Optimal. Leaf size=54 \[ -\frac {e^2 (d+e x)^{m+1} \, _2F_1\left (3,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^3} \]
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Rubi [A] time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 68} \[ -\frac {e^2 (d+e x)^{m+1} \, _2F_1\left (3,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 68
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^m}{(a+b x)^3} \, dx\\ &=-\frac {e^2 (d+e x)^{1+m} \, _2F_1\left (3,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 54, normalized size = 1.00 \[ \frac {e^2 (d+e x)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac {b (d+e x)}{a e-b d}\right )}{(m+1) (a e-b d)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right ) \left (e x +d \right )^{m}}{\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )} {\left (e x + d\right )}^{m}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\left (a+b\,x\right )\,{\left (d+e\,x\right )}^m}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^2} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{m}}{\left (a + b x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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