Optimal. Leaf size=652 \[ \frac {40 \sqrt {2} b \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{a+b x^2}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1}\right ),4 \sqrt {3}-7\right )}{9 \sqrt [4]{3} a^2 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}+\frac {40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}-\frac {40 b^2 x}{9 a^2 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )}-\frac {20 \sqrt {2+\sqrt {3}} b \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} a^2 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac {3}{a x^3 \sqrt [6]{a+b x^2}} \]
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Rubi [A] time = 0.65, antiderivative size = 652, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac {40 b^2 x}{9 a^2 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )}+\frac {40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}-\frac {10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac {40 \sqrt {2} b \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} a^2 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {20 \sqrt {2+\sqrt {3}} b \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} a^2 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}+\frac {3}{a x^3 \sqrt [6]{a+b x^2}} \]
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^4 \left (a+b x^2\right )^{7/6}} \, dx &=\frac {3}{a x^3 \sqrt [6]{a+b x^2}}+\frac {10 \int \frac {1}{x^4 \sqrt [6]{a+b x^2}} \, dx}{a}\\ &=\frac {3}{a x^3 \sqrt [6]{a+b x^2}}-\frac {10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}-\frac {(40 b) \int \frac {1}{x^2 \sqrt [6]{a+b x^2}} \, dx}{9 a^2}\\ &=\frac {3}{a x^3 \sqrt [6]{a+b x^2}}-\frac {10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac {40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}-\frac {\left (80 b^2\right ) \int \frac {1}{\sqrt [6]{a+b x^2}} \, dx}{27 a^3}\\ &=\frac {3}{a x^3 \sqrt [6]{a+b x^2}}-\frac {40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac {10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac {40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}+\frac {\left (40 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{7/6}} \, dx}{27 a^2}\\ &=\frac {3}{a x^3 \sqrt [6]{a+b x^2}}-\frac {40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac {10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac {40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}+\frac {\left (40 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^2 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\\ &=\frac {3}{a x^3 \sqrt [6]{a+b x^2}}-\frac {40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac {10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac {40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}-\frac {\left (20 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^2 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=\frac {3}{a x^3 \sqrt [6]{a+b x^2}}-\frac {40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac {10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac {40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}+\frac {\left (20 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^2 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}-\frac {\left (20 \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^2 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=\frac {3}{a x^3 \sqrt [6]{a+b x^2}}-\frac {40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac {10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac {40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}+\frac {40 b \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt {-1+\frac {a}{a+b x^2}}}{9 a^2 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 {20 \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^2 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 {40 \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^2 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 = 54, normalized size = 0.08 \[ -\frac {\sqrt [6]{\frac {b x^2}{a}+1} \, _2F_1\left (-\frac {3}{2},\frac {7}{6};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 a x^3 \sqrt [6]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {5}{6}}}{b^{2} x^{8} + 2 \, a b x^{6} + a^{2} x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{6}} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {7}{6}} x^{4}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {7}{6}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^{7/6}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.45, size = 32, normalized size = 0.05 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {7}{6} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac {7}{6}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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