Optimal. Leaf size=158 \[ \frac {3}{b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}-\frac {5 \sqrt {b \sqrt [3]{x}+a x}}{b^2 x^{2/3}}-\frac {5 a^{3/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{9/4} \sqrt {b \sqrt [3]{x}+a x}} \]
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Rubi [A]
time = 0.14, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2043, 2048,
2050, 2036, 335, 226} \begin {gather*} -\frac {5 a^{3/4} \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 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 b^{9/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {5 \sqrt {a x+b \sqrt [3]{x}}}{b^2 x^{2/3}}+\frac {3}{b \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 335
Rule 2036
Rule 2043
Rule 2048
Rule 2050
Rubi steps
\begin {align*} \int \frac {1}{x \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \text {Subst}\left (\int \frac {1}{x \left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}+\frac {15 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac {3}{b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}-\frac {5 \sqrt {b \sqrt [3]{x}+a x}}{b^2 x^{2/3}}-\frac {(5 a) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b^2}\\ &=\frac {3}{b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}-\frac {5 \sqrt {b \sqrt [3]{x}+a x}}{b^2 x^{2/3}}-\frac {\left (5 a \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{2 b^2 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}-\frac {5 \sqrt {b \sqrt [3]{x}+a x}}{b^2 x^{2/3}}-\frac {\left (5 a \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{b^2 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}-\frac {5 \sqrt {b \sqrt [3]{x}+a x}}{b^2 x^{2/3}}-\frac {5 a^{3/4} \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 )}{2 b^{9/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.05, size = 62, normalized size = 0.39 \begin {gather*} -\frac {2 \sqrt {1+\frac {a x^{2/3}}{b}} \, _2F_1\left (-\frac {3}{4},\frac {3}{2};\frac {1}{4};-\frac {a x^{2/3}}{b}\right )}{b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 180, normalized size = 1.14
method | result | size |
derivativedivides | \(-\frac {3 x^{\frac {1}{3}} a}{b^{2} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}-\frac {2 \sqrt {b \,x^{\frac {1}{3}}+a x}}{b^{2} x^{\frac {2}{3}}}-\frac {5 \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 b^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(162\) |
default | \(-\frac {5 x^{\frac {2}{3}} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \sqrt {-a b}+6 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x a +4 x^{\frac {1}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, b +4 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a x}{2 b^{2} x \left (b +a \,x^{\frac {2}{3}}\right )}\) | \(180\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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