Optimal. Leaf size=70 \[ -\frac {3 a^3 \sqrt [3]{a+b x}}{b^4}+\frac {9 a^2 (a+b x)^{4/3}}{4 b^4}-\frac {9 a (a+b x)^{7/3}}{7 b^4}+\frac {3 (a+b x)^{10/3}}{10 b^4} \]
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Rubi [A]
time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45}
\begin {gather*} -\frac {3 a^3 \sqrt [3]{a+b x}}{b^4}+\frac {9 a^2 (a+b x)^{4/3}}{4 b^4}+\frac {3 (a+b x)^{10/3}}{10 b^4}-\frac {9 a (a+b x)^{7/3}}{7 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x)^{2/3}} \, dx &=\int \left (-\frac {a^3}{b^3 (a+b x)^{2/3}}+\frac {3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac {3 a (a+b x)^{4/3}}{b^3}+\frac {(a+b x)^{7/3}}{b^3}\right ) \, dx\\ &=-\frac {3 a^3 \sqrt [3]{a+b x}}{b^4}+\frac {9 a^2 (a+b x)^{4/3}}{4 b^4}-\frac {9 a (a+b x)^{7/3}}{7 b^4}+\frac {3 (a+b x)^{10/3}}{10 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.66 \begin {gather*} \frac {3 \sqrt [3]{a+b x} \left (-81 a^3+27 a^2 b x-18 a b^2 x^2+14 b^3 x^3\right )}{140 b^4} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(319\) vs. \(2(70)=140\).
time = 15.19, size = 299, normalized size = 4.27 \begin {gather*} \frac {3 a^{\frac {1}{3}} \left (81 a^9 \left (1-\left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+27 a^8 b x \left (18-17 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+9 a^7 b^2 x^2 \left (135-119 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+a^6 b^3 x^3 \left (1620-1309 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+7 a^3 b^4 x^4 \left (-123 a^2-33 a b x+13 b^2 x^2\right ) \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}+1215 a^5 b^4 x^4+6 a b^5 x^5 \left (81 a^3+11 b^3 x^3 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+3 a^2 b^6 x^6 \left (27 a+43 b x \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )+14 b^9 x^9 \left (\frac {a+b x}{a}\right )^{\frac {1}{3}}\right )}{140 b^4 \left (a^6+6 a^5 b x+15 a^4 b^2 x^2+20 a^3 b^3 x^3+15 a^2 b^4 x^4+6 a b^5 x^5+b^6 x^6\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.11, size = 50, normalized size = 0.71
method | result | size |
gosper | \(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-14 b^{3} x^{3}+18 a \,b^{2} x^{2}-27 a^{2} b x +81 a^{3}\right )}{140 b^{4}}\) | \(43\) |
trager | \(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-14 b^{3} x^{3}+18 a \,b^{2} x^{2}-27 a^{2} b x +81 a^{3}\right )}{140 b^{4}}\) | \(43\) |
risch | \(-\frac {3 \left (b x +a \right )^{\frac {1}{3}} \left (-14 b^{3} x^{3}+18 a \,b^{2} x^{2}-27 a^{2} b x +81 a^{3}\right )}{140 b^{4}}\) | \(43\) |
derivativedivides | \(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {9 a \left (b x +a \right )^{\frac {7}{3}}}{7}+\frac {9 a^{2} \left (b x +a \right )^{\frac {4}{3}}}{4}-3 a^{3} \left (b x +a \right )^{\frac {1}{3}}}{b^{4}}\) | \(50\) |
default | \(\frac {\frac {3 \left (b x +a \right )^{\frac {10}{3}}}{10}-\frac {9 a \left (b x +a \right )^{\frac {7}{3}}}{7}+\frac {9 a^{2} \left (b x +a \right )^{\frac {4}{3}}}{4}-3 a^{3} \left (b x +a \right )^{\frac {1}{3}}}{b^{4}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 56, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x + a\right )}^{\frac {10}{3}}}{10 \, b^{4}} - \frac {9 \, {\left (b x + a\right )}^{\frac {7}{3}} a}{7 \, b^{4}} + \frac {9 \, {\left (b x + a\right )}^{\frac {4}{3}} a^{2}}{4 \, b^{4}} - \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{3}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.29, size = 42, normalized size = 0.60 \begin {gather*} \frac {3 \, {\left (14 \, b^{3} x^{3} - 18 \, a b^{2} x^{2} + 27 \, a^{2} b x - 81 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{140 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1640 vs.
\(2 (66) = 132\).
time = 1.30, size = 1640, normalized size = 23.43
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 85, normalized size = 1.21 \begin {gather*} \frac {3 \left (\frac {1}{10} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{3}-\frac {3}{7} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right )^{2} a+\frac {3}{4} \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) a^{2}-\left (a+b x\right )^{\frac {1}{3}} a^{3}\right )}{b b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 56, normalized size = 0.80 \begin {gather*} \frac {3\,{\left (a+b\,x\right )}^{10/3}}{10\,b^4}-\frac {3\,a^3\,{\left (a+b\,x\right )}^{1/3}}{b^4}+\frac {9\,a^2\,{\left (a+b\,x\right )}^{4/3}}{4\,b^4}-\frac {9\,a\,{\left (a+b\,x\right )}^{7/3}}{7\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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