3.1216 \(\int \frac{a+b \tan ^{-1}(c x)}{x^4 (d+e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=249 \[ \frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}-\frac{b \left (c^4 d^2+4 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{3 d^3 \sqrt{c^2 d-e}}+\frac{b c \left (c^2 d+4 e\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{b c \sqrt{d+e x^2}}{6 d^2 x^2}+\frac{b c e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{6 d^{5/2}} \]

[Out]

-(b*c*Sqrt[d + e*x^2])/(6*d^2*x^2) - (a + b*ArcTan[c*x])/(3*d*x^3*Sqrt[d + e*x^2]) + (4*e*(a + b*ArcTan[c*x]))
/(3*d^2*x*Sqrt[d + e*x^2]) + (8*e^2*x*(a + b*ArcTan[c*x]))/(3*d^3*Sqrt[d + e*x^2]) + (b*c*e*ArcTanh[Sqrt[d + e
*x^2]/Sqrt[d]])/(6*d^(5/2)) + (b*c*(c^2*d + 4*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*d^(5/2)) - (b*(c^4*d^2 +
4*c^2*d*e - 8*e^2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^3*Sqrt[c^2*d - e])

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Rubi [A]  time = 0.873621, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {271, 191, 4976, 12, 6725, 266, 51, 63, 208, 444} \[ \frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}-\frac{b \left (c^4 d^2+4 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{3 d^3 \sqrt{c^2 d-e}}+\frac{b c \left (c^2 d+4 e\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{b c \sqrt{d+e x^2}}{6 d^2 x^2}+\frac{b c e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{6 d^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x])/(x^4*(d + e*x^2)^(3/2)),x]

[Out]

-(b*c*Sqrt[d + e*x^2])/(6*d^2*x^2) - (a + b*ArcTan[c*x])/(3*d*x^3*Sqrt[d + e*x^2]) + (4*e*(a + b*ArcTan[c*x]))
/(3*d^2*x*Sqrt[d + e*x^2]) + (8*e^2*x*(a + b*ArcTan[c*x]))/(3*d^3*Sqrt[d + e*x^2]) + (b*c*e*ArcTanh[Sqrt[d + e
*x^2]/Sqrt[d]])/(6*d^(5/2)) + (b*c*(c^2*d + 4*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*d^(5/2)) - (b*(c^4*d^2 +
4*c^2*d*e - 8*e^2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^3*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 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 266

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, 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]

Rule 444

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

Rubi steps

\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-(b c) \int \frac{-d^2+4 d e x^2+8 e^2 x^4}{3 d^3 x^3 \left (1+c^2 x^2\right ) \sqrt{d+e x^2}} \, dx\\ &=-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{(b c) \int \frac{-d^2+4 d e x^2+8 e^2 x^4}{x^3 \left (1+c^2 x^2\right ) \sqrt{d+e x^2}} \, dx}{3 d^3}\\ &=-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{(b c) \int \left (-\frac{d^2}{x^3 \sqrt{d+e x^2}}+\frac{d \left (c^2 d+4 e\right )}{x \sqrt{d+e x^2}}-\frac{\left (c^4 d^2+4 c^2 d e-8 e^2\right ) x}{\left (1+c^2 x^2\right ) \sqrt{d+e x^2}}\right ) \, dx}{3 d^3}\\ &=-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{(b c) \int \frac{1}{x^3 \sqrt{d+e x^2}} \, dx}{3 d}-\frac{\left (b c \left (c^2 d+4 e\right )\right ) \int \frac{1}{x \sqrt{d+e x^2}} \, dx}{3 d^2}+\frac{\left (b c \left (c^4 d^2+4 c^2 d e-8 e^2\right )\right ) \int \frac{x}{\left (1+c^2 x^2\right ) \sqrt{d+e x^2}} \, dx}{3 d^3}\\ &=-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{d+e x}} \, dx,x,x^2\right )}{6 d}-\frac{\left (b c \left (c^2 d+4 e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{6 d^2}+\frac{\left (b c \left (c^4 d^2+4 c^2 d e-8 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+c^2 x\right ) \sqrt{d+e x}} \, dx,x,x^2\right )}{6 d^3}\\ &=-\frac{b c \sqrt{d+e x^2}}{6 d^2 x^2}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}-\frac{(b c e) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )}{12 d^2}-\frac{\left (b c \left (c^2 d+4 e\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{3 d^2 e}+\frac{\left (b c \left (c^4 d^2+4 c^2 d e-8 e^2\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 )}{3 d^3 e}\\ &=-\frac{b c \sqrt{d+e x^2}}{6 d^2 x^2}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{b c \left (c^2 d+4 e\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{b \left (c^4 d^2+4 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{3 d^3 \sqrt{c^2 d-e}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )}{6 d^2}\\ &=-\frac{b c \sqrt{d+e x^2}}{6 d^2 x^2}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}+\frac{4 e \left (a+b \tan ^{-1}(c x)\right )}{3 d^2 x \sqrt{d+e x^2}}+\frac{8 e^2 x \left (a+b \tan ^{-1}(c x)\right )}{3 d^3 \sqrt{d+e x^2}}+\frac{b c e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{6 d^{5/2}}+\frac{b c \left (c^2 d+4 e\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{3 d^{5/2}}-\frac{b \left (c^4 d^2+4 c^2 d e-8 e^2\right ) \tanh ^{-1}\left (\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right )}{3 d^3 \sqrt{c^2 d-e}}\\ \end{align*}

Mathematica [C]  time = 0.711319, size = 405, normalized size = 1.63 \[ -\frac{\frac{2 a \left (d^2-4 d e x^2-8 e^2 x^4\right )+b c d x \left (d+e x^2\right )}{x^3 \sqrt{d+e x^2}}+\frac{b \left (c^4 d^2+4 c^2 d e-8 e^2\right ) \log \left (\frac{12 c d^3 \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right )}{b (c x+i) \sqrt{c^2 d-e} \left (c^4 d^2+4 c^2 d e-8 e^2\right )}\right )}{\sqrt{c^2 d-e}}+\frac{b \left (c^4 d^2+4 c^2 d e-8 e^2\right ) \log \left (\frac{12 c d^3 \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right )}{b (c x-i) \sqrt{c^2 d-e} \left (c^4 d^2+4 c^2 d e-8 e^2\right )}\right )}{\sqrt{c^2 d-e}}-b c \sqrt{d} \left (2 c^2 d+9 e\right ) \log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )+b c \sqrt{d} \log (x) \left (2 c^2 d+9 e\right )+\frac{2 b \tan ^{-1}(c x) \left (d^2-4 d e x^2-8 e^2 x^4\right )}{x^3 \sqrt{d+e x^2}}}{6 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTan[c*x])/(x^4*(d + e*x^2)^(3/2)),x]

[Out]

-((b*c*d*x*(d + e*x^2) + 2*a*(d^2 - 4*d*e*x^2 - 8*e^2*x^4))/(x^3*Sqrt[d + e*x^2]) + (2*b*(d^2 - 4*d*e*x^2 - 8*
e^2*x^4)*ArcTan[c*x])/(x^3*Sqrt[d + e*x^2]) + b*c*Sqrt[d]*(2*c^2*d + 9*e)*Log[x] - b*c*Sqrt[d]*(2*c^2*d + 9*e)
*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + (b*(c^4*d^2 + 4*c^2*d*e - 8*e^2)*Log[(12*c*d^3*(c*d - I*e*x + Sqrt[c^2*d -
e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(c^4*d^2 + 4*c^2*d*e - 8*e^2)*(I + c*x))])/Sqrt[c^2*d - e] + (b*(c^4*
d^2 + 4*c^2*d*e - 8*e^2)*Log[(12*c*d^3*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(c^
4*d^2 + 4*c^2*d*e - 8*e^2)*(-I + c*x))])/Sqrt[c^2*d - e])/(6*d^3)

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Maple [F]  time = 0.598, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{4}} \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^4/(e*x^2+d)^(3/2),x)

[Out]

int((a+b*arctan(c*x))/x^4/(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^4/(e*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.39429, size = 3976, normalized size = 15.97 \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^4/(e*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

[-1/12*(((b*c^4*d^2*e + 4*b*c^2*d*e^2 - 8*b*e^3)*x^5 + (b*c^4*d^3 + 4*b*c^2*d^2*e - 8*b*d*e^2)*x^3)*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^5*d^2*e + 7*b*c^3*d*e^2 - 9*b*c*e
^3)*x^5 + (2*b*c^5*d^3 + 7*b*c^3*d^2*e - 9*b*c*d*e^2)*x^3)*sqrt(d)*log(-(e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(d) + 2
*d)/x^2) + 2*(2*a*c^2*d^3 - 16*(a*c^2*d*e^2 - a*e^3)*x^4 - 2*a*d^2*e + (b*c^3*d^2*e - b*c*d*e^2)*x^3 - 8*(a*c^
2*d^2*e - a*d*e^2)*x^2 + (b*c^3*d^3 - b*c*d^2*e)*x + 2*(b*c^2*d^3 - 8*(b*c^2*d*e^2 - b*e^3)*x^4 - b*d^2*e - 4*
(b*c^2*d^2*e - b*d*e^2)*x^2)*arctan(c*x))*sqrt(e*x^2 + d))/((c^2*d^4*e - d^3*e^2)*x^5 + (c^2*d^5 - d^4*e)*x^3)
, -1/12*(2*((b*c^4*d^2*e + 4*b*c^2*d*e^2 - 8*b*e^3)*x^5 + (b*c^4*d^3 + 4*b*c^2*d^2*e - 8*b*d*e^2)*x^3)*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^5*d^2*e + 7*b*c^3*d*e^2 - 9*b*c*e^3)*x^5 + (2*b*c^5*d^3 + 7*b*c^3*d^2*e - 9*b*c*d*e^2)
*x^3)*sqrt(d)*log(-(e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(d) + 2*d)/x^2) + 2*(2*a*c^2*d^3 - 16*(a*c^2*d*e^2 - a*e^3)*
x^4 - 2*a*d^2*e + (b*c^3*d^2*e - b*c*d*e^2)*x^3 - 8*(a*c^2*d^2*e - a*d*e^2)*x^2 + (b*c^3*d^3 - b*c*d^2*e)*x +
2*(b*c^2*d^3 - 8*(b*c^2*d*e^2 - b*e^3)*x^4 - b*d^2*e - 4*(b*c^2*d^2*e - b*d*e^2)*x^2)*arctan(c*x))*sqrt(e*x^2
+ d))/((c^2*d^4*e - d^3*e^2)*x^5 + (c^2*d^5 - d^4*e)*x^3), -1/12*(2*((2*b*c^5*d^2*e + 7*b*c^3*d*e^2 - 9*b*c*e^
3)*x^5 + (2*b*c^5*d^3 + 7*b*c^3*d^2*e - 9*b*c*d*e^2)*x^3)*sqrt(-d)*arctan(sqrt(-d)/sqrt(e*x^2 + d)) + ((b*c^4*
d^2*e + 4*b*c^2*d*e^2 - 8*b*e^3)*x^5 + (b*c^4*d^3 + 4*b*c^2*d^2*e - 8*b*d*e^2)*x^3)*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*(2*a*c^2*d^3 - 16*(a*c^2*d*e^2 - a*e^3)*x^4 - 2*a*d^2*e
+ (b*c^3*d^2*e - b*c*d*e^2)*x^3 - 8*(a*c^2*d^2*e - a*d*e^2)*x^2 + (b*c^3*d^3 - b*c*d^2*e)*x + 2*(b*c^2*d^3 -
8*(b*c^2*d*e^2 - b*e^3)*x^4 - b*d^2*e - 4*(b*c^2*d^2*e - b*d*e^2)*x^2)*arctan(c*x))*sqrt(e*x^2 + d))/((c^2*d^4
*e - d^3*e^2)*x^5 + (c^2*d^5 - d^4*e)*x^3), -1/6*(((b*c^4*d^2*e + 4*b*c^2*d*e^2 - 8*b*e^3)*x^5 + (b*c^4*d^3 +
4*b*c^2*d^2*e - 8*b*d*e^2)*x^3)*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^5*d^2*e + 7*b*c^3*d*e^2 - 9*b*c*e^3)*x^5 + (2*b
*c^5*d^3 + 7*b*c^3*d^2*e - 9*b*c*d*e^2)*x^3)*sqrt(-d)*arctan(sqrt(-d)/sqrt(e*x^2 + d)) + (2*a*c^2*d^3 - 16*(a*
c^2*d*e^2 - a*e^3)*x^4 - 2*a*d^2*e + (b*c^3*d^2*e - b*c*d*e^2)*x^3 - 8*(a*c^2*d^2*e - a*d*e^2)*x^2 + (b*c^3*d^
3 - b*c*d^2*e)*x + 2*(b*c^2*d^3 - 8*(b*c^2*d*e^2 - b*e^3)*x^4 - b*d^2*e - 4*(b*c^2*d^2*e - b*d*e^2)*x^2)*arcta
n(c*x))*sqrt(e*x^2 + d))/((c^2*d^4*e - d^3*e^2)*x^5 + (c^2*d^5 - d^4*e)*x^3)]

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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**4/(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^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/x^4/(e*x^2+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arctan(c*x) + a)/((e*x^2 + d)^(3/2)*x^4), x)