Optimal. Leaf size=290 \[ \frac {2 \sqrt {2-\sqrt {3}} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),4 \sqrt {3}-7\right )}{9 \sqrt [4]{3} a x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 b \sqrt [3]{a+b x^2}}{9 a x}-\frac {\sqrt [3]{a+b x^2}}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {277, 325, 236, 219} \[ \frac {2 \sqrt {2-\sqrt {3}} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} a x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 b \sqrt [3]{a+b x^2}}{9 a x}-\frac {\sqrt [3]{a+b x^2}}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 219
Rule 236
Rule 277
Rule 325
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x^2}}{x^4} \, dx &=-\frac {\sqrt [3]{a+b x^2}}{3 x^3}+\frac {1}{9} (2 b) \int \frac {1}{x^2 \left (a+b x^2\right )^{2/3}} \, dx\\ &=-\frac {\sqrt [3]{a+b x^2}}{3 x^3}-\frac {2 b \sqrt [3]{a+b x^2}}{9 a x}-\frac {\left (2 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{2/3}} \, dx}{27 a}\\ &=-\frac {\sqrt [3]{a+b x^2}}{3 x^3}-\frac {2 b \sqrt [3]{a+b x^2}}{9 a x}-\frac {\left (b \sqrt {b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a+b x^2}\right )}{9 a x}\\ &=-\frac {\sqrt [3]{a+b x^2}}{3 x^3}-\frac {2 b \sqrt [3]{a+b x^2}}{9 a x}+\frac {2 \sqrt {2-\sqrt {3}} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} a x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 51, normalized size = 0.18 \[ -\frac {\sqrt [3]{a+b x^2} \, _2F_1\left (-\frac {3}{2},-\frac {1}{3};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 x^3 \sqrt [3]{\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+a \right )^{\frac {1}{3}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^2+a\right )}^{1/3}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.95, size = 34, normalized size = 0.12 \[ - \frac {\sqrt [3]{a} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {1}{3} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________