3.886 \(\int (c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=209 \[ \frac{45}{32} c^2 \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right )+\frac{3}{8} c^2 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{a^2 c x^2+c}},x\right )+\frac{5}{16} c \text{Unintegrable}\left (\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)},x\right )+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}-\frac{5 \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}-\frac{15 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{16 a} \]

[Out]

(-15*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(16*a) - (5*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(24*a) + (3
*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/4 + (45*c^2*Uninte
grable[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/32 + (5*c*Unintegrable[Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]
, x])/16 + (3*c^2*Unintegrable[ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x])/8

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Rubi [A]  time = 0.179346, 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 = {} \[ \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2),x]

[Out]

(-15*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(16*a) - (5*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(24*a) + (3
*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/4 + (45*c^2*Defer[
Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/32 + (5*c*Defer[Int][Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]
)/16 + (3*c^2*Defer[Int][ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x])/8

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx &=-\frac{5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac{1}{16} (5 c) \int \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{4} (3 c) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx\\ &=-\frac{15 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{16 a}-\frac{5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac{1}{16} (5 c) \int \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{8} \left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{32} \left (45 c^2\right ) \int \frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx\\ \end{align*}

Mathematica [A]  time = 1.51022, size = 0, normalized size = 0. \[ \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2),x]

[Out]

Integrate[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x]

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Maple [A]  time = 0.741, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{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((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(5/2),x)

[Out]

int((a^2*c*x^2+c)^(3/2)*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((a^2*c*x^2+c)^(3/2)*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((a^2*c*x^2+c)^(3/2)*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((a**2*c*x**2+c)**(3/2)*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{3}{2}} \arctan \left (a x\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate((a^2*c*x^2 + c)^(3/2)*arctan(a*x)^(5/2), x)