Optimal. Leaf size=383 \[ \frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}+\frac {77 a^{5/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 b^4 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}-\frac {77 a^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {77 a^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{10 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}} \]
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Rubi [A]
time = 0.34, antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2043, 2048,
2050, 2057, 335, 311, 226, 1210} \begin {gather*} \frac {77 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 )}{10 b^{15/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {77 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^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {77 a^{5/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 b^4 \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}-\frac {77 a^2 \sqrt {a x+b \sqrt [3]{x}}}{5 b^4 \sqrt [3]{x}}+\frac {77 a \sqrt {a x+b \sqrt [3]{x}}}{15 b^3 x}-\frac {11 \sqrt {a x+b \sqrt [3]{x}}}{3 b^2 x^{5/3}}+\frac {3}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2043
Rule 2048
Rule 2050
Rule 2057
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \text {Subst}\left (\int \frac {1}{x^4 \left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}+\frac {33 \text {Subst}\left (\int \frac {1}{x^5 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}-\frac {(77 a) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{6 b^2}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}+\frac {\left (77 a^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 b^3}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}+\frac {\left (77 a^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 b^4}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}+\frac {\left (77 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 )}{10 b^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}+\frac {\left (77 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^4 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}+\frac {\left (77 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 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}}-\frac {\left (77 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 b^{7/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}+\frac {77 a^{5/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 b^4 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}-\frac {77 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^{15/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {77 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 )}{10 b^{15/4} \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 = 64, normalized size = 0.17 \begin {gather*} -\frac {2 \sqrt {1+\frac {a x^{2/3}}{b}} \, _2F_1\left (-\frac {9}{4},\frac {3}{2};-\frac {5}{4};-\frac {a x^{2/3}}{b}\right )}{3 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 339, normalized size = 0.89
method | result | size |
derivativedivides | \(-\frac {2 \sqrt {b \,x^{\frac {1}{3}}+a x}}{3 b^{2} x^{\frac {5}{3}}}+\frac {32 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{15 b^{3} x}-\frac {62 \left (b +a \,x^{\frac {2}{3}}\right ) a^{2}}{5 b^{4} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}-\frac {3 x^{\frac {2}{3}} a^{3}}{b^{4} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}+\frac {77 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{10 b^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(267\) |
default | \(-\frac {-462 a^{2} b \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+231 a^{2} b \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+462 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {10}{3}} a^{3}+372 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {8}{3}} a^{2} b -44 x^{2} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a \,b^{2}-64 x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} b +20 x^{\frac {4}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, b^{3}}{30 x^{3} \left (b +a \,x^{\frac {2}{3}}\right ) b^{4}}\) | \(339\) |
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 {1}{x^{2} \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 = -0.00 \begin {gather*} \int \frac {1}{x^2\,{\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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