\(\int \frac {x (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx\) [1203]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5094, 399, 223, 212, 385, 209} \[ \int \frac {x (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{e}-\frac {b \sqrt {c^2 d-e} \arctan \left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{c e}-\frac {b \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c \sqrt {e}} \]

[In]

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

[Out]

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

Rule 209

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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 385

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

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 5094

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{e}-\frac {(b c) \int \frac {\sqrt {d+e x^2}}{1+c^2 x^2} \, dx}{e} \\ & = \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{e}-\frac {b \int \frac {1}{\sqrt {d+e x^2}} \, dx}{c}+\frac {\left (b \left (-c^2 d+e\right )\right ) \int \frac {1}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{c e} \\ & = \frac {\sqrt {d+e x^2} (a+b \arctan (c x))}{e}-\frac {b \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c}+\frac {\left (b \left (-c^2 d+e\right )\right ) \text {Subst}\left (\int \frac {1}{1-\left (-c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c e} \\ & = \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}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.44 \[ \int \frac {x (a+b \arctan (c x))}{\sqrt {d+e x^2}} \, dx=\frac {2 a c \sqrt {d+e x^2}+2 b c \sqrt {d+e x^2} \arctan (c x)-i b \sqrt {c^2 d-e} \log \left (\frac {4 c^2 e \left (-i c d+e x-i \sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{3/2} (-i+c x)}\right )+i b \sqrt {c^2 d-e} \log \left (\frac {4 c^2 e \left (i c d+e x+i \sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{3/2} (i+c x)}\right )-2 b \sqrt {e} \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{2 c e} \]

[In]

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

[Out]

(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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (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 ] \]

[In]

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

[Out]

[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*arc
tan(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
*sqrt(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)*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)) - 2*b*sqrt(-e)*ar
ctan(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]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e-c^2*d>0)', see `assume?` for
more detail

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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