Optimal. Leaf size=239 \[ -\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {663 b^{15/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{154 a^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \]
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Rubi [A]
time = 0.25, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2043, 2047,
2049, 2036, 335, 226} \begin {gather*} \frac {663 b^{15/4} \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 )}{154 a^{21/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {663 b^3 \sqrt {a x+b \sqrt [3]{x}}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{55 a^3}+\frac {17 x^2 \sqrt {a x+b \sqrt [3]{x}}}{5 a^2}-\frac {3 x^3}{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 2047
Rule 2049
Rubi steps
\begin {align*} \int \frac {x^3}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \text {Subst}\left (\int \frac {x^{11}}{\left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {51 \text {Subst}\left (\int \frac {x^8}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 a}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}-\frac {(221 b) \text {Subst}\left (\int \frac {x^6}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 a^2}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (1989 b^2\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{110 a^3}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}-\frac {\left (1989 b^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 a^4}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (663 b^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 a^5}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (663 b^4 \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 )}{154 a^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {\left (663 b^4 \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 )}{77 a^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {3 x^3}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {663 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^5}+\frac {1989 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^4}-\frac {221 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^3}+\frac {17 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a^2}+\frac {663 b^{15/4} \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 )}{154 a^{21/4} \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 = 10.08, size = 124, normalized size = 0.52 \begin {gather*} \frac {\sqrt {b \sqrt [3]{x}+a x} \left (-3315 b^4-1326 a b^3 x^{2/3}+442 a^2 b^2 x^{4/3}-238 a^3 b x^2+154 a^4 x^{8/3}+3315 b^4 \sqrt {1+\frac {a x^{2/3}}{b}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {a x^{2/3}}{b}\right )\right )}{385 a^5 \left (b+a x^{2/3}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 260, normalized size = 1.09
method | result | size |
derivativedivides | \(-\frac {3 x^{\frac {1}{3}} b^{4}}{a^{5} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}+\frac {2 x^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{5 a^{2}}-\frac {56 b \,x^{\frac {4}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{55 a^{3}}+\frac {834 b^{2} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{385 a^{4}}-\frac {432 b^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{77 a^{5}}+\frac {663 b^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{154 a^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(228\) |
default | \(-\frac {-884 x^{\frac {5}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{3} b^{2}+476 x^{\frac {7}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{4} b -3315 \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, b^{4}+2652 x \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} b^{3}-308 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{5} x^{3}+2310 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {1}{3}} a \,b^{4}+4320 x^{\frac {1}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a \,b^{4}}{770 x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right ) a^{6}}\) | \(260\) |
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 {x^{3}}{\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 {x^3}{{\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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