Optimal. Leaf size=32 \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{e x}{d}+1\right )}{c d m} \]
[Out]
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Rubi [A] time = 0.0461447, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{e x}{d}+1\right )}{c d m} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m/(c*d*x + c*e*x^2),x]
[Out]
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Rubi in Sympy [A] time = 9.3979, size = 22, normalized size = 0.69 \[ - \frac{\left (d + e x\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} 1, m \\ m + 1 \end{matrix}\middle |{1 + \frac{e x}{d}} \right )}}{c d m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m/(c*e*x**2+c*d*x),x)
[Out]
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Mathematica [A] time = 0.0715492, size = 57, normalized size = 1.78 \[ -\frac{(d+e x)^m \left (1-\left (\frac{d}{e x}+1\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{d}{e x}\right )\right )}{c d m} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^m/(c*d*x + c*e*x^2),x]
[Out]
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Maple [F] time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ce{x}^{2}+cdx}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m/(c*e*x^2+c*d*x),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{c e x^{2} + c d x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*e*x^2 + c*d*x),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{c e x^{2} + c d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*e*x^2 + c*d*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{\left (d + e x\right )^{m}}{d x + e x^{2}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**m/(c*e*x**2+c*d*x),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{c e x^{2} + c d x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*e*x^2 + c*d*x),x, algorithm="giac")
[Out]