Optimal. Leaf size=388 \[ -\frac {77 a^{13/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{65 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {154 a^{13/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{65 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {154 a^{7/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{65 b^4 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {154 a^3 \sqrt {a x+b \sqrt [3]{x}}}{65 b^4 \sqrt [3]{x}}-\frac {154 a^2 \sqrt {a x+b \sqrt [3]{x}}}{195 b^3 x}+\frac {22 a \sqrt {a x+b \sqrt [3]{x}}}{39 b^2 x^{5/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}} \]
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Rubi [A] time = 0.48, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2018, 2025, 2032, 329, 305, 220, 1196} \[ -\frac {154 a^{7/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{65 b^4 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}-\frac {77 a^{13/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{65 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {154 a^{13/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{65 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {154 a^3 \sqrt {a x+b \sqrt [3]{x}}}{65 b^4 \sqrt [3]{x}}-\frac {154 a^2 \sqrt {a x+b \sqrt [3]{x}}}{195 b^3 x}+\frac {22 a \sqrt {a x+b \sqrt [3]{x}}}{39 b^2 x^{5/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2018
Rule 2025
Rule 2032
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {b \sqrt [3]{x}+a x}} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x^7 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{13 b x^{7/3}}-\frac {(33 a) \operatorname {Subst}\left (\int \frac {1}{x^5 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{13 b}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac {22 a \sqrt {b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}+\frac {\left (77 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{39 b^2}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac {22 a \sqrt {b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac {154 a^2 \sqrt {b \sqrt [3]{x}+a x}}{195 b^3 x}-\frac {\left (77 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{65 b^3}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac {22 a \sqrt {b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac {154 a^2 \sqrt {b \sqrt [3]{x}+a x}}{195 b^3 x}+\frac {154 a^3 \sqrt {b \sqrt [3]{x}+a x}}{65 b^4 \sqrt [3]{x}}-\frac {\left (77 a^4\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{65 b^4}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac {22 a \sqrt {b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac {154 a^2 \sqrt {b \sqrt [3]{x}+a x}}{195 b^3 x}+\frac {154 a^3 \sqrt {b \sqrt [3]{x}+a x}}{65 b^4 \sqrt [3]{x}}-\frac {\left (77 a^4 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{65 b^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac {22 a \sqrt {b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac {154 a^2 \sqrt {b \sqrt [3]{x}+a x}}{195 b^3 x}+\frac {154 a^3 \sqrt {b \sqrt [3]{x}+a x}}{65 b^4 \sqrt [3]{x}}-\frac {\left (154 a^4 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{65 b^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac {22 a \sqrt {b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac {154 a^2 \sqrt {b \sqrt [3]{x}+a x}}{195 b^3 x}+\frac {154 a^3 \sqrt {b \sqrt [3]{x}+a x}}{65 b^4 \sqrt [3]{x}}-\frac {\left (154 a^{7/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{65 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (154 a^{7/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {a} x^2}{\sqrt {b}}}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{65 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {154 a^{7/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{65 b^4 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{13 b x^{7/3}}+\frac {22 a \sqrt {b \sqrt [3]{x}+a x}}{39 b^2 x^{5/3}}-\frac {154 a^2 \sqrt {b \sqrt [3]{x}+a x}}{195 b^3 x}+\frac {154 a^3 \sqrt {b \sqrt [3]{x}+a x}}{65 b^4 \sqrt [3]{x}}+\frac {154 a^{13/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{65 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {77 a^{13/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{65 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 59, normalized size = 0.15 \[ -\frac {6 \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (-\frac {13}{4},\frac {1}{2};-\frac {9}{4};-\frac {a x^{2/3}}{b}\right )}{13 x^2 \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 7.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} - a b x^{\frac {4}{3}} + b^{2} x^{\frac {2}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{3} x^{6} + b^{3} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 365, normalized size = 0.94 \[ -\frac {462 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{3} b \,x^{\frac {10}{3}} \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-231 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{3} b \,x^{\frac {10}{3}} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-462 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{4} x^{4}-462 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{3} b \,x^{\frac {10}{3}}+154 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{3} b \,x^{\frac {10}{3}}+44 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b^{2} x^{\frac {8}{3}}-20 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a \,b^{3} x^{2}+90 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{4} x^{\frac {4}{3}}}{195 \left (a \,x^{\frac {2}{3}}+b \right ) b^{4} x^{\frac {11}{3}}} \]
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 \[ \int \frac {1}{\sqrt {a x + b x^{\frac {1}{3}}} x^{3}}\,{d x} \]
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 \[ \int \frac {1}{x^3\,\sqrt {a\,x+b\,x^{1/3}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \sqrt {a x + b \sqrt [3]{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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