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

Optimal. Leaf size=30 \[ \text {Int}\left (\frac {(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{(a+b x)^2+1}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.14, 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 = {} \[ \int \frac {(a+b x)^2 \cot ^{-1}(a+b x)}{\sqrt [3]{1+a^2+2 a b x+b^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Subst][Defer[Int][(x^2*ArcCot[x])/(1 + 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]{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*}

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Mathematica [A]  time = 1.00, size = 198, normalized size = 6.60 \[ \frac {3 \left (5 \sqrt [3]{2} \sqrt {\pi } \Gamma \left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{a^2+2 b x a+b^2 x^2+1}\right )+\Gamma \left (\frac {11}{6}\right ) \Gamma \left (\frac {7}{3}\right ) \left (5 \left ((a+b x)^2+1\right ) \left (3 \left ((a+b x)^2+7\right )+4 (a+b x) \left ((a+b x)^2-2\right ) \cot ^{-1}(a+b x)\right )-24 (a+b x) \, _2F_1\left (1,\frac {4}{3};\frac {11}{6};\frac {1}{a^2+2 b x a+b^2 x^2+1}\right ) \cot ^{-1}(a+b x)\right )\right )}{140 b \Gamma \left (\frac {11}{6}\right ) \Gamma \left (\frac {7}{3}\right ) \sqrt [3]{a^2+2 a b x+b^2 x^2+1} \left ((a+b x)^2+1\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*x)^2*ArcCot[a + b*x])/(1 + a^2 + 2*a*b*x + b^2*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*(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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fricas [A]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \operatorname {arccot}\left (b x + a\right )}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{2} \operatorname {arccot}\left (b x + a\right )}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 3.49, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{2} \mathrm {arccot}\left (b x +a \right )}{\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{2} \operatorname {arccot}\left (b x + a\right )}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acot}\left (a+b\,x\right )\,{\left (a+b\,x\right )}^2}{{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{1/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right )^{2} \operatorname {acot}{\left (a + b x \right )}}{\sqrt [3]{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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