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

Optimal. Leaf size=168 \[ \frac {\text {Int}\left (\frac {x}{\sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}},x\right )}{8 a^2}+\frac {5 \text {Int}\left (\frac {\sqrt {\tan ^{-1}(a x)}}{\sqrt {a^2 c x^2+c}},x\right )}{4 a^3}+\frac {x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{3 a^2 c}-\frac {2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{3 a^4 c}-\frac {x \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}{4 a^3 c} \]

[Out]

-2/3*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/a^4/c+1/3*x^2*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/a^2/c-1/4*x*(a^
2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/a^3/c+1/8*Unintegrable(x/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)/a^2+5/4*U
nintegrable(arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(1/2),x)/a^3

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Rubi [A]  time = 0.45, 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 = {} \[ \int \frac {x^3 \tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 3.73, size = 0, normalized size = 0.00 \[ \int \frac {x^3 \tan ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[(x^3*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^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^3*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [F(-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^3*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 9.12, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \arctan \left (a x \right )^{\frac {3}{2}}}{\sqrt {a^{2} c \,x^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(x^3*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),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^3*arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),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 \[ \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^{3/2}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*atan(a*x)^(3/2))/(c + a^2*c*x^2)^(1/2),x)

[Out]

int((x^3*atan(a*x)^(3/2))/(c + a^2*c*x^2)^(1/2), x)

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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**3*atan(a*x)**(3/2)/(a**2*c*x**2+c)**(1/2),x)

[Out]

Timed out

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