Optimal. Leaf size=526 \[ \frac {\sqrt [3]{a^3 x^3+b^2 x^2}}{a}+\frac {\left (3 a^2 b-b^2\right ) \log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-a x\right )}{3 a^3}+\frac {\left (b^2-3 a^2 b\right ) \log \left (a x \sqrt [3]{a^3 x^3+b^2 x^2}+\left (a^3 x^3+b^2 x^2\right )^{2/3}+a^2 x^2\right )}{6 a^3}+\frac {\left (3 a^2 b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3+b^2 x^2}+a x}\right )}{\sqrt {3} a^3}+\frac {\left (b \sqrt [3]{a^2-b}-i \sqrt {3} b \sqrt [3]{a^2-b}\right ) \log \left (\sqrt [3]{-1} \sqrt [3]{a^3 x^3+b^2 x^2}+\sqrt [3]{a} x \sqrt [3]{a^2-b}\right )}{2 a^{5/3}}+\frac {i \left (\sqrt {3} b \sqrt [3]{a^2-b}+i b \sqrt [3]{a^2-b}\right ) \log \left ((-1)^{2/3} \left (a^3 x^3+b^2 x^2\right )^{2/3}+a^{2/3} x^2 \left (a^2-b\right )^{2/3}-\sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3+b^2 x^2}\right )}{4 a^{5/3}}+\frac {\sqrt {-3-3 i \sqrt {3}} b \sqrt [3]{a^2-b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x \sqrt [3]{a^2-b}}{\sqrt [3]{a} x \sqrt [3]{a^2-b}-2 \sqrt [3]{-1} \sqrt [3]{a^3 x^3+b^2 x^2}}\right )}{\sqrt {2} a^{5/3}} \]
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Rubi [A] time = 0.27, antiderivative size = 524, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2056, 101, 157, 59, 91} \begin {gather*} \frac {\sqrt [3]{a^3 x^3+b^2 x^2}}{a}+\frac {b \left (3 a^2-b\right ) \sqrt [3]{a^3 x^3+b^2 x^2} \log \left (a^3 x+b^2\right )}{6 a^3 x^{2/3} \sqrt [3]{a^3 x+b^2}}+\frac {b \left (3 a^2-b\right ) \sqrt [3]{a^3 x^3+b^2 x^2} \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x+b^2}}-1\right )}{2 a^3 x^{2/3} \sqrt [3]{a^3 x+b^2}}+\frac {b \left (3 a^2-b\right ) \sqrt [3]{a^3 x^3+b^2 x^2} \tan ^{-1}\left (\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} a^3 x^{2/3} \sqrt [3]{a^3 x+b^2}}+\frac {b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3+b^2 x^2} \log (a x+b)}{2 a^{5/3} x^{2/3} \sqrt [3]{a^3 x+b^2}}-\frac {3 b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3+b^2 x^2} \log \left (\sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2-b}-\sqrt [3]{a^3 x+b^2}\right )}{2 a^{5/3} x^{2/3} \sqrt [3]{a^3 x+b^2}}-\frac {\sqrt {3} b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3+b^2 x^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2-b}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{a^{5/3} x^{2/3} \sqrt [3]{a^3 x+b^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 91
Rule 101
Rule 157
Rule 2056
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{b+a x} \, dx &=\frac {\sqrt [3]{b^2 x^2+a^3 x^3} \int \frac {x^{2/3} \sqrt [3]{b^2+a^3 x}}{b+a x} \, dx}{x^{2/3} \sqrt [3]{b^2+a^3 x}}\\ &=\frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{a}-\frac {\sqrt [3]{b^2 x^2+a^3 x^3} \int \frac {\frac {2 b^3}{3}+\frac {1}{3} a \left (3 a^2-b\right ) b x}{\sqrt [3]{x} (b+a x) \left (b^2+a^3 x\right )^{2/3}} \, dx}{a x^{2/3} \sqrt [3]{b^2+a^3 x}}\\ &=\frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{a}-\frac {\left (\left (3 a^2-b\right ) b \sqrt [3]{b^2 x^2+a^3 x^3}\right ) \int \frac {1}{\sqrt [3]{x} \left (b^2+a^3 x\right )^{2/3}} \, dx}{3 a x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {\left (\left (a^2-b\right ) b^2 \sqrt [3]{b^2 x^2+a^3 x^3}\right ) \int \frac {1}{\sqrt [3]{x} (b+a x) \left (b^2+a^3 x\right )^{2/3}} \, dx}{a x^{2/3} \sqrt [3]{b^2+a^3 x}}\\ &=\frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{a}+\frac {\left (3 a^2-b\right ) b \sqrt [3]{b^2 x^2+a^3 x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{b^2+a^3 x}}\right )}{\sqrt {3} a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}-\frac {\sqrt {3} \sqrt [3]{a^2-b} b \sqrt [3]{b^2 x^2+a^3 x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{b^2+a^3 x}}\right )}{a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {\sqrt [3]{a^2-b} b \sqrt [3]{b^2 x^2+a^3 x^3} \log (b+a x)}{2 a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {\left (3 a^2-b\right ) b \sqrt [3]{b^2 x^2+a^3 x^3} \log \left (b^2+a^3 x\right )}{6 a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {\left (3 a^2-b\right ) b \sqrt [3]{b^2 x^2+a^3 x^3} \log \left (-1+\frac {a \sqrt [3]{x}}{\sqrt [3]{b^2+a^3 x}}\right )}{2 a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}-\frac {3 \sqrt [3]{a^2-b} b \sqrt [3]{b^2 x^2+a^3 x^3} \log \left (\sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}-\sqrt [3]{b^2+a^3 x}\right )}{2 a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 162, normalized size = 0.31 \begin {gather*} \frac {3 x^2 \left (\left (a^3 b x+b^3\right ) \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-\frac {a^3 x}{b^2}\right )-\left (a^2 \left (a^3 x+b^2\right ) \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};-\frac {a^3 x}{b^2}\right )\right )+b^2 \left (a^2-b\right ) \sqrt [3]{\frac {a^3 x}{b^2}+1} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\left (a^3-a b\right ) x}{x a^3+b^2}\right )\right )}{2 a b \left (x^2 \left (a^3 x+b^2\right )\right )^{2/3} \sqrt [3]{\frac {a^3 x}{b^2}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.15, size = 571, normalized size = 1.09 \begin {gather*} \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{a}+\frac {\left (3 a^2 b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{\sqrt {3} a^3}+\frac {\sqrt {-3-3 i \sqrt {3}} \sqrt [3]{a^2-b} b \tan ^{-1}\left (\frac {3 \sqrt [3]{a} \sqrt [3]{a^2-b} x}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b} x-3 i \sqrt [3]{b^2 x^2+a^3 x^3}-\sqrt {3} \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{\sqrt {2} a^{5/3}}+\frac {\left (3 a^2 b-b^2\right ) \log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{3 a^3}+\frac {\left (\sqrt [3]{a^2-b} b-i \sqrt {3} \sqrt [3]{a^2-b} b\right ) \log \left (2 \sqrt [3]{a} \sqrt [3]{a^2-b} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a^{5/3}}+\frac {\left (-3 a^2 b+b^2\right ) \log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{6 a^3}+\frac {i \left (i \sqrt [3]{a^2-b} b+\sqrt {3} \sqrt [3]{a^2-b} b\right ) \log \left (-2 i a^{2/3} \left (a^2-b\right )^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{a^2-b} \left (i x-\sqrt {3} x\right ) \sqrt [3]{b^2 x^2+a^3 x^3}+\left (i+\sqrt {3}\right ) \left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 437, normalized size = 0.83 \begin {gather*} \frac {6 \, \sqrt {3} a^{2} b \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (a^{2} - b\right )} x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {2}{3}}}{3 \, {\left (a^{2} - b\right )} x}\right ) + 6 \, a^{2} b \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} \log \left (\frac {a x \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 3 \, a^{2} b \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} \log \left (\frac {a^{2} x^{2} \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {2}{3}} - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x \left (-\frac {a^{2} - b}{a^{2}}\right )^{\frac {1}{3}} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 2 \, \sqrt {3} {\left (3 \, a^{2} b - b^{2}\right )} \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) + 6 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a^{2} + 2 \, {\left (3 \, a^{2} b - b^{2}\right )} \log \left (-\frac {a x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (3 \, a^{2} b - b^{2}\right )} \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 126.67, size = 339, normalized size = 0.64 \begin {gather*} -\frac {{\left (a^{3} - a b\right )}^{\frac {1}{3}} {\left (a^{2} b - b^{2}\right )} \log \left ({\left | -{\left (a^{3} - a b\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a^{4} - a^{2} b} + \frac {\sqrt {3} {\left (a^{3} - a b\right )}^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} - a b\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} - a b\right )}^{\frac {1}{3}}}\right )}{a^{2}} + \frac {{\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} x}{a} + \frac {{\left (a^{3} - a b\right )}^{\frac {1}{3}} b \log \left ({\left (a^{3} - a b\right )}^{\frac {2}{3}} + {\left (a^{3} - a b\right )}^{\frac {1}{3}} {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{2 \, a^{2}} - \frac {\sqrt {3} {\left (3 \, a^{2} b - b^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{3 \, a^{3}} - \frac {{\left (3 \, a^{2} b - b^{2}\right )} \log \left (a^{2} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{6 \, a^{3}} + \frac {{\left (3 \, a^{2} b - b^{2}\right )} \log \left ({\left | -a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}{a x +b}\, dx\]
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*} \int \frac {{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x + b}\,{d x} \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 {{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}}{b+a\,x} \,d x \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 {\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )}}{a x + b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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