\(\int \frac {a+b \arctan (c x)}{(d+e x^2)^{5/2}} \, dx\) [1221]

Optimal result

Integrand size = 20, antiderivative size = 144 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b c}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {b \left (3 c^2 d-2 e\right ) \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^2 \left (c^2 d-e\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {198, 197, 5032, 6820, 12, 585, 79, 65, 214} \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {b \left (3 c^2 d-2 e\right ) \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^2 \left (c^2 d-e\right )^{3/2}}-\frac {b c}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}} \]

[In]

Int[(a + b*ArcTan[c*x])/(d + e*x^2)^(5/2),x]

[Out]

-1/3*(b*c)/(d*(c^2*d - e)*Sqrt[d + e*x^2]) + (x*(a + b*ArcTan[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*Arc
Tan[c*x]))/(3*d^2*Sqrt[d + e*x^2]) + (b*(3*c^2*d - 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^2*(
c^2*d - e)^(3/2))

Rule 12

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

Rule 65

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 197

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 198

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

Rule 214

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

Rule 585

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[(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] && EqQ[m - n + 1, 0]

Rule 5032

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

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}-(b c) \int \frac {\frac {x}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x}{3 d^2 \sqrt {d+e x^2}}}{1+c^2 x^2} \, dx \\ & = \frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}-(b c) \int \frac {x \left (3 d+2 e x^2\right )}{3 d^2 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx \\ & = \frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {(b c) \int \frac {x \left (3 d+2 e x^2\right )}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2} \\ & = \frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {(b c) \text {Subst}\left (\int \frac {3 d+2 e x}{\left (1+c^2 x\right ) (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 d^2} \\ & = -\frac {b c}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {\left (b c \left (3 c^2 d-2 e\right )\right ) \text {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) \sqrt {d+e x}} \, dx,x,x^2\right )}{6 d^2 \left (c^2 d-e\right )} \\ & = -\frac {b c}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}-\frac {\left (b c \left (3 c^2 d-2 e\right )\right ) \text {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^2 \left (c^2 d-e\right ) e} \\ & = -\frac {b c}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x (a+b \arctan (c x))}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x (a+b \arctan (c x))}{3 d^2 \sqrt {d+e x^2}}+\frac {b \left (3 c^2 d-2 e\right ) \text {arctanh}\left (\frac {c \sqrt {d+e x^2}}{\sqrt {c^2 d-e}}\right )}{3 d^2 \left (c^2 d-e\right )^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.20 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {c^2 d-e} \left (-b c d \left (d+e x^2\right )+a \left (c^2 d-e\right ) x \left (3 d+2 e x^2\right )\right )+2 b \left (c^2 d-e\right )^{3/2} x \left (3 d+2 e x^2\right ) \arctan (c x)+b \left (3 c^2 d-2 e\right ) \left (d+e x^2\right )^{3/2} \log \left (-\frac {12 c d^2 \sqrt {c^2 d-e} \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (3 c^2 d-2 e\right ) (i+c x)}\right )+b \left (3 c^2 d-2 e\right ) \left (d+e x^2\right )^{3/2} \log \left (-\frac {12 c d^2 \sqrt {c^2 d-e} \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (3 c^2 d-2 e\right ) (-i+c x)}\right )}{6 d^2 \left (c^2 d-e\right )^{3/2} \left (d+e x^2\right )^{3/2}} \]

[In]

Integrate[(a + b*ArcTan[c*x])/(d + e*x^2)^(5/2),x]

[Out]

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

Maple [F]

\[\int \frac {a +b \arctan \left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]

[In]

int((a+b*arctan(c*x))/(e*x^2+d)^(5/2),x)

[Out]

int((a+b*arctan(c*x))/(e*x^2+d)^(5/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (124) = 248\).

Time = 0.63 (sec) , antiderivative size = 864, normalized size of antiderivative = 6.00 \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {{\left (3 \, b c^{2} d^{3} + {\left (3 \, b c^{2} d e^{2} - 2 \, b e^{3}\right )} x^{4} - 2 \, b d^{2} e + 2 \, {\left (3 \, b c^{2} d^{2} e - 2 \, b d e^{2}\right )} x^{2}\right )} \sqrt {c^{2} d - e} \log \left (\frac {c^{4} e^{2} x^{4} + 8 \, c^{4} d^{2} - 8 \, c^{2} d e + 2 \, {\left (4 \, c^{4} d e - 3 \, c^{2} e^{2}\right )} x^{2} + 4 \, {\left (c^{3} e x^{2} + 2 \, c^{3} d - c e\right )} \sqrt {c^{2} d - e} \sqrt {e x^{2} + d} + e^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) - 4 \, {\left (b c^{3} d^{3} - b c d^{2} e - 2 \, {\left (a c^{4} d^{2} e - 2 \, a c^{2} d e^{2} + a e^{3}\right )} x^{3} + {\left (b c^{3} d^{2} e - b c d e^{2}\right )} x^{2} - 3 \, {\left (a c^{4} d^{3} - 2 \, a c^{2} d^{2} e + a d e^{2}\right )} x - {\left (2 \, {\left (b c^{4} d^{2} e - 2 \, b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{4} d^{3} - 2 \, b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{12 \, {\left (c^{4} d^{6} - 2 \, c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}, \frac {{\left (3 \, b c^{2} d^{3} + {\left (3 \, b c^{2} d e^{2} - 2 \, b e^{3}\right )} x^{4} - 2 \, b d^{2} e + 2 \, {\left (3 \, b c^{2} d^{2} e - 2 \, b d e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} d + e} \arctan \left (-\frac {{\left (c^{2} e x^{2} + 2 \, c^{2} d - e\right )} \sqrt {-c^{2} d + e} \sqrt {e x^{2} + d}}{2 \, {\left (c^{3} d^{2} - c d e + {\left (c^{3} d e - c e^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (b c^{3} d^{3} - b c d^{2} e - 2 \, {\left (a c^{4} d^{2} e - 2 \, a c^{2} d e^{2} + a e^{3}\right )} x^{3} + {\left (b c^{3} d^{2} e - b c d e^{2}\right )} x^{2} - 3 \, {\left (a c^{4} d^{3} - 2 \, a c^{2} d^{2} e + a d e^{2}\right )} x - {\left (2 \, {\left (b c^{4} d^{2} e - 2 \, b c^{2} d e^{2} + b e^{3}\right )} x^{3} + 3 \, {\left (b c^{4} d^{3} - 2 \, b c^{2} d^{2} e + b d e^{2}\right )} x\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{4} d^{6} - 2 \, c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{4} d^{4} e^{2} - 2 \, c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e - 2 \, c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}\right ] \]

[In]

integrate((a+b*arctan(c*x))/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

[1/12*((3*b*c^2*d^3 + (3*b*c^2*d*e^2 - 2*b*e^3)*x^4 - 2*b*d^2*e + 2*(3*b*c^2*d^2*e - 2*b*d*e^2)*x^2)*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*(b*c^3*d^3 - b*c*d^2*e - 2*(a*c^4*d^2*
e - 2*a*c^2*d*e^2 + a*e^3)*x^3 + (b*c^3*d^2*e - b*c*d*e^2)*x^2 - 3*(a*c^4*d^3 - 2*a*c^2*d^2*e + a*d*e^2)*x - (
2*(b*c^4*d^2*e - 2*b*c^2*d*e^2 + b*e^3)*x^3 + 3*(b*c^4*d^3 - 2*b*c^2*d^2*e + b*d*e^2)*x)*arctan(c*x))*sqrt(e*x
^2 + d))/(c^4*d^6 - 2*c^2*d^5*e + d^4*e^2 + (c^4*d^4*e^2 - 2*c^2*d^3*e^3 + d^2*e^4)*x^4 + 2*(c^4*d^5*e - 2*c^2
*d^4*e^2 + d^3*e^3)*x^2), 1/6*((3*b*c^2*d^3 + (3*b*c^2*d*e^2 - 2*b*e^3)*x^4 - 2*b*d^2*e + 2*(3*b*c^2*d^2*e - 2
*b*d*e^2)*x^2)*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^3 - b*c*d^2*e - 2*(a*c^4*d^2*e - 2*a*c^2*d*e^2 + a*e^3)*x^3 +
(b*c^3*d^2*e - b*c*d*e^2)*x^2 - 3*(a*c^4*d^3 - 2*a*c^2*d^2*e + a*d*e^2)*x - (2*(b*c^4*d^2*e - 2*b*c^2*d*e^2 +
b*e^3)*x^3 + 3*(b*c^4*d^3 - 2*b*c^2*d^2*e + b*d*e^2)*x)*arctan(c*x))*sqrt(e*x^2 + d))/(c^4*d^6 - 2*c^2*d^5*e
+ d^4*e^2 + (c^4*d^4*e^2 - 2*c^2*d^3*e^3 + d^2*e^4)*x^4 + 2*(c^4*d^5*e - 2*c^2*d^4*e^2 + d^3*e^3)*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*atan(c*x))/(e*x**2+d)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+b*arctan(c*x))/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+b*arctan(c*x))/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]

[In]

int((a + b*atan(c*x))/(d + e*x^2)^(5/2),x)

[Out]

int((a + b*atan(c*x))/(d + e*x^2)^(5/2), x)