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

Optimal. Leaf size=137 \[ -\frac {\text {Int}\left (\frac {x^2}{\sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}},x\right )}{6 a}+\frac {\text {Int}\left (\frac {1}{\sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}},x\right )}{3 a^3}+\frac {x^2 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}{3 a^2 c}-\frac {2 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}{3 a^4 c} \]

[Out]

-2/3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/a^4/c+1/3*x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/a^2/c+1/3*Unint
egrable(1/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)/a^3-1/6*Unintegrable(x^2/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1
/2),x)/a

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Rubi [A]  time = 0.33, 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 = {} \[ \int \frac {x^3 \sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*a^2*c) +
Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(3*a^3) - Defer[Int][x^2/(Sqrt[c + a^2*c*x^2]*Sqrt[A
rcTan[a*x]]), x]/(6*a)

Rubi steps

\begin {align*} \int \frac {x^3 \sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {x^2 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{3 a^2 c}-\frac {2 \int \frac {x \sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\int \frac {x^2}{\sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx}{6 a}\\ &=-\frac {2 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{3 a^2 c}+\frac {\int \frac {1}{\sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx}{3 a^3}-\frac {\int \frac {x^2}{\sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}} \, dx}{6 a}\\ \end {align*}

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Mathematica [A]  time = 4.50, size = 0, normalized size = 0.00 \[ \int \frac {x^3 \sqrt {\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[(x^3*Sqrt[ArcTan[a*x]])/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)^(1/2)/(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)^(1/2)/(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 = 10.35, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sqrt {\arctan \left (a x \right )}}{\sqrt {a^{2} c \,x^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \sqrt {\operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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