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

Optimal. Leaf size=93 \[ \frac{x^{m+1} (a B (m+1)+A b (2-m)) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{3 a^2 b (m+1)}+\frac{x^{m+1} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.135231, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x^{m+1} (a B (m+1)+A b (2-m)) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{3 a^2 b (m+1)}+\frac{x^{m+1} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.4381, size = 71, normalized size = 0.76 \[ \frac{x^{m + 1} \left (A b - B a\right )}{3 a b \left (a + b x^{3}\right )} + \frac{x^{m + 1} \left (A b \left (- m + 2\right ) + B a \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{3 a^{2} b \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0891434, size = 80, normalized size = 0.86 \[ \frac{x^{m+1} \left ((A b-a B) \, _2F_1\left (2,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )+a B \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )\right )}{a^2 b (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( B{x}^{3}+A \right ) }{ \left ( b{x}^{3}+a \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{3} + A\right )} x^{m}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^3 + A)*x^m/(b^2*x^6 + 2*a*b*x^3 + a^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(x**m*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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