3.9.86 \(\int (c+a^2 c x^2)^{3/2} \text {ArcTan}(a x)^{5/2} \, dx\) [886]

Optimal. Leaf size=212 \[ -\frac {15 c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{3/2}}{16 a}-\frac {5 \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{3/2}}{24 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^{5/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2}+\frac {45}{32} c^2 \text {Int}\left (\frac {\sqrt {\text {ArcTan}(a x)}}{\sqrt {c+a^2 c x^2}},x\right )+\frac {5}{16} c \text {Int}\left (\sqrt {c+a^2 c x^2} \sqrt {\text {ArcTan}(a x)},x\right )+\frac {3}{8} c^2 \text {Int}\left (\frac {\text {ArcTan}(a x)^{5/2}}{\sqrt {c+a^2 c x^2}},x\right ) \]

[Out]

-5/24*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(3/2)/a+1/4*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(5/2)-15/16*c*arctan(a*x)^
(3/2)*(a^2*c*x^2+c)^(1/2)/a+3/8*c*x*arctan(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)+3/8*c^2*Unintegrable(arctan(a*x)^(5/
2)/(a^2*c*x^2+c)^(1/2),x)+45/32*c^2*Unintegrable(arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(1/2),x)+5/16*c*Unintegrable(
(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2),x)

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Rubi [A]
time = 0.12, 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 = {} \begin {gather*} \int \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 1.10, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^{5/2} \, dx \end {gather*}

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 = 0.53, size = 0, normalized size = 0.00 \[\int \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3877 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)^(5/2)*(c + a^2*c*x^2)^(3/2),x)

[Out]

int(atan(a*x)^(5/2)*(c + a^2*c*x^2)^(3/2), x)

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