Optimal. Leaf size=129 \[ \frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right )}{g^2 (m+1) (e f-d g)}-\frac{(d+e x)^{m+1} (-b e g+c d g+c e f)}{e^2 g^2 (m+1)}+\frac{c (d+e x)^{m+2}}{e^2 g (m+2)} \]
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Rubi [A] time = 0.399742, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right )}{g^2 (m+1) (e f-d g)}-\frac{(d+e x)^{m+1} (-b e g+c d g+c e f)}{e^2 g^2 (m+1)}+\frac{c (d+e x)^{m+2}}{e^2 g (m+2)} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^m*(a + b*x + c*x^2))/(f + g*x),x]
[Out]
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Rubi in Sympy [A] time = 41.7642, size = 107, normalized size = 0.83 \[ \frac{c \left (d + e x\right )^{m + 2}}{e^{2} g \left (m + 2\right )} - \frac{\left (d + e x\right )^{m + 1} \left (a g^{2} - b f g + c f^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{2} \left (m + 1\right ) \left (d g - e f\right )} + \frac{\left (d + e x\right )^{m + 1} \left (b e g - c d g - c e f\right )}{e^{2} g^{2} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m*(c*x**2+b*x+a)/(g*x+f),x)
[Out]
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Mathematica [A] time = 1.15942, size = 166, normalized size = 1.29 \[ \frac{(d+e x)^m \left (\frac{\left (g (a g-b f)+c f^2\right ) \left (\frac{g (d+e x)}{e (f+g x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{e f-d g}{e f+e g x}\right )}{m}+\frac{g \left (b e g (m+2) (d+e x)+c \left (d^2 g \left (\left (\frac{e x}{d}+1\right )^{-m}-1\right )+d e (g m x-f (m+2))+e^2 x (g (m+1) x-f (m+2))\right )\right )}{e^2 (m+1) (m+2)}\right )}{g^3} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)^m*(a + b*x + c*x^2))/(f + g*x),x]
[Out]
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Maple [F] time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) }{gx+f}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m*(c*x^2+b*x+a)/(g*x+f),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}}{g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)^m/(g*x + f),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}}{g x + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)^m/(g*x + f),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{m} \left (a + b x + c x^{2}\right )}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**m*(c*x**2+b*x+a)/(g*x+f),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}}{g x + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)^m/(g*x + f),x, algorithm="giac")
[Out]