Optimal. Leaf size=166 \[ -\frac{2 c (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(m+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{2 c (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]
[Out]
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Rubi [A] time = 0.433108, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 c (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(m+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{2 c (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]
Antiderivative was successfully verified.
[In] Int[((b + 2*c*x)*(d + e*x)^m)/(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 51.8373, size = 150, normalized size = 0.9 \[ - \frac{2 c \left (d + e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{\left (m + 1\right ) \left (2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )\right )} - \frac{2 c \left (d + e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{\left (m + 1\right ) \left (2 c d - e \left (b - \sqrt{- 4 a c + b^{2}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x+b)*(e*x+d)**m/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.692261, size = 227, normalized size = 1.37 \[ \frac{2^{-m} (d+e x)^m \left (\left (\frac{c (d+e x)}{-\sqrt{e^2 \left (b^2-4 a c\right )}+b e+2 c e x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{-b e-2 c x e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )+\left (\frac{c (d+e x)}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e+2 c e x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{b e+2 c x e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{m} \]
Antiderivative was successfully verified.
[In] Integrate[((b + 2*c*x)*(d + e*x)^m)/(a + b*x + c*x^2),x]
[Out]
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Maple [F] time = 0.159, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2\,cx+b \right ) \left ( ex+d \right ) ^{m}}{c{x}^{2}+bx+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(e*x+d)^m/(c*x^2+b*x+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c x + b\right )}{\left (e x + d\right )}^{m}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)*(e*x + d)^m/(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (2 \, c x + b\right )}{\left (e x + d\right )}^{m}}{c x^{2} + b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)*(e*x + d)^m/(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)*(e*x+d)**m/(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, c x + b\right )}{\left (e x + d\right )}^{m}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)*(e*x + d)^m/(c*x^2 + b*x + a),x, algorithm="giac")
[Out]