Optimal. Leaf size=100 \[ -\frac{b^3 p}{2 a^3 \sqrt{x}}+\frac{b^2 p}{4 a^2 x}+\frac{b^4 p \log \left (a+b \sqrt{x}\right )}{2 a^4}-\frac{b^4 p \log (x)}{4 a^4}-\frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{2 x^2}-\frac{b p}{6 a x^{3/2}} \]
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Rubi [A] time = 0.0632133, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2454, 2395, 44} \[ -\frac{b^3 p}{2 a^3 \sqrt{x}}+\frac{b^2 p}{4 a^2 x}+\frac{b^4 p \log \left (a+b \sqrt{x}\right )}{2 a^4}-\frac{b^4 p \log (x)}{4 a^4}-\frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{2 x^2}-\frac{b p}{6 a x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2395
Rule 44
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{\log \left (c (a+b x)^p\right )}{x^5} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{2 x^2}+\frac{1}{2} (b p) \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{2 x^2}+\frac{1}{2} (b p) \operatorname{Subst}\left (\int \left (\frac{1}{a x^4}-\frac{b}{a^2 x^3}+\frac{b^2}{a^3 x^2}-\frac{b^3}{a^4 x}+\frac{b^4}{a^4 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{b p}{6 a x^{3/2}}+\frac{b^2 p}{4 a^2 x}-\frac{b^3 p}{2 a^3 \sqrt{x}}+\frac{b^4 p \log \left (a+b \sqrt{x}\right )}{2 a^4}-\frac{\log \left (c \left (a+b \sqrt{x}\right )^p\right )}{2 x^2}-\frac{b^4 p \log (x)}{4 a^4}\\ \end{align*}
Mathematica [A] time = 0.0436651, size = 90, normalized size = 0.9 \[ \frac{a b p \sqrt{x} \left (-2 a^2+3 a b \sqrt{x}-6 b^2 x\right )-6 a^4 \log \left (c \left (a+b \sqrt{x}\right )^p\right )+6 b^4 p x^2 \log \left (a+b \sqrt{x}\right )-3 b^4 p x^2 \log (x)}{12 a^4 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.27, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\ln \left ( c \left ( a+b\sqrt{x} \right ) ^{p} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06173, size = 103, normalized size = 1.03 \begin{align*} \frac{1}{12} \, b p{\left (\frac{6 \, b^{3} \log \left (b \sqrt{x} + a\right )}{a^{4}} - \frac{3 \, b^{3} \log \left (x\right )}{a^{4}} - \frac{6 \, b^{2} x - 3 \, a b \sqrt{x} + 2 \, a^{2}}{a^{3} x^{\frac{3}{2}}}\right )} - \frac{\log \left ({\left (b \sqrt{x} + a\right )}^{p} c\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20647, size = 208, normalized size = 2.08 \begin{align*} -\frac{6 \, b^{4} p x^{2} \log \left (\sqrt{x}\right ) - 3 \, a^{2} b^{2} p x + 6 \, a^{4} \log \left (c\right ) - 6 \,{\left (b^{4} p x^{2} - a^{4} p\right )} \log \left (b \sqrt{x} + a\right ) + 2 \,{\left (3 \, a b^{3} p x + a^{3} b p\right )} \sqrt{x}}{12 \, a^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30011, size = 313, normalized size = 3.13 \begin{align*} -\frac{\frac{6 \, b^{5} p \log \left (b \sqrt{x} + a\right )}{{\left (b \sqrt{x} + a\right )}^{4} - 4 \,{\left (b \sqrt{x} + a\right )}^{3} a + 6 \,{\left (b \sqrt{x} + a\right )}^{2} a^{2} - 4 \,{\left (b \sqrt{x} + a\right )} a^{3} + a^{4}} - \frac{6 \, b^{5} p \log \left (b \sqrt{x} + a\right )}{a^{4}} + \frac{6 \, b^{5} p \log \left (b \sqrt{x}\right )}{a^{4}} + \frac{6 \,{\left (b \sqrt{x} + a\right )}^{3} b^{5} p - 21 \,{\left (b \sqrt{x} + a\right )}^{2} a b^{5} p + 26 \,{\left (b \sqrt{x} + a\right )} a^{2} b^{5} p - 11 \, a^{3} b^{5} p + 6 \, a^{3} b^{5} \log \left (c\right )}{{\left (b \sqrt{x} + a\right )}^{4} a^{3} - 4 \,{\left (b \sqrt{x} + a\right )}^{3} a^{4} + 6 \,{\left (b \sqrt{x} + a\right )}^{2} a^{5} - 4 \,{\left (b \sqrt{x} + a\right )} a^{6} + a^{7}}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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