Optimal. Leaf size=633 \[ \frac {4 \sqrt {2} b \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{a+b x^2}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1}\right ),4 \sqrt {3}-7\right )}{9 \sqrt [4]{3} a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {4 b^2 x}{9 a^2 \sqrt [6]{a+b x^2}}+\frac {4 b \left (a+b x^2\right )^{5/6}}{9 a^2 x}-\frac {4 b^2 x}{9 a \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )}-\frac {2 \sqrt {2+\sqrt {3}} b \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.59, antiderivative size = 633, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {325, 238, 198, 235, 304, 219, 1879} \[ -\frac {4 b^2 x}{9 a^2 \sqrt [6]{a+b x^2}}+\frac {4 b \left (a+b x^2\right )^{5/6}}{9 a^2 x}-\frac {4 b^2 x}{9 a \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}+\frac {4 \sqrt {2} b \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2+\sqrt {3}} b \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 198
Rule 219
Rule 235
Rule 238
Rule 304
Rule 325
Rule 1879
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt [6]{a+b x^2}} \, dx &=-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}-\frac {(4 b) \int \frac {1}{x^2 \sqrt [6]{a+b x^2}} \, dx}{9 a}\\ &=-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}+\frac {4 b \left (a+b x^2\right )^{5/6}}{9 a^2 x}-\frac {\left (8 b^2\right ) \int \frac {1}{\sqrt [6]{a+b x^2}} \, dx}{27 a^2}\\ &=-\frac {4 b^2 x}{9 a^2 \sqrt [6]{a+b x^2}}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}+\frac {4 b \left (a+b x^2\right )^{5/6}}{9 a^2 x}+\frac {\left (4 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{7/6}} \, dx}{27 a}\\ &=-\frac {4 b^2 x}{9 a^2 \sqrt [6]{a+b x^2}}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}+\frac {4 b \left (a+b x^2\right )^{5/6}}{9 a^2 x}+\frac {\left (4 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-b x^2}} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{27 a \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\\ &=-\frac {4 b^2 x}{9 a^2 \sqrt [6]{a+b x^2}}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}+\frac {4 b \left (a+b x^2\right )^{5/6}}{9 a^2 x}-\frac {\left (2 b \sqrt {-\frac {b x^2}{a+b x^2}}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{\frac {a}{a+b x^2}}\right )}{9 a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=-\frac {4 b^2 x}{9 a^2 \sqrt [6]{a+b x^2}}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}+\frac {4 b \left (a+b x^2\right )^{5/6}}{9 a^2 x}+\frac {\left (2 b \sqrt {-\frac {b x^2}{a+b x^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{\frac {a}{a+b x^2}}\right )}{9 a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}-\frac {\left (2 \sqrt {2 \left (2+\sqrt {3}\right )} b \sqrt {-\frac {b x^2}{a+b x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{\frac {a}{a+b x^2}}\right )}{9 a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=-\frac {4 b^2 x}{9 a^2 \sqrt [6]{a+b x^2}}-\frac {\left (a+b x^2\right )^{5/6}}{3 a x^3}+\frac {4 b \left (a+b x^2\right )^{5/6}}{9 a^2 x}+\frac {4 b \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt {-1+\frac {a}{a+b x^2}}}{9 a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )}-\frac {2 \sqrt {2+\sqrt {3}} b \sqrt {-\frac {b x^2}{a+b x^2}} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} \sqrt {-1+\frac {a}{a+b x^2}}}+\frac {4 \sqrt {2} b \sqrt {-\frac {b x^2}{a+b x^2}} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} \sqrt {-1+\frac {a}{a+b x^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 51, normalized size = 0.08 \[ -\frac {\sqrt [6]{\frac {b x^2}{a}+1} \, _2F_1\left (-\frac {3}{2},\frac {1}{6};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 x^3 \sqrt [6]{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {5}{6}}}{b x^{6} + a x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{6}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{6}} x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{6}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^{1/6}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.12, size = 32, normalized size = 0.05 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{6} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 \sqrt [6]{a} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________