Optimal. Leaf size=97 \[ \frac {(b d-a e)^{10} \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{m+1} \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (m+1,-2 (p+5);m+2;\frac {b (d+e x)}{b d-a e}\right )}{e^{11} (m+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {646, 70, 69} \[ \frac {(b d-a e)^{10} \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{m+1} \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (m+1,-2 (p+5);m+2;\frac {b (d+e x)}{b d-a e}\right )}{e^{11} (m+1)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 646
Rubi steps
\begin {align*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5+p} \, dx &=\frac {\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 \left (a b+b^2 x\right )^{2 (5+p)} (d+e x)^m \, dx}{b^{10}}\\ &=\frac {\left (\left (-b^2 d+a b e\right )^{10} \left (\frac {e \left (a b+b^2 x\right )}{-b^2 d+a b e}\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int (d+e x)^m \left (-\frac {a e}{b d-a e}-\frac {b e x}{b d-a e}\right )^{2 (5+p)} \, dx}{b^{10} e^{10}}\\ &=\frac {(b d-a e)^{10} \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} (d+e x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (1+m,-2 (5+p);2+m;\frac {b (d+e x)}{b d-a e}\right )}{e^{11} (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 87, normalized size = 0.90 \[ \frac {(b d-a e)^{10} \left ((a+b x)^2\right )^p (d+e x)^{m+1} \left (\frac {e (a+b x)}{a e-b d}\right )^{-2 p} \, _2F_1\left (m+1,-2 (p+5);m+2;\frac {b (d+e x)}{b d-a e}\right )}{e^{11} (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p + 5} {\left (e x + d\right )}^{m}, x\right ) \]
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 \[ \int {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p + 5} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.35, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{m} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p +5}\, dx \]
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 \[ \int {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p + 5} {\left (e x + d\right )}^{m}\,{d x} \]
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 \[ \int {\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{p+5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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