3.12.88 \(\int \frac {(d+e x^2)^{3/2} (a+b \arctan (c x))}{x^2} \, dx\) [1188]

3.12.88.1 Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^2} \, dx=\frac {3}{2} a e x \sqrt {d+e x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{x}+\frac {3}{2} a d \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+b \text {Int}\left (\frac {\left (d+e x^2\right )^{3/2} \arctan (c x)}{x^2},x\right ) \]

output

-a*(e*x^2+d)^(3/2)/x+3/2*a*d*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))*e^(1/2)+3/ 
2*a*e*x*(e*x^2+d)^(1/2)+b*Unintegrable((e*x^2+d)^(3/2)*arctan(c*x)/x^2,x)
 

3.12.88.2 Mathematica [N/A]

Not integrable

Time = 11.62 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^2} \, dx=\int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^2} \, dx \]

input

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

output

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

3.12.88.3 Rubi [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5517, 247, 211, 224, 219, 5560}

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.

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

\(\Big \downarrow \) 5517

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

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5560

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

input

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

output

$Aborted
 

3.12.88.3.1 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 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, 
m, p, x]
 

rule 5517

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

rule 5560

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Unintegrab 
le[u*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || M 
atchQ[u, ((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_. 
)*x)^(m_.)*((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[ 
u, ((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x) 
^(m_.)*((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, f, m, q}, x]])
 

3.12.88.4 Maple [N/A] (verified)

Not integrable

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )}{x^{2}}d x\]

input

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

output

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

3.12.88.5 Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

input

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

output

integral((a*e*x^2 + a*d + (b*e*x^2 + b*d)*arctan(c*x))*sqrt(e*x^2 + d)/x^2 
, x)
 

3.12.88.6 Sympy [N/A]

Not integrable

Time = 13.83 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^2} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]

input

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

output

Integral((a + b*atan(c*x))*(d + e*x**2)**(3/2)/x**2, x)
 

3.12.88.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^2} \, dx=\text {Exception raised: ValueError} \]

input

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^2,x, algorithm="maxima")
 

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.12.88.8 Giac [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^2} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.12.88.9 Mupad [N/A]

Not integrable

Time = 1.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} (a+b \arctan (c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2}}{x^2} \,d x \]

input

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

output

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