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

Optimal. Leaf size=201 \[ \frac{5 a^3 \sqrt{e} (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{192 b^{3/2}}+\frac{5 a^2 (e x)^{3/2} \sqrt{a+b x^3} (8 A b-a B)}{192 b e}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{5/2} (8 A b-a B)}{72 b e}+\frac{5 a (e x)^{3/2} \left (a+b x^3\right )^{3/2} (8 A b-a B)}{288 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e} \]

[Out]

(5*a^2*(8*A*b - a*B)*(e*x)^(3/2)*Sqrt[a + b*x^3])/(192*b*e) + (5*a*(8*A*b - a*B)
*(e*x)^(3/2)*(a + b*x^3)^(3/2))/(288*b*e) + ((8*A*b - a*B)*(e*x)^(3/2)*(a + b*x^
3)^(5/2))/(72*b*e) + (B*(e*x)^(3/2)*(a + b*x^3)^(7/2))/(12*b*e) + (5*a^3*(8*A*b
- a*B)*Sqrt[e]*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(192*b^
(3/2))

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Rubi [A]  time = 0.386291, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{5 a^3 \sqrt{e} (8 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{192 b^{3/2}}+\frac{5 a^2 (e x)^{3/2} \sqrt{a+b x^3} (8 A b-a B)}{192 b e}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{5/2} (8 A b-a B)}{72 b e}+\frac{5 a (e x)^{3/2} \left (a+b x^3\right )^{3/2} (8 A b-a B)}{288 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e} \]

Antiderivative was successfully verified.

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

[Out]

(5*a^2*(8*A*b - a*B)*(e*x)^(3/2)*Sqrt[a + b*x^3])/(192*b*e) + (5*a*(8*A*b - a*B)
*(e*x)^(3/2)*(a + b*x^3)^(3/2))/(288*b*e) + ((8*A*b - a*B)*(e*x)^(3/2)*(a + b*x^
3)^(5/2))/(72*b*e) + (B*(e*x)^(3/2)*(a + b*x^3)^(7/2))/(12*b*e) + (5*a^3*(8*A*b
- a*B)*Sqrt[e]*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(192*b^
(3/2))

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Rubi in Sympy [A]  time = 33.5616, size = 177, normalized size = 0.88 \[ \frac{B \left (e x\right )^{\frac{3}{2}} \left (a + b x^{3}\right )^{\frac{7}{2}}}{12 b e} + \frac{5 a^{3} \sqrt{e} \left (8 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{192 b^{\frac{3}{2}}} + \frac{5 a^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (8 A b - B a\right )}{192 b e} + \frac{5 a \left (e x\right )^{\frac{3}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (8 A b - B a\right )}{288 b e} + \frac{\left (e x\right )^{\frac{3}{2}} \left (a + b x^{3}\right )^{\frac{5}{2}} \left (8 A b - B a\right )}{72 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(e*x)**(3/2)*(a + b*x**3)**(7/2)/(12*b*e) + 5*a**3*sqrt(e)*(8*A*b - B*a)*atanh
(sqrt(b)*(e*x)**(3/2)/(e**(3/2)*sqrt(a + b*x**3)))/(192*b**(3/2)) + 5*a**2*(e*x)
**(3/2)*sqrt(a + b*x**3)*(8*A*b - B*a)/(192*b*e) + 5*a*(e*x)**(3/2)*(a + b*x**3)
**(3/2)*(8*A*b - B*a)/(288*b*e) + (e*x)**(3/2)*(a + b*x**3)**(5/2)*(8*A*b - B*a)
/(72*b*e)

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Mathematica [A]  time = 0.273173, size = 144, normalized size = 0.72 \[ \frac{x \sqrt{e x} \left (\sqrt{b} \left (a+b x^3\right ) \left (15 a^3 B+2 a^2 b \left (132 A+59 B x^3\right )+8 a b^2 x^3 \left (26 A+17 B x^3\right )+16 b^3 x^6 \left (4 A+3 B x^3\right )\right )-15 a^3 \sqrt{\frac{a}{x^3}+b} (a B-8 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )\right )}{576 b^{3/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[e*x]*(Sqrt[b]*(a + b*x^3)*(15*a^3*B + 16*b^3*x^6*(4*A + 3*B*x^3) + 8*a*b
^2*x^3*(26*A + 17*B*x^3) + 2*a^2*b*(132*A + 59*B*x^3)) - 15*a^3*(-8*A*b + a*B)*S
qrt[b + a/x^3]*ArcTanh[Sqrt[b + a/x^3]/Sqrt[b]]))/(576*b^(3/2)*Sqrt[a + b*x^3])

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Maple [C]  time = 0.047, size = 7702, normalized size = 38.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.67271, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (48 \, B b^{3} x^{10} + 8 \,{\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{7} + 2 \,{\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x^{4} + 3 \,{\left (5 \, B a^{3} + 88 \, A a^{2} b\right )} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{2304 \, b}, -\frac{15 \,{\left (B a^{4} - 8 \, A a^{3} b\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right ) - 2 \,{\left (48 \, B b^{3} x^{10} + 8 \,{\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{7} + 2 \,{\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x^{4} + 3 \,{\left (5 \, B a^{3} + 88 \, A a^{2} b\right )} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{1152 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2304*(15*(B*a^4 - 8*A*a^3*b)*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*x^1
0 + 8*(17*B*a*b^2 + 8*A*b^3)*x^7 + 2*(59*B*a^2*b + 104*A*a*b^2)*x^4 + 3*(5*B*a^3
+ 88*A*a^2*b)*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b, -1/1152*(15*(B*a^4 - 8*A*a^3*b)*
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*x^10 + 8*(17*B*a*b^2 + 8*A*b^3)*x^7 + 2*(59*B*a^2*b + 104*A*a*b^2)*x^4
+ 3*(5*B*a^3 + 88*A*a^2*b)*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out