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

Optimal. Leaf size=212 \[ \frac {9}{32} c^2 \text {Int}\left (\frac {1}{\sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}},x\right )+\frac {3}{8} c^2 \text {Int}\left (\frac {\tan ^{-1}(a x)^{3/2}}{\sqrt {a^2 c x^2+c}},x\right )+\frac {1}{16} c \text {Int}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {\tan ^{-1}(a x)}},x\right )+\frac {3}{8} c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}-\frac {9 c \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}{16 a}+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}-\frac {\left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}{8 a} \]

[Out]

1/4*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^(3/2)+3/8*c*x*arctan(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)-1/8*(a^2*c*x^2+c)^(3
/2)*arctan(a*x)^(1/2)/a-9/16*c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^(1/2)/a+3/8*c^2*Unintegrable(arctan(a*x)^(3/2)/
(a^2*c*x^2+c)^(1/2),x)+9/32*c^2*Unintegrable(1/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)+1/16*c*Unintegrable((a
^2*c*x^2+c)^(1/2)/arctan(a*x)^(1/2),x)

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Rubi [A]  time = 0.18, 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 = {} \[ \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-9*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(16*a) - ((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(8*a) + (3*c*x
*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/4 + (9*c^2*Defer[Int][
1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/32 + (c*Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/1
6 + (3*c^2*Defer[Int][ArcTan[a*x]^(3/2)/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)^{3/2} \, dx &=-\frac {\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}{8 a}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}+\frac {1}{16} c \int \frac {\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)^{3/2} \, dx\\ &=-\frac {9 c \sqrt {c+a^2 c x^2} \sqrt {\tan ^{-1}(a x)}}{16 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}{8 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2}+\frac {1}{16} c \int \frac {\sqrt {c+a^2 c x^2}}{\sqrt {\tan ^{-1}(a x)}} \, dx+\frac {1}{32} \left (9 c^2\right ) \int \frac {1}{\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)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 1.49, size = 0, normalized size = 0.00 \[ \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^{3/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 1.50, size = 0, normalized size = 0.00 \[ \int \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^(3/2),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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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*c*x**2+c)**(3/2)*atan(a*x)**(3/2),x)

[Out]

Timed out

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