Optimal. Leaf size=413 \[ -\frac {154 a^{17/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {308 a^{17/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {308 a^{9/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 b^4 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {308 a^4 \sqrt {a x+b \sqrt [3]{x}}}{1105 b^4 \sqrt [3]{x}}-\frac {308 a^3 \sqrt {a x+b \sqrt [3]{x}}}{3315 b^3 x}+\frac {44 a^2 \sqrt {a x+b \sqrt [3]{x}}}{663 b^2 x^{5/3}}-\frac {12 a \sqrt {a x+b \sqrt [3]{x}}}{221 b x^{7/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{17 x^3} \]
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Rubi [A] time = 0.54, antiderivative size = 413, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2018, 2020, 2025, 2032, 329, 305, 220, 1196} \[ -\frac {308 a^{9/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 b^4 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {44 a^2 \sqrt {a x+b \sqrt [3]{x}}}{663 b^2 x^{5/3}}-\frac {154 a^{17/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {308 a^{17/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {308 a^4 \sqrt {a x+b \sqrt [3]{x}}}{1105 b^4 \sqrt [3]{x}}-\frac {308 a^3 \sqrt {a x+b \sqrt [3]{x}}}{3315 b^3 x}-\frac {12 a \sqrt {a x+b \sqrt [3]{x}}}{221 b x^{7/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{17 x^3} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2018
Rule 2020
Rule 2025
Rule 2032
Rubi steps
\begin {align*} \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^4} \, dx &=3 \operatorname {Subst}\left (\int \frac {\sqrt {b x+a x^3}}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}+\frac {1}{17} (6 a) \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}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}-\frac {\left (66 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^5 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{221 b}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}+\frac {\left (154 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{663 b^2}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}-\frac {\left (154 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1105 b^3}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {308 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {\left (154 a^5\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1105 b^4}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {308 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {\left (154 a^5 \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 )}{1105 b^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {308 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {\left (308 a^5 \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 )}{1105 b^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {308 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}-\frac {\left (308 a^{9/2} \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 )}{1105 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (308 a^{9/2} \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 )}{1105 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {308 a^{9/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{1105 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}}{17 x^3}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{221 b x^{7/3}}+\frac {44 a^2 \sqrt {b \sqrt [3]{x}+a x}}{663 b^2 x^{5/3}}-\frac {308 a^3 \sqrt {b \sqrt [3]{x}+a x}}{3315 b^3 x}+\frac {308 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1105 b^4 \sqrt [3]{x}}+\frac {308 a^{17/4} \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 )}{1105 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {154 a^{17/4} \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 )}{1105 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 59, normalized size = 0.14 \[ -\frac {6 \sqrt {a x+b \sqrt [3]{x}} \, _2F_1\left (-\frac {17}{4},-\frac {1}{2};-\frac {13}{4};-\frac {a x^{2/3}}{b}\right )}{17 x^3 \sqrt {\frac {a x^{2/3}}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 8.29, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x^{4}}, x\right ) \]
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 = 0.08, size = 281, normalized size = 0.68 \[ \frac {308 \left (a \,x^{\frac {2}{3}}+b \right ) a^{4}}{1105 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{4}}-\frac {154 \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right ) a^{4}}{1105 \sqrt {a x +b \,x^{\frac {1}{3}}}\, b^{4}}-\frac {308 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{3}}{3315 b^{3} x}+\frac {44 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a^{2}}{663 b^{2} x^{\frac {5}{3}}}-\frac {12 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a}{221 b \,x^{\frac {7}{3}}}-\frac {6 \sqrt {a x +b \,x^{\frac {1}{3}}}}{17 x^{3}} \]
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 \[ \int \frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x^{4}}\,{d x} \]
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 \[ \int \frac {\sqrt {a\,x+b\,x^{1/3}}}{x^4} \,d x \]
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 \[ \int \frac {\sqrt {a x + b \sqrt [3]{x}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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