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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]