3.851 \(\int x (c+a^2 c x^2)^3 \tan ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=309 \[ -\frac {9 c^2 \text {Int}\left (\frac {a^2 c x^2+c}{\sqrt {\tan ^{-1}(a x)}},x\right )}{896 a}-\frac {5 c \text {Int}\left (\frac {\left (a^2 c x^2+c\right )^2}{\sqrt {\tan ^{-1}(a x)}},x\right )}{896 a}-\frac {3 c^3 \text {Int}\left (\frac {1}{\sqrt {\tan ^{-1}(a x)}},x\right )}{112 a}-\frac {c^3 \text {Int}\left (\tan ^{-1}(a x)^{3/2},x\right )}{7 a}+\frac {c^3 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^{5/2}}{8 a^2}-\frac {5 c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^{3/2}}{112 a}+\frac {5 c^3 \left (a^2 x^2+1\right )^3 \sqrt {\tan ^{-1}(a x)}}{448 a^2}-\frac {3 c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}{56 a}+\frac {9 c^3 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}{448 a^2}-\frac {c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}{14 a}+\frac {3 c^3 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}{56 a^2} \]

[Out]

-1/14*c^3*x*(a^2*x^2+1)*arctan(a*x)^(3/2)/a-3/56*c^3*x*(a^2*x^2+1)^2*arctan(a*x)^(3/2)/a-5/112*c^3*x*(a^2*x^2+
1)^3*arctan(a*x)^(3/2)/a+1/8*c^3*(a^2*x^2+1)^4*arctan(a*x)^(5/2)/a^2+3/56*c^3*(a^2*x^2+1)*arctan(a*x)^(1/2)/a^
2+9/448*c^3*(a^2*x^2+1)^2*arctan(a*x)^(1/2)/a^2+5/448*c^3*(a^2*x^2+1)^3*arctan(a*x)^(1/2)/a^2-1/7*c^3*Unintegr
able(arctan(a*x)^(3/2),x)/a-3/112*c^3*Unintegrable(1/arctan(a*x)^(1/2),x)/a-9/896*c^2*Unintegrable((a^2*c*x^2+
c)/arctan(a*x)^(1/2),x)/a-5/896*c*Unintegrable((a^2*c*x^2+c)^2/arctan(a*x)^(1/2),x)/a

________________________________________________________________________________________

Rubi [A]  time = 0.19, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2),x]

[Out]

(3*c^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(56*a^2) + (9*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(448*a^2) + (5*c^
3*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(448*a^2) - (c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(14*a) - (3*c^3*x*(1
+ a^2*x^2)^2*ArcTan[a*x]^(3/2))/(56*a) - (5*c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/(112*a) + (c^3*(1 + a^2*x
^2)^4*ArcTan[a*x]^(5/2))/(8*a^2) - (3*c^3*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/(112*a) - (9*c^2*Defer[Int][(c +
a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/(896*a) - (5*c*Defer[Int][(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x])/(896*a)
- (c^3*Defer[Int][ArcTan[a*x]^(3/2), x])/(7*a)

Rubi steps

\begin {align*} \int x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2} \, dx &=\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^{5/2}}{8 a^2}-\frac {5 \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{3/2} \, dx}{16 a}\\ &=\frac {5 c^3 \left (1+a^2 x^2\right )^3 \sqrt {\tan ^{-1}(a x)}}{448 a^2}-\frac {5 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}}{112 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^{5/2}}{8 a^2}-\frac {(5 c) \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\tan ^{-1}(a x)}} \, dx}{896 a}-\frac {(15 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx}{56 a}\\ &=\frac {9 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}{448 a^2}+\frac {5 c^3 \left (1+a^2 x^2\right )^3 \sqrt {\tan ^{-1}(a x)}}{448 a^2}-\frac {3 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{56 a}-\frac {5 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}}{112 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^{5/2}}{8 a^2}-\frac {(5 c) \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\tan ^{-1}(a x)}} \, dx}{896 a}-\frac {\left (9 c^2\right ) \int \frac {c+a^2 c x^2}{\sqrt {\tan ^{-1}(a x)}} \, dx}{896 a}-\frac {\left (3 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx}{14 a}\\ &=\frac {3 c^3 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{56 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}{448 a^2}+\frac {5 c^3 \left (1+a^2 x^2\right )^3 \sqrt {\tan ^{-1}(a x)}}{448 a^2}-\frac {c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}{14 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{56 a}-\frac {5 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{3/2}}{112 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^{5/2}}{8 a^2}-\frac {(5 c) \int \frac {\left (c+a^2 c x^2\right )^2}{\sqrt {\tan ^{-1}(a x)}} \, dx}{896 a}-\frac {\left (9 c^2\right ) \int \frac {c+a^2 c x^2}{\sqrt {\tan ^{-1}(a x)}} \, dx}{896 a}-\frac {\left (3 c^3\right ) \int \frac {1}{\sqrt {\tan ^{-1}(a x)}} \, dx}{112 a}-\frac {c^3 \int \tan ^{-1}(a x)^{3/2} \, dx}{7 a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.65, size = 0, normalized size = 0.00 \[ \int x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2),x]

[Out]

Integrate[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), 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(x*(a^2*c*x^2+c)^3*arctan(a*x)^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [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)^3*arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 2.95, size = 0, normalized size = 0.00 \[ \int x \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a^2*c*x^2+c)^3*arctan(a*x)^(5/2),x)

[Out]

int(x*(a^2*c*x^2+c)^3*arctan(a*x)^(5/2),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(x*(a^2*c*x^2+c)^3*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.

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*atan(a*x)^(5/2)*(c + a^2*c*x^2)^3,x)

[Out]

int(x*atan(a*x)^(5/2)*(c + a^2*c*x^2)^3, x)

________________________________________________________________________________________

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)**3*atan(a*x)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________