Optimal. Leaf size=276 \[ -\frac {1326 a^{23/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{33649 b^{21/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {2652 a^5 \sqrt {a x+b \sqrt [3]{x}}}{33649 b^5 x^{2/3}}+\frac {7956 a^4 \sqrt {a x+b \sqrt [3]{x}}}{168245 b^4 x^{4/3}}-\frac {884 a^3 \sqrt {a x+b \sqrt [3]{x}}}{24035 b^3 x^2}+\frac {68 a^2 \sqrt {a x+b \sqrt [3]{x}}}{2185 b^2 x^{8/3}}-\frac {12 a \sqrt {a x+b \sqrt [3]{x}}}{437 b x^{10/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{23 x^4} \]
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Rubi [A] time = 0.41, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2018, 2020, 2025, 2011, 329, 220} \[ -\frac {2652 a^5 \sqrt {a x+b \sqrt [3]{x}}}{33649 b^5 x^{2/3}}+\frac {7956 a^4 \sqrt {a x+b \sqrt [3]{x}}}{168245 b^4 x^{4/3}}-\frac {884 a^3 \sqrt {a x+b \sqrt [3]{x}}}{24035 b^3 x^2}+\frac {68 a^2 \sqrt {a x+b \sqrt [3]{x}}}{2185 b^2 x^{8/3}}-\frac {1326 a^{23/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{33649 b^{21/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {12 a \sqrt {a x+b \sqrt [3]{x}}}{437 b x^{10/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{23 x^4} \]
Antiderivative was successfully verified.
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Rule 220
Rule 329
Rule 2011
Rule 2018
Rule 2020
Rule 2025
Rubi steps
\begin {align*} \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^5} \, dx &=3 \operatorname {Subst}\left (\int \frac {\sqrt {b x+a x^3}}{x^{13}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{23 x^4}+\frac {1}{23} (6 a) \operatorname {Subst}\left (\int \frac {1}{x^{10} \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{23 x^4}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{437 b x^{10/3}}-\frac {\left (102 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^8 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{437 b}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{23 x^4}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac {68 a^2 \sqrt {b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}+\frac {\left (442 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^6 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2185 b^2}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{23 x^4}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac {68 a^2 \sqrt {b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac {884 a^3 \sqrt {b \sqrt [3]{x}+a x}}{24035 b^3 x^2}-\frac {\left (3978 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{24035 b^3}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{23 x^4}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac {68 a^2 \sqrt {b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac {884 a^3 \sqrt {b \sqrt [3]{x}+a x}}{24035 b^3 x^2}+\frac {7956 a^4 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}+\frac {\left (3978 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{33649 b^4}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{23 x^4}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac {68 a^2 \sqrt {b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac {884 a^3 \sqrt {b \sqrt [3]{x}+a x}}{24035 b^3 x^2}+\frac {7956 a^4 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {2652 a^5 \sqrt {b \sqrt [3]{x}+a x}}{33649 b^5 x^{2/3}}-\frac {\left (1326 a^6\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{33649 b^5}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{23 x^4}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac {68 a^2 \sqrt {b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac {884 a^3 \sqrt {b \sqrt [3]{x}+a x}}{24035 b^3 x^2}+\frac {7956 a^4 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {2652 a^5 \sqrt {b \sqrt [3]{x}+a x}}{33649 b^5 x^{2/3}}-\frac {\left (1326 a^6 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{33649 b^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{23 x^4}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac {68 a^2 \sqrt {b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac {884 a^3 \sqrt {b \sqrt [3]{x}+a x}}{24035 b^3 x^2}+\frac {7956 a^4 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {2652 a^5 \sqrt {b \sqrt [3]{x}+a x}}{33649 b^5 x^{2/3}}-\frac {\left (2652 a^6 \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 )}{33649 b^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{23 x^4}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac {68 a^2 \sqrt {b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac {884 a^3 \sqrt {b \sqrt [3]{x}+a x}}{24035 b^3 x^2}+\frac {7956 a^4 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {2652 a^5 \sqrt {b \sqrt [3]{x}+a x}}{33649 b^5 x^{2/3}}-\frac {1326 a^{23/4} \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 )}{33649 b^{21/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 59, normalized size = 0.21 \[ -\frac {6 \sqrt {a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac {23}{4},-\frac {1}{2};-\frac {19}{4};-\frac {a x^{2/3}}{b}\right )}{23 x^4 \sqrt {\frac {a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x^{5}}, x\right ) \]
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 = 0.08, size = 245, normalized size = 0.89 \[ -\frac {1326 \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, a^{5} \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{33649 \sqrt {a x +b \,x^{\frac {1}{3}}}\, b^{5}}-\frac {2652 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{5}}{33649 b^{5} x^{\frac {2}{3}}}+\frac {7956 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{4}}{168245 b^{4} x^{\frac {4}{3}}}-\frac {884 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{3}}{24035 b^{3} x^{2}}+\frac {68 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{2}}{2185 b^{2} x^{\frac {8}{3}}}-\frac {12 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a}{437 b \,x^{\frac {10}{3}}}-\frac {6 \sqrt {a x +b \,x^{\frac {1}{3}}}}{23 x^{4}} \]
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 \[ \int \frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x^{5}}\,{d x} \]
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 \[ \int \frac {\sqrt {a\,x+b\,x^{1/3}}}{x^5} \,d x \]
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 \[ \int \frac {\sqrt {a x + b \sqrt [3]{x}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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