3.1203 \(\int \frac{x (a+b \tan ^{-1}(c x))}{\sqrt{d+e x^2}} \, dx\)

Optimal. Leaf size=103 \[ \frac{\sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{e}-\frac{b \sqrt{c^2 d-e} \tan ^{-1}\left (\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right )}{c e}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{c \sqrt{e}} \]

[Out]

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

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Rubi [A]  time = 0.0986641, antiderivative size = 103, 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 = {4974, 402, 217, 206, 377, 203} \[ \frac{\sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{e}-\frac{b \sqrt{c^2 d-e} \tan ^{-1}\left (\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right )}{c e}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{c \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4974

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]

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

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

Rule 377

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 203

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

Rubi steps

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

Mathematica [C]  time = 0.371652, size = 251, normalized size = 2.44 \[ \frac{2 a c \sqrt{d+e x^2}-i b \sqrt{c^2 d-e} \log \left (\frac{4 c^2 e \left (-i \sqrt{c^2 d-e} \sqrt{d+e x^2}-i c d+e x\right )}{b (c x-i) \left (c^2 d-e\right )^{3/2}}\right )+i b \sqrt{c^2 d-e} \log \left (\frac{4 c^2 e \left (i \sqrt{c^2 d-e} \sqrt{d+e x^2}+i c d+e x\right )}{b (c x+i) \left (c^2 d-e\right )^{3/2}}\right )+2 b c \tan ^{-1}(c x) \sqrt{d+e x^2}-2 b \sqrt{e} \log \left (\sqrt{e} \sqrt{d+e x^2}+e x\right )}{2 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.806, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b\arctan \left ( cx \right ) \right ){\frac{1}{\sqrt{e{x}^{2}+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),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(x*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.57512, size = 1461, normalized size = 14.18 \begin{align*} \left [\frac{2 \, b \sqrt{e} \log \left (-2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) + \sqrt{-c^{2} d + e} b \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 \, \sqrt{e x^{2} + d}{\left (b c \arctan \left (c x\right ) + a c\right )}}{4 \, c e}, -\frac{\sqrt{c^{2} d - e} b \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 ) - b \sqrt{e} \log \left (-2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x - d\right ) - 2 \, \sqrt{e x^{2} + d}{\left (b c \arctan \left (c x\right ) + a c\right )}}{2 \, c e}, \frac{4 \, b \sqrt{-e} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) + \sqrt{-c^{2} d + e} b \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 \, \sqrt{e x^{2} + d}{\left (b c \arctan \left (c x\right ) + a c\right )}}{4 \, c e}, -\frac{\sqrt{c^{2} d - e} b \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 \, b \sqrt{-e} \arctan \left (\frac{\sqrt{-e} x}{\sqrt{e x^{2} + d}}\right ) - 2 \, \sqrt{e x^{2} + d}{\left (b c \arctan \left (c x\right ) + a c\right )}}{2 \, c e}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

[1/4*(2*b*sqrt(e)*log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + sqrt(-c^2*d + e)*b*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*sqrt(e*x^2 + d)*(b*c*arctan(c*x) + a*c))/(c*e), -1/2*(sqrt(c^2*d - e)*b*arc
tan(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)) - b
*sqrt(e)*log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) - 2*sqrt(e*x^2 + d)*(b*c*arctan(c*x) + a*c))/(c*e), 1
/4*(4*b*sqrt(-e)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + sqrt(-c^2*d + e)*b*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*sqrt(e*x^2 + d)*(b*c*arctan(c*x) + a*c))/(c*e), -1/2*(sqrt(c^2*d - e)*b*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*b*sqrt(-e)*ar
ctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - 2*sqrt(e*x^2 + d)*(b*c*arctan(c*x) + a*c))/(c*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*atan(c*x))/(e*x**2+d)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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