3.1214 \(\int \frac{a+b \tan ^{-1}(c x)}{x^2 (d+e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=135 \[ -\frac{2 e x \left (a+b \tan ^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{a+b \tan ^{-1}(c x)}{d x \sqrt{d+e x^2}}+\frac{b \left (c^2 d-2 e\right ) \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{d^2 \sqrt{c^2 d-e}}-\frac{b c \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.229967, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {271, 191, 4976, 12, 573, 156, 63, 208} \[ -\frac{2 e x \left (a+b \tan ^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{a+b \tan ^{-1}(c x)}{d x \sqrt{d+e x^2}}+\frac{b \left (c^2 d-2 e\right ) \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{d^2 \sqrt{c^2 d-e}}-\frac{b c \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{d^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 4976

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^
2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && ((IGtQ[q, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m +
2*q + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[q, 0] && GtQ[m + 2*q + 3, 0])) || (ILtQ[(m + 2*q + 1)/2, 0] &&
!ILtQ[(m - 1)/2, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 573

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n],
x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

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

Mathematica [C]  time = 0.6409, size = 306, normalized size = 2.27 \[ \frac{-\frac{2 a \left (d+2 e x^2\right )}{x \sqrt{d+e x^2}}+\frac{b \left (c^2 d-2 e\right ) \log \left (-\frac{4 c d^2 \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right )}{b (c x+i) \left (c^2 d-2 e\right ) \sqrt{c^2 d-e}}\right )}{\sqrt{c^2 d-e}}+\frac{b \left (c^2 d-2 e\right ) \log \left (-\frac{4 c d^2 \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right )}{b (c x-i) \left (c^2 d-2 e\right ) \sqrt{c^2 d-e}}\right )}{\sqrt{c^2 d-e}}-2 b c \sqrt{d} \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )-\frac{2 b \tan ^{-1}(c x) \left (d+2 e x^2\right )}{x \sqrt{d+e x^2}}+2 b c \sqrt{d} \log (x)}{2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.599, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{2}} \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.60906, size = 2758, normalized size = 20.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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