\(\int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx\) [128]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx=\text {Int}\left ((a+b x)^2 \sqrt {\cot ^{-1}(a+b x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx=\int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 4.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx=\int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.45 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

\[\int \left (b x +a \right )^{2} \sqrt {\operatorname {arccot}\left (b x +a \right )}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [N/A]

Not integrable

Time = 3.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx=\int \left (a + b x\right )^{2} \sqrt {\operatorname {acot}{\left (a + b x \right )}}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx=\int { {\left (b x + a\right )}^{2} \sqrt {\operatorname {arccot}\left (b x + a\right )} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 1.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (a+b x)^2 \sqrt {\cot ^{-1}(a+b x)} \, dx=\int \sqrt {\mathrm {acot}\left (a+b\,x\right )}\,{\left (a+b\,x\right )}^2 \,d x \]

[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)