Optimal. Leaf size=51 \[ \frac {c \left (a^2+2 a b x+b^2 x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)} \]
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Rubi [A] time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {767} \begin {gather*} \frac {c \left (a^2+2 a b x+b^2 x^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 767
Rubi steps
\begin {align*} \int (a c+b c x) (d+e x)^{-3-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\frac {c (d+e x)^{-2 (1+p)} \left (a^2+2 a b x+b^2 x^2\right )^{1+p}}{2 (b d-a e) (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.82 \begin {gather*} \frac {c \left ((a+b x)^2\right )^{p+1} (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int (a c+b c x) (d+e x)^{-3-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.46, size = 99, normalized size = 1.94 \begin {gather*} \frac {{\left (b^{2} c e x^{3} + a^{2} c d + {\left (b^{2} c d + 2 \, a b c e\right )} x^{2} + {\left (2 \, a b c d + a^{2} c e\right )} x\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}}{2 \, {\left (b d - a e + {\left (b d - a e\right )} p\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 305, normalized size = 5.98 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{2} c x^{3} e^{\left (-2 \, p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 1\right )} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{2} c d x^{2} e^{\left (-2 \, p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right )\right )} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b c x^{2} e^{\left (-2 \, p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 1\right )} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b c d x e^{\left (-2 \, p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right )\right )} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} c x e^{\left (-2 \, p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right ) + 1\right )} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} c d e^{\left (-2 \, p \log \left (x e + d\right ) - 3 \, \log \left (x e + d\right )\right )}}{2 \, {\left (b d p - a p e + b d - a e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 59, normalized size = 1.16 \begin {gather*} -\frac {\left (b x +a \right )^{2} c \left (e x +d \right )^{-2 p -2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{2 \left (a e p -b d p +a e -b d \right )} \end {gather*}
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*} \int {\left (b c x + a c\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.25, size = 178, normalized size = 3.49 \begin {gather*} -{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {a^2\,c\,d}{2\,\left (a\,e-b\,d\right )\,\left (p+1\right )\,{\left (d+e\,x\right )}^{2\,p+3}}+\frac {a\,c\,x\,\left (a\,e+2\,b\,d\right )}{2\,\left (a\,e-b\,d\right )\,\left (p+1\right )\,{\left (d+e\,x\right )}^{2\,p+3}}+\frac {b\,c\,x^2\,\left (2\,a\,e+b\,d\right )}{2\,\left (a\,e-b\,d\right )\,\left (p+1\right )\,{\left (d+e\,x\right )}^{2\,p+3}}+\frac {b^2\,c\,e\,x^3}{2\,\left (a\,e-b\,d\right )\,\left (p+1\right )\,{\left (d+e\,x\right )}^{2\,p+3}}\right ) \end {gather*}
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 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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