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 = 1.07, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 68
Rule 740
Rule 830
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*}
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Mathematica [A] time = 1.15, size = 339, normalized size = 0.80 \[ \frac {(d+e x)^{m+1} \left (-\frac {c \left (\frac {4 c e (a e (m-1)+b d)-b^2 e^2 m-4 c^2 d^2}{\sqrt {b^2-4 a c}}+e m (2 c d-b e)\right ) \, _2F_1\left (1,m+1;m+2;\frac {2 c (d+e x)}{2 c d+\left (\sqrt {b^2-4 a c}-b\right ) e}\right )}{(m+1) \left (e \left (\sqrt {b^2-4 a c}-b\right )+2 c d\right )}-\frac {c \left (\frac {-4 c e (a e (m-1)+b d)+b^2 e^2 m+4 c^2 d^2}{\sqrt {b^2-4 a c}}+e m (2 c d-b e)\right ) \, _2F_1\left (1,m+1;m+2;\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{(m+1) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}+\frac {-2 c (a e+c d x)+b^2 e+b c (e x-d)}{a+x (b+c x)}\right )}{\left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.44, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{m}}{\left (c \,x^{2}+b x +a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (c\,x^2+b\,x+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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