\(\int \frac {(c+a^2 c x^2)^{5/2} \arctan (a x)}{x} \, dx\) [220]

Optimal result

Integrand size = 22, antiderivative size = 329 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x} \, dx=-\frac {29}{120} a c^2 x \sqrt {c+a^2 c x^2}-\frac {1}{20} a c x \left (c+a^2 c x^2\right )^{3/2}+c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)-\frac {2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149}{120} c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}} \] Output:

-29/120*a*c^2*x*(a^2*c*x^2+c)^(1/2)-1/20*a*c*x*(a^2*c*x^2+c)^(3/2)+c^2*(a^ 
2*c*x^2+c)^(1/2)*arctan(a*x)+1/3*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)+1/5*(a^ 
2*c*x^2+c)^(5/2)*arctan(a*x)-2*c^3*(a^2*x^2+1)^(1/2)*arctan(a*x)*arctanh(( 
1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^(1/2)-149/120*c^(5/2)*arctan 
h(a*c^(1/2)*x/(a^2*c*x^2+c)^(1/2))+I*c^3*(a^2*x^2+1)^(1/2)*polylog(2,-(1+I 
*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^(1/2)-I*c^3*(a^2*x^2+1)^(1/2)*p 
olylog(2,(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.86 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x} \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (-35 a x \sqrt {1+a^2 x^2}-6 a^3 x^3 \sqrt {1+a^2 x^2}+184 \sqrt {1+a^2 x^2} \arctan (a x)+88 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)+24 a^4 x^4 \sqrt {1+a^2 x^2} \arctan (a x)+120 \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-120 \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+29 \log \left (-a x+\sqrt {1+a^2 x^2}\right )+120 \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )-\sin \left (\frac {1}{2} \arctan (a x)\right )\right )-120 \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )+\sin \left (\frac {1}{2} \arctan (a x)\right )\right )+120 i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-120 i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )}{120 \sqrt {1+a^2 x^2}} \] Input:

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

Output:

(c^2*Sqrt[c + a^2*c*x^2]*(-35*a*x*Sqrt[1 + a^2*x^2] - 6*a^3*x^3*Sqrt[1 + a 
^2*x^2] + 184*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 88*a^2*x^2*Sqrt[1 + a^2*x^2] 
*ArcTan[a*x] + 24*a^4*x^4*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 120*ArcTan[a*x]* 
Log[1 - E^(I*ArcTan[a*x])] - 120*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + 
29*Log[-(a*x) + Sqrt[1 + a^2*x^2]] + 120*Log[Cos[ArcTan[a*x]/2] - Sin[ArcT 
an[a*x]/2]] - 120*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (120*I)*P 
olyLog[2, -E^(I*ArcTan[a*x])] - (120*I)*PolyLog[2, E^(I*ArcTan[a*x])]))/(1 
20*Sqrt[1 + a^2*x^2])
 

Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.21, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {5485, 5465, 211, 211, 224, 219, 5485, 5465, 211, 224, 219, 5481, 224, 219, 5493, 5489}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{x} \, dx\)

\(\Big \downarrow \) 5485

\(\displaystyle a^2 c \int x \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)dx+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{x}dx\)

\(\Big \downarrow \) 5465

\(\displaystyle a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\int \left (a^2 c x^2+c\right )^{3/2}dx}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{x}dx\)

\(\Big \downarrow \) 211

\(\displaystyle a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \int \sqrt {a^2 c x^2+c}dx+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{x}dx\)

\(\Big \downarrow \) 211

\(\displaystyle a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{x}dx\)

\(\Big \downarrow \) 224

\(\displaystyle a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{x}dx\)

\(\Big \downarrow \) 219

\(\displaystyle c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{x}dx+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )\)

\(\Big \downarrow \) 5485

\(\displaystyle c \left (a^2 c \int x \sqrt {a^2 c x^2+c} \arctan (a x)dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )\)

\(\Big \downarrow \) 5465

\(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\int \sqrt {a^2 c x^2+c}dx}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )\)

\(\Big \downarrow \) 211

\(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )\)

\(\Big \downarrow \) 224

\(\displaystyle c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )\)

\(\Big \downarrow \) 5481

\(\displaystyle c \left (c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-a c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )\)

\(\Big \downarrow \) 224

\(\displaystyle c \left (c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-a c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle c \left (c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )\)

\(\Big \downarrow \) 5493

\(\displaystyle c \left (c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )\)

\(\Big \downarrow \) 5489

\(\displaystyle a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{5 a^2 c}-\frac {\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}}{5 a}\right )+c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\right )\)

Input:

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

Output:

a^2*c*(((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(5*a^2*c) - ((x*(c + a^2*c*x^2) 
^(3/2))/4 + (3*c*((x*Sqrt[c + a^2*c*x^2])/2 + (Sqrt[c]*ArcTanh[(a*Sqrt[c]* 
x)/Sqrt[c + a^2*c*x^2]])/(2*a)))/4)/(5*a)) + c*(a^2*c*(((c + a^2*c*x^2)^(3 
/2)*ArcTan[a*x])/(3*a^2*c) - ((x*Sqrt[c + a^2*c*x^2])/2 + (Sqrt[c]*ArcTanh 
[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(2*a))/(3*a)) + c*(Sqrt[c + a^2*c*x^2 
]*ArcTan[a*x] - Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + (c*Sq 
rt[1 + a^2*x^2]*(-2*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + 
 I*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[1 + 
I*a*x]/Sqrt[1 - I*a*x]]))/Sqrt[c + a^2*c*x^2]))
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 219

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

rule 224

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 5465

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] - Simp[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 5481

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

rule 5485

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

rule 5489

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_ 
Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTan[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sq 
rt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 
 - I*c*x]], x] - Simp[I*(b/Sqrt[d])*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c 
*x]], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
 

rule 5493

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

Maple [A] (verified)

Time = 2.30 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.60

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (24 x^{4} \arctan \left (a x \right ) a^{4}-6 a^{3} x^{3}+88 x^{2} a^{2} \arctan \left (a x \right )-35 a x +184 \arctan \left (a x \right )\right )}{120}-\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (60 \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-149 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-60 i \operatorname {dilog}\left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-60 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{60 \sqrt {a^{2} x^{2}+1}}\)	\(198\)

Input:

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

Output:

1/120*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*(24*x^4*arctan(a*x)*a^4-6*a^3*x^3+88*x 
^2*a^2*arctan(a*x)-35*a*x+184*arctan(a*x))-1/60*c^2*(c*(a*x-I)*(a*x+I))^(1 
/2)*(60*arctan(a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-149*I*arctan((1+I*a* 
x)/(a^2*x^2+1)^(1/2))-60*I*dilog(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-60*I*dilog 
((1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)
 

Fricas [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )}{x} \,d x } \] Input:

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

Output:

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)*arctan(a* 
x)/x, x)
 

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}{\left (a x \right )}}{x}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )}{x} \,d x } \] Input:

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

Output:

2/3*(a^2*c^2*x^2 + c^2)*sqrt(a^2*x^2 + 1)*sqrt(c)*arctan(a*x) - 1/3*(a^4*x 
^4 + 10*a^2*x^2 + 9)^(1/4)*(a*c^2*x*cos(1/2*arctan2(4*a*x, -a^2*x^2 + 3)) 
+ 2*c^2*sin(1/2*arctan2(4*a*x, -a^2*x^2 + 3)))*sqrt(c) - 1/120*((a*(3*(2*( 
a^2*x^2 + 1)^(3/2)*x/a^2 - sqrt(a^2*x^2 + 1)*x/a^2 - arcsinh(a*x)/a^3)/a^2 
 - 8*(sqrt(a^2*x^2 + 1)*x + arcsinh(a*x)/a)/a^4) - 8*(3*(a^2*x^2 + 1)^(3/2 
)*x^2/a^2 - 2*(a^2*x^2 + 1)^(3/2)/a^4)*arctan(a*x))*a^4*c^2 - 20*c^2*arcta 
n2((a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*sin(1/2*arctan2(4*a*x, a^2*x^2 - 3)) + 
 2, a*x + (a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*cos(1/2*arctan2(4*a*x, a^2*x^2 
- 3))) - 20*c^2*arctan2((a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*sin(1/2*arctan2(4 
*a*x, a^2*x^2 - 3)) - 2, -a*x + (a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*cos(1/2*a 
rctan2(4*a*x, a^2*x^2 - 3))) - 120*c^2*integrate(sqrt(a^2*x^2 + 1)*arctan( 
a*x)/x, x))*sqrt(c)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x} \, dx=\frac {\sqrt {c}\, c^{2} \left (24 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right ) a^{4} x^{4}+88 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right ) a^{2} x^{2}+64 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )-6 \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}-35 \sqrt {a^{2} x^{2}+1}\, a x +120 \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )}{x}d x \right )-29 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x \right )\right )}{120} \] Input:

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

Output:

(sqrt(c)*c**2*(24*sqrt(a**2*x**2 + 1)*atan(a*x)*a**4*x**4 + 88*sqrt(a**2*x 
**2 + 1)*atan(a*x)*a**2*x**2 + 64*sqrt(a**2*x**2 + 1)*atan(a*x) - 6*sqrt(a 
**2*x**2 + 1)*a**3*x**3 - 35*sqrt(a**2*x**2 + 1)*a*x + 120*int((sqrt(a**2* 
x**2 + 1)*atan(a*x))/x,x) - 29*log(sqrt(a**2*x**2 + 1) + a*x)))/120