\(\int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{7/2}} \, dx\) [266]

Optimal result

Integrand size = 27, antiderivative size = 529 \[ \int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{7/2}} \, dx=\frac {4 \left (4 B c d^2+A c d e-5 a B e^2\right ) \sqrt {a-c x^2}}{15 e^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {4 c \left (4 B c d^3+A c d^2 e-8 a B d e^2+3 a A e^3\right ) \sqrt {a-c x^2}}{15 e^2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {2 (4 B d+A e+5 B e x) \sqrt {a-c x^2}}{5 e^2 (d+e x)^{5/2}}-\frac {4 \sqrt {a} c^{3/2} \left (4 B c d^3+A c d^2 e-8 a B d e^2+3 a A e^3\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{15 e^3 \left (c d^2-a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}+\frac {4 \sqrt {a} \sqrt {c} \left (4 B c d^2+A c d e-5 a B e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{15 e^3 \left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

4/15*(A*c*d*e-5*B*a*e^2+4*B*c*d^2)*(-c*x^2+a)^(1/2)/e^2/(-a*e^2+c*d^2)/(e* 
x+d)^(3/2)+4/15*c*(3*A*a*e^3+A*c*d^2*e-8*B*a*d*e^2+4*B*c*d^3)*(-c*x^2+a)^( 
1/2)/e^2/(-a*e^2+c*d^2)^2/(e*x+d)^(1/2)-2/5*(5*B*e*x+A*e+4*B*d)*(-c*x^2+a) 
^(1/2)/e^2/(e*x+d)^(5/2)-4/15*a^(1/2)*c^(3/2)*(3*A*a*e^3+A*c*d^2*e-8*B*a*d 
*e^2+4*B*c*d^3)*(e*x+d)^(1/2)*(1-c*x^2/a)^(1/2)*EllipticE(1/2*(1-c^(1/2)*x 
/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*e/(c^(1/2)*d+a^(1/2)*e))^(1/2))/e 
^3/(-a*e^2+c*d^2)^2/(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)/(-c*x^2+ 
a)^(1/2)+4/15*a^(1/2)*c^(1/2)*(A*c*d*e-5*B*a*e^2+4*B*c*d^2)*(c^(1/2)*(e*x+ 
d)/(c^(1/2)*d+a^(1/2)*e))^(1/2)*(1-c*x^2/a)^(1/2)*EllipticF(1/2*(1-c^(1/2) 
*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*e/(c^(1/2)*d+a^(1/2)*e))^(1/2)) 
/e^3/(-a*e^2+c*d^2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.13 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{7/2}} \, dx=\frac {2 \sqrt {a-c x^2} \left (\frac {3 (B d-A e) \left (c d^2-a e^2\right )^2}{(d+e x)^2}-\frac {\left (c d^2-a e^2\right ) \left (7 B c d^2-2 A c d e-5 a B e^2\right )}{d+e x}+\frac {2 i c^{3/2} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (4 B c d^3+A c d^2 e-8 a B d e^2+3 a A e^3\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}+\frac {2 i \sqrt {a} c \left (\sqrt {c} d-\sqrt {a} e\right ) \left (4 B c d^2+3 \sqrt {a} B \sqrt {c} d e+A c d e-5 a B e^2-3 \sqrt {a} A \sqrt {c} e^2\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{15 \left (c d^2 e-a e^3\right )^2 \sqrt {d+e x}} \] Input:

Integrate[((A + B*x)*Sqrt[a - c*x^2])/(d + e*x)^(7/2),x]
 

Output:

(2*Sqrt[a - c*x^2]*((3*(B*d - A*e)*(c*d^2 - a*e^2)^2)/(d + e*x)^2 - ((c*d^ 
2 - a*e^2)*(7*B*c*d^2 - 2*A*c*d*e - 5*a*B*e^2))/(d + e*x) + ((2*I)*c^(3/2) 
*(Sqrt[c]*d - Sqrt[a]*e)*(4*B*c*d^3 + A*c*d^2*e - 8*a*B*d*e^2 + 3*a*A*e^3) 
*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/Sqrt[c] - e 
*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d + (Sqrt[a]*e)/ 
Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sqrt[a]*e)]) 
/(e^2*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2)) + ((2*I)*Sqrt[a]*c*(Sqr 
t[c]*d - Sqrt[a]*e)*(4*B*c*d^2 + 3*Sqrt[a]*B*Sqrt[c]*d*e + A*c*d*e - 5*a*B 
*e^2 - 3*Sqrt[a]*A*Sqrt[c]*e^2)*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]* 
Sqrt[-(((Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I 
*ArcSinh[Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[ 
a]*e)/(Sqrt[c]*d - Sqrt[a]*e)])/(e*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c* 
x^2))))/(15*(c*d^2*e - a*e^3)^2*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 524, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {680, 25, 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.

\(\displaystyle \int \frac {\sqrt {a-c x^2} (A+B x)}{(d+e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 680

\(\displaystyle \frac {2 \int -\frac {c \left (3 a e (B d-A e)+\left (4 B c d^2+A c e d-5 a B e^2\right ) x\right )}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \int \frac {c \left (3 a e (B d-A e)+\left (4 B c d^2+A c e d-5 a B e^2\right ) x\right )}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 c \int \frac {3 a e (B d-A e)+\left (4 B c d^2+A c e d-5 a B e^2\right ) x}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 688

\(\displaystyle -\frac {2 c \left (\frac {2 \int -\frac {a e \left (B c d^2+4 A c e d-5 a B e^2\right )+c \left (4 B c d^3+A c e d^2-8 a B e^2 d+3 a A e^3\right ) x}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 c \left (-\frac {\int \frac {a e \left (B c d^2+4 A c e d-5 a B e^2\right )+c \left (4 B c d^3+A c e d^2-8 a B e^2 d+3 a A e^3\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 600

\(\displaystyle -\frac {2 c \left (-\frac {\frac {c \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2+A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 509

\(\displaystyle -\frac {2 c \left (-\frac {\frac {c \sqrt {1-\frac {c x^2}{a}} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2+A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 508

\(\displaystyle -\frac {2 c \left (-\frac {-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2+A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2 c \left (-\frac {-\frac {\left (c d^2-a e^2\right ) \left (-5 a B e^2+A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 512

\(\displaystyle -\frac {2 c \left (-\frac {-\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \left (-5 a B e^2+A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 511

\(\displaystyle -\frac {2 c \left (-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-5 a B e^2+A c d e+4 B c d^2\right ) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2 c \left (-\frac {\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (-5 a B e^2+A c d e+4 B c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{c d^2-a e^2}-\frac {2 \sqrt {a-c x^2} \left (3 a A e^3-8 a B d e^2+A c d^2 e+4 B c d^3\right )}{\sqrt {d+e x} \left (c d^2-a e^2\right )}\right )}{15 e^2 \left (c d^2-a e^2\right )}-\frac {2 \sqrt {a-c x^2} \left (e x \left (-5 a B e^2-2 A c d e+7 B c d^2\right )-3 a A e^3-2 a B d e^2+A c d^2 e+4 B c d^3\right )}{15 e^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\)

Input:

Int[((A + B*x)*Sqrt[a - c*x^2])/(d + e*x)^(7/2),x]
 

Output:

(-2*(4*B*c*d^3 + A*c*d^2*e - 2*a*B*d*e^2 - 3*a*A*e^3 + e*(7*B*c*d^2 - 2*A* 
c*d*e - 5*a*B*e^2)*x)*Sqrt[a - c*x^2])/(15*e^2*(c*d^2 - a*e^2)*(d + e*x)^( 
5/2)) - (2*c*((-2*(4*B*c*d^3 + A*c*d^2*e - 8*a*B*d*e^2 + 3*a*A*e^3)*Sqrt[a 
 - c*x^2])/((c*d^2 - a*e^2)*Sqrt[d + e*x]) - ((-2*Sqrt[a]*Sqrt[c]*(4*B*c*d 
^3 + A*c*d^2*e - 8*a*B*d*e^2 + 3*a*A*e^3)*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a 
]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c] 
*d)/Sqrt[a] + e)])/(e*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sq 
rt[a - c*x^2]) + (2*Sqrt[a]*(c*d^2 - a*e^2)*(4*B*c*d^2 + A*c*d*e - 5*a*B*e 
^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[1 - (c*x^2)/a]* 
EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d 
)/Sqrt[a] + e)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[a - c*x^2]))/(c*d^2 - a*e^2 
)))/(15*e^2*(c*d^2 - a*e^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 680

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + c*x^2)^p/(e^2*(m + 1)*(m 
+ 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e* 
f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Sim 
p[p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2))   Int[(d + e*x)^(m + 2)*(a + c*x^ 
2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1) - e*f 
*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, 
 g}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3 
, 0]
 

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])
 

Maple [A] (verified)

Time = 5.62 (sec) , antiderivative size = 890, normalized size of antiderivative = 1.68

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (-\frac {2 \left (A e -B d \right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{5 e^{5} \left (x +\frac {d}{e}\right )^{3}}-\frac {2 \left (2 A c d e +5 B a \,e^{2}-7 B c \,d^{2}\right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{15 \left (a \,e^{2}-c \,d^{2}\right ) e^{4} \left (x +\frac {d}{e}\right )^{2}}+\frac {4 \left (-c e \,x^{2}+a e \right ) c \left (3 A a \,e^{3}+A c \,d^{2} e -8 B a d \,e^{2}+4 B c \,d^{3}\right )}{15 e^{3} \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (-c e \,x^{2}+a e \right )}}+\frac {2 \left (-\frac {B c}{e^{3}}+\frac {c \left (2 A c d e +5 B a \,e^{2}-7 B c \,d^{2}\right )}{15 e^{3} \left (a \,e^{2}-c \,d^{2}\right )}+\frac {2 c^{2} d \left (3 A a \,e^{3}+A c \,d^{2} e -8 B a d \,e^{2}+4 B c \,d^{3}\right )}{15 e^{3} \left (a \,e^{2}-c \,d^{2}\right )^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {4 c^{2} \left (3 A a \,e^{3}+A c \,d^{2} e -8 B a d \,e^{2}+4 B c \,d^{3}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{15 e^{2} \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(890\)
default	\(\text {Expression too large to display}\)	\(6994\)

Input:

int((B*x+A)*(-c*x^2+a)^(1/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

((e*x+d)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)*(-2/5*(A*e-B*d)/ 
e^5*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^3-2/15*(2*A*c*d*e+5*B*a*e^2 
-7*B*c*d^2)/(a*e^2-c*d^2)/e^4*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^2 
+4/15*(-c*e*x^2+a*e)/e^3/(a*e^2-c*d^2)^2*c*(3*A*a*e^3+A*c*d^2*e-8*B*a*d*e^ 
2+4*B*c*d^3)/((x+d/e)*(-c*e*x^2+a*e))^(1/2)+2*(-B*c/e^3+1/15*c*(2*A*c*d*e+ 
5*B*a*e^2-7*B*c*d^2)/e^3/(a*e^2-c*d^2)+2/15*c^2/e^3*d*(3*A*a*e^3+A*c*d^2*e 
-8*B*a*d*e^2+4*B*c*d^3)/(a*e^2-c*d^2)^2)*(d/e-1/c*(a*c)^(1/2))*((x+d/e)/(d 
/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1 
/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2)))^(1/2)/(-c*e*x^3-c*d*x^2+a 
*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c 
*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))+4/15*c^2/e^2*(3*A*a*e^3+A*c*d 
^2*e-8*B*a*d*e^2+4*B*c*d^3)/(a*e^2-c*d^2)^2*(d/e-1/c*(a*c)^(1/2))*((x+d/e) 
/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2))) 
^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2)))^(1/2)/(-c*e*x^3-c*d*x^ 
2+a*e*x+a*d)^(1/2)*((-d/e-1/c*(a*c)^(1/2))*EllipticE(((x+d/e)/(d/e-1/c*(a* 
c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))+1/ 
c*(a*c)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c*( 
a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 903, normalized size of antiderivative = 1.71 \[ \int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{7/2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)*(-c*x^2+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="fricas")
 

Output:

2/45*(2*(4*B*c^2*d^7 + A*c^2*d^6*e - 11*B*a*c*d^5*e^2 - 9*A*a*c*d^4*e^3 + 
15*B*a^2*d^3*e^4 + (4*B*c^2*d^4*e^3 + A*c^2*d^3*e^4 - 11*B*a*c*d^2*e^5 - 9 
*A*a*c*d*e^6 + 15*B*a^2*e^7)*x^3 + 3*(4*B*c^2*d^5*e^2 + A*c^2*d^4*e^3 - 11 
*B*a*c*d^3*e^4 - 9*A*a*c*d^2*e^5 + 15*B*a^2*d*e^6)*x^2 + 3*(4*B*c^2*d^6*e 
+ A*c^2*d^5*e^2 - 11*B*a*c*d^4*e^3 - 9*A*a*c*d^3*e^4 + 15*B*a^2*d^2*e^5)*x 
)*sqrt(-c*e)*weierstrassPInverse(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d 
^3 - 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 6*(4*B*c^2*d^6*e + A*c^2*d^5 
*e^2 - 8*B*a*c*d^4*e^3 + 3*A*a*c*d^3*e^4 + (4*B*c^2*d^3*e^4 + A*c^2*d^2*e^ 
5 - 8*B*a*c*d*e^6 + 3*A*a*c*e^7)*x^3 + 3*(4*B*c^2*d^4*e^3 + A*c^2*d^3*e^4 
- 8*B*a*c*d^2*e^5 + 3*A*a*c*d*e^6)*x^2 + 3*(4*B*c^2*d^5*e^2 + A*c^2*d^4*e^ 
3 - 8*B*a*c*d^3*e^4 + 3*A*a*c*d^2*e^5)*x)*sqrt(-c*e)*weierstrassZeta(4/3*( 
c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), weierstrassPI 
nverse(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), 1 
/3*(3*e*x + d)/e)) + 3*(4*B*c^2*d^5*e^2 + A*c^2*d^4*e^3 - 10*B*a*c*d^3*e^4 
 + 10*A*a*c*d^2*e^5 - 2*B*a^2*d*e^6 - 3*A*a^2*e^7 + 2*(4*B*c^2*d^3*e^4 + A 
*c^2*d^2*e^5 - 8*B*a*c*d*e^6 + 3*A*a*c*e^7)*x^2 + (9*B*c^2*d^4*e^3 + 6*A*c 
^2*d^3*e^4 - 20*B*a*c*d^2*e^5 + 10*A*a*c*d*e^6 - 5*B*a^2*e^7)*x)*sqrt(-c*x 
^2 + a)*sqrt(e*x + d))/(c^2*d^7*e^4 - 2*a*c*d^5*e^6 + a^2*d^3*e^8 + (c^2*d 
^4*e^7 - 2*a*c*d^2*e^9 + a^2*e^11)*x^3 + 3*(c^2*d^5*e^6 - 2*a*c*d^3*e^8 + 
a^2*d*e^10)*x^2 + 3*(c^2*d^6*e^5 - 2*a*c*d^4*e^7 + a^2*d^2*e^9)*x)
 

Sympy [F]

\[ \int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a - c x^{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((B*x+A)*(-c*x**2+a)**(1/2)/(e*x+d)**(7/2),x)
 

Output:

Integral((A + B*x)*sqrt(a - c*x**2)/(d + e*x)**(7/2), x)
 

Maxima [F]

\[ \int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{7/2}} \, dx=\int { \frac {\sqrt {-c x^{2} + a} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((B*x+A)*(-c*x^2+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(-c*x^2 + a)*(B*x + A)/(e*x + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{7/2}} \, dx=\int { \frac {\sqrt {-c x^{2} + a} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((B*x+A)*(-c*x^2+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="giac")
 

Output:

integrate(sqrt(-c*x^2 + a)*(B*x + A)/(e*x + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\sqrt {a-c\,x^2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \] Input:

int(((a - c*x^2)^(1/2)*(A + B*x))/(d + e*x)^(7/2),x)
 

Output:

int(((a - c*x^2)^(1/2)*(A + B*x))/(d + e*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {(A+B x) \sqrt {a-c x^2}}{(d+e x)^{7/2}} \, dx=\text {too large to display} \] Input:

int((B*x+A)*(-c*x^2+a)^(1/2)/(e*x+d)^(7/2),x)
 

Output:

(2*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*e - 2*sqrt(d + e*x)*sqrt(a - c*x**2) 
*b*c*d*x - 3*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**4 + 4*a*d**3* 
e*x + 6*a*d**2*e**2*x**2 + 4*a*d*e**3*x**3 + a*e**4*x**4 - c*d**4*x**2 - 4 
*c*d**3*e*x**3 - 6*c*d**2*e**2*x**4 - 4*c*d*e**3*x**5 - c*e**4*x**6),x)*a* 
b*c*d**3*e**2 - 9*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**4 + 4*a* 
d**3*e*x + 6*a*d**2*e**2*x**2 + 4*a*d*e**3*x**3 + a*e**4*x**4 - c*d**4*x** 
2 - 4*c*d**3*e*x**3 - 6*c*d**2*e**2*x**4 - 4*c*d*e**3*x**5 - c*e**4*x**6), 
x)*a*b*c*d**2*e**3*x - 9*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d**4 
 + 4*a*d**3*e*x + 6*a*d**2*e**2*x**2 + 4*a*d*e**3*x**3 + a*e**4*x**4 - c*d 
**4*x**2 - 4*c*d**3*e*x**3 - 6*c*d**2*e**2*x**4 - 4*c*d*e**3*x**5 - c*e**4 
*x**6),x)*a*b*c*d*e**4*x**2 - 3*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/ 
(a*d**4 + 4*a*d**3*e*x + 6*a*d**2*e**2*x**2 + 4*a*d*e**3*x**3 + a*e**4*x** 
4 - c*d**4*x**2 - 4*c*d**3*e*x**3 - 6*c*d**2*e**2*x**4 - 4*c*d*e**3*x**5 - 
 c*e**4*x**6),x)*a*b*c*e**5*x**3 - int((sqrt(d + e*x)*sqrt(a - c*x**2)*x** 
2)/(a*d**4 + 4*a*d**3*e*x + 6*a*d**2*e**2*x**2 + 4*a*d*e**3*x**3 + a*e**4* 
x**4 - c*d**4*x**2 - 4*c*d**3*e*x**3 - 6*c*d**2*e**2*x**4 - 4*c*d*e**3*x** 
5 - c*e**4*x**6),x)*a*c**2*d**4*e - 3*int((sqrt(d + e*x)*sqrt(a - c*x**2)* 
x**2)/(a*d**4 + 4*a*d**3*e*x + 6*a*d**2*e**2*x**2 + 4*a*d*e**3*x**3 + a*e* 
*4*x**4 - c*d**4*x**2 - 4*c*d**3*e*x**3 - 6*c*d**2*e**2*x**4 - 4*c*d*e**3* 
x**5 - c*e**4*x**6),x)*a*c**2*d**3*e**2*x - 3*int((sqrt(d + e*x)*sqrt(a...