Optimal. Leaf size=191 \[ -\frac {2 c (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1+m)}+\frac {2 c (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+m)} \]
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Rubi [A]
time = 0.26, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {725, 70}
\begin {gather*} \frac {2 c (d+e x)^{m+1} \, _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) \sqrt {b^2-4 a c} \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}-\frac {2 c (d+e x)^{m+1} \, _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) \sqrt {b^2-4 a c} \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 725
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{a+b x+c x^2} \, dx &=\int \left (\frac {2 c (d+e x)^m}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}-\frac {2 c (d+e x)^m}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \, dx\\ &=\frac {(2 c) \int \frac {(d+e x)^m}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {(d+e x)^m}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}\\ &=-\frac {2 c (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 )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (1+m)}+\frac {2 c (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 )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 245, normalized size = 1.28 \begin {gather*} \frac {2^{-m} e (d+e x)^m \left (\left (\frac {c (d+e x)}{b e-\sqrt {\left (b^2-4 a c\right ) e^2}+2 c e x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{-b e+\sqrt {\left (b^2-4 a c\right ) e^2}-2 c e x}\right )-\left (\frac {c (d+e x)}{b e+\sqrt {\left (b^2-4 a c\right ) e^2}+2 c e x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{b e+\sqrt {\left (b^2-4 a c\right ) e^2}+2 c e x}\right )\right )}{\sqrt {\left (b^2-4 a c\right ) e^2} m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.42, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{m}}{c \,x^{2}+b x +a}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{m}}{a + b x + c x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^m}{c\,x^2+b\,x+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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