Optimal. Leaf size=614 \[ -\frac {4 \sqrt {2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{a+b x^2}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1}\right ),4 \sqrt {3}-7\right )}{\sqrt [4]{3} a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}+\frac {4 b x}{a \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )}+\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {2 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}} \]
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Rubi [A] time = 0.57, antiderivative size = 614, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {290, 325, 238, 198, 235, 304, 219, 1879} \[ \frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}+\frac {4 b x}{a \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )}+\frac {3}{a x \sqrt [6]{a+b x^2}}-\frac {4 \sqrt {2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}+\frac {2 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 198
Rule 219
Rule 235
Rule 238
Rule 290
Rule 304
Rule 325
Rule 1879
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right )^{7/6}} \, dx &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 \int \frac {1}{x^2 \sqrt [6]{a+b x^2}} \, dx}{a}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}+\frac {(8 b) \int \frac {1}{\sqrt [6]{a+b x^2}} \, dx}{3 a^2}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}-\frac {(4 b) \int \frac {1}{\left (a+b x^2\right )^{7/6}} \, dx}{3 a}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-b x^2}} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{3 a \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}+\frac {\left (2 \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 )}{a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}-\frac {\left (2 \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 )}{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 )} \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 )}{a x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=\frac {3}{a x \sqrt [6]{a+b x^2}}+\frac {4 b x}{a^2 \sqrt [6]{a+b x^2}}-\frac {4 \left (a+b x^2\right )^{5/6}}{a^2 x}-\frac {4 \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt {-1+\frac {a}{a+b x^2}}}{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 [4]{3} \sqrt {2+\sqrt {3}} \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 )}{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} \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 )}{\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*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.08 \[ -\frac {\sqrt [6]{\frac {b x^2}{a}+1} \, _2F_1\left (-\frac {1}{2},\frac {7}{6};\frac {1}{2};-\frac {b x^2}{a}\right )}{a x \sqrt [6]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {5}{6}}}{b^{2} x^{6} + 2 \, a b x^{4} + a^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{6}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {7}{6}} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{6}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.12, size = 40, normalized size = 0.07 \[ -\frac {3\,{\left (\frac {a}{b\,x^2}+1\right )}^{7/6}\,{{}}_2{\mathrm {F}}_1\left (\frac {7}{6},\frac {5}{3};\ \frac {8}{3};\ -\frac {a}{b\,x^2}\right )}{10\,x\,{\left (b\,x^2+a\right )}^{7/6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 27, normalized size = 0.04 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{6} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac {7}{6}} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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