3.128 \(\int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx\)

Optimal. Leaf size=21 \[ \text {Int}\left ((a+b x)^2 \sqrt {\cot ^{-1}(a+b x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, 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 = {} \[ \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*x)^2*Sqrt[ArcCot[a + b*x]],x]

[Out]

Defer[Int][(a + b*x)^2*Sqrt[ArcCot[a + b*x]], x]

Rubi steps

\begin {align*} \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx &=\int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx\\ \end {align*}

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Mathematica [A]  time = 9.26, size = 0, normalized size = 0.00 \[ \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*x)^2*Sqrt[ArcCot[a + b*x]],x]

[Out]

Integrate[(a + b*x)^2*Sqrt[ArcCot[a + b*x]], x]

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^2*sqrt(arccot(b*x + a)), x)

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maple [A]  time = 3.41, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{2} \sqrt {\mathrm {arccot}\left (b x +a \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \sqrt {\mathrm {acot}\left (a+b\,x\right )}\,{\left (a+b\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*x)**2*sqrt(acot(a + b*x)), x)

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