Optimal. Leaf size=211 \[ -\frac{(d+e x)^{m+1} \left (B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(m+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{(d+e x)^{m+1} \left (\frac{b B-2 A c}{\sqrt{b^2-4 a c}}+B\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]
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Rubi [A] time = 0.676599, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{(d+e x)^{m+1} \left (B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(m+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{(d+e x)^{m+1} \left (\frac{b B-2 A c}{\sqrt{b^2-4 a c}}+B\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^m)/(a + b*x + c*x^2),x]
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Rubi in Sympy [A] time = 65.6956, size = 211, normalized size = 1. \[ \frac{\left (d + e x\right )^{m + 1} \left (2 A c - B b + B \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{\left (m + 1\right ) \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d - e \sqrt{- 4 a c + b^{2}}\right )} + \frac{\left (d + e x\right )^{m + 1} \left (2 A c - B b - B \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{\left (m + 1\right ) \sqrt{- 4 a c + b^{2}} \left (2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**m/(c*x**2+b*x+a),x)
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Mathematica [A] time = 0.848389, size = 305, normalized size = 1.45 \[ \frac{2^{-m-1} (d+e x)^m \left (\left (B \sqrt{e^2 \left (b^2-4 a c\right )}+2 A c e-b B e\right ) \left (\frac{c (d+e x)}{-\sqrt{e^2 \left (b^2-4 a c\right )}+b e+2 c e x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{-b e-2 c x e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )+\left (B \sqrt{e^2 \left (b^2-4 a c\right )}-2 A c e+b B e\right ) \left (\frac{c (d+e x)}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e+2 c e x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{b e+2 c x e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{c m \sqrt{e^2 \left (b^2-4 a c\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^m)/(a + b*x + c*x^2),x]
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Maple [F] time = 0.437, size = 0, normalized size = 0. \[ \int{\frac{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{m}}{c{x}^{2}+bx+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(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. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(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. \[{\rm integral}\left (\frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{c x^{2} + b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(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((B*x+A)*(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. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^m/(c*x^2 + b*x + a),x, algorithm="giac")
[Out]