\(\int \frac {x^5}{(c+d x)^{5/2} (a-b x^2)^{3/2}} \, dx\) [1570]

Optimal result

Integrand size = 25, antiderivative size = 568 \[ \int \frac {x^5}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {a^2 (c-d x)}{b^2 \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}+\frac {2 c \left (b^2 c^4+3 a^2 d^4\right ) \sqrt {a-b x^2}}{3 b^2 d^2 \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}-\frac {\left (10 b^3 c^6-30 a b^2 c^4 d^2-9 a^2 b c^2 d^4-3 a^3 d^6\right ) \sqrt {a-b x^2}}{3 b^2 d^2 \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}+\frac {\sqrt {a} \left (16 b^3 c^6-48 a b^2 c^4 d^2+9 a^2 b c^2 d^4-9 a^3 d^6\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} d^3 \left (b c^2-a d^2\right )^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {4 \sqrt {a} c \left (4 b^2 c^4-9 a b c^2 d^2+3 a^2 d^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 b^{3/2} d^3 \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

a^2*(-d*x+c)/b^2/(-a*d^2+b*c^2)/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2)+2/3*c*(3*a^ 
2*d^4+b^2*c^4)*(-b*x^2+a)^(1/2)/b^2/d^2/(-a*d^2+b*c^2)^2/(d*x+c)^(3/2)-1/3 
*(-3*a^3*d^6-9*a^2*b*c^2*d^4-30*a*b^2*c^4*d^2+10*b^3*c^6)*(-b*x^2+a)^(1/2) 
/b^2/d^2/(-a*d^2+b*c^2)^3/(d*x+c)^(1/2)+1/3*a^(1/2)*(-9*a^3*d^6+9*a^2*b*c^ 
2*d^4-48*a*b^2*c^4*d^2+16*b^3*c^6)*(d*x+c)^(1/2)*(1-b*x^2/a)^(1/2)*Ellipti 
cE(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)/d^3/(-a*d^2+b*c^2)^3/(b^(1/2)*(d*x+c)/(b^(1/2)*c 
+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-4/3*a^(1/2)*c*(3*a^2*d^4-9*a*b*c^2*d^2 
+4*b^2*c^4)*(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^(3/2)/d^3/(-a*d^2+b*c^2)^2/(d*x+c)^(1/2)/(-b* 
x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.74 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.18 \[ \int \frac {x^5}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (16 b^3 c^6-48 a b^2 c^4 d^2+9 a^2 b c^2 d^4-9 a^3 d^6-10 b^2 c^4 \left (b c^2-3 a d^2\right )+\frac {2 b^2 c^5 \left (b c^2-a d^2\right )}{c+d x}+\frac {3 a^2 b d^2 (c+d x) \left (-b c^2 (c-3 d x)+a d^2 (-3 c+d x)\right )}{-a+b x^2}+\frac {i b \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (16 b^3 c^6-48 a b^2 c^4 d^2+9 a^2 b c^2 d^4-9 a^3 d^6\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \left (-a+b x^2\right )}-\frac {i \sqrt {a} \sqrt {b} \left (16 b^3 c^6-4 \sqrt {a} b^{5/2} c^5 d-48 a b^2 c^4 d^2+39 a^{3/2} b^{3/2} c^3 d^3+9 a^2 b c^2 d^4-3 a^{5/2} \sqrt {b} c d^5-9 a^3 d^6\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{3 b^2 d^2 \left (b c^2-a d^2\right )^3 \sqrt {c+d x}} \] Input:

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

Output:

(Sqrt[a - b*x^2]*(16*b^3*c^6 - 48*a*b^2*c^4*d^2 + 9*a^2*b*c^2*d^4 - 9*a^3* 
d^6 - 10*b^2*c^4*(b*c^2 - 3*a*d^2) + (2*b^2*c^5*(b*c^2 - a*d^2))/(c + d*x) 
 + (3*a^2*b*d^2*(c + d*x)*(-(b*c^2*(c - 3*d*x)) + a*d^2*(-3*c + d*x)))/(-a 
 + b*x^2) + (I*b*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(16*b^3*c^6 - 48*a*b^2*c^4 
*d^2 + 9*a^2*b*c^2*d^4 - 9*a^3*d^6)*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)*Ellipti 
cE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + S 
qrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d^2*(-a + b*x^2)) - (I*Sqrt[a]*Sqrt[b 
]*(16*b^3*c^6 - 4*Sqrt[a]*b^(5/2)*c^5*d - 48*a*b^2*c^4*d^2 + 39*a^(3/2)*b^ 
(3/2)*c^3*d^3 + 9*a^2*b*c^2*d^4 - 3*a^(5/2)*Sqrt[b]*c*d^5 - 9*a^3*d^6)*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))))/(3*b^2*d^2*(b*c^2 - a*d^2)^ 
3*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {602, 27, 2182, 27, 2182, 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.

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

\(\Big \downarrow \) 602

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {\frac {2 \int -\frac {a \left (a c \left (\frac {4 b c^4}{d}-39 a d c^2+\frac {3 a^2 d^3}{b}\right )-\left (-\frac {16 b^2 c^6}{d^2}+48 a b c^4-9 a^2 d^2 c^2+\frac {9 a^3 d^4}{b}\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}-\frac {2 a d \sqrt {a-b x^2} \left (-\frac {3 a^3 d^3}{b^2}-\frac {9 a^2 c^2 d}{b}-\frac {30 a c^4}{d}+\frac {10 b c^6}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {3 a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {a \int \frac {a c \left (\frac {4 b c^4}{d}-39 a d c^2+\frac {3 a^2 d^3}{b}\right )-\left (-\frac {16 b^2 c^6}{d^2}+48 a b c^4-9 a^2 d^2 c^2+\frac {9 a^3 d^4}{b}\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b c^2-a d^2}-\frac {2 a d \sqrt {a-b x^2} \left (-\frac {3 a^3 d^3}{b^2}-\frac {9 a^2 c^2 d}{b}-\frac {30 a c^4}{d}+\frac {10 b c^6}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {3 a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {\frac {-\frac {a \left (-\frac {4 c \left (b c^2-a d^2\right ) \left (3 a^2 d^4-9 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d^3}-\frac {\left (\frac {9 a^3 d^4}{b}-9 a^2 c^2 d^2+48 a b c^4-\frac {16 b^2 c^6}{d^2}\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{b c^2-a d^2}-\frac {2 a d \sqrt {a-b x^2} \left (-\frac {3 a^3 d^3}{b^2}-\frac {9 a^2 c^2 d}{b}-\frac {30 a c^4}{d}+\frac {10 b c^6}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {3 a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle \frac {\frac {-\frac {a \left (-\frac {4 c \left (b c^2-a d^2\right ) \left (3 a^2 d^4-9 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d^3}-\frac {\sqrt {1-\frac {b x^2}{a}} \left (\frac {9 a^3 d^4}{b}-9 a^2 c^2 d^2+48 a b c^4-\frac {16 b^2 c^6}{d^2}\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {2 a d \sqrt {a-b x^2} \left (-\frac {3 a^3 d^3}{b^2}-\frac {9 a^2 c^2 d}{b}-\frac {30 a c^4}{d}+\frac {10 b c^6}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {3 a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {\frac {-\frac {a \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (\frac {9 a^3 d^4}{b}-9 a^2 c^2 d^2+48 a b c^4-\frac {16 b^2 c^6}{d^2}\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {4 c \left (b c^2-a d^2\right ) \left (3 a^2 d^4-9 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d^3}\right )}{b c^2-a d^2}-\frac {2 a d \sqrt {a-b x^2} \left (-\frac {3 a^3 d^3}{b^2}-\frac {9 a^2 c^2 d}{b}-\frac {30 a c^4}{d}+\frac {10 b c^6}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {3 a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {-\frac {a \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (\frac {9 a^3 d^4}{b}-9 a^2 c^2 d^2+48 a b c^4-\frac {16 b^2 c^6}{d^2}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {4 c \left (b c^2-a d^2\right ) \left (3 a^2 d^4-9 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{b d^3}\right )}{b c^2-a d^2}-\frac {2 a d \sqrt {a-b x^2} \left (-\frac {3 a^3 d^3}{b^2}-\frac {9 a^2 c^2 d}{b}-\frac {30 a c^4}{d}+\frac {10 b c^6}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {3 a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle \frac {\frac {-\frac {a \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (\frac {9 a^3 d^4}{b}-9 a^2 c^2 d^2+48 a b c^4-\frac {16 b^2 c^6}{d^2}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {4 c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a^2 d^4-9 a b c^2 d^2+4 b^2 c^4\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{b d^3 \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {2 a d \sqrt {a-b x^2} \left (-\frac {3 a^3 d^3}{b^2}-\frac {9 a^2 c^2 d}{b}-\frac {30 a c^4}{d}+\frac {10 b c^6}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {3 a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {-\frac {a \left (\frac {8 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a^2 d^4-9 a b c^2 d^2+4 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{b^{3/2} d^3 \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (\frac {9 a^3 d^4}{b}-9 a^2 c^2 d^2+48 a b c^4-\frac {16 b^2 c^6}{d^2}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}-\frac {2 a d \sqrt {a-b x^2} \left (-\frac {3 a^3 d^3}{b^2}-\frac {9 a^2 c^2 d}{b}-\frac {30 a c^4}{d}+\frac {10 b c^6}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}+\frac {4 a c d \sqrt {a-b x^2} \left (\frac {3 a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}+\frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {a^2 (c-d x)}{b^2 \sqrt {a-b x^2} (c+d x)^{3/2} \left (b c^2-a d^2\right )}+\frac {\frac {4 a c d \sqrt {a-b x^2} \left (\frac {3 a^2 d}{b^2}+\frac {c^4}{d^3}\right )}{3 (c+d x)^{3/2} \left (b c^2-a d^2\right )}+\frac {-\frac {a \left (\frac {8 \sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \left (3 a^2 d^4-9 a b c^2 d^2+4 b^2 c^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{b^{3/2} d^3 \sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (\frac {9 a^3 d^4}{b}-9 a^2 c^2 d^2+48 a b c^4-\frac {16 b^2 c^6}{d^2}\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{b c^2-a d^2}-\frac {2 a d \sqrt {a-b x^2} \left (-\frac {3 a^3 d^3}{b^2}-\frac {9 a^2 c^2 d}{b}-\frac {30 a c^4}{d}+\frac {10 b c^6}{d^3}\right )}{\sqrt {c+d x} \left (b c^2-a d^2\right )}}{3 \left (b c^2-a d^2\right )}}{2 a \left (b c^2-a d^2\right )}\)

Input:

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

Output:

(a^2*(c - d*x))/(b^2*(b*c^2 - a*d^2)*(c + d*x)^(3/2)*Sqrt[a - b*x^2]) + (( 
4*a*c*d*(c^4/d^3 + (3*a^2*d)/b^2)*Sqrt[a - b*x^2])/(3*(b*c^2 - a*d^2)*(c + 
 d*x)^(3/2)) + ((-2*a*d*((10*b*c^6)/d^3 - (30*a*c^4)/d - (9*a^2*c^2*d)/b - 
 (3*a^3*d^3)/b^2)*Sqrt[a - b*x^2])/((b*c^2 - a*d^2)*Sqrt[c + d*x]) - (a*(( 
2*Sqrt[a]*(48*a*b*c^4 - (16*b^2*c^6)/d^2 - 9*a^2*c^2*d^2 + (9*a^3*d^4)/b)* 
Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sq 
rt[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^2*c^4 - 9*a*b*c^2*d^2 + 3*a^2*d^4)*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)])/(b^(3/2)* 
d^3*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/(b*c^2 - a*d^2))/(3*(b*c^2 - a*d^2))) 
/(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 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1012\) vs. \(2(494)=988\).

Time = 8.76 (sec) , antiderivative size = 1013, normalized size of antiderivative = 1.78

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1013\)
default	\(\text {Expression too large to display}\)	\(4013\)

Input:

int(x^5/(d*x+c)^(5/2)/(-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)*(2/3/d^4/(a*d^2- 
b*c^2)^2*c^5*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2-10/3*(-b*d*x^2+a 
*d)/d^3/(a*d^2-b*c^2)^3*c^4*(3*a*d^2-b*c^2)/((x+c/d)*(-b*d*x^2+a*d))^(1/2) 
-2*(-b*d*x-b*c)*(1/2*d*(a*d^2+3*b*c^2)*a^2/(a*d^2-b*c^2)^3/b^2*x-1/2*a^2*c 
*(3*a*d^2+b*c^2)/(a*d^2-b*c^2)^3/b^2)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2*(2/ 
b*c/d^3-1/3/d^3*b*c^5/(a*d^2-b*c^2)^2-5/3/d^3*b*c^5*(3*a*d^2-b*c^2)/(a*d^2 
-b*c^2)^3-2/(a*d^2-b*c^2)^2/b*a^2*c*d+1/2/b*d*a^2*c*(3*a*d^2+b*c^2)/(a*d^2 
-b*c^2)^3-1/b*c*d*(a*d^2+3*b*c^2)*a^2/(a*d^2-b*c^2)^3)*(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*(-1/b/d^2-5/3 
*c^4/d^2*b*(3*a*d^2-b*c^2)/(a*d^2-b*c^2)^3-1/2*a^2*d^2*(a*d^2+3*b*c^2)/b/( 
a*d^2-b*c^2)^3)*(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. 1185 vs. \(2 (496) = 992\).

Time = 0.22 (sec) , antiderivative size = 1185, normalized size of antiderivative = 2.09 \[ \int \frac {x^5}{(c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/9*(2*(8*a*b^3*c^9 - 30*a^2*b^2*c^7*d^2 + 63*a^3*b*c^5*d^4 - 9*a^4*c^3*d 
^6 - (8*b^4*c^7*d^2 - 30*a*b^3*c^5*d^4 + 63*a^2*b^2*c^3*d^6 - 9*a^3*b*c*d^ 
8)*x^4 - 2*(8*b^4*c^8*d - 30*a*b^3*c^6*d^3 + 63*a^2*b^2*c^4*d^5 - 9*a^3*b* 
c^2*d^7)*x^3 - (8*b^4*c^9 - 38*a*b^3*c^7*d^2 + 93*a^2*b^2*c^5*d^4 - 72*a^3 
*b*c^3*d^6 + 9*a^4*c*d^8)*x^2 + 2*(8*a*b^3*c^8*d - 30*a^2*b^2*c^6*d^3 + 63 
*a^3*b*c^4*d^5 - 9*a^4*c^2*d^7)*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*(16*a*b^3*c^8*d - 48*a^2*b^2*c^6*d^3 + 9*a^3*b*c^4*d^5 - 9*a^4*c^2* 
d^7 - (16*b^4*c^6*d^3 - 48*a*b^3*c^4*d^5 + 9*a^2*b^2*c^2*d^7 - 9*a^3*b*d^9 
)*x^4 - 2*(16*b^4*c^7*d^2 - 48*a*b^3*c^5*d^4 + 9*a^2*b^2*c^3*d^6 - 9*a^3*b 
*c*d^8)*x^3 - (16*b^4*c^8*d - 64*a*b^3*c^6*d^3 + 57*a^2*b^2*c^4*d^5 - 18*a 
^3*b*c^2*d^7 + 9*a^4*d^9)*x^2 + 2*(16*a*b^3*c^7*d^2 - 48*a^2*b^2*c^5*d^4 + 
 9*a^3*b*c^3*d^6 - 9*a^4*c*d^8)*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), 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*(8*a*b^3*c^7*d^2 - 31*a^2*b^2*c^5*d^4 - 9*a^3*b*c^3*d^6 - 
(10*b^4*c^6*d^3 - 30*a*b^3*c^4*d^5 - 9*a^2*b^2*c^2*d^7 - 3*a^3*b*d^9)*x^3 
- (8*b^4*c^7*d^2 - 28*a*b^3*c^5*d^4 - 15*a^2*b^2*c^3*d^6 + 3*a^3*b*c*d^8)* 
x^2 + (10*a*b^3*c^6*d^3 - 27*a^2*b^2*c^4*d^5 - 15*a^3*b*c^2*d^7)*x)*sqrt(- 
b*x^2 + a)*sqrt(d*x + c))/(a*b^5*c^8*d^4 - 3*a^2*b^4*c^6*d^6 + 3*a^3*b^...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(12*sqrt(c + d*x)*sqrt(a - b*x**2)*a**2*c*d**2 + 18*sqrt(c + d*x)*sqrt(a - 
 b*x**2)*a**2*d**3*x + 160*sqrt(c + d*x)*sqrt(a - b*x**2)*a*b*c**3 + 240*s 
qrt(c + d*x)*sqrt(a - b*x**2)*a*b*c**2*d*x - 96*sqrt(c + d*x)*sqrt(a - b*x 
**2)*b**2*c**3*x**2 - 24*sqrt(c + d*x)*sqrt(a - b*x**2)*b**2*c**2*d*x**3 - 
 27*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c**3 + 3*a**2*c**2*d*x 
 + 3*a**2*c*d**2*x**2 + a**2*d**3*x**3 - 2*a*b*c**3*x**2 - 6*a*b*c**2*d*x* 
*3 - 6*a*b*c*d**2*x**4 - 2*a*b*d**3*x**5 + b**2*c**3*x**4 + 3*b**2*c**2*d* 
x**5 + 3*b**2*c*d**2*x**6 + b**2*d**3*x**7),x)*a**3*b*c**2*d**4 - 54*int(( 
sqrt(c + d*x)*sqrt(a - b*x**2)*x**3)/(a**2*c**3 + 3*a**2*c**2*d*x + 3*a**2 
*c*d**2*x**2 + a**2*d**3*x**3 - 2*a*b*c**3*x**2 - 6*a*b*c**2*d*x**3 - 6*a* 
b*c*d**2*x**4 - 2*a*b*d**3*x**5 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**5 + 3* 
b**2*c*d**2*x**6 + b**2*d**3*x**7),x)*a**3*b*c*d**5*x - 27*int((sqrt(c + d 
*x)*sqrt(a - b*x**2)*x**3)/(a**2*c**3 + 3*a**2*c**2*d*x + 3*a**2*c*d**2*x* 
*2 + a**2*d**3*x**3 - 2*a*b*c**3*x**2 - 6*a*b*c**2*d*x**3 - 6*a*b*c*d**2*x 
**4 - 2*a*b*d**3*x**5 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**5 + 3*b**2*c*d** 
2*x**6 + b**2*d**3*x**7),x)*a**3*b*d**6*x**2 - 324*int((sqrt(c + d*x)*sqrt 
(a - b*x**2)*x**3)/(a**2*c**3 + 3*a**2*c**2*d*x + 3*a**2*c*d**2*x**2 + a** 
2*d**3*x**3 - 2*a*b*c**3*x**2 - 6*a*b*c**2*d*x**3 - 6*a*b*c*d**2*x**4 - 2* 
a*b*d**3*x**5 + b**2*c**3*x**4 + 3*b**2*c**2*d*x**5 + 3*b**2*c*d**2*x**6 + 
 b**2*d**3*x**7),x)*a**2*b**2*c**4*d**2 - 648*int((sqrt(c + d*x)*sqrt(a...