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

Optimal. Leaf size=121 \[ \frac{a \sqrt{e} (4 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{12 b^{3/2}}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (4 A b-a B)}{12 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e} \]

[Out]

((4*A*b - a*B)*(e*x)^(3/2)*Sqrt[a + b*x^3])/(12*b*e) + (B*(e*x)^(3/2)*(a + b*x^3
)^(3/2))/(6*b*e) + (a*(4*A*b - a*B)*Sqrt[e]*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/
2)*Sqrt[a + b*x^3])])/(12*b^(3/2))

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Rubi [A]  time = 0.258785, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a \sqrt{e} (4 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{12 b^{3/2}}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (4 A b-a B)}{12 b e}+\frac{B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e} \]

Antiderivative was successfully verified.

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

[Out]

((4*A*b - a*B)*(e*x)^(3/2)*Sqrt[a + b*x^3])/(12*b*e) + (B*(e*x)^(3/2)*(a + b*x^3
)^(3/2))/(6*b*e) + (a*(4*A*b - a*B)*Sqrt[e]*ArcTanh[(Sqrt[b]*(e*x)^(3/2))/(e^(3/
2)*Sqrt[a + b*x^3])])/(12*b^(3/2))

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Rubi in Sympy [A]  time = 22.9875, size = 104, normalized size = 0.86 \[ \frac{B \left (e x\right )^{\frac{3}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}}}{6 b e} + \frac{a \sqrt{e} \left (4 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{12 b^{\frac{3}{2}}} + \frac{\left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (4 A b - B a\right )}{12 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(e*x)**(3/2)*(a + b*x**3)**(3/2)/(6*b*e) + a*sqrt(e)*(4*A*b - B*a)*atanh(sqrt(
b)*(e*x)**(3/2)/(e**(3/2)*sqrt(a + b*x**3)))/(12*b**(3/2)) + (e*x)**(3/2)*sqrt(a
+ b*x**3)*(4*A*b - B*a)/(12*b*e)

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Mathematica [A]  time = 0.238444, size = 97, normalized size = 0.8 \[ \frac{x \sqrt{e x} \left (\sqrt{b} \left (a+b x^3\right ) \left (B \left (a+2 b x^3\right )+4 A b\right )-a \sqrt{\frac{a}{x^3}+b} (a B-4 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )\right )}{12 b^{3/2} \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[e*x]*(Sqrt[b]*(a + b*x^3)*(4*A*b + B*(a + 2*b*x^3)) - a*(-4*A*b + a*B)*S
qrt[b + a/x^3]*ArcTanh[Sqrt[b + a/x^3]/Sqrt[b]]))/(12*b^(3/2)*Sqrt[a + b*x^3])

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Maple [C]  time = 0.063, size = 6858, normalized size = 56.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.68395, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (B a^{2} - 4 \, A a b\right )} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (2 \, B b x^{4} +{\left (B a + 4 \, A b\right )} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{48 \, b}, -\frac{{\left (B a^{2} - 4 \, A a b\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right ) - 2 \,{\left (2 \, B b x^{4} +{\left (B a + 4 \, A b\right )} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{24 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/48*((B*a^2 - 4*A*a*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*(2*B*b*x^4 + (B*a +
4*A*b)*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b, -1/24*((B*a^2 - 4*A*a*b)*sqrt(-e/b)*arct
an(2*sqrt(b*x^3 + a)*sqrt(e*x)*x/((2*b*x^3 + a)*sqrt(-e/b))) - 2*(2*B*b*x^4 + (B
*a + 4*A*b)*x)*sqrt(b*x^3 + a)*sqrt(e*x))/b]

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Sympy [A]  time = 14.0615, size = 201, normalized size = 1.66 \[ \frac{A \sqrt{a} \left (e x\right )^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 e} + \frac{A a \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{3 \sqrt{b}} + \frac{B a^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}}}{12 b e \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{B \sqrt{a} \left (e x\right )^{\frac{9}{2}}}{4 e^{4} \sqrt{1 + \frac{b x^{3}}{a}}} - \frac{B a^{2} \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{\sqrt{a} e^{\frac{3}{2}}} \right )}}{12 b^{\frac{3}{2}}} + \frac{B b \left (e x\right )^{\frac{15}{2}}}{6 \sqrt{a} e^{7} \sqrt{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*sqrt(a)*(e*x)**(3/2)*sqrt(1 + b*x**3/a)/(3*e) + A*a*sqrt(e)*asinh(sqrt(b)*(e*x
)**(3/2)/(sqrt(a)*e**(3/2)))/(3*sqrt(b)) + B*a**(3/2)*(e*x)**(3/2)/(12*b*e*sqrt(
1 + b*x**3/a)) + B*sqrt(a)*(e*x)**(9/2)/(4*e**4*sqrt(1 + b*x**3/a)) - B*a**2*sqr
t(e)*asinh(sqrt(b)*(e*x)**(3/2)/(sqrt(a)*e**(3/2)))/(12*b**(3/2)) + B*b*(e*x)**(
15/2)/(6*sqrt(a)*e**7*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)*sqrt(b*x^3 + a)*sqrt(e*x),x, algorithm="giac")

[Out]

Timed out