3.527 \(\int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=201 \[ -\frac{a^3 e^{7/2} (8 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{192 b^{5/2}}+\frac{a^2 e^2 (e x)^{3/2} \sqrt{a+b x^3} (8 A b-3 a B)}{192 b^2}+\frac{(e x)^{9/2} \left (a+b x^3\right )^{3/2} (8 A b-3 a B)}{72 b e}+\frac{a (e x)^{9/2} \sqrt{a+b x^3} (8 A b-3 a B)}{96 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e} \]

[Out]

(a^2*(8*A*b - 3*a*B)*e^2*(e*x)^(3/2)*Sqrt[a + b*x^3])/(192*b^2) + (a*(8*A*b - 3*
a*B)*(e*x)^(9/2)*Sqrt[a + b*x^3])/(96*b*e) + ((8*A*b - 3*a*B)*(e*x)^(9/2)*(a + b
*x^3)^(3/2))/(72*b*e) + (B*(e*x)^(9/2)*(a + b*x^3)^(5/2))/(12*b*e) - (a^3*(8*A*b
- 3*a*B)*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(192
*b^(5/2))

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Rubi [A]  time = 0.394775, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{a^3 e^{7/2} (8 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{192 b^{5/2}}+\frac{a^2 e^2 (e x)^{3/2} \sqrt{a+b x^3} (8 A b-3 a B)}{192 b^2}+\frac{(e x)^{9/2} \left (a+b x^3\right )^{3/2} (8 A b-3 a B)}{72 b e}+\frac{a (e x)^{9/2} \sqrt{a+b x^3} (8 A b-3 a B)}{96 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(7/2)*(a + b*x^3)^(3/2)*(A + B*x^3),x]

[Out]

(a^2*(8*A*b - 3*a*B)*e^2*(e*x)^(3/2)*Sqrt[a + b*x^3])/(192*b^2) + (a*(8*A*b - 3*
a*B)*(e*x)^(9/2)*Sqrt[a + b*x^3])/(96*b*e) + ((8*A*b - 3*a*B)*(e*x)^(9/2)*(a + b
*x^3)^(3/2))/(72*b*e) + (B*(e*x)^(9/2)*(a + b*x^3)^(5/2))/(12*b*e) - (a^3*(8*A*b
- 3*a*B)*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(192
*b^(5/2))

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Rubi in Sympy [A]  time = 34.8102, size = 182, normalized size = 0.91 \[ \frac{B \left (e x\right )^{\frac{9}{2}} \left (a + b x^{3}\right )^{\frac{5}{2}}}{12 b e} - \frac{a^{3} e^{\frac{7}{2}} \left (8 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{192 b^{\frac{5}{2}}} + \frac{a^{2} e^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (8 A b - 3 B a\right )}{192 b^{2}} + \frac{a \left (e x\right )^{\frac{9}{2}} \sqrt{a + b x^{3}} \left (8 A b - 3 B a\right )}{96 b e} + \frac{\left (e x\right )^{\frac{9}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (8 A b - 3 B a\right )}{72 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(7/2)*(b*x**3+a)**(3/2)*(B*x**3+A),x)

[Out]

B*(e*x)**(9/2)*(a + b*x**3)**(5/2)/(12*b*e) - a**3*e**(7/2)*(8*A*b - 3*B*a)*atan
h(sqrt(b)*(e*x)**(3/2)/(e**(3/2)*sqrt(a + b*x**3)))/(192*b**(5/2)) + a**2*e**2*(
e*x)**(3/2)*sqrt(a + b*x**3)*(8*A*b - 3*B*a)/(192*b**2) + a*(e*x)**(9/2)*sqrt(a
+ b*x**3)*(8*A*b - 3*B*a)/(96*b*e) + (e*x)**(9/2)*(a + b*x**3)**(3/2)*(8*A*b - 3
*B*a)/(72*b*e)

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Mathematica [A]  time = 0.291884, size = 147, normalized size = 0.73 \[ \frac{e^2 (e x)^{3/2} \left (3 a^3 \sqrt{\frac{a}{x^3}+b} (3 a B-8 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )-\sqrt{b} \left (a+b x^3\right ) \left (9 a^3 B-6 a^2 b \left (4 A+B x^3\right )-8 a b^2 x^3 \left (14 A+9 B x^3\right )-16 b^3 x^6 \left (4 A+3 B x^3\right )\right )\right )}{576 b^{5/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^(7/2)*(a + b*x^3)^(3/2)*(A + B*x^3),x]

[Out]

(e^2*(e*x)^(3/2)*(-(Sqrt[b]*(a + b*x^3)*(9*a^3*B - 6*a^2*b*(4*A + B*x^3) - 16*b^
3*x^6*(4*A + 3*B*x^3) - 8*a*b^2*x^3*(14*A + 9*B*x^3))) + 3*a^3*(-8*A*b + 3*a*B)*
Sqrt[b + a/x^3]*ArcTanh[Sqrt[b + a/x^3]/Sqrt[b]]))/(576*b^(5/2)*Sqrt[a + b*x^3])

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Maple [C]  time = 0.069, size = 7705, normalized size = 38.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(b*x^3+a)^(3/2)*(B*x^3+A),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*(e*x)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.699104, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{3} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e + 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (48 \, B b^{3} e^{3} x^{10} + 8 \,{\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} e^{3} x^{7} + 2 \,{\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} e^{3} x^{4} - 3 \,{\left (3 \, B a^{3} - 8 \, A a^{2} b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{2304 \, b^{2}}, \frac{3 \,{\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{3} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right ) + 2 \,{\left (48 \, B b^{3} e^{3} x^{10} + 8 \,{\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} e^{3} x^{7} + 2 \,{\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} e^{3} x^{4} - 3 \,{\left (3 \, B a^{3} - 8 \, A a^{2} b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{1152 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*(e*x)^(7/2),x, algorithm="fricas")

[Out]

[-1/2304*(3*(3*B*a^4 - 8*A*a^3*b)*e^3*sqrt(e/b)*log(-8*b^2*e*x^6 - 8*a*b*e*x^3 -
a^2*e + 4*(2*b^2*x^4 + a*b*x)*sqrt(b*x^3 + a)*sqrt(e*x)*sqrt(e/b)) - 4*(48*B*b^
3*e^3*x^10 + 8*(9*B*a*b^2 + 8*A*b^3)*e^3*x^7 + 2*(3*B*a^2*b + 56*A*a*b^2)*e^3*x^
4 - 3*(3*B*a^3 - 8*A*a^2*b)*e^3*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b^2, 1/1152*(3*(3*
B*a^4 - 8*A*a^3*b)*e^3*sqrt(-e/b)*arctan(2*sqrt(b*x^3 + a)*sqrt(e*x)*x/((2*b*x^3
+ a)*sqrt(-e/b))) + 2*(48*B*b^3*e^3*x^10 + 8*(9*B*a*b^2 + 8*A*b^3)*e^3*x^7 + 2*
(3*B*a^2*b + 56*A*a*b^2)*e^3*x^4 - 3*(3*B*a^3 - 8*A*a^2*b)*e^3*x)*sqrt(b*x^3 + a
)*sqrt(e*x))/b^2]

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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)**(7/2)*(b*x**3+a)**(3/2)*(B*x**3+A),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*(e*x)^(7/2),x, algorithm="giac")

[Out]

integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*(e*x)^(7/2), x)