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

Optimal. Leaf size=356 \[ -\frac{75 c^3 \text{Unintegrable}\left (\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )}{896 a}-\frac{25 c^2 \text{Unintegrable}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right )}{1344 a}-\frac{25 c^3 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right )}{224 a}-\frac{c \text{Unintegrable}\left (\frac{\left (a^2 c x^2+c\right )^{3/2}}{\sqrt{\tan ^{-1}(a x)}},x\right )}{112 a}-\frac{25 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{224 a}+\frac{75 c^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{448 a^2}+\frac{\left (a^2 c x^2+c\right )^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac{5 x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{84 a}+\frac{\left (a^2 c x^2+c\right )^{5/2} \sqrt{\tan ^{-1}(a x)}}{56 a^2}-\frac{25 c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{336 a}+\frac{25 c \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}{672 a^2} \]

[Out]

(75*c^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(448*a^2) + (25*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(672
*a^2) + ((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]])/(56*a^2) - (25*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/
(224*a) - (25*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(336*a) - (5*x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3
/2))/(84*a) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^(5/2))/(7*a^2*c) - (75*c^3*Unintegrable[1/(Sqrt[c + a^2*c*x^2
]*Sqrt[ArcTan[a*x]]), x])/(896*a) - (25*c^2*Unintegrable[Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/(1344*a) -
(c*Unintegrable[(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x])/(112*a) - (25*c^3*Unintegrable[ArcTan[a*x]^(3/2)
/Sqrt[c + a^2*c*x^2], x])/(224*a)

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Rubi [A]  time = 0.363545, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(75*c^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(448*a^2) + (25*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(672
*a^2) + ((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]])/(56*a^2) - (25*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/
(224*a) - (25*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(336*a) - (5*x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3
/2))/(84*a) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^(5/2))/(7*a^2*c) - (75*c^3*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*
Sqrt[ArcTan[a*x]]), x])/(896*a) - (25*c^2*Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/(1344*a) - (c*
Defer[Int][(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x])/(112*a) - (25*c^3*Defer[Int][ArcTan[a*x]^(3/2)/Sqrt[c
+ a^2*c*x^2], x])/(224*a)

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2} \, dx &=\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac{5 \int \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2} \, dx}{14 a}\\ &=\frac{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}}{56 a^2}-\frac{5 x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{84 a}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac{c \int \frac{\left (c+a^2 c x^2\right )^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx}{112 a}-\frac{(25 c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2} \, dx}{84 a}\\ &=\frac{25 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}{672 a^2}+\frac{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}}{56 a^2}-\frac{25 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{336 a}-\frac{5 x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{84 a}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac{c \int \frac{\left (c+a^2 c x^2\right )^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx}{112 a}-\frac{\left (25 c^2\right ) \int \frac{\sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx}{1344 a}-\frac{\left (25 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx}{112 a}\\ &=\frac{75 c^2 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}{448 a^2}+\frac{25 c \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)}}{672 a^2}+\frac{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}}{56 a^2}-\frac{25 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{224 a}-\frac{25 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{336 a}-\frac{5 x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{84 a}+\frac{\left (c+a^2 c x^2\right )^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac{c \int \frac{\left (c+a^2 c x^2\right )^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx}{112 a}-\frac{\left (25 c^2\right ) \int \frac{\sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx}{1344 a}-\frac{\left (75 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx}{896 a}-\frac{\left (25 c^3\right ) \int \frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx}{224 a}\\ \end{align*}

Mathematica [A]  time = 5.80239, size = 0, normalized size = 0. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.964, size = 0, normalized size = 0. \begin{align*} \int x \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(5/2),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*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(5/2),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*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x \arctan \left (a x\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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