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

Optimal. Leaf size=216 \[ -\frac {c \text {Int}\left (\frac {a^2 c x^2+c}{\sqrt {\tan ^{-1}(a x)}},x\right )}{64 a}-\frac {c^2 \text {Int}\left (\frac {1}{\sqrt {\tan ^{-1}(a x)}},x\right )}{24 a}-\frac {2 c^2 \text {Int}\left (\tan ^{-1}(a x)^{3/2},x\right )}{9 a}+\frac {c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac {c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^{3/2}}{12 a}+\frac {c^2 \left (a^2 x^2+1\right )^2 \sqrt {\tan ^{-1}(a x)}}{32 a^2}-\frac {c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^{3/2}}{9 a}+\frac {c^2 \left (a^2 x^2+1\right ) \sqrt {\tan ^{-1}(a x)}}{12 a^2} \]

[Out]

-1/9*c^2*x*(a^2*x^2+1)*arctan(a*x)^(3/2)/a-1/12*c^2*x*(a^2*x^2+1)^2*arctan(a*x)^(3/2)/a+1/6*c^2*(a^2*x^2+1)^3*
arctan(a*x)^(5/2)/a^2+1/12*c^2*(a^2*x^2+1)*arctan(a*x)^(1/2)/a^2+1/32*c^2*(a^2*x^2+1)^2*arctan(a*x)^(1/2)/a^2-
2/9*c^2*Unintegrable(arctan(a*x)^(3/2),x)/a-1/24*c^2*Unintegrable(1/arctan(a*x)^(1/2),x)/a-1/64*c*Unintegrable
((a^2*c*x^2+c)/arctan(a*x)^(1/2),x)/a

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Rubi [A]  time = 0.11, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(12*a^2) + (c^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(32*a^2) - (c^2*x*(1
+ a^2*x^2)*ArcTan[a*x]^(3/2))/(9*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(12*a) + (c^2*(1 + a^2*x^2)^3*
ArcTan[a*x]^(5/2))/(6*a^2) - (c^2*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/(24*a) - (c*Defer[Int][(c + a^2*c*x^2)/S
qrt[ArcTan[a*x]], x])/(64*a) - (2*c^2*Defer[Int][ArcTan[a*x]^(3/2), x])/(9*a)

Rubi steps

\begin {align*} \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2} \, dx &=\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac {5 \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{3/2} \, dx}{12 a}\\ &=\frac {c^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}{32 a^2}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{12 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac {c \int \frac {c+a^2 c x^2}{\sqrt {\tan ^{-1}(a x)}} \, dx}{64 a}-\frac {c \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{3/2} \, dx}{3 a}\\ &=\frac {c^2 \left (1+a^2 x^2\right ) \sqrt {\tan ^{-1}(a x)}}{12 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2 \sqrt {\tan ^{-1}(a x)}}{32 a^2}-\frac {c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^{3/2}}{9 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^{3/2}}{12 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac {c \int \frac {c+a^2 c x^2}{\sqrt {\tan ^{-1}(a x)}} \, dx}{64 a}-\frac {c^2 \int \frac {1}{\sqrt {\tan ^{-1}(a x)}} \, dx}{24 a}-\frac {\left (2 c^2\right ) \int \tan ^{-1}(a x)^{3/2} \, dx}{9 a}\\ \end {align*}

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Mathematica [A]  time = 1.57, size = 0, normalized size = 0.00 \[ \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

[Out]

Timed out

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maple [A]  time = 2.35, size = 0, normalized size = 0.00 \[ \int x \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(a^2*c*x^2+c)^2*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(x*(a^2*c*x^2+c)^2*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 = -0.00 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int 2 a^{2} x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx + \int a^{4} x^{5} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*(Integral(x*atan(a*x)**(5/2), x) + Integral(2*a**2*x**3*atan(a*x)**(5/2), x) + Integral(a**4*x**5*atan(a*
x)**(5/2), x))

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