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

Optimal. Leaf size=33 \[ -\frac{d (d x)^{m-1} \, _2F_1\left (2,m-1;m;-\frac{c x}{b}\right )}{b^2 (1-m)} \]

[Out]

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

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Rubi [A]  time = 0.0514536, antiderivative size = 33, 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 \[ -\frac{d (d x)^{m-1} \, _2F_1\left (2,m-1;m;-\frac{c x}{b}\right )}{b^2 (1-m)} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m/(b*x + c*x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.91172, size = 26, normalized size = 0.79 \[ - \frac{d \left (d x\right )^{m - 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m - 1 \\ m \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{b^{2} \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0329749, size = 30, normalized size = 0.91 \[ \frac{(d x)^m \, _2F_1\left (2,m-1;m;-\frac{c x}{b}\right )}{b^2 (m-1) x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( c{x}^{2}+bx \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m/(c*x^2+b*x)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*x)**m/(x**2*(b + c*x)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x)^2, x)