Optimal. Leaf size=211 \[ -\frac{(d+e x)^{m+2} \left (B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,m+2;m+3;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(m+2) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{(d+e x)^{m+2} \left (\frac{b B-2 A c}{\sqrt{b^2-4 a c}}+B\right ) \, _2F_1\left (1,m+2;m+3;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+2) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]
[Out]
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Rubi [A] time = 0.593315, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{(d+e x)^{m+2} \left (B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,m+2;m+3;\frac{2 c (d+e x)}{2 c d-b e+\sqrt{b^2-4 a c} e}\right )}{(m+2) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{(d+e x)^{m+2} \left (\frac{b B-2 A c}{\sqrt{b^2-4 a c}}+B\right ) \, _2F_1\left (1,m+2;m+3;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+2) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(1 + m))/(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 66.7094, size = 211, normalized size = 1. \[ \frac{\left (d + e x\right )^{m + 2} \left (2 A c - B b + B \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 2 \\ m + 3 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d - e \sqrt{- 4 a c + b^{2}}}} \right )}}{\left (m + 2\right ) \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d - e \sqrt{- 4 a c + b^{2}}\right )} + \frac{\left (d + e x\right )^{m + 2} \left (2 A c - B b - B \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 2 \\ m + 3 \end{matrix}\middle |{\frac{c \left (- 2 d - 2 e x\right )}{b e - 2 c d + e \sqrt{- 4 a c + b^{2}}}} \right )}}{\left (m + 2\right ) \sqrt{- 4 a c + b^{2}} \left (2 c d - e \left (b + \sqrt{- 4 a c + b^{2}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(1+m)/(c*x**2+b*x+a),x)
[Out]
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Mathematica [B] time = 3.41417, size = 1358, normalized size = 6.44 \[ -\frac{(d+e x)^m \left (B \left (-2^{-m} \left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right )^2 \, _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)}{b e+2 c x e-\sqrt{\left (b^2-4 a c\right ) e^2}}\right )^{-m}-2^{-m} \left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right )^2 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)}{b e+2 c x e-\sqrt{\left (b^2-4 a c\right ) e^2}}\right )^{-m}+2^{1-m} c d (m+1) \left (\frac{c (d+e x)}{b e+2 c x e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )^{-m} \left (\left (-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) \, _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)}{b e+2 c x e-\sqrt{\left (b^2-4 a c\right ) e^2}}\right )^m+\left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) \left (\frac{c (d+e x)}{b e+2 c x e+\sqrt{\left (b^2-4 a c\right ) e^2}}\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 ) \left (\frac{c (d+e x)}{b e+2 c x e-\sqrt{\left (b^2-4 a c\right ) e^2}}\right )^{-m}+2 c \left (2 c d-b e-\sqrt{\left (b^2-4 a c\right ) e^2}\right ) m (d+e x)-2 c \left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) m (d+e x)+2^{-m} \left (-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right )^2 \left (\frac{c (d+e x)}{b e+2 c x e+\sqrt{\left (b^2-4 a c\right ) e^2}}\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 )+2^{-m} \left (-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right )^2 m \left (\frac{c (d+e x)}{b e+2 c x e+\sqrt{\left (b^2-4 a c\right ) e^2}}\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 )-2^{1-m} A c e (m+1) \left (\frac{c (d+e x)}{b e+2 c x e-\sqrt{\left (b^2-4 a c\right ) e^2}}\right )^{-m} \left (\frac{c (d+e x)}{b e+2 c x e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )^{-m} \left (\left (-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) \, _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)}{b e+2 c x e-\sqrt{\left (b^2-4 a c\right ) e^2}}\right )^m+\left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) \left (\frac{c (d+e x)}{b e+2 c x e+\sqrt{\left (b^2-4 a c\right ) e^2}}\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 )\right )}{4 c^2 \sqrt{\left (b^2-4 a c\right ) e^2} m (m+1)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(1 + m))/(a + b*x + c*x^2),x]
[Out]
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Maple [F] time = 0.147, size = 0, normalized size = 0. \[ \int{\frac{ \left ( Bx+A \right ) \left ( ex+d \right ) ^{1+m}}{c{x}^{2}+bx+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(1+m)/(c*x^2+b*x+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m + 1}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(m + 1)/(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m + 1}}{c x^{2} + b x + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(m + 1)/(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)**(1+m)/(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{m + 1}}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(m + 1)/(c*x^2 + b*x + a),x, algorithm="giac")
[Out]