Optimal. Leaf size=61 \[ -\frac {(f+g x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{-2 p}}{2 c^3 (a+b x)^2 (b f-a g)} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {770, 23, 37} \begin {gather*} -\frac {(f+g x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{-2 p}}{2 c^3 (a+b x)^2 (b f-a g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 37
Rule 770
Rubi steps
\begin {align*} \int (a c+b c x)^{-3-2 p} (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)^{-3-2 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 \frac {f+g x}{(a c+b c x)^3} \, dx\\ &=-\frac {(a c+b c x)^{-2 p} (f+g x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 c^3 (b f-a g) (a+b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 0.80 \begin {gather*} -\frac {\left ((a+b x)^2\right )^p (c (a+b x))^{-2 p} (a g+b (f+2 g x))}{2 b^2 c^3 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.77, size = 49, normalized size = 0.80 \begin {gather*} -\frac {\left ((a+b x)^2\right )^p (c (a+b x))^{-2 p} (a g+b f+2 b g x)}{2 b^2 c^3 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 52, normalized size = 0.85 \begin {gather*} -\frac {{\left (2 \, b g x + b f + a g\right )} \frac {1}{c^{2}}^{p}}{2 \, {\left (b^{4} c^{3} x^{2} + 2 \, a b^{3} c^{3} x + a^{2} b^{2} c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 221, normalized size = 3.62 \begin {gather*} -\frac {2 \, {\left (b x + a\right )}^{2 \, p} b^{2} g x^{2} e^{\left (-2 \, p \log \left (b x + a\right ) - 2 \, p \log \relax (c) - 3 \, \log \left (b x + a\right ) - 3 \, \log \relax (c)\right )} + {\left (b x + a\right )}^{2 \, p} b^{2} f x e^{\left (-2 \, p \log \left (b x + a\right ) - 2 \, p \log \relax (c) - 3 \, \log \left (b x + a\right ) - 3 \, \log \relax (c)\right )} + 3 \, {\left (b x + a\right )}^{2 \, p} a b g x e^{\left (-2 \, p \log \left (b x + a\right ) - 2 \, p \log \relax (c) - 3 \, \log \left (b x + a\right ) - 3 \, \log \relax (c)\right )} + {\left (b x + a\right )}^{2 \, p} a b f e^{\left (-2 \, p \log \left (b x + a\right ) - 2 \, p \log \relax (c) - 3 \, \log \left (b x + a\right ) - 3 \, \log \relax (c)\right )} + {\left (b x + a\right )}^{2 \, p} a^{2} g e^{\left (-2 \, p \log \left (b x + a\right ) - 2 \, p \log \relax (c) - 3 \, \log \left (b x + a\right ) - 3 \, \log \relax (c)\right )}}{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 = 55, normalized size = 0.90 \begin {gather*} -\frac {\left (b x +a \right ) \left (2 b g x +a g +b f \right ) \left (b c x +a c \right )^{-2 p -3} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{2 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 101, normalized size = 1.66 \begin {gather*} -\frac {{\left (2 \, b x + a\right )} g}{2 \, {\left (b^{4} c^{2 \, p + 3} x^{2} + 2 \, a b^{3} c^{2 \, p + 3} x + a^{2} b^{2} c^{2 \, p + 3}\right )}} - \frac {f}{2 \, {\left (b^{3} c^{2 \, p + 3} x^{2} + 2 \, a b^{2} c^{2 \, p + 3} x + a^{2} b c^{2 \, p + 3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.14, size = 106, normalized size = 1.74 \begin {gather*} -\left (\frac {g\,a^2+b\,f\,a}{2\,b^2\,{\left (a\,c+b\,c\,x\right )}^{2\,p+3}}+\frac {g\,x^2}{{\left (a\,c+b\,c\,x\right )}^{2\,p+3}}+\frac {x\,\left (f\,b^2+3\,a\,g\,b\right )}{2\,b^2\,{\left (a\,c+b\,c\,x\right )}^{2\,p+3}}\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p \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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