Optimal. Leaf size=29 \[ \text{Unintegrable}\left (\frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{(a+b x)^2+1}},x\right ) \]
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Rubi [A] time = 0.137071, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \cot ^{-1}(x)}{\sqrt [3]{1+x^2}} \, dx,x,a+b x\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.920116, size = 198, normalized size = 6.83 \[ \frac{3 \left (5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{4}{3},\frac{4}{3}\right \},\left \{\frac{11}{6},\frac{7}{3}\right \},\frac{1}{a^2+2 a b x+b^2 x^2+1}\right )+\text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (5 \left ((a+b x)^2+1\right ) \left (3 \left ((a+b x)^2+7\right )+4 (a+b x) \left ((a+b x)^2-2\right ) \cot ^{-1}(a+b x)\right )-24 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right ) \cot ^{-1}(a+b x)\right )\right )}{140 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \sqrt [3]{a^2+2 a b x+b^2 x^2+1} \left ((a+b x)^2+1\right )} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 1.473, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{2}{\rm arccot} \left (bx+a\right ){\frac{1}{\sqrt [3]{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{2} \operatorname{arccot}\left (b x + a\right )}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \operatorname{arccot}\left (b x + a\right )}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{1}{3}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{2} \operatorname{acot}{\left (a + b x \right )}}{\sqrt [3]{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{2} \operatorname{arccot}\left (b x + a\right )}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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