Optimal. Leaf size=135 \[ -\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{5/3}}+\frac {b^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{5/3}}+\frac {b^2 \log (x)}{18 a^{5/3}}-\frac {b \sqrt [3]{a+b x^2}}{12 a x^2}-\frac {\sqrt [3]{a+b x^2}}{4 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {266, 47, 51, 57, 617, 204, 31} \[ -\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{5/3}}+\frac {b^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{5/3}}+\frac {b^2 \log (x)}{18 a^{5/3}}-\frac {b \sqrt [3]{a+b x^2}}{12 a x^2}-\frac {\sqrt [3]{a+b x^2}}{4 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 47
Rule 51
Rule 57
Rule 204
Rule 266
Rule 617
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x^2}}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [3]{a+b x^2}}{4 x^4}+\frac {1}{12} b \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{2/3}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [3]{a+b x^2}}{4 x^4}-\frac {b \sqrt [3]{a+b x^2}}{12 a x^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{2/3}} \, dx,x,x^2\right )}{18 a}\\ &=-\frac {\sqrt [3]{a+b x^2}}{4 x^4}-\frac {b \sqrt [3]{a+b x^2}}{12 a x^2}+\frac {b^2 \log (x)}{18 a^{5/3}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{12 a^{5/3}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}\\ &=-\frac {\sqrt [3]{a+b x^2}}{4 x^4}-\frac {b \sqrt [3]{a+b x^2}}{12 a x^2}+\frac {b^2 \log (x)}{18 a^{5/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{5/3}}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )}{6 a^{5/3}}\\ &=-\frac {\sqrt [3]{a+b x^2}}{4 x^4}-\frac {b \sqrt [3]{a+b x^2}}{12 a x^2}+\frac {b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{6 \sqrt {3} a^{5/3}}+\frac {b^2 \log (x)}{18 a^{5/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{5/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 39, normalized size = 0.29 \[ -\frac {3 b^2 \left (a+b x^2\right )^{4/3} \, _2F_1\left (\frac {4}{3},3;\frac {7}{3};\frac {b x^2}{a}+1\right )}{8 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 199, normalized size = 1.47 \[ \frac {2 \, \sqrt {3} a b^{2} x^{4} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} a - 2 \, \sqrt {3} {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}}}{3 \, a^{2}}\right ) + \left (-a^{2}\right )^{\frac {2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} a - \left (-a^{2}\right )^{\frac {1}{3}} a + {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a^{2}\right )^{\frac {2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} a - \left (-a^{2}\right )^{\frac {2}{3}}\right ) - 3 \, {\left (a^{2} b x^{2} + 3 \, a^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{36 \, a^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 2.37, size = 140, normalized size = 1.04 \[ \frac {\frac {2 \, \sqrt {3} b^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {5}{3}}} + \frac {b^{3} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {5}{3}}} - \frac {2 \, b^{3} \log \left ({\left | {\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {5}{3}}} - \frac {3 \, {\left ({\left (b x^{2} + a\right )}^{\frac {4}{3}} b^{3} + 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a b^{3}\right )}}{a b^{2} x^{4}}}{36 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+a \right )^{\frac {1}{3}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.91, size = 155, normalized size = 1.15 \[ \frac {\sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{18 \, a^{\frac {5}{3}}} + \frac {b^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{36 \, a^{\frac {5}{3}}} - \frac {b^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{18 \, a^{\frac {5}{3}}} - \frac {{\left (b x^{2} + a\right )}^{\frac {4}{3}} b^{2} + 2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a b^{2}}{12 \, {\left ({\left (b x^{2} + a\right )}^{2} a - 2 \, {\left (b x^{2} + a\right )} a^{2} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.11, size = 217, normalized size = 1.61 \[ \frac {b^2\,\ln \left (\frac {b^2}{2\,{\left (-a\right )}^{2/3}}-\frac {b^2\,{\left (b\,x^2+a\right )}^{1/3}}{2\,a}\right )}{18\,{\left (-a\right )}^{5/3}}-\frac {\ln \left (\frac {b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}}{4\,{\left (-a\right )}^{2/3}}+\frac {b^2\,{\left (b\,x^2+a\right )}^{1/3}}{2\,a}\right )\,\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )}{36\,{\left (-a\right )}^{5/3}}-\frac {\frac {b^2\,{\left (b\,x^2+a\right )}^{1/3}}{3}+\frac {b^2\,{\left (b\,x^2+a\right )}^{4/3}}{6\,a}}{2\,{\left (b\,x^2+a\right )}^2-4\,a\,\left (b\,x^2+a\right )+2\,a^2}+\frac {b^2\,\ln \left (\frac {b^2\,{\left (b\,x^2+a\right )}^{1/3}}{2\,a}-\frac {b^2\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,{\left (-a\right )}^{2/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18\,{\left (-a\right )}^{5/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.39, size = 42, normalized size = 0.31 \[ - \frac {\sqrt [3]{b} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{\frac {10}{3}} \Gamma \left (\frac {8}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________