Optimal. Leaf size=659 \[ \frac {81\ 3^{3/4} a^4 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{a+b x^2}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1}\right ),4 \sqrt {3}-7\right )}{448 \sqrt {2} b^4 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {243 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^4 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{1792 b^4 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {243 a^4 x}{896 b^3 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )}-\frac {243 a^3 x}{896 b^3 \sqrt [6]{a+b x^2}}+\frac {81 a^2 x \left (a+b x^2\right )^{5/6}}{448 b^3}-\frac {9 a x^3 \left (a+b x^2\right )^{5/6}}{56 b^2}+\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b} \]
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Rubi [A] time = 0.69, antiderivative size = 659, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {321, 238, 198, 235, 304, 219, 1879} \[ \frac {81 a^2 x \left (a+b x^2\right )^{5/6}}{448 b^3}-\frac {243 a^3 x}{896 b^3 \sqrt [6]{a+b x^2}}-\frac {243 a^4 x}{896 b^3 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )}+\frac {81\ 3^{3/4} a^4 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{448 \sqrt {2} b^4 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {243 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^4 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{1792 b^4 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {9 a x^3 \left (a+b x^2\right )^{5/6}}{56 b^2}+\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b} \]
Antiderivative was successfully verified.
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Rule 198
Rule 219
Rule 235
Rule 238
Rule 304
Rule 321
Rule 1879
Rubi steps
\begin {align*} \int \frac {x^6}{\sqrt [6]{a+b x^2}} \, dx &=\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}-\frac {(3 a) \int \frac {x^4}{\sqrt [6]{a+b x^2}} \, dx}{4 b}\\ &=-\frac {9 a x^3 \left (a+b x^2\right )^{5/6}}{56 b^2}+\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}+\frac {\left (27 a^2\right ) \int \frac {x^2}{\sqrt [6]{a+b x^2}} \, dx}{56 b^2}\\ &=\frac {81 a^2 x \left (a+b x^2\right )^{5/6}}{448 b^3}-\frac {9 a x^3 \left (a+b x^2\right )^{5/6}}{56 b^2}+\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}-\frac {\left (81 a^3\right ) \int \frac {1}{\sqrt [6]{a+b x^2}} \, dx}{448 b^3}\\ &=-\frac {243 a^3 x}{896 b^3 \sqrt [6]{a+b x^2}}+\frac {81 a^2 x \left (a+b x^2\right )^{5/6}}{448 b^3}-\frac {9 a x^3 \left (a+b x^2\right )^{5/6}}{56 b^2}+\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}+\frac {\left (81 a^4\right ) \int \frac {1}{\left (a+b x^2\right )^{7/6}} \, dx}{896 b^3}\\ &=-\frac {243 a^3 x}{896 b^3 \sqrt [6]{a+b x^2}}+\frac {81 a^2 x \left (a+b x^2\right )^{5/6}}{448 b^3}-\frac {9 a x^3 \left (a+b x^2\right )^{5/6}}{56 b^2}+\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}+\frac {\left (81 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-b x^2}} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{896 b^3 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\\ &=-\frac {243 a^3 x}{896 b^3 \sqrt [6]{a+b x^2}}+\frac {81 a^2 x \left (a+b x^2\right )^{5/6}}{448 b^3}-\frac {9 a x^3 \left (a+b x^2\right )^{5/6}}{56 b^2}+\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}-\frac {\left (243 a^4 \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 )}{1792 b^4 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=-\frac {243 a^3 x}{896 b^3 \sqrt [6]{a+b x^2}}+\frac {81 a^2 x \left (a+b x^2\right )^{5/6}}{448 b^3}-\frac {9 a x^3 \left (a+b x^2\right )^{5/6}}{56 b^2}+\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}+\frac {\left (243 a^4 \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 )}{1792 b^4 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}-\frac {\left (243 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )} a^4 \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 )}{896 b^4 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=-\frac {243 a^3 x}{896 b^3 \sqrt [6]{a+b x^2}}+\frac {81 a^2 x \left (a+b x^2\right )^{5/6}}{448 b^3}-\frac {9 a x^3 \left (a+b x^2\right )^{5/6}}{56 b^2}+\frac {3 x^5 \left (a+b x^2\right )^{5/6}}{20 b}+\frac {243 a^4 \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt {-1+\frac {a}{a+b x^2}}}{896 b^4 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 {243 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^4 \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 )}{1792 b^4 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 {81\ 3^{3/4} a^4 \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 )}{448 \sqrt {2} b^4 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 = 0.03, size = 90, normalized size = 0.14 \[ \frac {3 \left (-135 a^3 x \sqrt [6]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )+135 a^3 x+15 a^2 b x^3-8 a b^2 x^5+112 b^3 x^7\right )}{2240 b^3 \sqrt [6]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {1}{6}}}, x\right ) \]
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 \[ \int \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {1}{6}}}\,{d x} \]
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 \[ \int \frac {x^{6}}{\left (b \,x^{2}+a \right )^{\frac {1}{6}}}\, dx \]
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 \[ \int \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {1}{6}}}\,{d x} \]
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 \[ \int \frac {x^6}{{\left (b\,x^2+a\right )}^{1/6}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.11, size = 27, normalized size = 0.04 \[ \frac {x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{6}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7 \sqrt [6]{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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