Optimal. Leaf size=461 \[ \frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (-g^2 \left (a^2 e^2 g^2 (1-m) m-2 a b e g m (2 d g-e f (m+1))+b^2 \left (-\left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right )\right )+2 c g \left (a g \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )-b f \left (6 d^2 g^2-6 d e f g (m+2)+e^2 f^2 \left (m^2+5 m+6\right )\right )\right )+c^2 f^2 \left (12 d^2 g^2-8 d e f g (m+3)+e^2 f^2 \left (m^2+7 m+12\right )\right )\right )}{2 g^4 (m+1) (e f-d g)^3}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) (g (a e g (1-m)-b (4 d g-e f (m+3)))+c f (8 d g-e f (m+7)))}{2 g^4 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right )^2}{2 g^4 (f+g x)^2 (e f-d g)}-\frac{c (d+e x)^{m+1} (-2 b e g+c d g+3 c e f)}{e^2 g^4 (m+1)}+\frac{c^2 (d+e x)^{m+2}}{e^2 g^3 (m+2)} \]
[Out]
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Rubi [A] time = 4.35087, antiderivative size = 461, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (-g^2 \left (a^2 e^2 g^2 (1-m) m-2 a b e g m (2 d g-e f (m+1))+b^2 \left (-\left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right )\right )+2 c g \left (a g \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )-b f \left (6 d^2 g^2-6 d e f g (m+2)+e^2 f^2 \left (m^2+5 m+6\right )\right )\right )+c^2 f^2 \left (12 d^2 g^2-8 d e f g (m+3)+e^2 f^2 \left (m^2+7 m+12\right )\right )\right )}{2 g^4 (m+1) (e f-d g)^3}-\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) (g (-a e g (1-m)+4 b d g-b e f (m+3))-c f (8 d g-e f (m+7)))}{2 g^4 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right )^2}{2 g^4 (f+g x)^2 (e f-d g)}-\frac{c (d+e x)^{m+1} (-2 b e g+c d g+3 c e f)}{e^2 g^4 (m+1)}+\frac{c^2 (d+e x)^{m+2}}{e^2 g^3 (m+2)} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^m*(a + b*x + c*x^2)^2)/(f + g*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 112.77, size = 265, normalized size = 0.57 \[ \frac{c^{2} \left (d + e x\right )^{m + 2}}{e^{2} g^{3} \left (m + 2\right )} + \frac{c \left (d + e x\right )^{m + 1} \left (2 b e g - c d g - 3 c e f\right )}{e^{2} g^{4} \left (m + 1\right )} - \frac{e^{2} \left (d + e x\right )^{m + 1} \left (a g^{2} - b f g + c f^{2}\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} 3, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{4} \left (m + 1\right ) \left (d g - e f\right )^{3}} + \frac{2 e \left (d + e x\right )^{m + 1} \left (b g - 2 c f\right ) \left (a g^{2} - b f g + c f^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{4} \left (m + 1\right ) \left (d g - e f\right )^{2}} - \frac{\left (d + e x\right )^{m + 1} \left (2 a c g^{2} + b^{2} g^{2} - 6 b c f g + 6 c^{2} f^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{4} \left (m + 1\right ) \left (d g - e f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m*(c*x**2+b*x+a)**2/(g*x+f)**3,x)
[Out]
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Mathematica [A] time = 0.693177, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m \left (a+b x+c x^2\right )^2}{(f+g x)^3} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[((d + e*x)^m*(a + b*x + c*x^2)^2)/(f + g*x)^3,x]
[Out]
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Maple [F] time = 0.182, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{2}}{ \left ( gx+f \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m*(c*x^2+b*x+a)^2/(g*x+f)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{2}{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(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. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}{\left (e x + d\right )}^{m}}{g^{3} x^{3} + 3 \, f g^{2} x^{2} + 3 \, f^{2} g x + f^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(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)**2/(g*x+f)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{2}{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(e*x + d)^m/(g*x + f)^3,x, algorithm="giac")
[Out]