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

Optimal. Leaf size=37 \[ -\frac{d^2 (d x)^{m-2} \, _2F_1\left (3,m-2;m-1;-\frac{c x}{b}\right )}{b^3 (2-m)} \]

[Out]

-((d^2*(d*x)^(-2 + m)*Hypergeometric2F1[3, -2 + m, -1 + m, -((c*x)/b)])/(b^3*(2
- m)))

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Rubi [A]  time = 0.054208, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{d^2 (d x)^{m-2} \, _2F_1\left (3,m-2;m-1;-\frac{c x}{b}\right )}{b^3 (2-m)} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m/(b*x + c*x^2)^3,x]

[Out]

-((d^2*(d*x)^(-2 + m)*Hypergeometric2F1[3, -2 + m, -1 + m, -((c*x)/b)])/(b^3*(2
- m)))

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Rubi in Sympy [A]  time = 7.38248, size = 29, normalized size = 0.78 \[ - \frac{d^{2} \left (d x\right )^{m - 2}{{}_{2}F_{1}\left (\begin{matrix} 3, m - 2 \\ m - 1 \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{b^{3} \left (- m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**2*(d*x)**(m - 2)*hyper((3, m - 2), (m - 1,), -c*x/b)/(b**3*(-m + 2))

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Mathematica [B]  time = 0.176213, size = 123, normalized size = 3.32 \[ \frac{(d x)^m \left (\frac{b^3}{(m-2) x^2}+\frac{3 b^2 c}{x-m x}-\frac{6 c^3 x \, _2F_1\left (1,m+1;m+2;-\frac{c x}{b}\right )}{m+1}-\frac{3 c^3 x \, _2F_1\left (2,m+1;m+2;-\frac{c x}{b}\right )}{m+1}-\frac{c^3 x \, _2F_1\left (3,m+1;m+2;-\frac{c x}{b}\right )}{m+1}+\frac{6 b c^2}{m}\right )}{b^6} \]

Antiderivative was successfully verified.

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

[Out]

((d*x)^m*((6*b*c^2)/m + b^3/((-2 + m)*x^2) + (3*b^2*c)/(x - m*x) - (6*c^3*x*Hype
rgeometric2F1[1, 1 + m, 2 + m, -((c*x)/b)])/(1 + m) - (3*c^3*x*Hypergeometric2F1
[2, 1 + m, 2 + m, -((c*x)/b)])/(1 + m) - (c^3*x*Hypergeometric2F1[3, 1 + m, 2 +
m, -((c*x)/b)])/(1 + m)))/b^6

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( c{x}^{2}+bx \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m/(c*x^2+b*x)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (d x\right )^{m}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x)^m/(c^3*x^6 + 3*b*c^2*x^5 + 3*b^2*c*x^4 + b^3*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*x)**m/(x**3*(b + c*x)**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^3, x)