Optimal. Leaf size=75 \[ \frac {a^5 x^4}{4}+\frac {15}{13} a^4 b x^{13/3}+\frac {15}{7} a^3 b^2 x^{14/3}+2 a^2 b^3 x^5+\frac {15}{16} a b^4 x^{16/3}+\frac {3}{17} b^5 x^{17/3} \]
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Rubi [A] time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ 2 a^2 b^3 x^5+\frac {15}{7} a^3 b^2 x^{14/3}+\frac {15}{13} a^4 b x^{13/3}+\frac {a^5 x^4}{4}+\frac {15}{16} a b^4 x^{16/3}+\frac {3}{17} b^5 x^{17/3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \left (a+b \sqrt [3]{x}\right )^5 x^3 \, dx &=3 \operatorname {Subst}\left (\int x^{11} (a+b x)^5 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (a^5 x^{11}+5 a^4 b x^{12}+10 a^3 b^2 x^{13}+10 a^2 b^3 x^{14}+5 a b^4 x^{15}+b^5 x^{16}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {a^5 x^4}{4}+\frac {15}{13} a^4 b x^{13/3}+\frac {15}{7} a^3 b^2 x^{14/3}+2 a^2 b^3 x^5+\frac {15}{16} a b^4 x^{16/3}+\frac {3}{17} b^5 x^{17/3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 1.00 \[ \frac {a^5 x^4}{4}+\frac {15}{13} a^4 b x^{13/3}+\frac {15}{7} a^3 b^2 x^{14/3}+2 a^2 b^3 x^5+\frac {15}{16} a b^4 x^{16/3}+\frac {3}{17} b^5 x^{17/3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 69, normalized size = 0.92 \[ 2 \, a^{2} b^{3} x^{5} + \frac {1}{4} \, a^{5} x^{4} + \frac {3}{119} \, {\left (7 \, b^{5} x^{5} + 85 \, a^{3} b^{2} x^{4}\right )} x^{\frac {2}{3}} + \frac {15}{208} \, {\left (13 \, a b^{4} x^{5} + 16 \, a^{4} b x^{4}\right )} x^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 57, normalized size = 0.76 \[ \frac {3}{17} \, b^{5} x^{\frac {17}{3}} + \frac {15}{16} \, a b^{4} x^{\frac {16}{3}} + 2 \, a^{2} b^{3} x^{5} + \frac {15}{7} \, a^{3} b^{2} x^{\frac {14}{3}} + \frac {15}{13} \, a^{4} b x^{\frac {13}{3}} + \frac {1}{4} \, a^{5} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 58, normalized size = 0.77 \[ \frac {3 b^{5} x^{\frac {17}{3}}}{17}+\frac {15 a \,b^{4} x^{\frac {16}{3}}}{16}+2 a^{2} b^{3} x^{5}+\frac {15 a^{3} b^{2} x^{\frac {14}{3}}}{7}+\frac {15 a^{4} b \,x^{\frac {13}{3}}}{13}+\frac {a^{5} x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.89, size = 200, normalized size = 2.67 \[ \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{17}}{17 \, b^{12}} - \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )}^{16} a}{16 \, b^{12}} + \frac {11 \, {\left (b x^{\frac {1}{3}} + a\right )}^{15} a^{2}}{b^{12}} - \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )}^{14} a^{3}}{14 \, b^{12}} + \frac {990 \, {\left (b x^{\frac {1}{3}} + a\right )}^{13} a^{4}}{13 \, b^{12}} - \frac {231 \, {\left (b x^{\frac {1}{3}} + a\right )}^{12} a^{5}}{2 \, b^{12}} + \frac {126 \, {\left (b x^{\frac {1}{3}} + a\right )}^{11} a^{6}}{b^{12}} - \frac {99 \, {\left (b x^{\frac {1}{3}} + a\right )}^{10} a^{7}}{b^{12}} + \frac {55 \, {\left (b x^{\frac {1}{3}} + a\right )}^{9} a^{8}}{b^{12}} - \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{8} a^{9}}{8 \, b^{12}} + \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7} a^{10}}{7 \, b^{12}} - \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{6} a^{11}}{2 \, b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 57, normalized size = 0.76 \[ \frac {a^5\,x^4}{4}+\frac {3\,b^5\,x^{17/3}}{17}+\frac {15\,a^4\,b\,x^{13/3}}{13}+\frac {15\,a\,b^4\,x^{16/3}}{16}+2\,a^2\,b^3\,x^5+\frac {15\,a^3\,b^2\,x^{14/3}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.61, size = 73, normalized size = 0.97 \[ \frac {a^{5} x^{4}}{4} + \frac {15 a^{4} b x^{\frac {13}{3}}}{13} + \frac {15 a^{3} b^{2} x^{\frac {14}{3}}}{7} + 2 a^{2} b^{3} x^{5} + \frac {15 a b^{4} x^{\frac {16}{3}}}{16} + \frac {3 b^{5} x^{\frac {17}{3}}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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