3.1159 \(\int \frac{x (a+b \tan ^{-1}(c x))}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=91 \[ -\frac{a+b \tan ^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac{b c^2 \tan ^{-1}(c x)}{2 e \left (c^2 d-e\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{e} \left (c^2 d-e\right )} \]

[Out]

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

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Rubi [A]  time = 0.0662689, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {4974, 391, 203, 205} \[ -\frac{a+b \tan ^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac{b c^2 \tan ^{-1}(c x)}{2 e \left (c^2 d-e\right )}-\frac{b c \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{e} \left (c^2 d-e\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4974

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]

Rule 391

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

Rule 203

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

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.148354, size = 98, normalized size = 1.08 \[ \frac{a \sqrt{d} \left (c^2 d-e\right )-b \sqrt{d} e \left (c^2 x^2+1\right ) \tan ^{-1}(c x)+b c \sqrt{e} \left (d+e x^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} e \left (e-c^2 d\right ) \left (d+e x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[d]*(c^2*d - e) - b*Sqrt[d]*e*(1 + c^2*x^2)*ArcTan[c*x] + b*c*Sqrt[e]*(d + e*x^2)*ArcTan[(Sqrt[e]*x)/Sq
rt[d]])/(2*Sqrt[d]*e*(-(c^2*d) + e)*(d + e*x^2))

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Maple [A]  time = 0.043, size = 109, normalized size = 1.2 \begin{align*} -{\frac{{c}^{2}a}{2\,e \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}-{\frac{{c}^{2}b\arctan \left ( cx \right ) }{2\,e \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}-{\frac{cb}{2\,{c}^{2}d-2\,e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{{c}^{2}b\arctan \left ( cx \right ) }{ \left ( 2\,{c}^{2}d-2\,e \right ) e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.88815, size = 501, normalized size = 5.51 \begin{align*} \left [-\frac{2 \, a c^{2} d^{2} - 2 \, a d e -{\left (b c e x^{2} + b c d\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^{2} d e x^{2} + b d e\right )} \arctan \left (c x\right )}{4 \,{\left (c^{2} d^{3} e - d^{2} e^{2} +{\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}, -\frac{a c^{2} d^{2} - a d e +{\left (b c e x^{2} + b c d\right )} \sqrt{d e} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) -{\left (b c^{2} d e x^{2} + b d e\right )} \arctan \left (c x\right )}{2 \,{\left (c^{2} d^{3} e - d^{2} e^{2} +{\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.28288, size = 157, normalized size = 1.73 \begin{align*} -\frac{b c \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{2 \,{\left (c^{2} d - e\right )} \sqrt{d}} - \frac{a e^{\left (-1\right )}}{2 \,{\left (x^{2} e + d\right )}} + \frac{b c^{2} x^{2} \arctan \left (c x\right ) e - 2 \, a c^{2} d + b \arctan \left (c x\right ) e + 2 \, a e}{2 \,{\left (c^{2} d x^{2} e^{2} + c^{2} d^{2} e - x^{2} e^{3} - d e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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