Optimal. Leaf size=93 \[ \frac {c (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {c (d+e x)}{c d-b e}\right )}{b (c d-b e) (1+m)}-\frac {(d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;1+\frac {e x}{d}\right )}{b d (1+m)} \]
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Rubi [A]
time = 0.05, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {725, 67, 70}
\begin {gather*} \frac {c (d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {c (d+e x)}{c d-b e}\right )}{b (m+1) (c d-b e)}-\frac {(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {e x}{d}+1\right )}{b d (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 70
Rule 725
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{b x+c x^2} \, dx &=\int \left (\frac {(d+e x)^m}{b x}-\frac {c (d+e x)^m}{b (b+c x)}\right ) \, dx\\ &=\frac {\int \frac {(d+e x)^m}{x} \, dx}{b}-\frac {c \int \frac {(d+e x)^m}{b+c x} \, dx}{b}\\ &=\frac {c (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {c (d+e x)}{c d-b e}\right )}{b (c d-b e) (1+m)}-\frac {(d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;1+\frac {e x}{d}\right )}{b d (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 86, normalized size = 0.92 \begin {gather*} -\frac {(d+e x)^{1+m} \left (c d \, _2F_1\left (1,1+m;2+m;\frac {c (d+e x)}{c d-b e}\right )+(-c d+b e) \, _2F_1\left (1,1+m;2+m;1+\frac {e x}{d}\right )\right )}{b d (-c d+b e) (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (e x +d \right )^{m}}{c \,x^{2}+b x}\, dx\]
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}}{x \left (b + c x\right )}\, 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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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