Optimal. Leaf size=100 \[ \frac {g \left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{m+2}}{b^2 c^2 (m+2 p+2)}+\frac {(b f-a g) \left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{m+1}}{b^2 c (m+2 p+1)} \]
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Rubi [A] time = 0.08, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {770, 23, 43} \begin {gather*} \frac {g \left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{m+2}}{b^2 c^2 (m+2 p+2)}+\frac {(b f-a g) \left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{m+1}}{b^2 c (m+2 p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 43
Rule 770
Rubi steps
\begin {align*} \int (a c+b c x)^m (f+g x) \left (a^2+2 a b x+b^2 x^2\right )^p \, 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 \left (a b+b^2 x\right )^{2 p} (a c+b c x)^m (f+g x) \, dx\\ &=\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)^{m+2 p} (f+g x) \, dx\\ &=\left ((a c+b c x)^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (\frac {(b f-a g) (a c+b c x)^{m+2 p}}{b}+\frac {g (a c+b c x)^{1+m+2 p}}{b c}\right ) \, dx\\ &=\frac {(b f-a g) (a c+b c x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^p}{b^2 c (1+m+2 p)}+\frac {g (a c+b c x)^{2+m} \left (a^2+2 a b x+b^2 x^2\right )^p}{b^2 c^2 (2+m+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 67, normalized size = 0.67 \begin {gather*} \frac {(a+b x) \left ((a+b x)^2\right )^p (c (a+b x))^m (-a g+b f (m+2 p+2)+b g x (m+2 p+1))}{b^2 (m+2 p+1) (m+2 p+2)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int (a c+b c x)^m (f+g x) \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 = 0.43, size = 155, normalized size = 1.55 \begin {gather*} \frac {{\left (a b f m + 2 \, a b f p + 2 \, a b f - a^{2} g + {\left (b^{2} g m + 2 \, b^{2} g p + b^{2} g\right )} x^{2} + {\left (2 \, b^{2} f + {\left (b^{2} f + a b g\right )} m + 2 \, {\left (b^{2} f + a b g\right )} p\right )} x\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^{2} m^{2} + 4 \, b^{2} p^{2} + 3 \, b^{2} m + 2 \, b^{2} + 2 \, {\left (2 \, b^{2} m + 3 \, b^{2}\right )} p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 404, normalized size = 4.04 \begin {gather*} \frac {{\left (b x + a\right )}^{2 \, p} b^{2} g m x^{2} e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + 2 \, {\left (b x + a\right )}^{2 \, p} b^{2} g p x^{2} e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + {\left (b x + a\right )}^{2 \, p} b^{2} f m x e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + {\left (b x + a\right )}^{2 \, p} a b g m x e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + 2 \, {\left (b x + a\right )}^{2 \, p} b^{2} f p x e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + 2 \, {\left (b x + a\right )}^{2 \, p} a b g p x e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + {\left (b x + a\right )}^{2 \, p} b^{2} g x^{2} e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + {\left (b x + a\right )}^{2 \, p} a b f m e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + 2 \, {\left (b x + a\right )}^{2 \, p} a b f p e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + 2 \, {\left (b x + a\right )}^{2 \, p} b^{2} f x e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} + 2 \, {\left (b x + a\right )}^{2 \, p} a b f e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )} - {\left (b x + a\right )}^{2 \, p} a^{2} g e^{\left (m \log \left (b x + a\right ) + m \log \relax (c)\right )}}{b^{2} m^{2} + 4 \, b^{2} m p + 4 \, b^{2} p^{2} + 3 \, b^{2} m + 6 \, b^{2} p + 2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 96, normalized size = 0.96 \begin {gather*} -\frac {\left (-b g m x -2 b g p x -b f m -2 b f p -b g x +a g -2 b f \right ) \left (b x +a \right ) \left (b c x +a c \right )^{m} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{\left (m^{2}+4 m p +4 p^{2}+3 m +6 p +2\right ) b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 128, normalized size = 1.28 \begin {gather*} \frac {{\left (b c^{m} x + a c^{m}\right )} f 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 )} g 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^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 178, normalized size = 1.78 \begin {gather*} {\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {g\,x^2\,{\left (a\,c+b\,c\,x\right )}^m\,\left (m+2\,p+1\right )}{m^2+4\,m\,p+3\,m+4\,p^2+6\,p+2}+\frac {a\,{\left (a\,c+b\,c\,x\right )}^m\,\left (2\,b\,f-a\,g+b\,f\,m+2\,b\,f\,p\right )}{b^2\,\left (m^2+4\,m\,p+3\,m+4\,p^2+6\,p+2\right )}+\frac {x\,{\left (a\,c+b\,c\,x\right )}^m\,\left (2\,b\,f+a\,g\,m+b\,f\,m+2\,a\,g\,p+2\,b\,f\,p\right )}{b\,\left (m^2+4\,m\,p+3\,m+4\,p^2+6\,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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