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

Optimal. Leaf size=170 \[ -\frac{5 x}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)}{3 a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{a^6 c^{5/2}}-\frac{x^3}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}} \]

[Out]

-x^3/(9*a^3*c*(c + a^2*c*x^2)^(3/2)) - (5*x)/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x])/(3*a^4*c*(c +
a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x])/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^6*
c^3) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^6*c^(5/2))

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Rubi [A]  time = 0.433126, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4964, 4930, 217, 206, 191, 4938} \[ -\frac{5 x}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)}{3 a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{a^6 c^{5/2}}-\frac{x^3}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^3/(9*a^3*c*(c + a^2*c*x^2)^(3/2)) - (5*x)/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x])/(3*a^4*c*(c +
a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x])/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^6*
c^3) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^6*c^(5/2))

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/e, Int[
x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d/e, Int[x^(m - 2)*(d + e*x^2)^q*(a + b*Arc
Tan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && IGtQ[m
, 1] && NeQ[p, -1]

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[((d + e*x^2)^
(q + 1)*(a + b*ArcTan[c*x])^p)/(2*e*(q + 1)), x] - Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan[c
*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 4938

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(b*(f*x
)^m*(d + e*x^2)^(q + 1))/(c*d*m^2), x] + (Dist[(f^2*(m - 1))/(c^2*d*m), Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*
(a + b*ArcTan[c*x]), x], x] - Simp[(f*(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]))/(c^2*d*m), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[q, -1]

Rubi steps

\begin{align*} \int \frac{x^5 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac{\int \frac{x^3 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac{\int \frac{x^3 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c}\\ &=-\frac{x^3}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{\int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{a^4 c^2}-\frac{2 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^4 c}-\frac{\int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c}\\ &=-\frac{x^3}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)}{3 a^6 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^6 c^3}-\frac{\int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{a^5 c^2}-\frac{2 \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^5 c}-\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^5 c}\\ &=-\frac{x^3}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 x}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{x^2 \tan ^{-1}(a x)}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)}{3 a^6 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^6 c^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{a^5 c^2}\\ &=-\frac{x^3}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 x}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{x^2 \tan ^{-1}(a x)}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)}{3 a^6 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^6 c^3}-\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a^6 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.202126, size = 131, normalized size = 0.77 \[ -\frac{a x \left (16 a^2 x^2+15\right ) \sqrt{a^2 c x^2+c}+9 \sqrt{c} \left (a^2 x^2+1\right )^2 \log \left (\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right )-3 \left (3 a^4 x^4+12 a^2 x^2+8\right ) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{9 a^6 c^3 \left (a^2 x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*x*(15 + 16*a^2*x^2)*Sqrt[c + a^2*c*x^2] - 3*Sqrt[c + a^2*c*x^2]*(8 + 12*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x] +
9*Sqrt[c]*(1 + a^2*x^2)^2*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(9*a^6*c^3*(1 + a^2*x^2)^2)

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Maple [C]  time = 1.507, size = 386, normalized size = 2.3 \begin{align*}{\frac{ \left ( i+3\,\arctan \left ( ax \right ) \right ) \left ( i{x}^{3}{a}^{3}+3\,{a}^{2}{x}^{2}-3\,iax-1 \right ) }{72\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{c}^{3}{a}^{6}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( 7\,\arctan \left ( ax \right ) +7\,i \right ) \left ( 1+iax \right ) }{8\,{c}^{3}{a}^{6} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( -7+7\,iax \right ) \left ( \arctan \left ( ax \right ) -i \right ) }{8\,{c}^{3}{a}^{6} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( i{x}^{3}{a}^{3}-3\,{a}^{2}{x}^{2}-3\,iax+1 \right ) \left ( -i+3\,\arctan \left ( ax \right ) \right ) }{72\,{c}^{3}{a}^{6} \left ({a}^{4}{x}^{4}+2\,{a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{\arctan \left ( ax \right ) }{{c}^{3}{a}^{6}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{1}{{c}^{3}{a}^{6}}\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-i \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{\frac{1}{{c}^{3}{a}^{6}}\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+i \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(I+3*arctan(a*x))*(I*x^3*a^3+3*a^2*x^2-3*I*a*x-1)*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)^2/c^3/a^6+7/8*(ar
ctan(a*x)+I)*(1+I*a*x)*(c*(a*x-I)*(a*x+I))^(1/2)/a^6/c^3/(a^2*x^2+1)-7/8*(c*(a*x-I)*(a*x+I))^(1/2)*(-1+I*a*x)*
(arctan(a*x)-I)/a^6/c^3/(a^2*x^2+1)-1/72*(c*(a*x-I)*(a*x+I))^(1/2)*(I*x^3*a^3-3*a^2*x^2-3*I*a*x+1)*(-I+3*arcta
n(a*x))/a^6/c^3/(a^4*x^4+2*a^2*x^2+1)+arctan(a*x)*(c*(a*x-I)*(a*x+I))^(1/2)/c^3/a^6+ln((1+I*a*x)/(a^2*x^2+1)^(
1/2)-I)/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(a*x+I))^(1/2)/a^6/c^3-ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+I)/(a^2*x^2+1)^(1/2
)*(c*(a*x-I)*(a*x+I))^(1/2)/a^6/c^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.49651, size = 316, normalized size = 1.86 \begin{align*} \frac{9 \,{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \sqrt{c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt{a^{2} c x^{2} + c} a \sqrt{c} x - c\right ) - 2 \,{\left (16 \, a^{3} x^{3} + 15 \, a x - 3 \,{\left (3 \, a^{4} x^{4} + 12 \, a^{2} x^{2} + 8\right )} \arctan \left (a x\right )\right )} \sqrt{a^{2} c x^{2} + c}}{18 \,{\left (a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(9*(a^4*x^4 + 2*a^2*x^2 + 1)*sqrt(c)*log(-2*a^2*c*x^2 + 2*sqrt(a^2*c*x^2 + c)*a*sqrt(c)*x - c) - 2*(16*a^
3*x^3 + 15*a*x - 3*(3*a^4*x^4 + 12*a^2*x^2 + 8)*arctan(a*x))*sqrt(a^2*c*x^2 + c))/(a^10*c^3*x^4 + 2*a^8*c^3*x^
2 + a^6*c^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \operatorname{atan}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5*atan(a*x)/(c*(a**2*x**2 + 1))**(5/2), x)

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Giac [A]  time = 1.24713, size = 177, normalized size = 1.04 \begin{align*} -\frac{x{\left (\frac{16 \, x^{2}}{a^{3} c} + \frac{15}{a^{5} c}\right )}}{9 \,{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}} + \frac{{\left (3 \, \sqrt{a^{2} c x^{2} + c} + \frac{6 \,{\left (a^{2} c x^{2} + c\right )} c - c^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\right )} \arctan \left (a x\right )}{3 \, a^{6} c^{3}} + \frac{\log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} + c} \right |}\right )}{a^{5} c^{\frac{5}{2}}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*x*(16*x^2/(a^3*c) + 15/(a^5*c))/(a^2*c*x^2 + c)^(3/2) + 1/3*(3*sqrt(a^2*c*x^2 + c) + (6*(a^2*c*x^2 + c)*c
- c^2)/(a^2*c*x^2 + c)^(3/2))*arctan(a*x)/(a^6*c^3) + log(abs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 + c)))/(a^5*c^(
5/2)*abs(a))