Optimal. Leaf size=461 \[ \frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (-g^2 \left (a^2 e^2 g^2 (1-m) m-2 a b e g m (2 d g-e f (m+1))+b^2 \left (-\left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right )\right )+2 c g \left (a g \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )-b f \left (6 d^2 g^2-6 d e f g (m+2)+e^2 f^2 \left (m^2+5 m+6\right )\right )\right )+c^2 f^2 \left (12 d^2 g^2-8 d e f g (m+3)+e^2 f^2 \left (m^2+7 m+12\right )\right )\right )}{2 g^4 (m+1) (e f-d g)^3}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) (g (a e g (1-m)-b (4 d g-e f (m+3)))+c f (8 d g-e f (m+7)))}{2 g^4 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right )^2}{2 g^4 (f+g x)^2 (e f-d g)}-\frac{c (d+e x)^{m+1} (-2 b e g+c d g+3 c e f)}{e^2 g^4 (m+1)}+\frac{c^2 (d+e x)^{m+2}}{e^2 g^3 (m+2)} \]
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Rubi [A] time = 1.48709, antiderivative size = 461, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {949, 1621, 951, 80, 68} \[ \frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (-g^2 \left (a^2 e^2 g^2 (1-m) m-2 a b e g m (2 d g-e f (m+1))+b^2 \left (-\left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right )\right )+2 c g \left (a g \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )-b f \left (6 d^2 g^2-6 d e f g (m+2)+e^2 f^2 \left (m^2+5 m+6\right )\right )\right )+c^2 f^2 \left (12 d^2 g^2-8 d e f g (m+3)+e^2 f^2 \left (m^2+7 m+12\right )\right )\right )}{2 g^4 (m+1) (e f-d g)^3}-\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) (g (-a e g (1-m)+4 b d g-b e f (m+3))-c f (8 d g-e f (m+7)))}{2 g^4 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right )^2}{2 g^4 (f+g x)^2 (e f-d g)}-\frac{c (d+e x)^{m+1} (-2 b e g+c d g+3 c e f)}{e^2 g^4 (m+1)}+\frac{c^2 (d+e x)^{m+2}}{e^2 g^3 (m+2)} \]
Antiderivative was successfully verified.
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Rule 949
Rule 1621
Rule 951
Rule 80
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^m \left (a+b x+c x^2\right )^2}{(f+g x)^3} \, dx &=\frac{\left (c f^2-b f g+a g^2\right )^2 (d+e x)^{1+m}}{2 g^4 (e f-d g) (f+g x)^2}+\frac{\int \frac{(d+e x)^m \left (\frac{c^2 f^3 (2 d g-e f (1+m))-2 c f g (b f-a g) (2 d g-e f (1+m))+g^2 \left (a^2 e g^2 (1-m)+b^2 f (2 d g-e f (1+m))-2 a b g (2 d g-e f (1+m))\right )}{g^4}+\frac{2 (e f-d g) \left (c^2 f^2+b^2 g^2-2 c g (b f-a g)\right ) x}{g^3}-\frac{2 c (c f-2 b g) (e f-d g) x^2}{g^2}-2 c^2 \left (d-\frac{e f}{g}\right ) x^3\right )}{(f+g x)^2} \, dx}{2 (e f-d g)}\\ &=\frac{\left (c f^2-b f g+a g^2\right )^2 (d+e x)^{1+m}}{2 g^4 (e f-d g) (f+g x)^2}-\frac{\left (c f^2-b f g+a g^2\right ) (g (4 b d g-a e g (1-m)-b e f (3+m))-c f (8 d g-e f (7+m))) (d+e x)^{1+m}}{2 g^4 (e f-d g)^2 (f+g x)}+\frac{\int \frac{(d+e x)^m \left (\frac{c^2 f^2 \left (6 d^2 g^2-4 d e f g (3+2 m)+e^2 f^2 \left (6+7 m+m^2\right )\right )-g^2 \left (a^2 e^2 g^2 (1-m) m-2 a b e g m (2 d g-e f (1+m))-b^2 \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )\right )+2 c g \left (a g \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )-b f \left (4 d^2 g^2-2 d e f g (4+3 m)+e^2 f^2 \left (4+5 m+m^2\right )\right )\right )}{g^4}-\frac{4 c (c f-b g) (e f-d g)^2 x}{g^3}+\frac{2 c^2 (e f-d g)^2 x^2}{g^2}\right )}{f+g x} \, dx}{2 (e f-d g)^2}\\ &=\frac{c^2 (d+e x)^{2+m}}{e^2 g^3 (2+m)}+\frac{\left (c f^2-b f g+a g^2\right )^2 (d+e x)^{1+m}}{2 g^4 (e f-d g) (f+g x)^2}-\frac{\left (c f^2-b f g+a g^2\right ) (g (4 b d g-a e g (1-m)-b e f (3+m))-c f (8 d g-e f (7+m))) (d+e x)^{1+m}}{2 g^4 (e f-d g)^2 (f+g x)}+\frac{\int \frac{(d+e x)^m \left (\frac{e (2+m) \left (c^2 f \left (10 d^2 e f g^2-2 d^3 g^3-2 d e^2 f^2 g (7+4 m)+e^3 f^3 \left (6+7 m+m^2\right )\right )-e g^2 \left (a^2 e^2 g^2 (1-m) m-2 a b e g m (2 d g-e f (1+m))-b^2 \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )\right )+2 c e g \left (a g \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )-b f \left (4 d^2 g^2-2 d e f g (4+3 m)+e^2 f^2 \left (4+5 m+m^2\right )\right )\right )\right )}{g^3}-\frac{2 c e (e f-d g)^2 (3 c e f+c d g-2 b e g) (2+m) x}{g^2}\right )}{f+g x} \, dx}{2 e^2 g (e f-d g)^2 (2+m)}\\ &=-\frac{c (3 c e f+c d g-2 b e g) (d+e x)^{1+m}}{e^2 g^4 (1+m)}+\frac{c^2 (d+e x)^{2+m}}{e^2 g^3 (2+m)}+\frac{\left (c f^2-b f g+a g^2\right )^2 (d+e x)^{1+m}}{2 g^4 (e f-d g) (f+g x)^2}-\frac{\left (c f^2-b f g+a g^2\right ) (g (4 b d g-a e g (1-m)-b e f (3+m))-c f (8 d g-e f (7+m))) (d+e x)^{1+m}}{2 g^4 (e f-d g)^2 (f+g x)}+\frac{\left (c^2 f^2 \left (12 d^2 g^2-8 d e f g (3+m)+e^2 f^2 \left (12+7 m+m^2\right )\right )-g^2 \left (a^2 e^2 g^2 (1-m) m-2 a b e g m (2 d g-e f (1+m))-b^2 \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )\right )+2 c g \left (a g \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )-b f \left (6 d^2 g^2-6 d e f g (2+m)+e^2 f^2 \left (6+5 m+m^2\right )\right )\right )\right ) \int \frac{(d+e x)^m}{f+g x} \, dx}{2 g^4 (e f-d g)^2}\\ &=-\frac{c (3 c e f+c d g-2 b e g) (d+e x)^{1+m}}{e^2 g^4 (1+m)}+\frac{c^2 (d+e x)^{2+m}}{e^2 g^3 (2+m)}+\frac{\left (c f^2-b f g+a g^2\right )^2 (d+e x)^{1+m}}{2 g^4 (e f-d g) (f+g x)^2}-\frac{\left (c f^2-b f g+a g^2\right ) (g (4 b d g-a e g (1-m)-b e f (3+m))-c f (8 d g-e f (7+m))) (d+e x)^{1+m}}{2 g^4 (e f-d g)^2 (f+g x)}+\frac{\left (c^2 f^2 \left (12 d^2 g^2-8 d e f g (3+m)+e^2 f^2 \left (12+7 m+m^2\right )\right )-g^2 \left (a^2 e^2 g^2 (1-m) m-2 a b e g m (2 d g-e f (1+m))-b^2 \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )\right )+2 c g \left (a g \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )-b f \left (6 d^2 g^2-6 d e f g (2+m)+e^2 f^2 \left (6+5 m+m^2\right )\right )\right )\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac{g (d+e x)}{e f-d g}\right )}{2 g^4 (e f-d g)^3 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.411693, size = 257, normalized size = 0.56 \[ \frac{(d+e x)^{m+1} \left (\frac{\left (2 c g (a g-3 b f)+b^2 g^2+6 c^2 f^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right )}{(m+1) (e f-d g)}+\frac{e^2 \left (g (a g-b f)+c f^2\right )^2 \, _2F_1\left (3,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right )}{(m+1) (e f-d g)^3}-\frac{2 e (2 c f-b g) \left (g (a g-b f)+c f^2\right ) \, _2F_1\left (2,m+1;m+2;\frac{g (d+e x)}{d g-e f}\right )}{(m+1) (e f-d g)^2}-\frac{c (-2 b e g+c d g+3 c e f)}{e^2 (m+1)}+\frac{c^2 g (d+e x)}{e^2 (m+2)}\right )}{g^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.459, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{2} \left ( ex+d \right ) ^{m}}{ \left ( gx+f \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{m} \left (a + b x + c x^{2}\right )^{2}}{\left (f + g x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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