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

Optimal. Leaf size=259 \[ \frac {9}{280} c^2 \text {Int}\left (\frac {a^2 c x^2+c}{\sqrt {\tan ^{-1}(a x)}},x\right )+\frac {1}{56} c \text {Int}\left (\frac {\left (a^2 c x^2+c\right )^2}{\sqrt {\tan ^{-1}(a x)}},x\right )+\frac {3}{35} c^3 \text {Int}\left (\frac {1}{\sqrt {\tan ^{-1}(a x)}},x\right )+\frac {16}{35} c^3 \text {Int}\left (\tan ^{-1}(a x)^{3/2},x\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^{3/2}+\frac {6}{35} c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}+\frac {8}{35} c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}-\frac {c^3 \left (a^2 x^2+1\right )^3 \sqrt {\tan ^{-1}(a x)}}{28 a}-\frac {9 c^3 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}{140 a}-\frac {6 c^3 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}{35 a} \]

[Out]

8/35*c^3*x*(a^2*x^2+1)*arctan(a*x)^(3/2)+6/35*c^3*x*(a^2*x^2+1)^2*arctan(a*x)^(3/2)+1/7*c^3*x*(a^2*x^2+1)^3*ar
ctan(a*x)^(3/2)-6/35*c^3*(a^2*x^2+1)*arctan(a*x)^(1/2)/a-9/140*c^3*(a^2*x^2+1)^2*arctan(a*x)^(1/2)/a-1/28*c^3*
(a^2*x^2+1)^3*arctan(a*x)^(1/2)/a+16/35*c^3*Unintegrable(arctan(a*x)^(3/2),x)+3/35*c^3*Unintegrable(1/arctan(a
*x)^(1/2),x)+9/280*c^2*Unintegrable((a^2*c*x^2+c)/arctan(a*x)^(1/2),x)+1/56*c*Unintegrable((a^2*c*x^2+c)^2/arc
tan(a*x)^(1/2),x)

________________________________________________________________________________________

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

Verification is Not applicable to the result.

[In]

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

[Out]

(-6*c^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(35*a) - (9*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(140*a) - (c^3*(1
+ a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(28*a) + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/35 + (6*c^3*x*(1 + a^2*x^2)
^2*ArcTan[a*x]^(3/2))/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/7 + (3*c^3*Defer[Int][1/Sqrt[ArcTan[a*x]]
, x])/35 + (9*c^2*Defer[Int][(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/280 + (c*Defer[Int][(c + a^2*c*x^2)^2/Sqrt
[ArcTan[a*x]], x])/56 + (16*c^3*Defer[Int][ArcTan[a*x]^(3/2), x])/35

Rubi steps

\begin {align*} \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2} \, dx &=-\frac {c^3 \left (1+a^2 x^2\right )^3 \sqrt {\tan ^{-1}(a x)}}{28 a}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}+\frac {1}{56} c \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\tan ^{-1}(a x)}} \, dx+\frac {1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx\\ &=-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}{140 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \sqrt {\tan ^{-1}(a x)}}{28 a}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}+\frac {1}{56} c \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\tan ^{-1}(a x)}} \, dx+\frac {1}{280} \left (9 c^2\right ) \int \frac {c+a^2 c x^2}{\sqrt {\tan ^{-1}(a x)}} \, dx+\frac {1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx\\ &=-\frac {6 c^3 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{35 a}-\frac {9 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}{140 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \sqrt {\tan ^{-1}(a x)}}{28 a}+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}+\frac {1}{56} c \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\tan ^{-1}(a x)}} \, dx+\frac {1}{280} \left (9 c^2\right ) \int \frac {c+a^2 c x^2}{\sqrt {\tan ^{-1}(a x)}} \, dx+\frac {1}{35} \left (3 c^3\right ) \int \frac {1}{\sqrt {\tan ^{-1}(a x)}} \, dx+\frac {1}{35} \left (16 c^3\right ) \int \tan ^{-1}(a x)^{3/2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 2.41, size = 0, normalized size = 0.00 \[ \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

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)^3*arctan(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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)^3*arctan(a*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 2.80, size = 0, normalized size = 0.00 \[ \int \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)^3*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.

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{3} \left (\int 3 a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int a^{6} x^{6} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx + \int \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(3*a**2*x**2*atan(a*x)**(3/2), x) + Integral(3*a**4*x**4*atan(a*x)**(3/2), x) + Integral(a**6*x*
*6*atan(a*x)**(3/2), x) + Integral(atan(a*x)**(3/2), x))

________________________________________________________________________________________