3.731 \(\int x (c+a^2 c x^2)^{5/2} \sqrt {\tan ^{-1}(a x)} \, dx\)

Optimal. Leaf size=66 \[ \frac {\left (a^2 c x^2+c\right )^{7/2} \sqrt {\tan ^{-1}(a x)}}{7 a^2 c}-\frac {\text {Int}\left (\frac {\left (a^2 c x^2+c\right )^{5/2}}{\sqrt {\tan ^{-1}(a x)}},x\right )}{14 a} \]

[Out]

1/7*(a^2*c*x^2+c)^(7/2)*arctan(a*x)^(1/2)/a^2/c-1/14*Unintegrable((a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x)/a

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Rubi [A]  time = 0.12, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]],x]

[Out]

((c + a^2*c*x^2)^(7/2)*Sqrt[ArcTan[a*x]])/(7*a^2*c) - Defer[Int][(c + a^2*c*x^2)^(5/2)/Sqrt[ArcTan[a*x]], x]/(
14*a)

Rubi steps

\begin {align*} \int x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)} \, dx &=\frac {\left (c+a^2 c x^2\right )^{7/2} \sqrt {\tan ^{-1}(a x)}}{7 a^2 c}-\frac {\int \frac {\left (c+a^2 c x^2\right )^{5/2}}{\sqrt {\tan ^{-1}(a x)}} \, dx}{14 a}\\ \end {align*}

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Mathematica [A]  time = 7.84, size = 0, normalized size = 0.00 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]],x]

[Out]

Integrate[x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*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(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(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 [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(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 3.30, size = 0, normalized size = 0.00 \[ \int x \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {\arctan \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(1/2),x)

[Out]

int(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(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(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(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 = -0.02 \[ \int x\,\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*atan(a*x)^(1/2)*(c + a^2*c*x^2)^(5/2),x)

[Out]

int(x*atan(a*x)^(1/2)*(c + a^2*c*x^2)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a**2*c*x**2+c)**(5/2)*atan(a*x)**(1/2),x)

[Out]

Timed out

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