\(\int \frac {x (a+b \arctan (c x))}{(d+e x^2)^3} \, dx\) [1167]

Optimal result

Integrand size = 19, antiderivative size = 131 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {b c^4 \arctan (c x)}{4 \left (c^2 d-e\right )^2 e}-\frac {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} \left (c^2 d-e\right )^2 \sqrt {e}} \]

[Out]

-1/8*b*c*x/d/(c^2*d-e)/(e*x^2+d)+1/4*b*c^4*arctan(c*x)/(c^2*d-e)^2/e+1/4*(-a-b*arctan(c*x))/e/(e*x^2+d)^2-1/8*
b*c*(3*c^2*d-e)*arctan(x*e^(1/2)/d^(1/2))/d^(3/2)/(c^2*d-e)^2/e^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5094, 425, 536, 209, 211} \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} \sqrt {e} \left (c^2 d-e\right )^2}+\frac {b c^4 \arctan (c x)}{4 e \left (c^2 d-e\right )^2}-\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )} \]

[In]

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

[Out]

-1/8*(b*c*x)/(d*(c^2*d - e)*(d + e*x^2)) + (b*c^4*ArcTan[c*x])/(4*(c^2*d - e)^2*e) - (a + b*ArcTan[c*x])/(4*e*
(d + e*x^2)^2) - (b*c*(3*c^2*d - e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*(c^2*d - e)^2*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 211

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

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

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 {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{4 e} \\ & = -\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c) \int \frac {2 c^2 d-e-c^2 e x^2}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{8 d \left (c^2 d-e\right ) e} \\ & = -\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}-\frac {\left (b c \left (3 c^2 d-e\right )\right ) \int \frac {1}{d+e x^2} \, dx}{8 d \left (c^2 d-e\right )^2}+\frac {\left (b c^5\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 \left (c^2 d-e\right )^2 e} \\ & = -\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {b c^4 \arctan (c x)}{4 \left (c^2 d-e\right )^2 e}-\frac {a+b \arctan (c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} \left (c^2 d-e\right )^2 \sqrt {e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {1}{8} \left (-\frac {\frac {2 a}{e}+\frac {b c x \left (d+e x^2\right )}{d \left (c^2 d-e\right )}}{\left (d+e x^2\right )^2}+\frac {2 b \left (\frac {c^4}{\left (-c^2 d+e\right )^2}-\frac {1}{\left (d+e x^2\right )^2}\right ) \arctan (c x)}{e}-\frac {b c \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e} \left (-c^2 d+e\right )^2}\right ) \]

[In]

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

[Out]

(-(((2*a)/e + (b*c*x*(d + e*x^2))/(d*(c^2*d - e)))/(d + e*x^2)^2) + (2*b*(c^4/(-(c^2*d) + e)^2 - (d + e*x^2)^(
-2))*ArcTan[c*x])/e - (b*c*(3*c^2*d - e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]*(-(c^2*d) + e)^2))/8

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.20

method	result	size
parts	\(-\frac {a}{4 e \left (e \,x^{2}+d \right )^{2}}+\frac {b \left (-\frac {\arctan \left (c x \right ) c^{6}}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+\frac {c^{6} \left (-\frac {e \left (\frac {\left (c^{2} d -e \right ) x}{2 d c \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\left (3 c^{2} d -e \right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 d \,c^{3} \sqrt {e d}}\right )}{\left (c^{2} d -e \right )^{2}}+\frac {\arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}\right )}{4 e}\right )}{c^{2}}\)	\(157\)
derivativedivides	\(\frac {-\frac {a \,c^{6}}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+b \,c^{6} \left (-\frac {\arctan \left (c x \right )}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+\frac {-\frac {e \left (\frac {\left (c^{2} d -e \right ) x}{2 d c \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\left (3 c^{2} d -e \right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 d \,c^{3} \sqrt {e d}}\right )}{\left (c^{2} d -e \right )^{2}}+\frac {\arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}}{4 e}\right )}{c^{2}}\)	\(165\)
default	\(\frac {-\frac {a \,c^{6}}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+b \,c^{6} \left (-\frac {\arctan \left (c x \right )}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}+\frac {-\frac {e \left (\frac {\left (c^{2} d -e \right ) x}{2 d c \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {\left (3 c^{2} d -e \right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 d \,c^{3} \sqrt {e d}}\right )}{\left (c^{2} d -e \right )^{2}}+\frac {\arctan \left (c x \right )}{\left (c^{2} d -e \right )^{2}}}{4 e}\right )}{c^{2}}\)	\(165\)
risch	\(\frac {i b \ln \left (i c x +1\right )}{8 e \left (e \,x^{2}+d \right )^{2}}+\frac {c^{5} b x}{16 \left (c^{2} d -e \right )^{2} \left (-e \,c^{2} x^{2}-c^{2} d \right )}-\frac {c^{3} b e x}{16 \left (c^{2} d -e \right )^{2} \left (-e \,c^{2} x^{2}-c^{2} d \right ) d}+\frac {i c^{6} b \ln \left (-i c x +1\right ) d}{4 \left (-e \,c^{2} x^{2}-c^{2} d \right )^{2} \left (c^{2} d -e \right )^{2}}+\frac {i c b \,\operatorname {arctanh}\left (\frac {2 \left (-i c x +1\right ) e -2 e}{2 c \sqrt {e d}}\right ) e}{16 \left (c^{2} d -e \right )^{2} d \sqrt {e d}}-\frac {3 i c^{3} b \,\operatorname {arctanh}\left (\frac {2 \left (-i c x +1\right ) e -2 e}{2 c \sqrt {e d}}\right )}{16 \left (c^{2} d -e \right )^{2} \sqrt {e d}}-\frac {i c^{4} b \ln \left (-i c x +1\right ) e}{8 \left (-e \,c^{2} x^{2}-c^{2} d \right )^{2} \left (c^{2} d -e \right )^{2}}+\frac {i c^{8} b \ln \left (-i c x +1\right ) d \,x^{2}}{4 \left (-e \,c^{2} x^{2}-c^{2} d \right )^{2} \left (c^{2} d -e \right )^{2}}-\frac {i b \,c^{4} d}{16 e \left (c^{2} d -e \right )^{2} \left (e \,x^{2}+d \right )}+\frac {i b \,c^{2}}{16 \left (c^{2} d -e \right )^{2} \left (e \,x^{2}+d \right )}+\frac {i c^{8} b \ln \left (-i c x +1\right ) e \,x^{4}}{8 \left (-e \,c^{2} x^{2}-c^{2} d \right )^{2} \left (c^{2} d -e \right )^{2}}-\frac {i c^{4} b \ln \left (\left (-i c x +1\right )^{2} e -c^{2} d -2 \left (-i c x +1\right ) e +e \right )}{16 \left (c^{2} d -e \right )^{2} e}-\frac {c^{4} a}{4 e \left (-e \,c^{2} x^{2}-c^{2} d \right )^{2}}+\frac {i c^{4} b}{16 \left (c^{2} d -e \right )^{2} \left (-e \,c^{2} x^{2}-c^{2} d \right )}+\frac {b \,c^{4} \arctan \left (c x \right )}{8 \left (c^{2} d -e \right )^{2} e}-\frac {b \,c^{3} x}{16 \left (c^{2} d -e \right )^{2} \left (e \,x^{2}+d \right )}+\frac {e b c x}{16 \left (c^{2} d -e \right )^{2} \left (e \,x^{2}+d \right ) d}-\frac {i b \,c^{4} \ln \left (c^{2} x^{2}+1\right )}{16 e \left (c^{2} d -e \right )^{2}}-\frac {i c^{6} b d}{16 \left (c^{2} d -e \right )^{2} \left (-e \,c^{2} x^{2}-c^{2} d \right ) e}+\frac {i b \,c^{4} \ln \left (e \,x^{2}+d \right )}{16 e \left (c^{2} d -e \right )^{2}}-\frac {3 b \,c^{3} \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{16 \left (c^{2} d -e \right )^{2} \sqrt {e d}}+\frac {e b c \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{16 \left (c^{2} d -e \right )^{2} d \sqrt {e d}}\)	\(802\)

[In]

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

[Out]

-1/4*a/e/(e*x^2+d)^2+b/c^2*(-1/4*arctan(c*x)*c^6/e/(c^2*e*x^2+c^2*d)^2+1/4*c^6/e*(-1/(c^2*d-e)^2*e*(1/2*(c^2*d
-e)/d/c*x/(c^2*e*x^2+c^2*d)+1/2*(3*c^2*d-e)/d/c^3/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2)))+1/(c^2*d-e)^2*arctan(c*
x)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (115) = 230\).

Time = 0.35 (sec) , antiderivative size = 637, normalized size of antiderivative = 4.86 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\left [-\frac {4 \, a c^{4} d^{4} - 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + 2 \, {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} - {\left (3 \, b c^{3} d^{3} - b c d^{2} e + {\left (3 \, b c^{3} d e^{2} - b c e^{3}\right )} x^{4} + 2 \, {\left (3 \, b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) + 2 \, {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x - 4 \, {\left (b c^{4} d^{2} e^{2} x^{4} + 2 \, b c^{4} d^{3} e x^{2} + 2 \, b c^{2} d^{3} e - b d^{2} e^{2}\right )} \arctan \left (c x\right )}{16 \, {\left (c^{4} d^{6} e - 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} - 4 \, a c^{2} d^{3} e + 2 \, a d^{2} e^{2} + {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} + {\left (3 \, b c^{3} d^{3} - b c d^{2} e + {\left (3 \, b c^{3} d e^{2} - b c e^{3}\right )} x^{4} + 2 \, {\left (3 \, b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x - 2 \, {\left (b c^{4} d^{2} e^{2} x^{4} + 2 \, b c^{4} d^{3} e x^{2} + 2 \, b c^{2} d^{3} e - b d^{2} e^{2}\right )} \arctan \left (c x\right )}{8 \, {\left (c^{4} d^{6} e - 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}\right ] \]

[In]

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

[Out]

[-1/16*(4*a*c^4*d^4 - 8*a*c^2*d^3*e + 4*a*d^2*e^2 + 2*(b*c^3*d^2*e^2 - b*c*d*e^3)*x^3 - (3*b*c^3*d^3 - b*c*d^2
*e + (3*b*c^3*d*e^2 - b*c*e^3)*x^4 + 2*(3*b*c^3*d^2*e - b*c*d*e^2)*x^2)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x
- d)/(e*x^2 + d)) + 2*(b*c^3*d^3*e - b*c*d^2*e^2)*x - 4*(b*c^4*d^2*e^2*x^4 + 2*b*c^4*d^3*e*x^2 + 2*b*c^2*d^3*
e - b*d^2*e^2)*arctan(c*x))/(c^4*d^6*e - 2*c^2*d^5*e^2 + d^4*e^3 + (c^4*d^4*e^3 - 2*c^2*d^3*e^4 + d^2*e^5)*x^4
+ 2*(c^4*d^5*e^2 - 2*c^2*d^4*e^3 + d^3*e^4)*x^2), -1/8*(2*a*c^4*d^4 - 4*a*c^2*d^3*e + 2*a*d^2*e^2 + (b*c^3*d^
2*e^2 - b*c*d*e^3)*x^3 + (3*b*c^3*d^3 - b*c*d^2*e + (3*b*c^3*d*e^2 - b*c*e^3)*x^4 + 2*(3*b*c^3*d^2*e - b*c*d*e
^2)*x^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) + (b*c^3*d^3*e - b*c*d^2*e^2)*x - 2*(b*c^4*d^2*e^2*x^4 + 2*b*c^4*d^3*
e*x^2 + 2*b*c^2*d^3*e - b*d^2*e^2)*arctan(c*x))/(c^4*d^6*e - 2*c^2*d^5*e^2 + d^4*e^3 + (c^4*d^4*e^3 - 2*c^2*d^
3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 - 2*c^2*d^4*e^3 + d^3*e^4)*x^2)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(x*(a+b*arctan(c*x))/(e*x^2+d)^3,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>0)', see `assume?` for more
details)Is e

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 2.88 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.53 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b\,c\,x}{8\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}-\frac {b\,\mathrm {atan}\left (c\,x\right )}{4\,e\,{\left (e\,x^2+d\right )}^2}-\frac {a}{4\,e\,{\left (e\,x^2+d\right )}^2}+\frac {b\,c^4\,\mathrm {atan}\left (c\,x\right )}{4\,e\,{\left (e-c^2\,d\right )}^2}+\frac {b\,c\,e\,x^3}{8\,d\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}+\frac {b\,c\,\mathrm {atan}\left (\frac {x\,\sqrt {-d^3\,e}\,1{}\mathrm {i}}{d^2}\right )\,\sqrt {-d^3\,e}\,1{}\mathrm {i}}{8\,d^3\,{\left (e-c^2\,d\right )}^2}-\frac {b\,c^3\,\mathrm {atan}\left (\frac {x\,\sqrt {-d^3\,e}\,1{}\mathrm {i}}{d^2}\right )\,\sqrt {-d^3\,e}\,3{}\mathrm {i}}{8\,d^2\,e\,{\left (e-c^2\,d\right )}^2} \]

[In]

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

[Out]

(b*c*x)/(8*(e - c^2*d)*(d + e*x^2)^2) - (b*atan(c*x))/(4*e*(d + e*x^2)^2) - a/(4*e*(d + e*x^2)^2) + (b*c^4*ata
n(c*x))/(4*e*(e - c^2*d)^2) + (b*c*atan((x*(-d^3*e)^(1/2)*1i)/d^2)*(-d^3*e)^(1/2)*1i)/(8*d^3*(e - c^2*d)^2) -
(b*c^3*atan((x*(-d^3*e)^(1/2)*1i)/d^2)*(-d^3*e)^(1/2)*3i)/(8*d^2*e*(e - c^2*d)^2) + (b*c*e*x^3)/(8*d*(e - c^2*
d)*(d + e*x^2)^2)