\(\int \frac {(c+a^2 c x^2)^{5/2} \arctan (a x)}{x^4} \, dx\) [223]

Optimal result

Integrand size = 22, antiderivative size = 372 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^4} \, dx=-\frac {1}{2} a^3 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2}}{6 x^2}-\frac {2 a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{x}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{3 x^3}-\frac {5 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {13}{6} a^3 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {5 i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {5 i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}} \] Output:

-1/2*a^3*c^2*(a^2*c*x^2+c)^(1/2)-1/6*a*c^2*(a^2*c*x^2+c)^(1/2)/x^2-2*a^2*c 
^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/x+1/2*a^4*c^2*x*(a^2*c*x^2+c)^(1/2)*arc 
tan(a*x)-1/3*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)/x^3-5*I*a^3*c^3*(a^2*x^2+1) 
^(1/2)*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^( 
1/2)-13/6*a^3*c^(5/2)*arctanh((a^2*c*x^2+c)^(1/2)/c^(1/2))+5/2*I*a^3*c^3*( 
a^2*x^2+1)^(1/2)*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+ 
c)^(1/2)-5/2*I*a^3*c^3*(a^2*x^2+1)^(1/2)*polylog(2,I*(1+I*a*x)^(1/2)/(1-I* 
a*x)^(1/2))/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.84 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^4} \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (-a x \sqrt {1+a^2 x^2}-3 a^3 x^3 \sqrt {1+a^2 x^2}-2 \sqrt {1+a^2 x^2} \arctan (a x)-14 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)+3 a^4 x^4 \sqrt {1+a^2 x^2} \arctan (a x)-a^3 x^3 \text {arctanh}\left (\sqrt {1+a^2 x^2}\right )+15 a^3 x^3 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )-15 a^3 x^3 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )-12 a^3 x^3 \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )\right )+12 a^3 x^3 \log \left (\sin \left (\frac {1}{2} \arctan (a x)\right )\right )+15 i a^3 x^3 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-15 i a^3 x^3 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )}{6 x^3 \sqrt {1+a^2 x^2}} \] Input:

Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^4,x]
 

Output:

(c^2*Sqrt[c + a^2*c*x^2]*(-(a*x*Sqrt[1 + a^2*x^2]) - 3*a^3*x^3*Sqrt[1 + a^ 
2*x^2] - 2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - 14*a^2*x^2*Sqrt[1 + a^2*x^2]*Ar 
cTan[a*x] + 3*a^4*x^4*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - a^3*x^3*ArcTanh[Sqrt 
[1 + a^2*x^2]] + 15*a^3*x^3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] - 15* 
a^3*x^3*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] - 12*a^3*x^3*Log[Cos[ArcT 
an[a*x]/2]] + 12*a^3*x^3*Log[Sin[ArcTan[a*x]/2]] + (15*I)*a^3*x^3*PolyLog[ 
2, (-I)*E^(I*ArcTan[a*x])] - (15*I)*a^3*x^3*PolyLog[2, I*E^(I*ArcTan[a*x]) 
]))/(6*x^3*Sqrt[1 + a^2*x^2])
 

Rubi [A] (verified)

Time = 3.30 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.89, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {5485, 5485, 5413, 5425, 5421, 5479, 243, 51, 73, 221, 5485, 5425, 5421, 5479, 243, 73, 221}

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.

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5413

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5421

\(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^4}dx\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )\)

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 243

\(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+c \left (\frac {1}{6} a \int \frac {\sqrt {a^2 c x^2+c}}{x^4}dx^2-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )\)

\(\Big \downarrow \) 51

\(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+c \left (\frac {1}{6} a \left (\frac {1}{2} a^2 c \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx^2-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )\)

\(\Big \downarrow \) 73

\(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+c \left (\frac {1}{6} a \left (\int \frac {1}{\frac {x^4}{a^2 c}-\frac {1}{a^2}}d\sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )\)

\(\Big \downarrow \) 221

\(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^2}dx+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )\)

\(\Big \downarrow \) 5485

\(\displaystyle c \left (a^2 c \left (a^2 c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \left (a^2 c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )\)

\(\Big \downarrow \) 5425

\(\displaystyle c \left (a^2 c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )\right )+a^2 c \left (c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )\)

\(\Big \downarrow \) 5421

\(\displaystyle c \left (c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (c \left (c \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )\)

\(\Big \downarrow \) 5479

\(\displaystyle c \left (c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \left (a \int \frac {1}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (c \left (c \left (a \int \frac {1}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )\)

\(\Big \downarrow \) 243

\(\displaystyle c \left (c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx^2-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (c \left (c \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx^2-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )\)

\(\Big \downarrow \) 73

\(\displaystyle c \left (c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \left (\frac {\int \frac {1}{\frac {x^4}{a^2 c}-\frac {1}{a^2}}d\sqrt {a^2 c x^2+c}}{a c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (c \left (c \left (\frac {\int \frac {1}{\frac {x^4}{a^2 c}-\frac {1}{a^2}}d\sqrt {a^2 c x^2+c}}{a c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )\)

\(\Big \downarrow \) 221

\(\displaystyle a^2 c \left (c \left (c \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )\right )+c \left (c \left (\frac {1}{6} a \left (a^2 \left (-\sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {\sqrt {a^2 c x^2+c}}{x^2}\right )-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 c x^3}\right )+a^2 c \left (c \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{\sqrt {a^2 c x^2+c}}\right )\right )\)

Input:

Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^4,x]
 

Output:

a^2*c*(a^2*c*(-1/2*Sqrt[c + a^2*c*x^2]/a + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a 
*x])/2 + (c*Sqrt[1 + a^2*x^2]*(((-2*I)*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/ 
Sqrt[1 - I*a*x]])/a + (I*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x] 
])/a - (I*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a))/(2*Sqrt[c + 
 a^2*c*x^2])) + c*(c*(-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c*x)) - (a*ArcT 
anh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c]) + (a^2*c*Sqrt[1 + a^2*x^2]*(((- 
2*I)*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/a + (I*PolyLog[2 
, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a - (I*PolyLog[2, (I*Sqrt[1 + I 
*a*x])/Sqrt[1 - I*a*x]])/a))/Sqrt[c + a^2*c*x^2])) + c*(c*(-1/3*((c + a^2* 
c*x^2)^(3/2)*ArcTan[a*x])/(c*x^3) + (a*(-(Sqrt[c + a^2*c*x^2]/x^2) - a^2*S 
qrt[c]*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]]))/6) + a^2*c*(c*(-((Sqrt[c + a 
^2*c*x^2]*ArcTan[a*x])/(c*x)) - (a*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/S 
qrt[c]) + (a^2*c*Sqrt[1 + a^2*x^2]*(((-2*I)*ArcTan[a*x]*ArcTan[Sqrt[1 + I* 
a*x]/Sqrt[1 - I*a*x]])/a + (I*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I 
*a*x]])/a - (I*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a))/Sqrt[c 
 + a^2*c*x^2]))
 

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 5413

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

rule 5421

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

rule 5425

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]   Int[(a + b*ArcTan[c*x])^ 
p/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]
 

rule 5479

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + 
 b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1)))   Int[(f*x) 
^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, 
c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] 
&& NeQ[m, -1]
 

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]))
 

Maple [A] (verified)

Time = 4.64 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.73

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 x^{4} \arctan \left (a x \right ) a^{4}-3 a^{3} x^{3}-14 x^{2} a^{2} \arctan \left (a x \right )-a x -2 \arctan \left (a x \right )\right )}{6 x^{3}}+\frac {i c^{2} a^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (15 i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-15 i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-13 i \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )+13 i \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+15 \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-15 \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{6 \sqrt {a^{2} x^{2}+1}}\)	\(270\)

Input:

int((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^4,x,method=_RETURNVERBOSE)
 

Output:

1/6*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*(3*x^4*arctan(a*x)*a^4-3*a^3*x^3-14*x^2* 
a^2*arctan(a*x)-a*x-2*arctan(a*x))/x^3+1/6*I*c^2*a^3*(c*(a*x-I)*(a*x+I))^( 
1/2)*(15*I*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-15*I*arctan(a*x 
)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-13*I*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)- 
1)+13*I*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+15*dilog(1+I*(1+I*a*x)/(a^2*x^2+ 
1)^(1/2))-15*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)
                                                                                    
                                                                                    
 

Fricas [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )}{x^{4}} \,d x } \] Input:

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^4,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^4, x)
 

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^4} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}{\left (a x \right )}}{x^{4}}\, dx \] Input:

integrate((a**2*c*x**2+c)**(5/2)*atan(a*x)/x**4,x)
 

Output:

Integral((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)/x**4, x)
 

Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )}{x^{4}} \,d x } \] Input:

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^4,x, algorithm="maxima")
 

Output:

integrate((a^2*c*x^2 + c)^(5/2)*arctan(a*x)/x^4, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)/x^4,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. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^4} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^4} \,d x \] Input:

int((atan(a*x)*(c + a^2*c*x^2)^(5/2))/x^4,x)
 

Output:

int((atan(a*x)*(c + a^2*c*x^2)^(5/2))/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^4} \, dx=\frac {\sqrt {c}\, c^{2} \left (4 \mathit {atan} \left (\sqrt {a^{2} x^{2}+1}+a x \right ) a^{3} x^{3}-2 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right ) a^{2} x^{2}-2 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )-2 \mathit {atan} \left (a x \right ) a^{3} x^{3}-\sqrt {a^{2} x^{2}+1}\, a x +12 \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )}{x^{2}}d x \right ) a^{2} x^{3}+6 \left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )d x \right ) a^{4} x^{3}+\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x -1\right ) a^{3} x^{3}-\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x +1\right ) a^{3} x^{3}\right )}{6 x^{3}} \] Input:

int((a^2*c*x^2+c)^(5/2)*atan(a*x)/x^4,x)
 

Output:

(sqrt(c)*c**2*(4*atan(sqrt(a**2*x**2 + 1) + a*x)*a**3*x**3 - 2*sqrt(a**2*x 
**2 + 1)*atan(a*x)*a**2*x**2 - 2*sqrt(a**2*x**2 + 1)*atan(a*x) - 2*atan(a* 
x)*a**3*x**3 - sqrt(a**2*x**2 + 1)*a*x + 12*int((sqrt(a**2*x**2 + 1)*atan( 
a*x))/x**2,x)*a**2*x**3 + 6*int(sqrt(a**2*x**2 + 1)*atan(a*x),x)*a**4*x**3 
 + log(sqrt(a**2*x**2 + 1) + a*x - 1)*a**3*x**3 - log(sqrt(a**2*x**2 + 1) 
+ a*x + 1)*a**3*x**3))/(6*x**3)