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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]