Optimal. Leaf size=144 \[ 4 a \sqrt {b \sqrt [3]{x}+a x}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+\frac {4 a^{3/4} b^{3/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {b \sqrt [3]{x}+a x}} \]
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Rubi [A]
time = 0.14, antiderivative size = 144, 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, 2045,
2046, 2036, 335, 226} \begin {gather*} \frac {4 a^{3/4} b^{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 )}{\sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{x}+4 a \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 2045
Rule 2046
Rubi steps
\begin {align*} \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^2} \, dx &=3 \text {Subst}\left (\int \frac {\left (b x+a x^3\right )^{3/2}}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+(6 a) \text {Subst}\left (\int \frac {\sqrt {b x+a x^3}}{x} \, dx,x,\sqrt [3]{x}\right )\\ &=4 a \sqrt {b \sqrt [3]{x}+a x}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+(4 a b) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=4 a \sqrt {b \sqrt [3]{x}+a x}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+\frac {\left (4 a b \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 )}{\sqrt {b \sqrt [3]{x}+a x}}\\ &=4 a \sqrt {b \sqrt [3]{x}+a x}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+\frac {\left (8 a b \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 )}{\sqrt {b \sqrt [3]{x}+a x}}\\ &=4 a \sqrt {b \sqrt [3]{x}+a x}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{x}+\frac {4 a^{3/4} b^{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 )}{\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 = 60, normalized size = 0.42 \begin {gather*} -\frac {2 b \sqrt {b \sqrt [3]{x}+a x} \, _2F_1\left (-\frac {3}{2},-\frac {3}{4};\frac {1}{4};-\frac {a x^{2/3}}{b}\right )}{\sqrt {1+\frac {a x^{2/3}}{b}} x^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 130, normalized size = 0.90
method | result | size |
default | \(\frac {4 x^{\frac {1}{3}} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, b +2 x^{\frac {4}{3}} a^{2}-2 b^{2}}{x^{\frac {1}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\) | \(130\) |
derivativedivides | \(-\frac {2 b \sqrt {b \,x^{\frac {1}{3}}+a x}}{x^{\frac {2}{3}}}+2 a \sqrt {b \,x^{\frac {1}{3}}+a x}+\frac {4 b \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(147\) |
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 {\left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}{x^{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 {{\left (a\,x+b\,x^{1/3}\right )}^{3/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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