Optimal. Leaf size=652 \[ \frac {3}{a x^3 \sqrt [6]{a+b x^2}}-\frac {40 b^2 x}{9 a^3 \sqrt [6]{a+b x^2}}-\frac {10 \left (a+b x^2\right )^{5/6}}{3 a^2 x^3}+\frac {40 b \left (a+b x^2\right )^{5/6}}{9 a^3 x}-\frac {40 b^2 x}{9 a^2 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )}-\frac {20 \sqrt {2+\sqrt {3}} b \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{3\ 3^{3/4} a^2 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}}+\frac {40 \sqrt {2} b \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} a^2 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}}} \]
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Rubi [A]
time = 0.47, 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 = {296, 331, 244,
204, 241, 310, 225, 1893} \begin {gather*} -\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 \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 (\text {ArcSin}\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 (\text {ArcSin}\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 {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 {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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 225
Rule 241
Rule 244
Rule 296
Rule 310
Rule 331
Rule 1893
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.01, size = 54, normalized size = 0.08 \begin {gather*} -\frac {\sqrt [6]{1+\frac {b x^2}{a}} \, _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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (b \,x^{2}+a \right )^{\frac {7}{6}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.71, size = 32, normalized size = 0.05 \begin {gather*} - \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}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^{7/6}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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