\(\int \frac {x (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx\) [1202]

Optimal result

Integrand size = 21, antiderivative size = 103 \[ \int \frac {x (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{e}-\frac {b \sqrt {c^2 d-e} \arctan \left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{c e}-\frac {b \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c \sqrt {e}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.44 \[ \int \frac {x (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\frac {2 a c \sqrt {d+e x^2}+2 b c \sqrt {d+e x^2} \arctan (c x)-i b \sqrt {c^2 d-e} \log \left (\frac {4 c^2 e \left (-i c d+e x-i \sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{3/2} (-i+c x)}\right )+i b \sqrt {c^2 d-e} \log \left (\frac {4 c^2 e \left (i c d+e x+i \sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{3/2} (i+c x)}\right )-2 b \sqrt {e} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{2 c e} \] Input:

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

Output:

(2*a*c*Sqrt[d + e*x^2] + 2*b*c*Sqrt[d + e*x^2]*ArcTan[c*x] - I*b*Sqrt[c^2* 
d - e]*Log[(4*c^2*e*((-I)*c*d + e*x - I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/ 
(b*(c^2*d - e)^(3/2)*(-I + c*x))] + I*b*Sqrt[c^2*d - e]*Log[(4*c^2*e*(I*c* 
d + e*x + I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(3/2)*(I + c* 
x))] - 2*b*Sqrt[e]*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(2*c*e)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5509, 301, 224, 219, 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.

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

\(\Big \downarrow \) 5509

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/e - (b*c*((Sqrt[c^2*d - e]*ArcTan[(S 
qrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/c^2 + (Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqr 
t[d + e*x^2]])/c^2))/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 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 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 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 647, normalized size of antiderivative = 6.28 \[ \int \frac {x (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\left [\frac {2 \, b \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + \sqrt {-c^{2} d + e} b \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 \, \sqrt {e x^{2} + d} {\left (b c \arctan \left (c x\right ) + a c\right )}}{4 \, c e}, -\frac {\sqrt {c^{2} d - e} b \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 ) - b \sqrt {e} \log \left (-2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) - 2 \, \sqrt {e x^{2} + d} {\left (b c \arctan \left (c x\right ) + a c\right )}}{2 \, c e}, \frac {4 \, b \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + \sqrt {-c^{2} d + e} b \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 \, \sqrt {e x^{2} + d} {\left (b c \arctan \left (c x\right ) + a c\right )}}{4 \, c e}, -\frac {\sqrt {c^{2} d - e} b \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 \, b \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - 2 \, \sqrt {e x^{2} + d} {\left (b c \arctan \left (c x\right ) + a c\right )}}{2 \, c e}\right ] \] Input:

integrate(x*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),x, algorithm="fricas")
 

Output:

[1/4*(2*b*sqrt(e)*log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + sqrt(- 
c^2*d + e)*b*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*sqrt(e*x^2 + d)*(b*c*arctan(c*x) + a*c))/ 
(c*e), -1/2*(sqrt(c^2*d - e)*b*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)) - b*sqr 
t(e)*log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) - 2*sqrt(e*x^2 + d)*( 
b*c*arctan(c*x) + a*c))/(c*e), 1/4*(4*b*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e* 
x^2 + d)) + sqrt(-c^2*d + e)*b*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)*sqr 
t(e*x^2 + d) + d^2)/(c^4*x^4 + 2*c^2*x^2 + 1)) + 4*sqrt(e*x^2 + d)*(b*c*ar 
ctan(c*x) + a*c))/(c*e), -1/2*(sqrt(c^2*d - e)*b*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*b*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - 2*sqrt(e*x^ 
2 + d)*(b*c*arctan(c*x) + a*c))/(c*e)]
 

Sympy [F]

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

integrate(x*(a+b*atan(c*x))/(e*x**2+d)**(1/2),x)
 

Output:

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

Maxima [F(-2)]

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

integrate(x*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),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]

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

integrate(x*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((x*(a + b*atan(c*x)))/(d + e*x^2)^(1/2),x)
 

Output:

int((x*(a + b*atan(c*x)))/(d + e*x^2)^(1/2), x)
 

Reduce [F]

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

int(x*(a+b*atan(c*x))/(e*x^2+d)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

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