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

Optimal. Leaf size=25 \[ \frac{(d x)^m \, _2F_1\left (1,m;m+1;-\frac{c x}{b}\right )}{b m} \]

[Out]

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

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Rubi [A]  time = 0.0415594, antiderivative size = 25, 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 \[ \frac{(d x)^m \, _2F_1\left (1,m;m+1;-\frac{c x}{b}\right )}{b m} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m/(b*x + c*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.03627, size = 17, normalized size = 0.68 \[ \frac{\left (d x\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} 1, m \\ m + 1 \end{matrix}\middle |{- \frac{c x}{b}} \right )}}{b m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0417549, size = 42, normalized size = 1.68 \[ \frac{(d x)^m \left (-c m x \, _2F_1\left (1,m+1;m+2;-\frac{c x}{b}\right )+b m+b\right )}{b^2 m (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx \right ) ^{m}}{c{x}^{2}+bx}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m/(c*x^2+b*x),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x)^m/(c*x^2 + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*x)**m/(x*(b + c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x)^m/(c*x^2 + b*x), x)