3.44 \(\int \frac{\left (a+b x^3\right )^m}{\left (c+d x^3\right )^2} \, dx\)

Optimal. Leaf size=57 \[ \frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} F_1\left (\frac{1}{3};-m,2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c^2} \]

[Out]

(x*(a + b*x^3)^m*AppellF1[1/3, -m, 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(c^2*(1
+ (b*x^3)/a)^m)

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Rubi [A]  time = 0.0716967, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} F_1\left (\frac{1}{3};-m,2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^3)^m*AppellF1[1/3, -m, 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(c^2*(1
+ (b*x^3)/a)^m)

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Rubi in Sympy [A]  time = 19.708, size = 44, normalized size = 0.77 \[ \frac{x \left (1 + \frac{b x^{3}}{a}\right )^{- m} \left (a + b x^{3}\right )^{m} \operatorname{appellf_{1}}{\left (\frac{1}{3},2,- m,\frac{4}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + b*x**3/a)**(-m)*(a + b*x**3)**m*appellf1(1/3, 2, -m, 4/3, -d*x**3/c, -b*x
**3/a)/c**2

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Mathematica [B]  time = 0.297776, size = 162, normalized size = 2.84 \[ -\frac{4 a c x \left (a+b x^3\right )^m F_1\left (\frac{1}{3};-m,2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{\left (c+d x^3\right )^2 \left (-3 x^3 \left (b c m F_1\left (\frac{4}{3};1-m,2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-2 a d F_1\left (\frac{4}{3};-m,3;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-4 a c F_1\left (\frac{1}{3};-m,2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-4*a*c*x*(a + b*x^3)^m*AppellF1[1/3, -m, 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(
(c + d*x^3)^2*(-4*a*c*AppellF1[1/3, -m, 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)] - 3*
x^3*(b*c*m*AppellF1[4/3, 1 - m, 2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] - 2*a*d*Appe
llF1[4/3, -m, 3, 7/3, -((b*x^3)/a), -((d*x^3)/c)])))

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{3}+a \right ) ^{m}}{ \left ( d{x}^{3}+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^m/(d*x^3+c)^2,x)

[Out]

int((b*x^3+a)^m/(d*x^3+c)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^m/(d*x^3 + c)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^3 + a)^m/(d^2*x^6 + 2*c*d*x^3 + c^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((b*x**3+a)**m/(d*x**3+c)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^m/(d*x^3 + c)^2, x)