Optimal. Leaf size=59 \[ \frac {a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 b^3}+\frac {3 \left (a+b \sqrt [3]{x}\right )^8}{8 b^3}-\frac {6 a \left (a+b \sqrt [3]{x}\right )^7}{7 b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac {a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 b^3}+\frac {3 \left (a+b \sqrt [3]{x}\right )^8}{8 b^3}-\frac {6 a \left (a+b \sqrt [3]{x}\right )^7}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rubi steps
\begin {align*} \int \left (a+b \sqrt [3]{x}\right )^5 \, dx &=3 \operatorname {Subst}\left (\int x^2 (a+b x)^5 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {a^2 (a+b x)^5}{b^2}-\frac {2 a (a+b x)^6}{b^2}+\frac {(a+b x)^7}{b^2}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 b^3}-\frac {6 a \left (a+b \sqrt [3]{x}\right )^7}{7 b^3}+\frac {3 \left (a+b \sqrt [3]{x}\right )^8}{8 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.69 \[ \frac {\left (a+b \sqrt [3]{x}\right )^6 \left (a^2-6 a b \sqrt [3]{x}+21 b^2 x^{2/3}\right )}{56 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 61, normalized size = 1.03 \[ 5 \, a^{2} b^{3} x^{2} + a^{5} x + \frac {3}{8} \, {\left (b^{5} x^{2} + 16 \, a^{3} b^{2} x\right )} x^{\frac {2}{3}} + \frac {15}{28} \, {\left (4 \, a b^{4} x^{2} + 7 \, a^{4} b x\right )} x^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 54, normalized size = 0.92 \[ \frac {3}{8} \, b^{5} x^{\frac {8}{3}} + \frac {15}{7} \, a b^{4} x^{\frac {7}{3}} + 5 \, a^{2} b^{3} x^{2} + 6 \, a^{3} b^{2} x^{\frac {5}{3}} + \frac {15}{4} \, a^{4} b x^{\frac {4}{3}} + a^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 55, normalized size = 0.93 \[ \frac {3 b^{5} x^{\frac {8}{3}}}{8}+\frac {15 a \,b^{4} x^{\frac {7}{3}}}{7}+5 a^{2} b^{3} x^{2}+6 a^{3} b^{2} x^{\frac {5}{3}}+\frac {15 a^{4} b \,x^{\frac {4}{3}}}{4}+a^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 54, normalized size = 0.92 \[ \frac {3}{8} \, b^{5} x^{\frac {8}{3}} + \frac {15}{7} \, a b^{4} x^{\frac {7}{3}} + 5 \, a^{2} b^{3} x^{2} + 6 \, a^{3} b^{2} x^{\frac {5}{3}} + \frac {15}{4} \, a^{4} b x^{\frac {4}{3}} + a^{5} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 54, normalized size = 0.92 \[ a^5\,x+\frac {3\,b^5\,x^{8/3}}{8}+\frac {15\,a^4\,b\,x^{4/3}}{4}+\frac {15\,a\,b^4\,x^{7/3}}{7}+5\,a^2\,b^3\,x^2+6\,a^3\,b^2\,x^{5/3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.08, size = 68, normalized size = 1.15 \[ a^{5} x + \frac {15 a^{4} b x^{\frac {4}{3}}}{4} + 6 a^{3} b^{2} x^{\frac {5}{3}} + 5 a^{2} b^{3} x^{2} + \frac {15 a b^{4} x^{\frac {7}{3}}}{7} + \frac {3 b^{5} x^{\frac {8}{3}}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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