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

Optimal. Leaf size=131 \[ -\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {b c^4 \text {ArcTan}(c x)}{4 \left (c^2 d-e\right )^2 e}-\frac {a+b \text {ArcTan}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} \left (c^2 d-e\right )^2 \sqrt {e}} \]

[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)

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Rubi [A]
time = 0.08, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5094, 425, 536, 209, 211} \begin {gather*} -\frac {a+b \text {ArcTan}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \text {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 \text {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 )} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.76, size = 131, normalized size = 1.00 \begin {gather*} \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 ) \text {ArcTan}(c x)}{e}-\frac {b c \left (3 c^2 d-e\right ) \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e} \left (-c^2 d+e\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[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

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Maple [A]
time = 0.34, size = 222, normalized size = 1.69

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{6}}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}-\frac {b \,c^{6} \arctan \left (c x \right )}{4 \left (e \,c^{2} x^{2}+c^{2} d \right )^{2} e}-\frac {b \,c^{7} x}{8 \left (c^{2} d -e \right )^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {b \,c^{5} e x}{8 \left (c^{2} d -e \right )^{2} d \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {3 b \,c^{5} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \left (c^{2} d -e \right )^{2} \sqrt {d e}}+\frac {b \,c^{3} e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \left (c^{2} d -e \right )^{2} d \sqrt {d e}}+\frac {b \,c^{6} \arctan \left (c x \right )}{4 e \left (c^{2} d -e \right )^{2}}}{c^{2}}\)	\(222\)
default	\(\frac {-\frac {a \,c^{6}}{4 e \left (e \,c^{2} x^{2}+c^{2} d \right )^{2}}-\frac {b \,c^{6} \arctan \left (c x \right )}{4 \left (e \,c^{2} x^{2}+c^{2} d \right )^{2} e}-\frac {b \,c^{7} x}{8 \left (c^{2} d -e \right )^{2} \left (e \,c^{2} x^{2}+c^{2} d \right )}+\frac {b \,c^{5} e x}{8 \left (c^{2} d -e \right )^{2} d \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {3 b \,c^{5} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \left (c^{2} d -e \right )^{2} \sqrt {d e}}+\frac {b \,c^{3} e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \left (c^{2} d -e \right )^{2} d \sqrt {d e}}+\frac {b \,c^{6} \arctan \left (c x \right )}{4 e \left (c^{2} d -e \right )^{2}}}{c^{2}}\)	\(222\)
risch	\(\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^{6} b d}{16 \left (c^{2} d -e \right )^{2} e \left (-e \,c^{2} x^{2}-c^{2} d \right )}+\frac {c^{5} b x}{16 \left (c^{2} d -e \right )^{2} \left (-e \,c^{2} x^{2}-c^{2} d \right )}-\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 {b \,c^{4} \arctan \left (c x \right )}{8 \left (c^{2} d -e \right )^{2} e}+\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 {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 b \,c^{4} \ln \left (c^{2} x^{2}+1\right )}{16 e \left (c^{2} d -e \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 {c^{4} a}{4 e \left (-e \,c^{2} x^{2}-c^{2} d \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^{4} \ln \left (e \,x^{2}+d \right )}{16 e \left (c^{2} d -e \right )^{2}}+\frac {i b \ln \left (i c x +1\right )}{8 e \left (e \,x^{2}+d \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 {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 {3 i c^{3} b \arctanh \left (\frac {2 \left (-i c x +1\right ) e -2 e}{2 c \sqrt {d e}}\right )}{16 \left (c^{2} d -e \right )^{2} \sqrt {d e}}+\frac {i c b \arctanh \left (\frac {2 \left (-i c x +1\right ) e -2 e}{2 c \sqrt {d e}}\right ) e}{16 \left (c^{2} d -e \right )^{2} d \sqrt {d e}}+\frac {i b \,c^{2}}{16 \left (c^{2} d -e \right )^{2} \left (e \,x^{2}+d \right )}-\frac {3 b \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{16 \left (c^{2} d -e \right )^{2} \sqrt {d e}}+\frac {e b c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{16 \left (c^{2} d -e \right )^{2} d \sqrt {d e}}\)	\(802\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(-1/4*a*c^6/e/(c^2*e*x^2+c^2*d)^2-1/4*b*c^6/(c^2*e*x^2+c^2*d)^2*arctan(c*x)/e-1/8*b*c^7/(c^2*d-e)^2*x/(c
^2*e*x^2+c^2*d)+1/8*b*c^5*e/(c^2*d-e)^2*x/d/(c^2*e*x^2+c^2*d)-3/8*b*c^5/(c^2*d-e)^2/(d*e)^(1/2)*arctan(e*x/(d*
e)^(1/2))+1/8*b*c^3*e/(c^2*d-e)^2/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+1/4*b*c^6/e/(c^2*d-e)^2*arctan(c*x))

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Maxima [A]
time = 0.47, size = 183, normalized size = 1.40 \begin {gather*} \frac {1}{8} \, {\left ({\left (\frac {2 \, c^{3} \arctan \left (c x\right )}{c^{4} d^{2} e - 2 \, c^{2} d e^{2} + e^{3}} - \frac {{\left (3 \, c^{2} d - e\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{{\left (c^{4} d^{3} - 2 \, c^{2} d^{2} e + d e^{2}\right )} \sqrt {d}} - \frac {x}{c^{2} d^{3} + {\left (c^{2} d^{2} e - d e^{2}\right )} x^{2} - d^{2} e}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{4} e^{3} + 2 \, d x^{2} e^{2} + d^{2} e}\right )} b - \frac {a}{4 \, {\left (x^{4} e^{3} + 2 \, d x^{2} e^{2} + d^{2} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (117) = 234\).
time = 2.50, size = 626, normalized size = 4.78 \begin {gather*} \left [-\frac {4 \, a c^{4} d^{4} - 2 \, b c d x^{3} e^{3} - {\left (3 \, b c^{3} d^{3} - b c x^{4} e^{3} + {\left (3 \, b c^{3} d x^{4} - 2 \, b c d x^{2}\right )} e^{2} + {\left (6 \, b c^{3} d^{2} x^{2} - b c d^{2}\right )} e\right )} \sqrt {-d e} \log \left (\frac {x^{2} e - 2 \, \sqrt {-d e} x - d}{x^{2} e + d}\right ) - 4 \, {\left ({\left (b c^{4} d^{2} x^{4} - b d^{2}\right )} e^{2} + 2 \, {\left (b c^{4} d^{3} x^{2} + b c^{2} d^{3}\right )} e\right )} \arctan \left (c x\right ) + 2 \, {\left (b c^{3} d^{2} x^{3} - b c d^{2} x + 2 \, a d^{2}\right )} e^{2} + 2 \, {\left (b c^{3} d^{3} x - 4 \, a c^{2} d^{3}\right )} e}{16 \, {\left (c^{4} d^{6} e + d^{2} x^{4} e^{5} - 2 \, {\left (c^{2} d^{3} x^{4} - d^{3} x^{2}\right )} e^{4} + {\left (c^{4} d^{4} x^{4} - 4 \, c^{2} d^{4} x^{2} + d^{4}\right )} e^{3} + 2 \, {\left (c^{4} d^{5} x^{2} - c^{2} d^{5}\right )} e^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} - b c d x^{3} e^{3} + {\left (3 \, b c^{3} d^{3} - b c x^{4} e^{3} + {\left (3 \, b c^{3} d x^{4} - 2 \, b c d x^{2}\right )} e^{2} + {\left (6 \, b c^{3} d^{2} x^{2} - b c d^{2}\right )} e\right )} \sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}} - 2 \, {\left ({\left (b c^{4} d^{2} x^{4} - b d^{2}\right )} e^{2} + 2 \, {\left (b c^{4} d^{3} x^{2} + b c^{2} d^{3}\right )} e\right )} \arctan \left (c x\right ) + {\left (b c^{3} d^{2} x^{3} - b c d^{2} x + 2 \, a d^{2}\right )} e^{2} + {\left (b c^{3} d^{3} x - 4 \, a c^{2} d^{3}\right )} e}{8 \, {\left (c^{4} d^{6} e + d^{2} x^{4} e^{5} - 2 \, {\left (c^{2} d^{3} x^{4} - d^{3} x^{2}\right )} e^{4} + {\left (c^{4} d^{4} x^{4} - 4 \, c^{2} d^{4} x^{2} + d^{4}\right )} e^{3} + 2 \, {\left (c^{4} d^{5} x^{2} - c^{2} d^{5}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [B]
time = 2.61, size = 201, normalized size = 1.53 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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