Optimal. Leaf size=75 \[ \frac {b \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 14} \[ \frac {b \log (x) \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 1355
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^3+b^2 x^6}}{x^4} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {a b+b^2 x^3}{x^4} \, dx}{a b+b^2 x^3}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a b}{x^4}+\frac {b^2}{x}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6}}{3 x^3 \left (a+b x^3\right )}+\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6} \log (x)}{a+b x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 39, normalized size = 0.52 \[ -\frac {\sqrt {\left (a+b x^3\right )^2} \left (a-3 b x^3 \log (x)\right )}{3 x^3 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 17, normalized size = 0.23 \[ \frac {3 \, b x^{3} \log \relax (x) - a}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.35, size = 43, normalized size = 0.57 \[ b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a \mathrm {sgn}\left (b x^{3} + a\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 38, normalized size = 0.51 \[ \frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (3 b \,x^{3} \ln \relax (x )-a \right )}{3 \left (b \,x^{3}+a \right ) x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.46, size = 99, normalized size = 1.32 \[ \frac {1}{3} \, \left (-1\right )^{2 \, b^{2} x^{3} + 2 \, a b} b \log \left (2 \, b^{2} x^{3} + 2 \, a b\right ) - \frac {1}{3} \, \left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right ) - \frac {\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.38, size = 112, normalized size = 1.49 \[ \frac {\ln \left (a\,b+\sqrt {{\left (b\,x^3+a\right )}^2}\,\sqrt {b^2}+b^2\,x^3\right )\,\sqrt {b^2}}{3}-\frac {\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{3\,x^3}-\frac {a\,b\,\ln \left (a\,b+\frac {a^2}{x^3}+\frac {\sqrt {a^2}\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{x^3}\right )}{3\,\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.16, size = 10, normalized size = 0.13 \[ - \frac {a}{3 x^{3}} + b \log {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________