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

Optimal. Leaf size=161 \[ \frac{a^2 \sqrt{e} (6 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{24 b^{3/2}}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{3/2} (6 A b-a B)}{36 b e}+\frac{a (e x)^{3/2} \sqrt{a+b x^3} (6 A b-a B)}{24 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e} \]

[Out]

(a*(6*A*b - a*B)*(e*x)^(3/2)*Sqrt[a + b*x^3])/(24*b*e) + ((6*A*b - a*B)*(e*x)^(3
/2)*(a + b*x^3)^(3/2))/(36*b*e) + (B*(e*x)^(3/2)*(a + b*x^3)^(5/2))/(9*b*e) + (a
^2*(6*A*b - a*B)*Sqrt[e]*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])
])/(24*b^(3/2))

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Rubi [A]  time = 0.324454, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a^2 \sqrt{e} (6 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{24 b^{3/2}}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{3/2} (6 A b-a B)}{36 b e}+\frac{a (e x)^{3/2} \sqrt{a+b x^3} (6 A b-a B)}{24 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e} \]

Antiderivative was successfully verified.

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

[Out]

(a*(6*A*b - a*B)*(e*x)^(3/2)*Sqrt[a + b*x^3])/(24*b*e) + ((6*A*b - a*B)*(e*x)^(3
/2)*(a + b*x^3)^(3/2))/(36*b*e) + (B*(e*x)^(3/2)*(a + b*x^3)^(5/2))/(9*b*e) + (a
^2*(6*A*b - a*B)*Sqrt[e]*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])
])/(24*b^(3/2))

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Rubi in Sympy [A]  time = 27.7504, size = 138, normalized size = 0.86 \[ \frac{B \left (e x\right )^{\frac{3}{2}} \left (a + b x^{3}\right )^{\frac{5}{2}}}{9 b e} + \frac{a^{2} \sqrt{e} \left (6 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{24 b^{\frac{3}{2}}} + \frac{a \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (6 A b - B a\right )}{24 b e} + \frac{\left (e x\right )^{\frac{3}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (6 A b - B a\right )}{36 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(e*x)**(3/2)*(a + b*x**3)**(5/2)/(9*b*e) + a**2*sqrt(e)*(6*A*b - B*a)*atanh(sq
rt(b)*(e*x)**(3/2)/(e**(3/2)*sqrt(a + b*x**3)))/(24*b**(3/2)) + a*(e*x)**(3/2)*s
qrt(a + b*x**3)*(6*A*b - B*a)/(24*b*e) + (e*x)**(3/2)*(a + b*x**3)**(3/2)*(6*A*b
- B*a)/(36*b*e)

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Mathematica [A]  time = 0.273415, size = 123, normalized size = 0.76 \[ \frac{x \sqrt{e x} \left (\sqrt{b} \left (a+b x^3\right ) \left (3 a^2 B+2 a b \left (15 A+7 B x^3\right )+4 b^2 x^3 \left (3 A+2 B x^3\right )\right )-3 a^2 \sqrt{\frac{a}{x^3}+b} (a B-6 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )\right )}{72 b^{3/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[e*x]*(Sqrt[b]*(a + b*x^3)*(3*a^2*B + 4*b^2*x^3*(3*A + 2*B*x^3) + 2*a*b*(
15*A + 7*B*x^3)) - 3*a^2*(-6*A*b + a*B)*Sqrt[b + a/x^3]*ArcTanh[Sqrt[b + a/x^3]/
Sqrt[b]]))/(72*b^(3/2)*Sqrt[a + b*x^3])

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Maple [C]  time = 0.045, size = 7290, normalized size = 45.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.651003, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (8 \, B b^{2} x^{7} + 2 \,{\left (7 \, B a b + 6 \, A b^{2}\right )} x^{4} + 3 \,{\left (B a^{2} + 10 \, A a b\right )} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{288 \, b}, -\frac{3 \,{\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right ) - 2 \,{\left (8 \, B b^{2} x^{7} + 2 \,{\left (7 \, B a b + 6 \, A b^{2}\right )} x^{4} + 3 \,{\left (B a^{2} + 10 \, A a b\right )} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{144 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/288*(3*(B*a^3 - 6*A*a^2*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*(8*B*b^2*x^7 +
2*(7*B*a*b + 6*A*b^2)*x^4 + 3*(B*a^2 + 10*A*a*b)*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b
, -1/144*(3*(B*a^3 - 6*A*a^2*b)*sqrt(-e/b)*arctan(2*sqrt(b*x^3 + a)*sqrt(e*x)*x/
((2*b*x^3 + a)*sqrt(-e/b))) - 2*(8*B*b^2*x^7 + 2*(7*B*a*b + 6*A*b^2)*x^4 + 3*(B*
a^2 + 10*A*a*b)*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b]

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Sympy [A]  time = 71.4413, size = 335, normalized size = 2.08 \[ \frac{A a^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 e} + \frac{A a^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}}}{12 e \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{A \sqrt{a} b \left (e x\right )^{\frac{9}{2}}}{4 e^{4} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{A a^{2} \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{4 \sqrt{b}} + \frac{A b^{2} \left (e x\right )^{\frac{15}{2}}}{6 \sqrt{a} e^{7} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{B a^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}}}{24 b e \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{17 B a^{\frac{3}{2}} \left (e x\right )^{\frac{9}{2}}}{72 e^{4} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{11 B \sqrt{a} b \left (e x\right )^{\frac{15}{2}}}{36 e^{7} \sqrt{1 + \frac{b x^{3}}{a}}} - \frac{B a^{3} \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{24 b^{\frac{3}{2}}} + \frac{B b^{2} \left (e x\right )^{\frac{21}{2}}}{9 \sqrt{a} e^{10} \sqrt{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**(3/2)*(e*x)**(3/2)*sqrt(1 + b*x**3/a)/(3*e) + A*a**(3/2)*(e*x)**(3/2)/(12*e
*sqrt(1 + b*x**3/a)) + A*sqrt(a)*b*(e*x)**(9/2)/(4*e**4*sqrt(1 + b*x**3/a)) + A*
a**2*sqrt(e)*asinh(sqrt(b)*(e*x)**(3/2)/(sqrt(a)*e**(3/2)))/(4*sqrt(b)) + A*b**2
*(e*x)**(15/2)/(6*sqrt(a)*e**7*sqrt(1 + b*x**3/a)) + B*a**(5/2)*(e*x)**(3/2)/(24
*b*e*sqrt(1 + b*x**3/a)) + 17*B*a**(3/2)*(e*x)**(9/2)/(72*e**4*sqrt(1 + b*x**3/a
)) + 11*B*sqrt(a)*b*(e*x)**(15/2)/(36*e**7*sqrt(1 + b*x**3/a)) - B*a**3*sqrt(e)*
asinh(sqrt(b)*(e*x)**(3/2)/(sqrt(a)*e**(3/2)))/(24*b**(3/2)) + B*b**2*(e*x)**(21
/2)/(9*sqrt(a)*e**10*sqrt(1 + b*x**3/a))

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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)^(3/2)*sqrt(e*x),x, algorithm="giac")

[Out]

Timed out