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3.517 \(\int (e x)^{5/2} \sqrt{a+b x^3} \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=324 \[ -\frac{3^{3/4} a^{5/3} e^2 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (16 A b-7 a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{640 b^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{3 a e^2 \sqrt{e x} \sqrt{a+b x^3} (16 A b-7 a B)}{320 b^2}+\frac{(e x)^{7/2} \sqrt{a+b x^3} (16 A b-7 a B)}{80 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{8 b e} \]

[Out]

(3*a*(16*A*b - 7*a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^3])/(320*b^2) + ((16*A*b - 7*a*
B)*(e*x)^(7/2)*Sqrt[a + b*x^3])/(80*b*e) + (B*(e*x)^(7/2)*(a + b*x^3)^(3/2))/(8*
b*e) - (3^(3/4)*a^(5/3)*(16*A*b - 7*a*B)*e^2*Sqrt[e*x]*(a^(1/3) + b^(1/3)*x)*Sqr
t[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x
)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3
])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(640*b^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x)
)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi [A]  time = 0.73923, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{3^{3/4} a^{5/3} e^2 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (16 A b-7 a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{640 b^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{3 a e^2 \sqrt{e x} \sqrt{a+b x^3} (16 A b-7 a B)}{320 b^2}+\frac{(e x)^{7/2} \sqrt{a+b x^3} (16 A b-7 a B)}{80 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{3/2}}{8 b e} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(5/2)*Sqrt[a + b*x^3]*(A + B*x^3),x]

[Out]

(3*a*(16*A*b - 7*a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^3])/(320*b^2) + ((16*A*b - 7*a*
B)*(e*x)^(7/2)*Sqrt[a + b*x^3])/(80*b*e) + (B*(e*x)^(7/2)*(a + b*x^3)^(3/2))/(8*
b*e) - (3^(3/4)*a^(5/3)*(16*A*b - 7*a*B)*e^2*Sqrt[e*x]*(a^(1/3) + b^(1/3)*x)*Sqr
t[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x
)^2]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3
])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(640*b^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x)
)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi in Sympy [A]  time = 36.3921, size = 292, normalized size = 0.9 \[ \frac{B \left (e x\right )^{\frac{7}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}}}{8 b e} - \frac{3^{\frac{3}{4}} a^{\frac{5}{3}} e^{2} \sqrt{e x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (16 A b - 7 B a\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{640 b^{2} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{3 a e^{2} \sqrt{e x} \sqrt{a + b x^{3}} \left (16 A b - 7 B a\right )}{320 b^{2}} + \frac{\left (e x\right )^{\frac{7}{2}} \sqrt{a + b x^{3}} \left (16 A b - 7 B a\right )}{80 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(5/2)*(B*x**3+A)*(b*x**3+a)**(1/2),x)

[Out]

B*(e*x)**(7/2)*(a + b*x**3)**(3/2)/(8*b*e) - 3**(3/4)*a**(5/3)*e**2*sqrt(e*x)*sq
rt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1/3) + b**(1/3)*x*(1 +
sqrt(3)))**2)*(a**(1/3) + b**(1/3)*x)*(16*A*b - 7*B*a)*elliptic_f(acos((a**(1/3)
 + b**(1/3)*x*(-sqrt(3) + 1))/(a**(1/3) + b**(1/3)*x*(1 + sqrt(3)))), sqrt(3)/4
+ 1/2)/(640*b**2*sqrt(b**(1/3)*x*(a**(1/3) + b**(1/3)*x)/(a**(1/3) + b**(1/3)*x*
(1 + sqrt(3)))**2)*sqrt(a + b*x**3)) + 3*a*e**2*sqrt(e*x)*sqrt(a + b*x**3)*(16*A
*b - 7*B*a)/(320*b**2) + (e*x)**(7/2)*sqrt(a + b*x**3)*(16*A*b - 7*B*a)/(80*b*e)

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Mathematica [C]  time = 1.11357, size = 234, normalized size = 0.72 \[ \frac{e^2 \sqrt{e x} \left (-\sqrt [3]{-a} \left (a+b x^3\right ) \left (21 a^2 B-12 a b \left (4 A+B x^3\right )-8 b^2 x^3 \left (8 A+5 B x^3\right )\right )+i 3^{3/4} a^2 \sqrt [3]{b} x \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} x}} \sqrt{\frac{\frac{(-a)^{2/3}}{b^{2/3}}+\frac{\sqrt [3]{-a} x}{\sqrt [3]{b}}+x^2}{x^2}} (16 A b-7 a B) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{320 \sqrt [3]{-a} b^2 \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^(5/2)*Sqrt[a + b*x^3]*(A + B*x^3),x]

[Out]

(e^2*Sqrt[e*x]*(-((-a)^(1/3)*(a + b*x^3)*(21*a^2*B - 12*a*b*(4*A + B*x^3) - 8*b^
2*x^3*(8*A + 5*B*x^3))) + I*3^(3/4)*a^2*b^(1/3)*(16*A*b - 7*a*B)*x*Sqrt[((-1)^(5
/6)*((-a)^(1/3) - b^(1/3)*x))/(b^(1/3)*x)]*Sqrt[((-a)^(2/3)/b^(2/3) + ((-a)^(1/3
)*x)/b^(1/3) + x^2)/x^2]*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-a)^(1/3))/(b^(
1/3)*x)]/3^(1/4)], (-1)^(1/3)]))/(320*(-a)^(1/3)*b^2*Sqrt[a + b*x^3])

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Maple [C]  time = 0.082, size = 4175, normalized size = 12.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(5/2)*(B*x^3+A)*(b*x^3+a)^(1/2),x)

[Out]

1/320*e^2*(e*x)^(1/2)*(b*x^3+a)^(1/2)/b^3/(-a*b^2)^(1/3)*(48*I*A*((b*x^3+a)*e*x)
^(1/2)*(-a*b^2)^(1/3)*3^(1/2)*(1/b^2*e*x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^
2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3)))^
(1/2)*a*b^2+64*I*A*((b*x^3+a)*e*x)^(1/2)*(-a*b^2)^(1/3)*3^(1/2)*(1/b^2*e*x*(-b*x
+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*
b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3)))^(1/2)*x^3*b^3-192*I*A*(-a*b^2)^(1/3)*3^(1/2)*(
-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^
2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^
(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^
(1/2)*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),(
(I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*x*a^2*b^2*e+12*I
*B*((b*x^3+a)*e*x)^(1/2)*(-a*b^2)^(1/3)*3^(1/2)*(1/b^2*e*x*(-b*x+(-a*b^2)^(1/3))
*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x
-(-a*b^2)^(1/3)))^(1/2)*x^3*a*b^2+40*I*B*((b*x^3+a)*e*x)^(1/2)*(-a*b^2)^(1/3)*3^
(1/2)*(1/b^2*e*x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^
(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3)))^(1/2)*x^6*b^3-21*I*B*((b
*x^3+a)*e*x)^(1/2)*(-a*b^2)^(1/3)*3^(1/2)*(1/b^2*e*x*(-b*x+(-a*b^2)^(1/3))*(I*3^
(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b
^2)^(1/3)))^(1/2)*a^2*b-96*A*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1
/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x
+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1
/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/
(-b*x+(-a*b^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/
2)-3))^(1/2))*x^2*a^2*b^3*e-120*B*((b*x^3+a)*e*x)^(1/2)*(-a*b^2)^(1/3)*(1/b^2*e*
x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/
2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3)))^(1/2)*x^6*b^3-42*I*B*3^(1/2)*(-(I*3^(1/
2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+
2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a
*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*Ell
ipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),((I*3^(1/2
)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*x^2*a^3*b^2*e+42*B*(-(I*3
^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1
/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)
*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)
*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),((I*3^
(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*x^2*a^3*b^2*e+84*I*B*
(-a*b^2)^(1/3)*3^(1/2)*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^
(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b
^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)
/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+
(-a*b^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))
^(1/2))*x*a^3*b*e+192*A*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))
^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*
b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1
)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x
+(-a*b^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3)
)^(1/2))*(-a*b^2)^(1/3)*x*a^2*b^2*e-42*I*B*(-a*b^2)^(2/3)*3^(1/2)*(-(I*3^(1/2)-3
)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*
x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2
)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*Ellipti
cF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),((I*3^(1/2)+3)
*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*a^3*e-84*B*(-(I*3^(1/2)-3)*x*
b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-
a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1
/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((
-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*
3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-a*b^2)^(1/3)*x*a^3*b*e-96*A*(-(
I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)
^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1
/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1
/2)*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),((I
*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-a*b^2)^(2/3)*a^2
*b*e+42*B*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(
1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(
1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^
2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3
)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-a*b
^2)^(2/3)*a^3*e-192*A*((b*x^3+a)*e*x)^(1/2)*(-a*b^2)^(1/3)*(1/b^2*e*x*(-b*x+(-a*
b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^
(1/3)-2*b*x-(-a*b^2)^(1/3)))^(1/2)*x^3*b^3-36*B*((b*x^3+a)*e*x)^(1/2)*(-a*b^2)^(
1/3)*(1/b^2*e*x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(
1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3)))^(1/2)*x^3*a*b^2+96*I*A*(-
a*b^2)^(2/3)*3^(1/2)*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1
/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2
)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(
-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-
a*b^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(
1/2))*a^2*b*e+96*I*A*3^(1/2)*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1
/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x
+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1
/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/
(-b*x+(-a*b^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/
2)-3))^(1/2))*x^2*a^2*b^3*e-144*A*((b*x^3+a)*e*x)^(1/2)*(-a*b^2)^(1/3)*(1/b^2*e*
x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/
2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3)))^(1/2)*a*b^2+63*B*((b*x^3+a)*e*x)^(1/2)*
(-a*b^2)^(1/3)*(1/b^2*e*x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+
(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3)))^(1/2)*a^2*b)/((
b*x^3+a)*e*x)^(1/2)/(I*3^(1/2)-3)/(1/b^2*e*x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-
a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3
)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{3} + A\right )} \sqrt{b x^{3} + a} \left (e x\right )^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B e^{2} x^{5} + A e^{2} x^{2}\right )} \sqrt{b x^{3} + a} \sqrt{e x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^(5/2),x, algorithm="fricas")

[Out]

integral((B*e^2*x^5 + A*e^2*x^2)*sqrt(b*x^3 + a)*sqrt(e*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**(5/2)*(B*x**3+A)*(b*x**3+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{3} + A\right )} \sqrt{b x^{3} + a} \left (e x\right )^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^(5/2), x)

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