Optimal. Leaf size=245 \[ \frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (c \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )-e g m (a e g (1-m)-b (2 d g-e f (m+1)))\right )}{2 g^2 (m+1) (e f-d g)^3}+\frac{(d+e x)^{m+1} (g (a e g (1-m)-b (2 d g-e f (m+1)))+c f (4 d g-e f (m+3)))}{2 g^2 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a+\frac{f (c f-b g)}{g^2}\right )}{2 (f+g x)^2 (e f-d g)} \]
[Out]
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Rubi [A] time = 0.846946, antiderivative size = 243, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (e g m (-a e g (1-m)+2 b d g-b e f (m+1))+c \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right )}{2 g^2 (m+1) (e f-d g)^3}-\frac{(d+e x)^{m+1} (g (-a e g (1-m)+2 b d g-b e f (m+1))-c f (4 d g-e f (m+3)))}{2 g^2 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a+\frac{f (c f-b g)}{g^2}\right )}{2 (f+g x)^2 (e f-d g)} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^m*(a + b*x + c*x^2))/(f + g*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 51.2783, size = 155, normalized size = 0.63 \[ - \frac{c \left (d + e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{2} \left (m + 1\right ) \left (d g - e f\right )} - \frac{e^{2} \left (d + e x\right )^{m + 1} \left (a g^{2} - b f g + c f^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 3, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{2} \left (m + 1\right ) \left (d g - e f\right )^{3}} + \frac{e \left (d + e x\right )^{m + 1} \left (b g - 2 c f\right ){{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{2} \left (m + 1\right ) \left (d g - e f\right )^{2}} \]
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)**3,x)
[Out]
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Mathematica [A] time = 0.286995, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^3} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[((d + e*x)^m*(a + b*x + c*x^2))/(f + g*x)^3,x]
[Out]
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Maple [F] time = 0.11, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) }{ \left ( gx+f \right ) ^{3}}}\, dx \]
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)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{3}}\,{d x} \]
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)^3,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}}{g^{3} x^{3} + 3 \, f g^{2} x^{2} + 3 \, f^{2} g x + f^{3}}, x\right ) \]
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)^3,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((e*x+d)**m*(c*x**2+b*x+a)/(g*x+f)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{3}}\,{d x} \]
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)^3,x, algorithm="giac")
[Out]