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

Optimal. Leaf size=133 \[ \frac{\sqrt{\frac{b x^3}{a}+1} (e x)^{m+1} (2 a B (m+1)+A (b-2 b m)) \, _2F_1\left (\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{3 a b e (m+1) \sqrt{a+b x^3}}+\frac{2 (e x)^{m+1} (A b-a B)}{3 a b e \sqrt{a+b x^3}} \]

[Out]

(2*(A*b - a*B)*(e*x)^(1 + m))/(3*a*b*e*Sqrt[a + b*x^3]) + ((2*a*B*(1 + m) + A*(b
- 2*b*m))*(e*x)^(1 + m)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, (1 + m)/3, (
4 + m)/3, -((b*x^3)/a)])/(3*a*b*e*(1 + m)*Sqrt[a + b*x^3])

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Rubi [A]  time = 0.239268, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{\frac{b x^3}{a}+1} (e x)^{m+1} (2 a B (m+1)+A (b-2 b m)) \, _2F_1\left (\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{3 a b e (m+1) \sqrt{a+b x^3}}+\frac{2 (e x)^{m+1} (A b-a B)}{3 a b e \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*(e*x)^(1 + m))/(3*a*b*e*Sqrt[a + b*x^3]) + ((2*a*B*(1 + m) + A*(b
- 2*b*m))*(e*x)^(1 + m)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, (1 + m)/3, (
4 + m)/3, -((b*x^3)/a)])/(3*a*b*e*(1 + m)*Sqrt[a + b*x^3])

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Rubi in Sympy [A]  time = 18.8188, size = 112, normalized size = 0.84 \[ \frac{2 \left (e x\right )^{m + 1} \left (A b - B a\right )}{3 a b e \sqrt{a + b x^{3}}} + \frac{2 \left (e x\right )^{m + 1} \sqrt{a + b x^{3}} \left (\frac{A b \left (- 2 m + 1\right )}{2} + B a \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{3 a^{2} b e \sqrt{1 + \frac{b x^{3}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(e*x)**(m + 1)*(A*b - B*a)/(3*a*b*e*sqrt(a + b*x**3)) + 2*(e*x)**(m + 1)*sqrt(
a + b*x**3)*(A*b*(-2*m + 1)/2 + B*a*(m + 1))*hyper((1/2, m/3 + 1/3), (m/3 + 4/3,
), -b*x**3/a)/(3*a**2*b*e*sqrt(1 + b*x**3/a)*(m + 1))

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Mathematica [A]  time = 0.143381, size = 110, normalized size = 0.83 \[ \frac{x \sqrt{\frac{b x^3}{a}+1} (e x)^m \left ((A b-a B) \, _2F_1\left (\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )+a B \, _2F_1\left (\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )\right )}{a b (m+1) \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*Sqrt[1 + (b*x^3)/a]*(a*B*Hypergeometric2F1[1/2, (1 + m)/3, (4 + m)/3,
-((b*x^3)/a)] + (A*b - a*B)*Hypergeometric2F1[3/2, (1 + m)/3, (4 + m)/3, -((b*x
^3)/a)]))/(a*b*(1 + m)*Sqrt[a + b*x^3])

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{ \left ( ex \right ) ^{m} \left ( B{x}^{3}+A \right ) \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x)^m*(B*x^3+A)/(b*x^3+a)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{3} + A\right )} \left (e x\right )^{m}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^3 + A)*(e*x)^m/(b*x^3 + a)^(3/2), x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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