3.1469 \(\int \frac{1-3 x}{\sqrt{4+3 x} (1+x^2)} \, dx\)

Optimal. Leaf size=53 \[ \frac{\log \left (x+\sqrt{2} \sqrt{3 x+4}+3\right )}{\sqrt{2}}-\frac{\log \left (x-\sqrt{2} \sqrt{3 x+4}+3\right )}{\sqrt{2}} \]

[Out]

-(Log[3 + x - Sqrt[2]*Sqrt[4 + 3*x]]/Sqrt[2]) + Log[3 + x + Sqrt[2]*Sqrt[4 + 3*x]]/Sqrt[2]

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Rubi [A]  time = 0.0535199, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {827, 1164, 628} \[ \frac{\log \left (x+\sqrt{2} \sqrt{3 x+4}+3\right )}{\sqrt{2}}-\frac{\log \left (x-\sqrt{2} \sqrt{3 x+4}+3\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 3*x)/(Sqrt[4 + 3*x]*(1 + x^2)),x]

[Out]

-(Log[3 + x - Sqrt[2]*Sqrt[4 + 3*x]]/Sqrt[2]) + Log[3 + x + Sqrt[2]*Sqrt[4 + 3*x]]/Sqrt[2]

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1164

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e - b/c, 2]},
Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x
- x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] &&  !GtQ[b^2
- 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1-3 x}{\sqrt{4+3 x} \left (1+x^2\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{15-3 x^2}{25-8 x^2+x^4} \, dx,x,\sqrt{4+3 x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{3 \sqrt{2}+2 x}{-5-3 \sqrt{2} x-x^2} \, dx,x,\sqrt{4+3 x}\right )}{\sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{3 \sqrt{2}-2 x}{-5+3 \sqrt{2} x-x^2} \, dx,x,\sqrt{4+3 x}\right )}{\sqrt{2}}\\ &=-\frac{\log \left (3+x-\sqrt{2} \sqrt{4+3 x}\right )}{\sqrt{2}}+\frac{\log \left (3+x+\sqrt{2} \sqrt{4+3 x}\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0290855, size = 63, normalized size = 1.19 \[ \frac{1}{5} \left ((3+i) \sqrt{4-3 i} \tanh ^{-1}\left (\frac{\sqrt{3 x+4}}{\sqrt{4-3 i}}\right )+(3-i) \sqrt{4+3 i} \tanh ^{-1}\left (\frac{\sqrt{3 x+4}}{\sqrt{4+3 i}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 3*x)/(Sqrt[4 + 3*x]*(1 + x^2)),x]

[Out]

((3 + I)*Sqrt[4 - 3*I]*ArcTanh[Sqrt[4 + 3*x]/Sqrt[4 - 3*I]] + (3 - I)*Sqrt[4 + 3*I]*ArcTanh[Sqrt[4 + 3*x]/Sqrt
[4 + 3*I]])/5

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Maple [A]  time = 0.016, size = 48, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{2}}{2}\ln \left ( 9+3\,x-3\,\sqrt{2}\sqrt{4+3\,x} \right ) }+{\frac{\sqrt{2}}{2}\ln \left ( 9+3\,x+3\,\sqrt{2}\sqrt{4+3\,x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-3*x)/(x^2+1)/(4+3*x)^(1/2),x)

[Out]

-1/2*2^(1/2)*ln(9+3*x-3*2^(1/2)*(4+3*x)^(1/2))+1/2*2^(1/2)*ln(9+3*x+3*2^(1/2)*(4+3*x)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{3 \, x - 1}{{\left (x^{2} + 1\right )} \sqrt{3 \, x + 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-3*x)/(x^2+1)/(4+3*x)^(1/2),x, algorithm="maxima")

[Out]

-integrate((3*x - 1)/((x^2 + 1)*sqrt(3*x + 4)), x)

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Fricas [A]  time = 1.79271, size = 108, normalized size = 2.04 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{3 \, x + 4}{\left (x + 3\right )} + x^{2} + 12 \, x + 17}{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-3*x)/(x^2+1)/(4+3*x)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*log((2*sqrt(2)*sqrt(3*x + 4)*(x + 3) + x^2 + 12*x + 17)/(x^2 + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{3 x}{x^{2} \sqrt{3 x + 4} + \sqrt{3 x + 4}}\, dx - \int - \frac{1}{x^{2} \sqrt{3 x + 4} + \sqrt{3 x + 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-3*x)/(x**2+1)/(4+3*x)**(1/2),x)

[Out]

-Integral(3*x/(x**2*sqrt(3*x + 4) + sqrt(3*x + 4)), x) - Integral(-1/(x**2*sqrt(3*x + 4) + sqrt(3*x + 4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{3 \, x - 1}{{\left (x^{2} + 1\right )} \sqrt{3 \, x + 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-3*x)/(x^2+1)/(4+3*x)^(1/2),x, algorithm="giac")

[Out]

integrate(-(3*x - 1)/((x^2 + 1)*sqrt(3*x + 4)), x)