Optimal. Leaf size=126 \[ -\frac {(A b-a B) (d+e x)^{1+m}}{3 b (b d-a e) (a+b x)^3}-\frac {e^2 (b (3 B d-A e (2-m))-a B e (1+m)) (d+e x)^{1+m} \, _2F_1\left (3,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{3 b (b d-a e)^4 (1+m)} \]
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Rubi [A]
time = 0.05, antiderivative size = 125, normalized size of antiderivative = 0.99, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {27, 79, 70}
\begin {gather*} -\frac {e^2 (d+e x)^{m+1} (-a B e (m+1)-A b e (2-m)+3 b B d) \, _2F_1\left (3,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{3 b (m+1) (b d-a e)^4}-\frac {(A b-a B) (d+e x)^{m+1}}{3 b (a+b x)^3 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 70
Rule 79
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 {(A+B x) (d+e x)^m}{(a+b x)^4} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{1+m}}{3 b (b d-a e) (a+b x)^3}+\frac {(3 b B d-A b e (2-m)-a B e (1+m)) \int \frac {(d+e x)^m}{(a+b x)^3} \, dx}{3 b (b d-a e)}\\ &=-\frac {(A b-a B) (d+e x)^{1+m}}{3 b (b d-a e) (a+b x)^3}-\frac {e^2 (3 b B d-A b e (2-m)-a B e (1+m)) (d+e x)^{1+m} \, _2F_1\left (3,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{3 b (b d-a e)^4 (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 109, normalized size = 0.87 \begin {gather*} \frac {(d+e x)^{1+m} \left (\frac {-A b+a B}{(a+b x)^3}-\frac {e^2 (3 b B d+A b e (-2+m)-a B e (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)}\right )}{3 b (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.65, 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 \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 (A + B x\right ) \left (d + e x\right )^{m}}{\left (a + b x\right )^{4}}\, 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 (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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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