Optimal. Leaf size=72 \[ \frac {b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (2,1+2 p;2 (1+p);-\frac {e (a+b x)}{b d-a e}\right )}{(b d-a e)^2 (1+2 p)} \]
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Rubi [A]
time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {660, 70}
\begin {gather*} \frac {b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (2,2 p+1;2 (p+1);-\frac {e (a+b x)}{b d-a e}\right )}{(2 p+1) (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 660
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{(d+e x)^2} \, dx &=\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \frac {\left (a b+b^2 x\right )^{2 p}}{(d+e x)^2} \, dx\\ &=\frac {b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (2,1+2 p;2 (1+p);-\frac {e (a+b x)}{b d-a e}\right )}{(b d-a e)^2 (1+2 p)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 63, normalized size = 0.88 \begin {gather*} \frac {b (a+b x) \left ((a+b x)^2\right )^p \, _2F_1\left (2,1+2 p;2+2 p;-\frac {e (a+b x)}{b d-a e}\right )}{(b d-a e)^2 (1+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.34, size = 0, normalized size = 0.00 \[\int \frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{\left (e x +d \right )^{2}}\, dx\]
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 (\left (a + b x\right )^{2}\right )^{p}}{\left (d + e x\right )^{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 (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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