3.121 \(\int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{(1+a^2) c+2 a b c x+b^2 c x^2}} \, dx\)

Optimal. Leaf size=31 \[ \text{Unintegrable}\left (\frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{c (a+b x)^2+c}},x\right ) \]

[Out]

Unintegrable[((a + b*x)^2*ArcCot[a + b*x])/(c + c*(a + b*x)^2)^(1/3), x]

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Rubi [A]  time = 0.189707, 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 = {} \[ \int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((a + b*x)^2*ArcCot[a + b*x])/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

[Out]

Defer[Subst][Defer[Int][(x^2*ArcCot[x])/(c + c*x^2)^(1/3), x], x, a + b*x]/b

Rubi steps

\begin{align*} \int \frac{(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \cot ^{-1}(x)}{\sqrt [3]{c+c x^2}} \, dx,x,a+b x\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.204658, size = 200, normalized size = 6.45 \[ \frac{3 \left (5 \sqrt [3]{2} \sqrt{\pi } \text{Gamma}\left (\frac{5}{3}\right ) \text{HypergeometricPFQ}\left (\left \{1,\frac{4}{3},\frac{4}{3}\right \},\left \{\frac{11}{6},\frac{7}{3}\right \},\frac{1}{a^2+2 a b x+b^2 x^2+1}\right )+\text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left (5 \left ((a+b x)^2+1\right ) \left (3 \left ((a+b x)^2+7\right )+4 (a+b x) \left ((a+b x)^2-2\right ) \cot ^{-1}(a+b x)\right )-24 (a+b x) \, _2F_1\left (1,\frac{4}{3};\frac{11}{6};\frac{1}{a^2+2 b x a+b^2 x^2+1}\right ) \cot ^{-1}(a+b x)\right )\right )}{140 b \text{Gamma}\left (\frac{11}{6}\right ) \text{Gamma}\left (\frac{7}{3}\right ) \left ((a+b x)^2+1\right ) \sqrt [3]{c \left (a^2+2 a b x+b^2 x^2+1\right )}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*x)^2*ArcCot[a + b*x])/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

[Out]

(3*(Gamma[11/6]*Gamma[7/3]*(5*(1 + (a + b*x)^2)*(3*(7 + (a + b*x)^2) + 4*(a + b*x)*(-2 + (a + b*x)^2)*ArcCot[a
+ b*x]) - 24*(a + b*x)*ArcCot[a + b*x]*Hypergeometric2F1[1, 4/3, 11/6, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1)]) +
5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{1, 4/3, 4/3}, {11/6, 7/3}, (1 + a^2 + 2*a*b*x + b^2*x^2)^(-1
)]))/(140*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))^(1/3)*(1 + (a + b*x)^2)*Gamma[11/6]*Gamma[7/3])

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Maple [A]  time = 1.456, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{2}{\rm arccot} \left (bx+a\right ){\frac{1}{\sqrt [3]{ \left ({a}^{2}+1 \right ) c+2\,abcx+{b}^{2}c{x}^{2}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*arccot(b*x+a)/((a^2+1)*c+2*a*b*c*x+b^2*c*x^2)^(1/3),x)

[Out]

int((b*x+a)^2*arccot(b*x+a)/((a^2+1)*c+2*a*b*c*x+b^2*c*x^2)^(1/3),x)

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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} c x^{2} + 2 \, a b c x +{\left (a^{2} + 1\right )} c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*arccot(b*x+a)/((a^2+1)*c+2*a*b*c*x+c*x^2*b^2)^(1/3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^2*arccot(b*x + a)/(b^2*c*x^2 + 2*a*b*c*x + (a^2 + 1)*c)^(1/3), x)

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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} c x^{2} + 2 \, a b c x +{\left (a^{2} + 1\right )} c\right )}^{\frac{1}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*arccot(b*x+a)/((a^2+1)*c+2*a*b*c*x+c*x^2*b^2)^(1/3),x, algorithm="fricas")

[Out]

integral((b^2*x^2 + 2*a*b*x + a^2)*arccot(b*x + a)/(b^2*c*x^2 + 2*a*b*c*x + (a^2 + 1)*c)^(1/3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*acot(b*x+a)/((a**2+1)*c+2*a*b*c*x+c*x**2*b**2)**(1/3),x)

[Out]

Timed out

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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} c x^{2} + 2 \, a b c x +{\left (a^{2} + 1\right )} c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*arccot(b*x+a)/((a^2+1)*c+2*a*b*c*x+c*x^2*b^2)^(1/3),x, algorithm="giac")

[Out]

integrate((b*x + a)^2*arccot(b*x + a)/(b^2*c*x^2 + 2*a*b*c*x + (a^2 + 1)*c)^(1/3), x)