Optimal. Leaf size=204 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3 a^3+a^6+2 a b^3\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-a x\right )}{2 a}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3+b^2 x^2}+a x}\right )}{2 a}+\frac {\log \left (a x \sqrt [3]{a^3 x^3+b^2 x^2}+\left (a^3 x^3+b^2 x^2\right )^{2/3}+a^2 x^2\right )}{4 a} \]
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Rubi [B] time = 0.96, antiderivative size = 874, normalized size of antiderivative = 4.28, number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {2056, 6725, 59, 912, 91} \begin {gather*} -\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 a \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log (x)}{4 a \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt {b}-\sqrt {2} \sqrt {-a} x\right )}{8 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt {2} \sqrt {-a} x+\sqrt {b}\right )}{8 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{a \sqrt [3]{x}}-1\right )}{4 a \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 91
Rule 912
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {b+a x^2}{\left (b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {b+a x^2}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (b+2 a x^2\right )} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (\frac {1}{2 x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {b}{2 x^{2/3} \sqrt [3]{b^2+a^3 x} \left (b+2 a x^2\right )}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x}} \, dx}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (b x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (b+2 a x^2\right )} \, dx}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (b x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}-\sqrt {2} \sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}+\sqrt {2} \sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}-\sqrt {2} \sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}+\sqrt {2} \sqrt {-a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{4 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt {b}-\sqrt {2} \sqrt {-a} x\right )}{8 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt {b}+\sqrt {2} \sqrt {-a} x\right )}{8 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}}}\right )}{8 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}}}\right )}{8 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 145, normalized size = 0.71 \begin {gather*} \frac {3 x \left (2 \sqrt [3]{\frac {a^3 x}{b^2}+1} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {a^3 x}{b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a^3-\sqrt {2} \sqrt {-a} b^{3/2}\right ) x}{x a^3+b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a^3+\sqrt {2} \sqrt {-a} b^{3/2}\right ) x}{x a^3+b^2}\right )\right )}{4 \sqrt [3]{x^2 \left (a^3 x+b^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 208, normalized size = 1.02 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{2 a}-\frac {\log \left (a^2 x-a \sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a}+\frac {\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 )}{4 a}-\frac {1}{4} \text {RootSum}\left [a^6+2 a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 2180, normalized size = 10.69
result too large to display
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*} \int \frac {a x^{2} + b}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{2} + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}+b}{\left (2 a \,x^{2}+b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]
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 {a x^{2} + b}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{2} + b\right )}}\,{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 {a\,x^2+b}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (2\,a\,x^2+b\right )} \,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 {a x^{2} + b}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a x^{2} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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