Optimal. Leaf size=188 \[ -\frac {b}{6 a^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {3 b \log (x) \left (a+b x^3\right )}{a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {2 b}{3 a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {a+b x^3}{3 a^3 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 266, 44} \[ -\frac {b}{6 a^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {2 b}{3 a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {a+b x^3}{3 a^3 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {3 b \log (x) \left (a+b x^3\right )}{a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{x^4 \left (a b+b^2 x^3\right )^3} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (b^2 \left (a b+b^2 x^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a b+b^2 x\right )^3} \, dx,x,x^3\right )}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=\frac {\left (b^2 \left (a b+b^2 x^3\right )\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^3 b^3 x^2}-\frac {3}{a^4 b^2 x}+\frac {1}{a^2 b (a+b x)^3}+\frac {2}{a^3 b (a+b x)^2}+\frac {3}{a^4 b (a+b x)}\right ) \, dx,x,x^3\right )}{3 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ &=-\frac {2 b}{3 a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b}{6 a^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {a+b x^3}{3 a^3 x^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {3 b \left (a+b x^3\right ) \log (x)}{a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {b \left (a+b x^3\right ) \log \left (a+b x^3\right )}{a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 97, normalized size = 0.52 \[ \frac {-a \left (2 a^2+9 a b x^3+6 b^2 x^6\right )-18 b x^3 \log (x) \left (a+b x^3\right )^2+6 b x^3 \left (a+b x^3\right )^2 \log \left (a+b x^3\right )}{6 a^4 x^3 \left (a+b x^3\right ) \sqrt {\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 119, normalized size = 0.63 \[ -\frac {6 \, a b^{2} x^{6} + 9 \, a^{2} b x^{3} + 2 \, a^{3} - 6 \, {\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \left (b x^{3} + a\right ) + 18 \, {\left (b^{3} x^{9} + 2 \, a b^{2} x^{6} + a^{2} b x^{3}\right )} \log \relax (x)}{6 \, {\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 133, normalized size = 0.71 \[ -\frac {\left (18 b^{3} x^{9} \ln \relax (x )-6 b^{3} x^{9} \ln \left (b \,x^{3}+a \right )+36 a \,b^{2} x^{6} \ln \relax (x )-12 a \,b^{2} x^{6} \ln \left (b \,x^{3}+a \right )+6 a \,b^{2} x^{6}+18 a^{2} b \,x^{3} \ln \relax (x )-6 a^{2} b \,x^{3} \ln \left (b \,x^{3}+a \right )+9 a^{2} b \,x^{3}+2 a^{3}\right ) \left (b \,x^{3}+a \right )}{6 \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {3}{2}} a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.99, size = 117, normalized size = 0.62 \[ \frac {\left (-1\right )^{2 \, a b x^{3} + 2 \, a^{2}} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{x^{2} {\left | x \right |}}\right )}{a^{4}} - \frac {b}{\sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a^{3}} - \frac {1}{6 \, {\left (x^{3} + \frac {a}{b}\right )}^{2} a^{2} b} - \frac {1}{3 \, \sqrt {b^{2} x^{6} + 2 \, a b x^{3} + a^{2}} a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^4\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________