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

Optimal. Leaf size=360 \[ -\frac {75 c^3 \text {Int}\left (\frac {1}{\sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}},x\right )}{896 a}-\frac {25 c^3 \text {Int}\left (\frac {\tan ^{-1}(a x)^{3/2}}{\sqrt {a^2 c x^2+c}},x\right )}{224 a}-\frac {25 c^2 \text {Int}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {\tan ^{-1}(a x)}},x\right )}{1344 a}-\frac {c \text {Int}\left (\frac {\left (a^2 c x^2+c\right )^{3/2}}{\sqrt {\tan ^{-1}(a x)}},x\right )}{112 a}-\frac {25 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{224 a}+\frac {75 c^2 \sqrt {a^2 c x^2+c} \sqrt {\tan ^{-1}(a x)}}{448 a^2}+\frac {\left (a^2 c x^2+c\right )^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac {5 x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^{3/2}}{84 a}+\frac {\left (a^2 c x^2+c\right )^{5/2} \sqrt {\tan ^{-1}(a x)}}{56 a^2}-\frac {25 c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^{3/2}}{336 a}+\frac {25 c \left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}}{672 a^2} \]

[Out]

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

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Rubi [A]  time = 0.36, 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 = {} \[ \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2} \, dx \]

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*}

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Mathematica [A]  time = 6.05, size = 0, normalized size = 0.00 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^{5/2} \, dx \]

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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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(x*(a^2*c*x^2+c)^(5/2)*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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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(x*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^(5/2),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 = 2.98, size = 0, normalized size = 0.00 \[ \int x \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> 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 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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