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

Optimal. Leaf size=303 \[ \frac{75}{64} c^3 \text{Unintegrable}\left (\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right )+\frac{5}{16} c^3 \text{Unintegrable}\left (\frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{a^2 c x^2+c}},x\right )+\frac{25}{96} c^2 \text{Unintegrable}\left (\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)},x\right )+\frac{1}{8} c \text{Unintegrable}\left (\left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)},x\right )+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}-\frac{25 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{32 a}+\frac{5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{5/2}-\frac{25 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{144 a}+\frac{1}{6} x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^{5/2}-\frac{\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{12 a} \]

[Out]

(-25*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(32*a) - (25*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(144*a
) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/(12*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/16 + (5
*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/6 + (75*c^3*Uni
ntegrable[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/64 + (25*c^2*Unintegrable[Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan
[a*x]], x])/96 + (c*Unintegrable[(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x])/8 + (5*c^3*Unintegrable[ArcTan[a
*x]^(5/2)/Sqrt[c + a^2*c*x^2], x])/16

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Rubi [A]  time = 0.266523, 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 = {} \[ \int \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-25*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(32*a) - (25*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(144*a
) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/(12*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/16 + (5
*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/6 + (75*c^3*Def
er[Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/64 + (25*c^2*Defer[Int][Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x
]], x])/96 + (c*Defer[Int][(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x])/8 + (5*c^3*Defer[Int][ArcTan[a*x]^(5/2
)/Sqrt[c + a^2*c*x^2], x])/16

Rubi steps

\begin{align*} \int \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 )^{5/2} \tan ^{-1}(a x)^{3/2}}{12 a}+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}+\frac{1}{8} c \int \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2} \, dx\\ &=-\frac{25 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{144 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{12 a}+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}+\frac{1}{8} c \int \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{96} \left (25 c^2\right ) \int \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{8} \left (5 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx\\ &=-\frac{25 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{32 a}-\frac{25 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{144 a}-\frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{12 a}+\frac{5}{16} c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}+\frac{5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{5/2}+\frac{1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2}+\frac{1}{8} c \int \left (c+a^2 c x^2\right )^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{96} \left (25 c^2\right ) \int \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx+\frac{1}{16} \left (5 c^3\right ) \int \frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{64} \left (75 c^3\right ) \int \frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx\\ \end{align*}

Mathematica [A]  time = 0.484942, size = 0, normalized size = 0. \[ \int \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.844, size = 0, normalized size = 0. \begin{align*} \int \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((a^2*c*x^2+c)^(5/2)*arctan(a*x)^(5/2),x)

[Out]

int((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((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((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((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}} \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)^(5/2)*arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

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