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

Optimal. Leaf size=81 \[ \frac {5}{8} c \text {Int}\left (\sqrt {\tan ^{-1}(a x)},x\right )+\frac {2}{3} c \text {Int}\left (\tan ^{-1}(a x)^{5/2},x\right )+\frac {1}{3} c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{5/2}-\frac {5 c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}{12 a} \]

[Out]

-5/12*c*(a^2*x^2+1)*arctan(a*x)^(3/2)/a+1/3*c*x*(a^2*x^2+1)*arctan(a*x)^(5/2)+2/3*c*Unintegrable(arctan(a*x)^(
5/2),x)+5/8*c*Unintegrable(arctan(a*x)^(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 = {} \[ \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-5*c*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(12*a) + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^(5/2))/3 + (5*c*Defer[Int][Sqrt
[ArcTan[a*x]], x])/8 + (2*c*Defer[Int][ArcTan[a*x]^(5/2), x])/3

Rubi steps

\begin {align*} \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx &=-\frac {5 c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}{12 a}+\frac {1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{5/2}+\frac {1}{8} (5 c) \int \sqrt {\tan ^{-1}(a x)} \, dx+\frac {1}{3} (2 c) \int \tan ^{-1}(a x)^{5/2} \, dx\\ \end {align*}

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Mathematica [A]  time = 3.99, size = 0, normalized size = 0.00 \[ \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), 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((a^2*c*x^2+c)*arctan(a*x)^(5/2),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 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)*arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 1.60, size = 0, normalized size = 0.00 \[ \int \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a^2*c*x^2+c)*arctan(a*x)^(5/2),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((a^2*c*x^2+c)*arctan(a*x)^(5/2),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 = -0.01 \[ \int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,\left (c\,a^2\,x^2+c\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c \left (\int a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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