\(\int \frac {x (a+b \arctan (c x))}{(d+e x^2)^{3/2}} \, dx\) [1210]

Optimal result

Integrand size = 21, antiderivative size = 71 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {a+b \arctan (c x)}{e \sqrt {d+e x^2}}+\frac {b c \arctan \left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c^2 d-e} e} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.96 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {\frac {2 a}{\sqrt {d+e x^2}}+\frac {2 b \arctan (c x)}{\sqrt {d+e x^2}}+\frac {i b c \log \left (-\frac {4 i e \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \sqrt {c^2 d-e} (i+c x)}\right )}{\sqrt {c^2 d-e}}-\frac {i b c \log \left (\frac {4 i e \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \sqrt {c^2 d-e} (-i+c x)}\right )}{\sqrt {c^2 d-e}}}{2 e} \] Input:

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

Output:

-1/2*((2*a)/Sqrt[d + e*x^2] + (2*b*ArcTan[c*x])/Sqrt[d + e*x^2] + (I*b*c*L 
og[((-4*I)*e*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2* 
d - e]*(I + c*x))])/Sqrt[c^2*d - e] - (I*b*c*Log[((4*I)*e*(c*d + I*e*x + S 
qrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(-I + c*x))])/Sqrt[c^2 
*d - e])/e
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5509, 291, 216}

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.

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

\(\Big \downarrow \) 5509

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

-((a + b*ArcTan[c*x])/(e*Sqrt[d + e*x^2])) + (b*c*ArcTan[(Sqrt[c^2*d - e]* 
x)/Sqrt[d + e*x^2]])/(Sqrt[c^2*d - e]*e)
 

Defintions of rubi rules used

rule 216

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

rule 291

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

rule 5509

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

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (63) = 126\).

Time = 0.18 (sec) , antiderivative size = 379, normalized size of antiderivative = 5.34 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (b c e x^{2} + b c d\right )} \sqrt {-c^{2} d + e} \log \left (\frac {{\left (c^{4} d^{2} - 8 \, c^{2} d e + 8 \, e^{2}\right )} x^{4} - 2 \, {\left (3 \, c^{2} d^{2} - 4 \, d e\right )} x^{2} - 4 \, {\left ({\left (c^{2} d - 2 \, e\right )} x^{3} - d x\right )} \sqrt {-c^{2} d + e} \sqrt {e x^{2} + d} + d^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) + 4 \, {\left (a c^{2} d - a e + {\left (b c^{2} d - b e\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{4 \, {\left (c^{2} d^{2} e - d e^{2} + {\left (c^{2} d e^{2} - e^{3}\right )} x^{2}\right )}}, \frac {{\left (b c e x^{2} + b c d\right )} \sqrt {c^{2} d - e} \arctan \left (\frac {\sqrt {c^{2} d - e} {\left ({\left (c^{2} d - 2 \, e\right )} x^{2} - d\right )} \sqrt {e x^{2} + d}}{2 \, {\left ({\left (c^{2} d e - e^{2}\right )} x^{3} + {\left (c^{2} d^{2} - d e\right )} x\right )}}\right ) - 2 \, {\left (a c^{2} d - a e + {\left (b c^{2} d - b e\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{2 \, {\left (c^{2} d^{2} e - d e^{2} + {\left (c^{2} d e^{2} - e^{3}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[-1/4*((b*c*e*x^2 + b*c*d)*sqrt(-c^2*d + e)*log(((c^4*d^2 - 8*c^2*d*e + 8* 
e^2)*x^4 - 2*(3*c^2*d^2 - 4*d*e)*x^2 - 4*((c^2*d - 2*e)*x^3 - d*x)*sqrt(-c 
^2*d + e)*sqrt(e*x^2 + d) + d^2)/(c^4*x^4 + 2*c^2*x^2 + 1)) + 4*(a*c^2*d - 
 a*e + (b*c^2*d - b*e)*arctan(c*x))*sqrt(e*x^2 + d))/(c^2*d^2*e - d*e^2 + 
(c^2*d*e^2 - e^3)*x^2), 1/2*((b*c*e*x^2 + b*c*d)*sqrt(c^2*d - e)*arctan(1/ 
2*sqrt(c^2*d - e)*((c^2*d - 2*e)*x^2 - d)*sqrt(e*x^2 + d)/((c^2*d*e - e^2) 
*x^3 + (c^2*d^2 - d*e)*x)) - 2*(a*c^2*d - a*e + (b*c^2*d - b*e)*arctan(c*x 
))*sqrt(e*x^2 + d))/(c^2*d^2*e - d*e^2 + (c^2*d*e^2 - e^3)*x^2)]
 

Sympy [F]

\[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

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

integrate(x*(a+b*arctan(c*x))/(e*x^2+d)^(3/2),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-c^2*d>0)', see `assume?` for m 
ore detail
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, a +\left (\int \frac {\mathit {atan} \left (c x \right ) x}{\sqrt {e \,x^{2}+d}\, d +\sqrt {e \,x^{2}+d}\, e \,x^{2}}d x \right ) b d e +\left (\int \frac {\mathit {atan} \left (c x \right ) x}{\sqrt {e \,x^{2}+d}\, d +\sqrt {e \,x^{2}+d}\, e \,x^{2}}d x \right ) b \,e^{2} x^{2}}{e \left (e \,x^{2}+d \right )} \] Input:

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

Output:

( - sqrt(d + e*x**2)*a + int((atan(c*x)*x)/(sqrt(d + e*x**2)*d + sqrt(d + 
e*x**2)*e*x**2),x)*b*d*e + int((atan(c*x)*x)/(sqrt(d + e*x**2)*d + sqrt(d 
+ e*x**2)*e*x**2),x)*b*e**2*x**2)/(e*(d + e*x**2))