Optimal. Leaf size=566 \[ -\frac {55 b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{56 \sqrt {2} \sqrt [4]{3} a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {55 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{224 a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {55 b^{7/3} \sqrt {a+\frac {b}{x^3}}}{112 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}+\frac {55 b^2 x \sqrt {a+\frac {b}{x^3}}}{112 a^3}-\frac {11 b x^4 \sqrt {a+\frac {b}{x^3}}}{56 a^2}+\frac {x^7 \sqrt {a+\frac {b}{x^3}}}{7 a} \]
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Rubi [A] time = 0.34, antiderivative size = 566, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {335, 325, 303, 218, 1877} \[ \frac {55 b^2 x \sqrt {a+\frac {b}{x^3}}}{112 a^3}-\frac {55 b^{7/3} \sqrt {a+\frac {b}{x^3}}}{112 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}-\frac {55 b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{56 \sqrt {2} \sqrt [4]{3} a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {55 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{224 a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {11 b x^4 \sqrt {a+\frac {b}{x^3}}}{56 a^2}+\frac {x^7 \sqrt {a+\frac {b}{x^3}}}{7 a} \]
Antiderivative was successfully verified.
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Rule 218
Rule 303
Rule 325
Rule 335
Rule 1877
Rubi steps
\begin {align*} \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^8 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a+\frac {b}{x^3}} x^7}{7 a}+\frac {(11 b) \operatorname {Subst}\left (\int \frac {1}{x^5 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{14 a}\\ &=-\frac {11 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^7}{7 a}-\frac {\left (55 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{112 a^2}\\ &=\frac {55 b^2 \sqrt {a+\frac {b}{x^3}} x}{112 a^3}-\frac {11 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^7}{7 a}-\frac {\left (55 b^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{224 a^3}\\ &=\frac {55 b^2 \sqrt {a+\frac {b}{x^3}} x}{112 a^3}-\frac {11 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^7}{7 a}-\frac {\left (55 b^{8/3}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{224 a^3}-\frac {\left (55 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} b^{8/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{112 a^{8/3}}\\ &=-\frac {55 b^{7/3} \sqrt {a+\frac {b}{x^3}}}{112 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}+\frac {55 b^2 \sqrt {a+\frac {b}{x^3}} x}{112 a^3}-\frac {11 b \sqrt {a+\frac {b}{x^3}} x^4}{56 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^7}{7 a}+\frac {55 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{224 a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}-\frac {55 b^{7/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{56 \sqrt {2} \sqrt [4]{3} a^{8/3} \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 79, normalized size = 0.14 \[ \frac {8 a^2 x^7+11 b^2 x \sqrt {\frac {a x^3}{b}+1} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};-\frac {a x^3}{b}\right )-3 a b x^4-11 b^2 x}{56 a^2 \sqrt {a+\frac {b}{x^3}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{9} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{a x^{3} + b}, x\right ) \]
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 \[ \int \frac {x^{6}}{\sqrt {a + \frac {b}{x^{3}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 2806, normalized size = 4.96 \[ \text {output too large to display} \]
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 \[ \int \frac {x^{6}}{\sqrt {a + \frac {b}{x^{3}}}}\,{d x} \]
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 \[ \int \frac {x^6}{\sqrt {a+\frac {b}{x^3}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.31, size = 46, normalized size = 0.08 \[ - \frac {x^{7} \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt {a} \Gamma \left (- \frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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