\(\int \frac {x (a+b \arctan (c x))}{(d+e x^2)^{3/2}} \, dx\) [1211]

Optimal result

Integrand size = 21, antiderivative size = 71 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {a+b \arctan (c x)}{e \sqrt {d+e x^2}}+\frac {b c \arctan \left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c^2 d-e} e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5094, 385, 209} \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {b c \arctan \left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{e \sqrt {c^2 d-e}}-\frac {a+b \arctan (c x)}{e \sqrt {d+e x^2}} \]

[In]

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

[Out]

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

Rule 209

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

Rule 385

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

Rule 5094

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]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.96 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {\frac {2 a}{\sqrt {d+e x^2}}+\frac {2 b \arctan (c x)}{\sqrt {d+e x^2}}+\frac {i b c \log \left (-\frac {4 i e \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \sqrt {c^2 d-e} (i+c x)}\right )}{\sqrt {c^2 d-e}}-\frac {i b c \log \left (\frac {4 i e \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \sqrt {c^2 d-e} (-i+c x)}\right )}{\sqrt {c^2 d-e}}}{2 e} \]

[In]

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

[Out]

-1/2*((2*a)/Sqrt[d + e*x^2] + (2*b*ArcTan[c*x])/Sqrt[d + e*x^2] + (I*b*c*Log[((-4*I)*e*(c*d - I*e*x + Sqrt[c^2
*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(I + c*x))])/Sqrt[c^2*d - e] - (I*b*c*Log[((4*I)*e*(c*d + I*e*x +
Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(-I + c*x))])/Sqrt[c^2*d - e])/e

Maple [F]

\[\int \frac {x \left (a +b \arctan \left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (63) = 126\).

Time = 0.32 (sec) , antiderivative size = 379, normalized size of antiderivative = 5.34 \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (b c e x^{2} + b c d\right )} \sqrt {-c^{2} d + e} \log \left (\frac {{\left (c^{4} d^{2} - 8 \, c^{2} d e + 8 \, e^{2}\right )} x^{4} - 2 \, {\left (3 \, c^{2} d^{2} - 4 \, d e\right )} x^{2} - 4 \, {\left ({\left (c^{2} d - 2 \, e\right )} x^{3} - d x\right )} \sqrt {-c^{2} d + e} \sqrt {e x^{2} + d} + d^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) + 4 \, {\left (a c^{2} d - a e + {\left (b c^{2} d - b e\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{4 \, {\left (c^{2} d^{2} e - d e^{2} + {\left (c^{2} d e^{2} - e^{3}\right )} x^{2}\right )}}, \frac {{\left (b c e x^{2} + b c d\right )} \sqrt {c^{2} d - e} \arctan \left (\frac {\sqrt {c^{2} d - e} {\left ({\left (c^{2} d - 2 \, e\right )} x^{2} - d\right )} \sqrt {e x^{2} + d}}{2 \, {\left ({\left (c^{2} d e - e^{2}\right )} x^{3} + {\left (c^{2} d^{2} - d e\right )} x\right )}}\right ) - 2 \, {\left (a c^{2} d - a e + {\left (b c^{2} d - b e\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{2 \, {\left (c^{2} d^{2} e - d e^{2} + {\left (c^{2} d e^{2} - e^{3}\right )} x^{2}\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e-c^2*d>0)', see `assume?` for
more detail

Giac [F]

\[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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