Optimal. Leaf size=425 \[ \frac{c (d+e x)^{m+1} \left (-2 c e \left (d m \sqrt{b^2-4 a c}-2 a e (1-m)+2 b d\right )+b e^2 m \left (\sqrt{b^2-4 a c}+b\right )+4 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (b^2-4 a c\right )^{3/2} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac{c (d+e x)^{m+1} \left (\frac{-4 c e (b d-a e (1-m))+b^2 e^2 m+4 c^2 d^2}{\sqrt{b^2-4 a c}}+e m (2 c d-b e)\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (b^2-4 a c\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac{(d+e x)^{m+1} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 1.06873, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {740, 830, 68} \[ \frac{c (d+e x)^{m+1} \left (-2 c e \left (d m \sqrt{b^2-4 a c}-2 a e (1-m)+2 b d\right )+b e^2 m \left (\sqrt{b^2-4 a c}+b\right )+4 c^2 d^2\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (b^2-4 a c\right )^{3/2} \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac{c (d+e x)^{m+1} \left (\frac{-4 c e (b d-a e (1-m))+b^2 e^2 m+4 c^2 d^2}{\sqrt{b^2-4 a c}}+e m (2 c d-b e)\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (b^2-4 a c\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac{(d+e x)^{m+1} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 740
Rule 830
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{(d+e x)^{1+m} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac{\int \frac{(d+e x)^m \left (2 c^2 d^2+b^2 e^2 m+c e (2 a e (1-m)-b d (2+m))-c e (2 c d-b e) m x\right )}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{(d+e x)^{1+m} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac{\int \left (\frac{\left (-c e (2 c d-b e) m+\frac{c \left (4 c^2 d^2-4 b c d e+4 a c e^2+b^2 e^2 m-4 a c e^2 m\right )}{\sqrt{b^2-4 a c}}\right ) (d+e x)^m}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{\left (-c e (2 c d-b e) m-\frac{c \left (4 c^2 d^2-4 b c d e+4 a c e^2+b^2 e^2 m-4 a c e^2 m\right )}{\sqrt{b^2-4 a c}}\right ) (d+e x)^m}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{(d+e x)^{1+m} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac{\left (c \left (e (2 c d-b e) m-\frac{4 c^2 d^2-4 c e (b d-a e (1-m))+b^2 e^2 m}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{(d+e x)^m}{b-\sqrt{b^2-4 a c}+2 c x} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}+\frac{\left (c \left (e (2 c d-b e) m+\frac{4 c^2 d^2-4 c e (b d-a e (1-m))+b^2 e^2 m}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{(d+e x)^m}{b+\sqrt{b^2-4 a c}+2 c x} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{(d+e x)^{1+m} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac{c \left (4 c^2 d^2+b \left (b+\sqrt{b^2-4 a c}\right ) e^2 m-2 c e \left (2 b d-2 a e (1-m)+\sqrt{b^2-4 a c} d m\right )\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{2 c (d+e x)}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )}{\left (b^2-4 a c\right )^{3/2} \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right ) (1+m)}-\frac{c \left (e (2 c d-b e) m+\frac{4 c^2 d^2-4 c e (b d-a e (1-m))+b^2 e^2 m}{\sqrt{b^2-4 a c}}\right ) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\left (b^2-4 a c\right ) \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right ) (1+m)}\\ \end{align*}
Mathematica [A] time = 1.28903, size = 339, normalized size = 0.8 \[ \frac{(d+e x)^{m+1} \left (-\frac{c \left (\frac{4 c e (a e (m-1)+b d)-b^2 e^2 m-4 c^2 d^2}{\sqrt{b^2-4 a c}}+e m (2 c d-b e)\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )}{(m+1) \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right )}-\frac{c \left (\frac{-4 c e (a e (m-1)+b d)+b^2 e^2 m+4 c^2 d^2}{\sqrt{b^2-4 a c}}+e m (2 c d-b e)\right ) \, _2F_1\left (1,m+1;m+2;\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{(m+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}+\frac{-2 c (a e+c d x)+b^2 e+b c (e x-d)}{a+x (b+c x)}\right )}{\left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.241, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}\, 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 (e x + d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{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 (e x + d\right )}^{m}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (e x + d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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