Optimal. Leaf size=45 \[ \frac {\left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{m+2}}{b c^2 (m+2 p+2)} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {21, 644, 32} \begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{m+2}}{b c^2 (m+2 p+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rule 644
Rubi steps
\begin {align*} \int (a+b x) (a c+b c x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\frac {\int (a c+b c x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx}{c}\\ &=\frac {\left ((a c+b c x)^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int (a c+b c x)^{1+m+2 p} \, dx}{c}\\ &=\frac {(a c+b c x)^{2+m} \left (a^2+2 a b x+b^2 x^2\right )^p}{b c^2 (2+m+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.71 \begin {gather*} \frac {\left ((a+b x)^2\right )^{p+1} (c (a+b x))^m}{b (m+2 p+2)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (a c+b c x)^m \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 = 0.45, size = 60, normalized size = 1.33 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} {\left (b c x + a c\right )}^{m} e^{\left (2 \, p \log \left (b c x + a c\right ) + p \log \left (\frac {1}{c^{2}}\right )\right )}}{b m + 2 \, b p + 2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 100, normalized size = 2.22 \begin {gather*} \frac {{\left (b x + a\right )}^{2 \, p} b^{2} x^{2} e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + 2 \, {\left (b x + a\right )}^{2 \, p} a b x e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + {\left (b x + a\right )}^{2 \, p} a^{2} e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )}}{b m + 2 \, b p + 2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 48, normalized size = 1.07 \begin {gather*} \frac {\left (b x +a \right )^{2} \left (b c x +a c \right )^{m} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{\left (m +2 p +2\right ) b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 127, normalized size = 2.82 \begin {gather*} \frac {{\left (b c^{m} x + a c^{m}\right )} a e^{\left (m \log \left (b x + a\right ) + 2 \, p \log \left (b x + a\right )\right )}}{b {\left (m + 2 \, p + 1\right )}} + \frac {{\left (b^{2} c^{m} {\left (m + 2 \, p + 1\right )} x^{2} + a b c^{m} {\left (m + 2 \, p\right )} x - a^{2} c^{m}\right )} e^{\left (m \log \left (b x + a\right ) + 2 \, p \log \left (b x + a\right )\right )}}{{\left (m^{2} + m {\left (4 \, p + 3\right )} + 4 \, p^{2} + 6 \, p + 2\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 90, normalized size = 2.00 \begin {gather*} {\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {2\,a\,x\,{\left (a\,c+b\,c\,x\right )}^m}{m+2\,p+2}+\frac {b\,x^2\,{\left (a\,c+b\,c\,x\right )}^m}{m+2\,p+2}+\frac {a^2\,{\left (a\,c+b\,c\,x\right )}^m}{b\,\left (m+2\,p+2\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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