\(\int \frac {(A+B x) (a-c x^2)^{3/2}}{\sqrt {d+e x}} \, dx\) [268]

Optimal result

Integrand size = 27, antiderivative size = 493 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {4 \sqrt {d+e x} \left (32 B c d^3-36 A c d^2 e-33 a B d e^2+45 a A e^3-3 e \left (8 B c d^2-9 A c d e-7 a B e^2\right ) x\right ) \sqrt {a-c x^2}}{315 e^4}-\frac {2 \sqrt {d+e x} (8 B d-9 A e-7 B e x) \left (a-c x^2\right )^{3/2}}{63 e^2}+\frac {8 \sqrt {a} \left (36 A c d e \left (c d^2-2 a e^2\right )-B \left (32 c^2 d^4-57 a c d^2 e^2+21 a^2 e^4\right )\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{315 \sqrt {c} e^5 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}+\frac {8 \sqrt {a} \left (c d^2-a e^2\right ) \left (32 B c d^3-36 A c d^2 e-33 a B d e^2+45 a A e^3\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{315 \sqrt {c} e^5 \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

4/315*(e*x+d)^(1/2)*(32*B*c*d^3-36*A*c*d^2*e-33*B*a*d*e^2+45*A*a*e^3-3*e*( 
-9*A*c*d*e-7*B*a*e^2+8*B*c*d^2)*x)*(-c*x^2+a)^(1/2)/e^4-2/63*(e*x+d)^(1/2) 
*(-7*B*e*x-9*A*e+8*B*d)*(-c*x^2+a)^(3/2)/e^2+8/315*a^(1/2)*(36*A*c*d*e*(-2 
*a*e^2+c*d^2)-B*(21*a^2*e^4-57*a*c*d^2*e^2+32*c^2*d^4))*(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))/c^(1/2)/e^5/(c^(1/2)*(e*x+d)/(c^(1/ 
2)*d+a^(1/2)*e))^(1/2)/(-c*x^2+a)^(1/2)+8/315*a^(1/2)*(-a*e^2+c*d^2)*(45*A 
*a*e^3-36*A*c*d^2*e-33*B*a*d*e^2+32*B*c*d^3)*(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))/c^(1/2)/e^5/(e 
*x+d)^(1/2)/(-c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 27.33 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {\sqrt {a-c x^2} \left (\frac {2 (d+e x) \left (135 a A e^3+a B e^2 (-106 d+77 e x)-9 A c e \left (8 d^2-6 d e x+5 e^2 x^2\right )+B c \left (64 d^3-48 d^2 e x+40 d e^2 x^2-35 e^3 x^3\right )\right )}{e^4}+\frac {8 \left (e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-36 A c d e \left (c d^2-2 a e^2\right )+B \left (32 c^2 d^4-57 a c d^2 e^2+21 a^2 e^4\right )\right ) \left (-a+c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (36 A c d e \left (c d^2-2 a e^2\right )+B \left (-32 c^2 d^4+57 a c d^2 e^2-21 a^2 e^4\right )\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {a} \sqrt {c} e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (9 A \sqrt {c} e \left (4 c d^2+3 \sqrt {a} \sqrt {c} d e-5 a e^2\right )+B \left (-32 c^{3/2} d^3-24 \sqrt {a} c d^2 e+33 a \sqrt {c} d e^2+21 a^{3/2} e^3\right )\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )\right )}{c e^6 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (a-c x^2\right )}\right )}{315 \sqrt {d+e x}} \] Input:

Integrate[((A + B*x)*(a - c*x^2)^(3/2))/Sqrt[d + e*x],x]
 

Output:

(Sqrt[a - c*x^2]*((2*(d + e*x)*(135*a*A*e^3 + a*B*e^2*(-106*d + 77*e*x) - 
9*A*c*e*(8*d^2 - 6*d*e*x + 5*e^2*x^2) + B*c*(64*d^3 - 48*d^2*e*x + 40*d*e^ 
2*x^2 - 35*e^3*x^3)))/e^4 + (8*(e^2*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-36*A* 
c*d*e*(c*d^2 - 2*a*e^2) + B*(32*c^2*d^4 - 57*a*c*d^2*e^2 + 21*a^2*e^4))*(- 
a + c*x^2) + I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(36*A*c*d*e*(c*d^2 - 2*a*e^ 
2) + B*(-32*c^2*d^4 + 57*a*c*d^2*e^2 - 21*a^2*e^4))*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)] + I*Sqrt[a]*Sqrt[c]*e*( 
Sqrt[c]*d - Sqrt[a]*e)*(9*A*Sqrt[c]*e*(4*c*d^2 + 3*Sqrt[a]*Sqrt[c]*d*e - 5 
*a*e^2) + B*(-32*c^(3/2)*d^3 - 24*Sqrt[a]*c*d^2*e + 33*a*Sqrt[c]*d*e^2 + 2 
1*a^(3/2)*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)*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)]))/(c*e^6*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(a - c*x^2))))/(315 
*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {682, 27, 682, 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.

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

\(\Big \downarrow \) 682

\(\displaystyle -\frac {4 \int \frac {c \left (a e (B d-9 A e)+\left (8 B c d^2-9 A c e d-7 a B e^2\right ) x\right ) \sqrt {a-c x^2}}{2 \sqrt {d+e x}}dx}{21 c e^2}-\frac {2 \left (a-c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \int \frac {\left (a e (B d-9 A e)+\left (8 B c d^2-9 A c e d-7 a B e^2\right ) x\right ) \sqrt {a-c x^2}}{\sqrt {d+e x}}dx}{21 e^2}-\frac {2 \left (a-c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\)

\(\Big \downarrow \) 682

\(\displaystyle -\frac {2 \left (-\frac {4 \int \frac {c \left (a e \left (8 B c d^3-9 A c e d^2-12 a B e^2 d+45 a A e^3\right )-\left (36 A c d e \left (c d^2-2 a e^2\right )-B \left (32 c^2 d^4-57 a c e^2 d^2+21 a^2 e^4\right )\right ) x\right )}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{15 c e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-3 e x \left (-7 a B e^2-9 A c d e+8 B c d^2\right )+45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a-c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {2 \int \frac {a e \left (8 B c d^3-9 A c e d^2-12 a B e^2 d+45 a A e^3\right )-\left (36 A c d e \left (c d^2-2 a e^2\right )-B \left (32 c^2 d^4-57 a c e^2 d^2+21 a^2 e^4\right )\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-3 e x \left (-7 a B e^2-9 A c d e+8 B c d^2\right )+45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a-c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\)

\(\Big \downarrow \) 600

\(\displaystyle -\frac {2 \left (-\frac {2 \left (-\frac {\left (36 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (c d^2-a e^2\right ) \left (45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-3 e x \left (-7 a B e^2-9 A c d e+8 B c d^2\right )+45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a-c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\)

\(\Big \downarrow \) 509

\(\displaystyle -\frac {2 \left (-\frac {2 \left (-\frac {\sqrt {1-\frac {c x^2}{a}} \left (36 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (c d^2-a e^2\right ) \left (45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-3 e x \left (-7 a B e^2-9 A c d e+8 B c d^2\right )+45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a-c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\)

\(\Big \downarrow \) 508

\(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (36 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {\left (c d^2-a e^2\right ) \left (45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-3 e x \left (-7 a B e^2-9 A c d e+8 B c d^2\right )+45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a-c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (36 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {\left (c d^2-a e^2\right ) \left (45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-3 e x \left (-7 a B e^2-9 A c d e+8 B c d^2\right )+45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a-c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\)

\(\Big \downarrow \) 512

\(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (36 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}-\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \left (45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-3 e x \left (-7 a B e^2-9 A c d e+8 B c d^2\right )+45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a-c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\)

\(\Big \downarrow \) 511

\(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right ) \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (36 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-3 e x \left (-7 a B e^2-9 A c d e+8 B c d^2\right )+45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a-c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2 \left (-\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (36 A c d e \left (c d^2-2 a e^2\right )-B \left (21 a^2 e^4-57 a c d^2 e^2+32 c^2 d^4\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}+\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \left (45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} \left (-3 e x \left (-7 a B e^2-9 A c d e+8 B c d^2\right )+45 a A e^3-33 a B d e^2-36 A c d^2 e+32 B c d^3\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a-c x^2\right )^{3/2} \sqrt {d+e x} (-9 A e+8 B d-7 B e x)}{63 e^2}\)

Input:

Int[((A + B*x)*(a - c*x^2)^(3/2))/Sqrt[d + e*x],x]
 

Output:

(-2*Sqrt[d + e*x]*(8*B*d - 9*A*e - 7*B*e*x)*(a - c*x^2)^(3/2))/(63*e^2) - 
(2*((-2*Sqrt[d + e*x]*(32*B*c*d^3 - 36*A*c*d^2*e - 33*a*B*d*e^2 + 45*a*A*e 
^3 - 3*e*(8*B*c*d^2 - 9*A*c*d*e - 7*a*B*e^2)*x)*Sqrt[a - c*x^2])/(15*e^2) 
- (2*((2*Sqrt[a]*(36*A*c*d*e*(c*d^2 - 2*a*e^2) - B*(32*c^2*d^4 - 57*a*c*d^ 
2*e^2 + 21*a^2*e^4))*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a]*EllipticE[ArcSin[Sq 
rt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(S 
qrt[c]*e*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2] 
) + (2*Sqrt[a]*(c*d^2 - a*e^2)*(32*B*c*d^3 - 36*A*c*d^2*e - 33*a*B*d*e^2 + 
 45*a*A*e^3)*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])))/(15* 
e^2)))/(21*e^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 682

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1049\) vs. \(2(423)=846\).

Time = 7.00 (sec) , antiderivative size = 1050, normalized size of antiderivative = 2.13

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1050\)
risch	\(\text {Expression too large to display}\)	\(1117\)
default	\(\text {Expression too large to display}\)	\(2966\)

Input:

int((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((e*x+d)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)*(-2/9*B/e*c*x^3* 
(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)-2/7*(A*c^2-8/9*c^2*d/e*B)/c/e*x^2*(-c*e 
*x^3-c*d*x^2+a*e*x+a*d)^(1/2)-2/5*(-11/9*a*B*c-6/7*d/e*(A*c^2-8/9*c^2*d/e* 
B))/c/e*x*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)-2/3*(-2*A*a*c-4/5*d/e*(-11/9* 
a*B*c-6/7*d/e*(A*c^2-8/9*c^2*d/e*B))+5/7*a/c*(A*c^2-8/9*c^2*d/e*B)+2/3*a*c 
*d/e*B)/c/e*(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)+2*(a^2*A+2/5*a/c*d/e*(-11/9 
*a*B*c-6/7*d/e*(A*c^2-8/9*c^2*d/e*B))+1/3*a/c*(-2*A*a*c-4/5*d/e*(-11/9*a*B 
*c-6/7*d/e*(A*c^2-8/9*c^2*d/e*B))+5/7*a/c*(A*c^2-8/9*c^2*d/e*B)+2/3*a*c*d/ 
e*B))*(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))+2*(a^2*B+4/7*a/c*d/e*(A*c^2-8/9*c^2*d/e*B)+3/5*a/c*(-11/9*a*B*c- 
6/7*d/e*(A*c^2-8/9*c^2*d/e*B))-2/3*d/e*(-2*A*a*c-4/5*d/e*(-11/9*a*B*c-6/7* 
d/e*(A*c^2-8/9*c^2*d/e*B))+5/7*a/c*(A*c^2-8/9*c^2*d/e*B)+2/3*a*c*d/e*B))*( 
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))*El 
lipticE(((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*...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 417, normalized size of antiderivative = 0.85 \[ \int \frac {(A+B x) \left (a-c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (4 \, {\left (32 \, B c^{2} d^{5} - 36 \, A c^{2} d^{4} e - 81 \, B a c d^{3} e^{2} + 99 \, A a c d^{2} e^{3} + 57 \, B a^{2} d e^{4} - 135 \, A a^{2} e^{5}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (32 \, B c^{2} d^{4} e - 36 \, A c^{2} d^{3} e^{2} - 57 \, B a c d^{2} e^{3} + 72 \, A a c d e^{4} + 21 \, B a^{2} e^{5}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (35 \, B c^{2} e^{5} x^{3} - 64 \, B c^{2} d^{3} e^{2} + 72 \, A c^{2} d^{2} e^{3} + 106 \, B a c d e^{4} - 135 \, A a c e^{5} - 5 \, {\left (8 \, B c^{2} d e^{4} - 9 \, A c^{2} e^{5}\right )} x^{2} + {\left (48 \, B c^{2} d^{2} e^{3} - 54 \, A c^{2} d e^{4} - 77 \, B a c e^{5}\right )} x\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}\right )}}{945 \, c e^{6}} \] Input:

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

Output:

2/945*(4*(32*B*c^2*d^5 - 36*A*c^2*d^4*e - 81*B*a*c*d^3*e^2 + 99*A*a*c*d^2* 
e^3 + 57*B*a^2*d*e^4 - 135*A*a^2*e^5)*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) + 12*(32*B*c^2*d^4*e - 36*A*c^2*d^3*e^2 - 57*B*a*c*d^2*e^3 + 72*A*a* 
c*d*e^4 + 21*B*a^2*e^5)*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), 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)) 
 - 3*(35*B*c^2*e^5*x^3 - 64*B*c^2*d^3*e^2 + 72*A*c^2*d^2*e^3 + 106*B*a*c*d 
*e^4 - 135*A*a*c*e^5 - 5*(8*B*c^2*d*e^4 - 9*A*c^2*e^5)*x^2 + (48*B*c^2*d^2 
*e^3 - 54*A*c^2*d*e^4 - 77*B*a*c*e^5)*x)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/( 
c*e^6)
 

Sympy [F]

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

integrate((B*x+A)*(-c*x**2+a)**(3/2)/(e*x+d)**(1/2),x)
 

Output:

Integral((A + B*x)*(a - c*x**2)**(3/2)/sqrt(d + e*x), x)
 

Maxima [F]

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

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

Output:

integrate((-c*x^2 + a)^(3/2)*(B*x + A)/sqrt(e*x + d), x)
 

Giac [F]

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

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

Output:

integrate((-c*x^2 + a)^(3/2)*(B*x + A)/sqrt(e*x + d), x)
 

Mupad [F(-1)]

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

int(((a - c*x^2)^(3/2)*(A + B*x))/(d + e*x)^(1/2),x)
 

Output:

int(((a - c*x^2)^(3/2)*(A + B*x))/(d + e*x)^(1/2), x)
 

Reduce [F]

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

int((B*x+A)*(-c*x^2+a)^(3/2)/(e*x+d)^(1/2),x)
 

Output:

(2*( - 42*sqrt(d + e*x)*sqrt(a - c*x**2)*a**2*b*e**3 - 9*sqrt(d + e*x)*sqr 
t(a - c*x**2)*a**2*c*d*e**2 + 8*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*c*d**2* 
e + 77*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*c*d*e**2*x + 54*sqrt(d + e*x)*sq 
rt(a - c*x**2)*a*c**2*d**2*e*x - 45*sqrt(d + e*x)*sqrt(a - c*x**2)*a*c**2* 
d*e**2*x**2 - 48*sqrt(d + e*x)*sqrt(a - c*x**2)*b*c**2*d**3*x + 40*sqrt(d 
+ e*x)*sqrt(a - c*x**2)*b*c**2*d**2*e*x**2 - 35*sqrt(d + e*x)*sqrt(a - c*x 
**2)*b*c**2*d*e**2*x**3 - 63*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a* 
d + a*e*x - c*d*x**2 - c*e*x**3),x)*a**2*b*c*e**4 - 216*int((sqrt(d + e*x) 
*sqrt(a - c*x**2)*x**2)/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a**2*c**2*d 
*e**3 + 171*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d + a*e*x - c*d*x 
**2 - c*e*x**3),x)*a*b*c**2*d**2*e**2 + 108*int((sqrt(d + e*x)*sqrt(a - c* 
x**2)*x**2)/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a*c**3*d**3*e - 96*int( 
(sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d + a*e*x - c*d*x**2 - c*e*x**3), 
x)*b*c**3*d**4 + 21*int((sqrt(d + e*x)*sqrt(a - c*x**2))/(a*d + a*e*x - c* 
d*x**2 - c*e*x**3),x)*a**3*b*e**4 + 162*int((sqrt(d + e*x)*sqrt(a - c*x**2 
))/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a**3*c*d*e**3 - 81*int((sqrt(d + 
 e*x)*sqrt(a - c*x**2))/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a**2*b*c*d* 
*2*e**2 - 54*int((sqrt(d + e*x)*sqrt(a - c*x**2))/(a*d + a*e*x - c*d*x**2 
- c*e*x**3),x)*a**2*c**2*d**3*e + 48*int((sqrt(d + e*x)*sqrt(a - c*x**2))/ 
(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a*b*c**2*d**4))/(315*c*d*e**3)