Optimal. Leaf size=71 \[ \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);-\frac {e (a+b x)}{b d-a e}\right )}{(2 p+1) (b d-a e)} \]
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Rubi [A] time = 0.03, antiderivative size = 71, 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 = {646, 68} \[ \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1,2 p+1;2 (p+1);-\frac {e (a+b x)}{b d-a e}\right )}{(2 p+1) (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 646
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{d+e x} \, 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} \, dx\\ &=\frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1,1+2 p;2 (1+p);-\frac {e (a+b x)}{b d-a e}\right )}{(b d-a e) (1+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.87 \[ -\frac {(a+b x) \left ((a+b x)^2\right )^p \, _2F_1\left (1,2 p+1;2 p+2;\frac {e (a+b x)}{a e-b d}\right )}{(2 p+1) (a e-b d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{e x +d}\, dx \]
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 \[ \int \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p}{d+e\,x} \,d x \]
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 \[ \int \frac {\left (\left (a + b x\right )^{2}\right )^{p}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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