Optimal. Leaf size=64 \[ \frac{b^2 p \log \left (a+b x^3\right )}{6 a^2}-\frac{b^2 p \log (x)}{2 a^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6}-\frac{b p}{6 a x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.051364, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2454, 2395, 44} \[ \frac{b^2 p \log \left (a+b x^3\right )}{6 a^2}-\frac{b^2 p \log (x)}{2 a^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6}-\frac{b p}{6 a x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2454
Rule 2395
Rule 44
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (a+b x^3\right )^p\right )}{x^7} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\log \left (c (a+b x)^p\right )}{x^3} \, dx,x,x^3\right )\\ &=-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6}+\frac{1}{6} (b p) \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,x^3\right )\\ &=-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6}+\frac{1}{6} (b p) \operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{b p}{6 a x^3}-\frac{b^2 p \log (x)}{2 a^2}+\frac{b^2 p \log \left (a+b x^3\right )}{6 a^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6}\\ \end{align*}
Mathematica [A] time = 0.0373722, size = 56, normalized size = 0.88 \[ \frac{1}{6} b p \left (\frac{b \log \left (a+b x^3\right )}{a^2}-\frac{3 b \log (x)}{a^2}-\frac{1}{a x^3}\right )-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.273, size = 198, normalized size = 3.1 \begin{align*} -{\frac{\ln \left ( \left ( b{x}^{3}+a \right ) ^{p} \right ) }{6\,{x}^{6}}}-{\frac{6\,{b}^{2}p\ln \left ( x \right ){x}^{6}-2\,{b}^{2}p\ln \left ( -b{x}^{3}-a \right ){x}^{6}+i\pi \,{a}^{2}{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}-i\pi \,{a}^{2}{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \right ) -i\pi \,{a}^{2} \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{3}+i\pi \,{a}^{2} \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +2\,abp{x}^{3}+2\,\ln \left ( c \right ){a}^{2}}{12\,{a}^{2}{x}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.1582, size = 73, normalized size = 1.14 \begin{align*} \frac{1}{6} \, b p{\left (\frac{b \log \left (b x^{3} + a\right )}{a^{2}} - \frac{b \log \left (x^{3}\right )}{a^{2}} - \frac{1}{a x^{3}}\right )} - \frac{\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.22418, size = 134, normalized size = 2.09 \begin{align*} -\frac{3 \, b^{2} p x^{6} \log \left (x\right ) + a b p x^{3} + a^{2} \log \left (c\right ) -{\left (b^{2} p x^{6} - a^{2} p\right )} \log \left (b x^{3} + a\right )}{6 \, a^{2} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 80.932, size = 102, normalized size = 1.59 \begin{align*} \begin{cases} - \frac{p \log{\left (a + b x^{3} \right )}}{6 x^{6}} - \frac{\log{\left (c \right )}}{6 x^{6}} - \frac{b p}{6 a x^{3}} - \frac{b^{2} p \log{\left (x \right )}}{2 a^{2}} + \frac{b^{2} p \log{\left (a + b x^{3} \right )}}{6 a^{2}} & \text{for}\: a \neq 0 \\- \frac{p \log{\left (b \right )}}{6 x^{6}} - \frac{p \log{\left (x \right )}}{2 x^{6}} - \frac{p}{12 x^{6}} - \frac{\log{\left (c \right )}}{6 x^{6}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.32081, size = 178, normalized size = 2.78 \begin{align*} -\frac{\frac{b^{3} p \log \left (b x^{3} + a\right )}{{\left (b x^{3} + a\right )}^{2} - 2 \,{\left (b x^{3} + a\right )} a + a^{2}} - \frac{b^{3} p \log \left (b x^{3} + a\right )}{a^{2}} + \frac{b^{3} p \log \left (b x^{3}\right )}{a^{2}} + \frac{{\left (b x^{3} + a\right )} b^{3} p - a b^{3} p + a b^{3} \log \left (c\right )}{{\left (b x^{3} + a\right )}^{2} a - 2 \,{\left (b x^{3} + a\right )} a^{2} + a^{3}}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]