b}+\frac {28 a c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {4 a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b^2 d^2}-\frac {\frac {a d \left (-\frac {4 c \left (b c^2-a d^2\right ) \left (11 a d^2+4 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-63 a^2 d^4+32 a b c^2 d^2+16 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b}+\frac {28 a c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {4 a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b^2 d^2}-\frac {\frac {a d \left (-\frac {4 c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (11 a d^2+4 b c^2\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-63 a^2 d^4+32 a b c^2 d^2+16 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b}+\frac {28 a c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {4 a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b^2 d^2}-\frac {\frac {a d \left (\frac {8 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (11 a d^2+4 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-63 a^2 d^4+32 a b c^2 d^2+16 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b}+\frac {28 a c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {4 a \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}{5 b^2 d^2}-\frac {\frac {a d \left (\frac {8 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (11 a d^2+4 b c^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (-63 a^2 d^4+32 a b c^2 d^2+16 b^2 c^4\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{3 b}+\frac {28 a c d \sqrt {a-b x^2} \sqrt {c+d x} \left (b c^2-a d^2\right )}{3 b}}{5 b d^3}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x) \sqrt {c+d x}}{b^2 \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\)

Input:

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

Output:

(a^2*(c - d*x)*Sqrt[c + d*x])/(b^2*(b*c^2 - a*d^2)*Sqrt[a - b*x^2]) + ((4* 
a*(b*c^2 - a*d^2)*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/(5*b^2*d^2) - ((28*a*c* 
d*(b*c^2 - a*d^2)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(3*b) + (a*d*((-2*Sqrt[a] 
*(16*b^2*c^4 + 32*a*b*c^2*d^2 - 63*a^2*d^4)*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]) + (8*Sqrt[a]*c*(b*c^2 - a*d^2)*(4*b*c^2 + 11*a*d^ 
2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*E 
llipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c) 
/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(3*b))/(5*b*d^ 
3))/(2*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 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(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. \(763\) vs. \(2(367)=734\).

Time = 8.01 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.73

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (\frac {d \,a^{2} x}{2 \left (a \,d^{2}-b \,c^{2}\right ) b^{3}}-\frac {a^{2} c}{2 b^{3} \left (a \,d^{2}-b \,c^{2}\right )}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b^{2} d}-\frac {8 c \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{15 d^{2} b^{2}}+\frac {2 \left (-\frac {c d \,a^{2}}{2 b^{2} \left (a \,d^{2}-b \,c^{2}\right )}-\frac {2 a c}{15 b^{2} d}\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 {8 a}{5 b^{2}}-\frac {a^{2} d^{2}}{2 b^{2} \left (a \,d^{2}-b \,c^{2}\right )}-\frac {8 c^{2}}{15 d^{2} b}\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}}\)	\(764\)
risch	\(\text {Expression too large to display}\)	\(1258\)
default	\(\text {Expression too large to display}\)	\(1642\)

Input:

int(x^5/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-2*(-b*d*x-b*c) 
*(1/2*d/(a*d^2-b*c^2)/b^3*a^2*x-1/2*a^2*c/b^3/(a*d^2-b*c^2))/((x^2-a/b)*(- 
b*d*x-b*c))^(1/2)+2/5/b^2/d*x*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-8/15/d^2/ 
b^2*c*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(-1/2*c*d/b^2*a^2/(a*d^2-b*c^2) 
-2/15/b^2/d*a*c)*(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)*Elliptic 
F(((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*(-8/5*a/b^2-1/2*a^2*d^2/b^2/(a*d^2-b*c^2)-8/15/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 [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.06 \[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (8 \, a b^{2} c^{5} + 10 \, a^{2} b c^{3} d^{2} - 3 \, a^{3} c d^{4} - {\left (8 \, b^{3} c^{5} + 10 \, a b^{2} c^{3} d^{2} - 3 \, a^{2} b c d^{4}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (16 \, a b^{2} c^{4} d + 32 \, a^{2} b c^{2} d^{3} - 63 \, a^{3} d^{5} - {\left (16 \, b^{3} c^{4} d + 32 \, a b^{2} c^{2} d^{3} - 63 \, a^{2} b d^{5}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (8 \, a b^{2} c^{3} d^{2} - 23 \, a^{2} b c d^{4} + 6 \, {\left (b^{3} c^{2} d^{3} - a b^{2} d^{5}\right )} x^{3} - 8 \, {\left (b^{3} c^{3} d^{2} - a b^{2} c d^{4}\right )} x^{2} - 3 \, {\left (2 \, a b^{2} c^{2} d^{3} - 7 \, a^{2} b d^{5}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{45 \, {\left (a b^{4} c^{2} d^{4} - a^{2} b^{3} d^{6} - {\left (b^{5} c^{2} d^{4} - a b^{4} d^{6}\right )} x^{2}\right )}} \] Input:

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

Output:

-1/45*(2*(8*a*b^2*c^5 + 10*a^2*b*c^3*d^2 - 3*a^3*c*d^4 - (8*b^3*c^5 + 10*a 
*b^2*c^3*d^2 - 3*a^2*b*c*d^4)*x^2)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c 
^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/ 
d) + 3*(16*a*b^2*c^4*d + 32*a^2*b*c^2*d^3 - 63*a^3*d^5 - (16*b^3*c^4*d + 3 
2*a*b^2*c^2*d^3 - 63*a^2*b*d^5)*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), weierstrassPInvers 
e(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*(8*a*b^2*c^3*d^2 - 23*a^2*b*c*d^4 + 6*(b^3*c^2*d^3 - a*b 
^2*d^5)*x^3 - 8*(b^3*c^3*d^2 - a*b^2*c*d^4)*x^2 - 3*(2*a*b^2*c^2*d^3 - 7*a 
^2*b*d^5)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/(a*b^4*c^2*d^4 - a^2*b^3*d^6 
- (b^5*c^2*d^4 - a*b^4*d^6)*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^5/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(x^5/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^5}{\sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:

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

Output:

( - 63*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt( 
a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)* 
b*d**2*x**4),x)*a**4*c*d**4 + 32*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt( 
a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c** 
2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**3*b*c**3*d**2 + 63*sqrt(a - b 
*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d** 
2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a 
**3*b*c*d**4*x**2 + 16*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2 
)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + s 
qrt(a - b*x**2)*b*d**2*x**4),x)*a**2*b**2*c**5 - 32*sqrt(a - b*x**2)*int(s 
qrt(c + d*x)/(sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqr 
t(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*x**4),x)*a**2*b**2*c** 
3*d**2*x**2 - 16*sqrt(a - b*x**2)*int(sqrt(c + d*x)/(sqrt(a - b*x**2)*a*c* 
*2 - sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a 
- b*x**2)*b*d**2*x**4),x)*a*b**3*c**5*x**2 - 21*sqrt(a - b*x**2)*int((sqrt 
(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b 
*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a**2*b**2*c*d**3 - 4*sqrt(a - b*x* 
*2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c + a**2*d*x - 2*a*b*c 
*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a*b**3*c**3*d + 21*sq 
rt(a - b*x**2)*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c + a**2...