Optimal. Leaf size=32 \[ \text {Int}\left (\frac {(a+b x)^2 \text {ArcTan}(a+b x)}{\sqrt [3]{c+c (a+b x)^2}},x\right ) \]
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Rubi [A]
time = 0.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {(a+b x)^2 \text {ArcTan}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(a+b x)^2 \tan ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2 \tan ^{-1}(x)}{\sqrt [3]{c+c x^2}} \, dx,x,a+b x\right )}{b}\\ \end {align*}
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Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(225\) vs. \(2(32)=64\).
time = 0.63, size = 225, normalized size = 7.03 \begin {gather*} -\frac {3 \sqrt [3]{1+a^2+2 a b x+b^2 x^2} \left (1+(a+b x)^2\right )^{2/3} \left (\frac {5 \sqrt [3]{2} \sqrt {\pi } \text {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+(a+b x)^2}\right )}{\left (1+(a+b x)^2\right )^2}+\text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right ) \left (15+\frac {90}{1+(a+b x)^2}+\frac {24 (a+b x) \text {ArcTan}(a+b x) \, _2F_1\left (1,\frac {4}{3};\frac {11}{6};\frac {1}{1+(a+b x)^2}\right )}{\left (1+(a+b x)^2\right )^2}+5 \text {ArcTan}(a+b x) (-4 (a+b x)+6 \sin (2 \text {ArcTan}(a+b x)))\right )\right )}{140 b \sqrt [3]{c \left (1+a^2+2 a b x+b^2 x^2\right )} \text {Gamma}\left (\frac {11}{6}\right ) \text {Gamma}\left (\frac {7}{3}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.35, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{2} \arctan \left (b x +a \right )}{\left (\left (a^{2}+1\right ) c +2 a b c x +b^{2} c \,x^{2}\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{2} \operatorname {atan}{\left (a + b x \right )}}{\sqrt [3]{c \left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {atan}\left (a+b\,x\right )\,{\left (a+b\,x\right )}^2}{{\left (c\,b^2\,x^2+2\,a\,c\,b\,x+c\,\left (a^2+1\right )\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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