Optimal. Leaf size=324 \[ -\frac {81\ 3^{3/4} \sqrt {2-\sqrt {3}} a^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{a+b x^2}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1}\right ),4 \sqrt {3}-7\right )}{128 b^4 x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}+\frac {81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac {9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b} \]
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Rubi [A] time = 0.29, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {321, 241, 236, 219} \[ \frac {81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac {81\ 3^{3/4} \sqrt {2-\sqrt {3}} a^3 \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{128 b^4 x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b} \]
Antiderivative was successfully verified.
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Rule 219
Rule 236
Rule 241
Rule 321
Rubi steps
\begin {align*} \int \frac {x^6}{\left (a+b x^2\right )^{5/6}} \, dx &=\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac {(15 a) \int \frac {x^4}{\left (a+b x^2\right )^{5/6}} \, dx}{16 b}\\ &=-\frac {9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}+\frac {\left (27 a^2\right ) \int \frac {x^2}{\left (a+b x^2\right )^{5/6}} \, dx}{32 b^2}\\ &=\frac {81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac {9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac {\left (81 a^3\right ) \int \frac {1}{\left (a+b x^2\right )^{5/6}} \, dx}{128 b^3}\\ &=\frac {81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac {9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac {\left (81 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-b x^2\right )^{2/3}} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 b^3 \sqrt [3]{\frac {a}{a+b x^2}} \sqrt [3]{a+b x^2}}\\ &=\frac {81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac {9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}+\frac {\left (243 a^3 \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{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 )}{256 b^4 x \sqrt [3]{\frac {a}{a+b x^2}}}\\ &=\frac {81 a^2 x \sqrt [6]{a+b x^2}}{128 b^3}-\frac {9 a x^3 \sqrt [6]{a+b x^2}}{32 b^2}+\frac {3 x^5 \sqrt [6]{a+b x^2}}{16 b}-\frac {81\ 3^{3/4} \sqrt {2-\sqrt {3}} a^3 \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{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 )}{128 b^4 x \sqrt [3]{\frac {a}{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 = 0.03, size = 89, normalized size = 0.27 \[ \frac {3 x \left (-27 a^3 \left (\frac {b x^2}{a}+1\right )^{5/6} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};-\frac {b x^2}{a}\right )+27 a^3+15 a^2 b x^2-4 a b^2 x^4+8 b^3 x^6\right )}{128 b^3 \left (a+b x^2\right )^{5/6}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {5}{6}}}, x\right ) \]
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 \[ \int \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {5}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\left (b \,x^{2}+a \right )^{\frac {5}{6}}}\, dx \]
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 \[ \int \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {5}{6}}}\,{d x} \]
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 \[ \int \frac {x^6}{{\left (b\,x^2+a\right )}^{5/6}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.99, size = 27, normalized size = 0.08 \[ \frac {x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{6}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7 a^{\frac {5}{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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