3.765 \(\int (c+a^2 c x^2)^2 \tan ^{-1}(a x)^{3/2} \, dx\)

Optimal. Leaf size=172 \[ \frac {3}{80} c \text {Int}\left (\frac {a^2 c x^2+c}{\sqrt {\tan ^{-1}(a x)}},x\right )+\frac {1}{10} c^2 \text {Int}\left (\frac {1}{\sqrt {\tan ^{-1}(a x)}},x\right )+\frac {8}{15} c^2 \text {Int}\left (\tan ^{-1}(a x)^{3/2},x\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}+\frac {4}{15} c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}{40 a}-\frac {c^2 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}{5 a} \]

[Out]

4/15*c^2*x*(a^2*x^2+1)*arctan(a*x)^(3/2)+1/5*c^2*x*(a^2*x^2+1)^2*arctan(a*x)^(3/2)-1/5*c^2*(a^2*x^2+1)*arctan(
a*x)^(1/2)/a-3/40*c^2*(a^2*x^2+1)^2*arctan(a*x)^(1/2)/a+8/15*c^2*Unintegrable(arctan(a*x)^(3/2),x)+1/10*c^2*Un
integrable(1/arctan(a*x)^(1/2),x)+3/80*c*Unintegrable((a^2*c*x^2+c)/arctan(a*x)^(1/2),x)

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Rubi [A]  time = 0.07, 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 = {} \[ \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2),x]

[Out]

-(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(5*a) - (3*c^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(40*a) + (4*c^2*x*(1
+ a^2*x^2)*ArcTan[a*x]^(3/2))/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/5 + (c^2*Defer[Int][1/Sqrt[ArcTan
[a*x]], x])/10 + (3*c*Defer[Int][(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/80 + (8*c^2*Defer[Int][ArcTan[a*x]^(3/
2), x])/15

Rubi steps

\begin {align*} \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx &=-\frac {3 c^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}{40 a}+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}+\frac {1}{80} (3 c) \int \frac {c+a^2 c x^2}{\sqrt {\tan ^{-1}(a x)}} \, dx+\frac {1}{5} (4 c) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx\\ &=-\frac {c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{5 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}{40 a}+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}+\frac {1}{80} (3 c) \int \frac {c+a^2 c x^2}{\sqrt {\tan ^{-1}(a x)}} \, dx+\frac {1}{10} c^2 \int \frac {1}{\sqrt {\tan ^{-1}(a x)}} \, dx+\frac {1}{15} \left (8 c^2\right ) \int \tan ^{-1}(a x)^{3/2} \, dx\\ \end {align*}

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Mathematica [A]  time = 2.32, size = 0, normalized size = 0.00 \[ \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2),x]

[Out]

Integrate[(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), 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)^2*arctan(a*x)^(3/2),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)^2*arctan(a*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 2.15, size = 0, normalized size = 0.00 \[ \int \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*c*x^2+c)^2*arctan(a*x)^(3/2),x)

[Out]

int((a^2*c*x^2+c)^2*arctan(a*x)^(3/2),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)^2*arctan(a*x)^(3/2),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 \[ \int {\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)^(3/2)*(c + a^2*c*x^2)^2,x)

[Out]

int(atan(a*x)^(3/2)*(c + a^2*c*x^2)^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int 2 a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int a^{4} x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**2*atan(a*x)**(3/2),x)

[Out]

c**2*(Integral(2*a**2*x**2*atan(a*x)**(3/2), x) + Integral(a**4*x**4*atan(a*x)**(3/2), x) + Integral(atan(a*x)
**(3/2), x))

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