Optimal. Leaf size=350 \[ \frac {8 a^{5/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 b \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}-\frac {8 a^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {4 a^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {b \sqrt [3]{x}+a x}} \]
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Rubi [A]
time = 0.30, antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2043, 2045,
2050, 2057, 335, 311, 226, 1210} \begin {gather*} \frac {4 a^{9/4} \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 )}{5 b^{3/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {8 a^{9/4} \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 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {8 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}-\frac {8 a^2 \sqrt {a x+b \sqrt [3]{x}}}{5 b \sqrt [3]{x}}-\frac {2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{3 x^2}-\frac {4 a \sqrt {a x+b \sqrt [3]{x}}}{5 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2043
Rule 2045
Rule 2050
Rule 2057
Rubi steps
\begin {align*} \int \frac {\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^3} \, dx &=3 \text {Subst}\left (\int \frac {\left (b x+a x^3\right )^{3/2}}{x^7} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+(2 a) \text {Subst}\left (\int \frac {\sqrt {b x+a x^3}}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac {1}{5} \left (4 a^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac {\left (4 a^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{5 b}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac {\left (4 a^3 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{5 b \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac {\left (8 a^3 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{5 b \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}+\frac {\left (8 a^{5/2} \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 )}{5 \sqrt {b} \sqrt {b \sqrt [3]{x}+a x}}-\frac {\left (8 a^{5/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {a} x^2}{\sqrt {b}}}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{5 \sqrt {b} \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {8 a^{5/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 b \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {4 a \sqrt {b \sqrt [3]{x}+a x}}{5 x}-\frac {8 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b \sqrt [3]{x}}-\frac {2 \left (b \sqrt [3]{x}+a x\right )^{3/2}}{3 x^2}-\frac {8 a^{9/4} \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 )}{5 b^{3/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {4 a^{9/4} \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 )}{5 b^{3/4} \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 = 10.04, size = 62, normalized size = 0.18 \begin {gather*} -\frac {2 b \sqrt {b \sqrt [3]{x}+a x} \, _2F_1\left (-\frac {9}{4},-\frac {3}{2};-\frac {5}{4};-\frac {a x^{2/3}}{b}\right )}{3 \sqrt {1+\frac {a x^{2/3}}{b}} x^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 339, normalized size = 0.97
method | result | size |
derivativedivides | \(-\frac {2 b \sqrt {b \,x^{\frac {1}{3}}+a x}}{3 x^{\frac {5}{3}}}-\frac {22 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{15 x}-\frac {8 \left (b +a \,x^{\frac {2}{3}}\right ) a^{2}}{5 b \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}+\frac {4 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{5 b \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(235\) |
default | \(-\frac {2 \left (-12 a^{2} b \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+6 a^{2} b \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+12 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {10}{3}} a^{3}+12 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {8}{3}} a^{2} b +16 x^{2} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a \,b^{2}+11 x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} b +5 x^{\frac {4}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, b^{3}\right )}{15 b \,x^{3} \left (b +a \,x^{\frac {2}{3}}\right )}\) | \(339\) |
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^{3}}\, 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 = -0.00 \begin {gather*} \int \frac {{\left (a\,x+b\,x^{1/3}\right )}^{3/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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