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

Optimal. Leaf size=135 \[ \frac{\text{Unintegrable}\left (\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )}{3 a^3}-\frac{\text{Unintegrable}\left (\frac{x^2}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )}{6 a}+\frac{x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{3 a^2 c}-\frac{2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{3 a^4 c} \]

[Out]

(-2*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) +
Unintegrable[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(3*a^3) - Unintegrable[x^2/(Sqrt[c + a^2*c*x^2]*Sq
rt[ArcTan[a*x]]), x]/(6*a)

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Rubi [A]  time = 0.332401, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 4.24728, size = 0, normalized size = 0. \[ \int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx \]

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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Maple [A]  time = 4.891, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3}\sqrt{\arctan \left ( ax \right ) }{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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: UnboundLocalError

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \sqrt{\operatorname{atan}{\left (a x \right )}}}{\sqrt{c \left (a^{2} x^{2} + 1\right )}}\, dx \end{align*}

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \sqrt{\arctan \left (a x\right )}}{\sqrt{a^{2} c x^{2} + c}}\,{d x} \end{align*}

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]

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