3.1211 \(\int \frac{x (a+b \tan ^{-1}(c x))}{(d+e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=71 \[ \frac{b c \tan ^{-1}\left (\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right )}{e \sqrt{c^2 d-e}}-\frac{a+b \tan ^{-1}(c x)}{e \sqrt{d+e x^2}} \]

[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)

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Rubi [A]  time = 0.0739139, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4974, 377, 203} \[ \frac{b c \tan ^{-1}\left (\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right )}{e \sqrt{c^2 d-e}}-\frac{a+b \tan ^{-1}(c x)}{e \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

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

Mathematica [C]  time = 0.36672, size = 210, normalized size = 2.96 \[ -\frac{\frac{2 a}{\sqrt{d+e x^2}}+\frac{i b c \log \left (-\frac{4 i e \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right )}{b (c x+i) \sqrt{c^2 d-e}}\right )}{\sqrt{c^2 d-e}}-\frac{i b c \log \left (\frac{4 i e \left (\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right )}{b (c x-i) \sqrt{c^2 d-e}}\right )}{\sqrt{c^2 d-e}}+\frac{2 b \tan ^{-1}(c x)}{\sqrt{d+e x^2}}}{2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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 + Sqr
t[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(-I + c*x))])/Sqrt[c^2*d - e])/(2*e)

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Maple [F]  time = 0.611, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b\arctan \left ( cx \right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, 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)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.34946, size = 791, normalized size = 11.14 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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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 )}{\left (d + e x^{2}\right )^{\frac{3}{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)**(3/2),x)

[Out]

Integral(x*(a + b*atan(c*x))/(d + e*x**2)**(3/2), 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}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{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)^(3/2),x, algorithm="giac")

[Out]

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