3.535 \(\int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=241 \[ -\frac{a^4 e^{7/2} (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{384 b^{5/2}}+\frac{a^3 e^2 (e x)^{3/2} \sqrt{a+b x^3} (10 A b-3 a B)}{384 b^2}+\frac{a^2 (e x)^{9/2} \sqrt{a+b x^3} (10 A b-3 a B)}{192 b e}+\frac{(e x)^{9/2} \left (a+b x^3\right )^{5/2} (10 A b-3 a B)}{120 b e}+\frac{a (e x)^{9/2} \left (a+b x^3\right )^{3/2} (10 A b-3 a B)}{144 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e} \]

[Out]

(a^3*(10*A*b - 3*a*B)*e^2*(e*x)^(3/2)*Sqrt[a + b*x^3])/(384*b^2) + (a^2*(10*A*b
- 3*a*B)*(e*x)^(9/2)*Sqrt[a + b*x^3])/(192*b*e) + (a*(10*A*b - 3*a*B)*(e*x)^(9/2
)*(a + b*x^3)^(3/2))/(144*b*e) + ((10*A*b - 3*a*B)*(e*x)^(9/2)*(a + b*x^3)^(5/2)
)/(120*b*e) + (B*(e*x)^(9/2)*(a + b*x^3)^(7/2))/(15*b*e) - (a^4*(10*A*b - 3*a*B)
*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(384*b^(5/2))

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Rubi [A]  time = 0.471333, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{a^4 e^{7/2} (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{384 b^{5/2}}+\frac{a^3 e^2 (e x)^{3/2} \sqrt{a+b x^3} (10 A b-3 a B)}{384 b^2}+\frac{a^2 (e x)^{9/2} \sqrt{a+b x^3} (10 A b-3 a B)}{192 b e}+\frac{(e x)^{9/2} \left (a+b x^3\right )^{5/2} (10 A b-3 a B)}{120 b e}+\frac{a (e x)^{9/2} \left (a+b x^3\right )^{3/2} (10 A b-3 a B)}{144 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(7/2)*(a + b*x^3)^(5/2)*(A + B*x^3),x]

[Out]

(a^3*(10*A*b - 3*a*B)*e^2*(e*x)^(3/2)*Sqrt[a + b*x^3])/(384*b^2) + (a^2*(10*A*b
- 3*a*B)*(e*x)^(9/2)*Sqrt[a + b*x^3])/(192*b*e) + (a*(10*A*b - 3*a*B)*(e*x)^(9/2
)*(a + b*x^3)^(3/2))/(144*b*e) + ((10*A*b - 3*a*B)*(e*x)^(9/2)*(a + b*x^3)^(5/2)
)/(120*b*e) + (B*(e*x)^(9/2)*(a + b*x^3)^(7/2))/(15*b*e) - (a^4*(10*A*b - 3*a*B)
*e^(7/2)*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(384*b^(5/2))

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Rubi in Sympy [A]  time = 41.4365, size = 218, normalized size = 0.9 \[ \frac{B \left (e x\right )^{\frac{9}{2}} \left (a + b x^{3}\right )^{\frac{7}{2}}}{15 b e} - \frac{a^{4} e^{\frac{7}{2}} \left (10 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{384 b^{\frac{5}{2}}} + \frac{a^{3} e^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (10 A b - 3 B a\right )}{384 b^{2}} + \frac{a^{2} \left (e x\right )^{\frac{9}{2}} \sqrt{a + b x^{3}} \left (10 A b - 3 B a\right )}{192 b e} + \frac{a \left (e x\right )^{\frac{9}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (10 A b - 3 B a\right )}{144 b e} + \frac{\left (e x\right )^{\frac{9}{2}} \left (a + b x^{3}\right )^{\frac{5}{2}} \left (10 A b - 3 B a\right )}{120 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(7/2)*(b*x**3+a)**(5/2)*(B*x**3+A),x)

[Out]

B*(e*x)**(9/2)*(a + b*x**3)**(7/2)/(15*b*e) - a**4*e**(7/2)*(10*A*b - 3*B*a)*ata
nh(sqrt(b)*(e*x)**(3/2)/(e**(3/2)*sqrt(a + b*x**3)))/(384*b**(5/2)) + a**3*e**2*
(e*x)**(3/2)*sqrt(a + b*x**3)*(10*A*b - 3*B*a)/(384*b**2) + a**2*(e*x)**(9/2)*sq
rt(a + b*x**3)*(10*A*b - 3*B*a)/(192*b*e) + a*(e*x)**(9/2)*(a + b*x**3)**(3/2)*(
10*A*b - 3*B*a)/(144*b*e) + (e*x)**(9/2)*(a + b*x**3)**(5/2)*(10*A*b - 3*B*a)/(1
20*b*e)

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Mathematica [A]  time = 0.298754, size = 168, normalized size = 0.7 \[ \frac{e^2 (e x)^{3/2} \left (15 a^4 \sqrt{\frac{a}{x^3}+b} (3 a B-10 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )-\sqrt{b} \left (a+b x^3\right ) \left (45 a^4 B-30 a^3 b \left (5 A+B x^3\right )-4 a^2 b^2 x^3 \left (295 A+186 B x^3\right )-16 a b^3 x^6 \left (85 A+63 B x^3\right )-96 b^4 x^9 \left (5 A+4 B x^3\right )\right )\right )}{5760 b^{5/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^(7/2)*(a + b*x^3)^(5/2)*(A + B*x^3),x]

[Out]

(e^2*(e*x)^(3/2)*(-(Sqrt[b]*(a + b*x^3)*(45*a^4*B - 30*a^3*b*(5*A + B*x^3) - 96*
b^4*x^9*(5*A + 4*B*x^3) - 16*a*b^3*x^6*(85*A + 63*B*x^3) - 4*a^2*b^2*x^3*(295*A
+ 186*B*x^3))) + 15*a^4*(-10*A*b + 3*a*B)*Sqrt[b + a/x^3]*ArcTanh[Sqrt[b + a/x^3
]/Sqrt[b]]))/(5760*b^(5/2)*Sqrt[a + b*x^3])

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Maple [C]  time = 0.072, size = 8117, normalized size = 33.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(b*x^3+a)^(5/2)*(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)^(5/2)*(e*x)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.688862, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} e^{3} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e + 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (384 \, B b^{4} e^{3} x^{13} + 48 \,{\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} e^{3} x^{10} + 8 \,{\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} e^{3} x^{7} + 10 \,{\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} e^{3} x^{4} - 15 \,{\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{23040 \, b^{2}}, \frac{15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} e^{3} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right ) + 2 \,{\left (384 \, B b^{4} e^{3} x^{13} + 48 \,{\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} e^{3} x^{10} + 8 \,{\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} e^{3} x^{7} + 10 \,{\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} e^{3} x^{4} - 15 \,{\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{11520 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(7/2),x, algorithm="fricas")

[Out]

[-1/23040*(15*(3*B*a^5 - 10*A*a^4*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*(384*
B*b^4*e^3*x^13 + 48*(21*B*a*b^3 + 10*A*b^4)*e^3*x^10 + 8*(93*B*a^2*b^2 + 170*A*a
*b^3)*e^3*x^7 + 10*(3*B*a^3*b + 118*A*a^2*b^2)*e^3*x^4 - 15*(3*B*a^4 - 10*A*a^3*
b)*e^3*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b^2, 1/11520*(15*(3*B*a^5 - 10*A*a^4*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
*(384*B*b^4*e^3*x^13 + 48*(21*B*a*b^3 + 10*A*b^4)*e^3*x^10 + 8*(93*B*a^2*b^2 + 1
70*A*a*b^3)*e^3*x^7 + 10*(3*B*a^3*b + 118*A*a^2*b^2)*e^3*x^4 - 15*(3*B*a^4 - 10*
A*a^3*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)**(5/2)*(B*x**3+A),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(7/2),x, algorithm="giac")

[Out]

integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(7/2), x)