3.1166 \(\int \frac{x^3 (a+b \tan ^{-1}(c x))}{(d+e x^2)^3} \, dx\)

Optimal. Leaf size=130 \[ \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac{b c \left (c^2 d-3 e\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{d} e^{3/2} \left (c^2 d-e\right )^2}+\frac{b c x}{8 e \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac{b \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.185947, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {264, 4976, 12, 470, 522, 205} \[ \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac{b c \left (c^2 d-3 e\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 \sqrt{d} e^{3/2} \left (c^2 d-e\right )^2}+\frac{b c x}{8 e \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac{b \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 264

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

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 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

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

Mathematica [A]  time = 3.18357, size = 158, normalized size = 1.22 \[ \frac{\frac{-4 a c^2 d+4 a e+b c e x}{\left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac{2 a d}{\left (d+e x^2\right )^2}+\frac{2 b c^2 \left (c^2 d-2 e\right ) \tan ^{-1}(c x)}{\left (e-c^2 d\right )^2}-\frac{b c \sqrt{e} \left (c^2 d-3 e\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \left (e-c^2 d\right )^2}-\frac{2 b \tan ^{-1}(c x) \left (d+2 e x^2\right )}{\left (d+e x^2\right )^2}}{8 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.058, size = 297, normalized size = 2.3 \begin{align*}{\frac{a{c}^{4}d}{4\,{e}^{2} \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) ^{2}}}-{\frac{{c}^{2}a}{2\,{e}^{2} \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}+{\frac{b{c}^{4}\arctan \left ( cx \right ) d}{4\,{e}^{2} \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) ^{2}}}-{\frac{{c}^{2}b\arctan \left ( cx \right ) }{2\,{e}^{2} \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}+{\frac{{c}^{5}bdx}{8\,e \left ({c}^{2}d-e \right ) ^{2} \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}-{\frac{{c}^{3}bx}{8\, \left ({c}^{2}d-e \right ) ^{2} \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}-{\frac{{c}^{3}bd}{8\,e \left ({c}^{2}d-e \right ) ^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,cb}{8\, \left ({c}^{2}d-e \right ) ^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{b{c}^{4}\arctan \left ( cx \right ) d}{4\,{e}^{2} \left ({c}^{2}d-e \right ) ^{2}}}-{\frac{{c}^{2}b\arctan \left ( cx \right ) }{2\,e \left ({c}^{2}d-e \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c^4*a/e^2*d/(c^2*e*x^2+c^2*d)^2-1/2*c^2*a/e^2/(c^2*e*x^2+c^2*d)+1/4*c^4*b*arctan(c*x)/e^2*d/(c^2*e*x^2+c^2
*d)^2-1/2*c^2*b*arctan(c*x)/e^2/(c^2*e*x^2+c^2*d)+1/8*c^5*b/e*d/(c^2*d-e)^2*x/(c^2*e*x^2+c^2*d)-1/8*c^3*b/(c^2
*d-e)^2*x/(c^2*e*x^2+c^2*d)-1/8*c^3*b/e*d/(c^2*d-e)^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+3/8*c*b/(c^2*d-e)^2/
(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+1/4*c^4*b/e^2*d/(c^2*d-e)^2*arctan(c*x)-1/2*c^2*b/e/(c^2*d-e)^2*arctan(c*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(x^3*(a+b*arctan(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.74923, size = 1418, normalized size = 10.91 \begin{align*} \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} + 8 \,{\left (a c^{4} d^{3} e - 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} -{\left (b c^{3} d^{3} - 3 \, b c d^{2} e +{\left (b c^{3} d e^{2} - 3 \, b c e^{3}\right )} x^{4} + 2 \,{\left (b c^{3} d^{2} e - 3 \, 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 (2 \, b d e^{3} x^{2} + b d^{2} e^{2} -{\left (b c^{4} d^{2} e^{2} - 2 \, b c^{2} d e^{3}\right )} x^{4}\right )} \arctan \left (c x\right )}{16 \,{\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} +{\left (c^{4} d^{3} e^{4} - 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \,{\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\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} + 4 \,{\left (a c^{4} d^{3} e - 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} +{\left (b c^{3} d^{3} - 3 \, b c d^{2} e +{\left (b c^{3} d e^{2} - 3 \, b c e^{3}\right )} x^{4} + 2 \,{\left (b c^{3} d^{2} e - 3 \, 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 (2 \, b d e^{3} x^{2} + b d^{2} e^{2} -{\left (b c^{4} d^{2} e^{2} - 2 \, b c^{2} d e^{3}\right )} x^{4}\right )} \arctan \left (c x\right )}{8 \,{\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} +{\left (c^{4} d^{3} e^{4} - 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \,{\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(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 + 8*(a*c^4*d^3*e - 2*a*c
^2*d^2*e^2 + a*d*e^3)*x^2 - (b*c^3*d^3 - 3*b*c*d^2*e + (b*c^3*d*e^2 - 3*b*c*e^3)*x^4 + 2*(b*c^3*d^2*e - 3*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*(2*
b*d*e^3*x^2 + b*d^2*e^2 - (b*c^4*d^2*e^2 - 2*b*c^2*d*e^3)*x^4)*arctan(c*x))/(c^4*d^5*e^2 - 2*c^2*d^4*e^3 + d^3
*e^4 + (c^4*d^3*e^4 - 2*c^2*d^2*e^5 + d*e^6)*x^4 + 2*(c^4*d^4*e^3 - 2*c^2*d^3*e^4 + d^2*e^5)*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 + 4*(a*c^4*d^3*e - 2*a*c^2*d^2*e^2 + a*
d*e^3)*x^2 + (b*c^3*d^3 - 3*b*c*d^2*e + (b*c^3*d*e^2 - 3*b*c*e^3)*x^4 + 2*(b*c^3*d^2*e - 3*b*c*d*e^2)*x^2)*sqr
t(d*e)*arctan(sqrt(d*e)*x/d) - (b*c^3*d^3*e - b*c*d^2*e^2)*x + 2*(2*b*d*e^3*x^2 + b*d^2*e^2 - (b*c^4*d^2*e^2 -
2*b*c^2*d*e^3)*x^4)*arctan(c*x))/(c^4*d^5*e^2 - 2*c^2*d^4*e^3 + d^3*e^4 + (c^4*d^3*e^4 - 2*c^2*d^2*e^5 + d*e^
6)*x^4 + 2*(c^4*d^4*e^3 - 2*c^2*d^3*e^4 + d^2*e^5)*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**3*(a+b*atan(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

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Giac [B]  time = 9.2829, size = 506, normalized size = 3.89 \begin{align*} -\frac{{\left (b c^{3} d - 3 \, b c e\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{1}{2}\right )}}{8 \,{\left (c^{4} d^{2} e - 2 \, c^{2} d e^{2} + e^{3}\right )} \sqrt{d}} + \frac{b c^{4} d x^{4} \arctan \left (c x\right ) e^{2} - 4 \, a c^{4} d^{2} x^{2} e - 2 \, b c^{2} x^{4} \arctan \left (c x\right ) e^{3} + b c^{3} d x^{3} e^{2} - 2 \, a c^{4} d^{3} + b c^{3} d^{2} x e + 8 \, a c^{2} d x^{2} e^{2} - b c x^{3} e^{3} + 4 \, a c^{2} d^{2} e - 2 \, b x^{2} \arctan \left (c x\right ) e^{3} - b c d x e^{2} - 4 \, a x^{2} e^{3} - b d \arctan \left (c x\right ) e^{2} - 2 \, a d e^{2}}{4 \,{\left (c^{4} d^{2} x^{4} e^{4} + 2 \, c^{4} d^{3} x^{2} e^{3} + c^{4} d^{4} e^{2} - 2 \, c^{2} d x^{4} e^{5} - 4 \, c^{2} d^{2} x^{2} e^{4} - 2 \, c^{2} d^{3} e^{3} + x^{4} e^{6} + 2 \, d x^{2} e^{5} + d^{2} e^{4}\right )}} - \frac{4 \, a c^{2} d x^{2} e - b c x^{3} e^{2} + 2 \, a c^{2} d^{2} - b c d x e - 4 \, a x^{2} e^{2} - 2 \, a d e}{8 \,{\left (c^{2} d e^{2} - e^{3}\right )}{\left (x^{2} e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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