3.15.69 \(\int \frac {1-3 x}{\sqrt {4+3 x} (1+x^2)} \, dx\) [1469]

3.15.69.1 Optimal result

Integrand size = 22, antiderivative size = 53 \[ \int \frac {1-3 x}{\sqrt {4+3 x} \left (1+x^2\right )} \, dx=-\frac {\log \left (3+x-\sqrt {2} \sqrt {4+3 x}\right )}{\sqrt {2}}+\frac {\log \left (3+x+\sqrt {2} \sqrt {4+3 x}\right )}{\sqrt {2}} \]

output

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

3.15.69.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.42 \[ \int \frac {1-3 x}{\sqrt {4+3 x} \left (1+x^2\right )} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {8+6 x}}{3+x}\right ) \]

input

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

output

Sqrt[2]*ArcTanh[Sqrt[8 + 6*x]/(3 + x)]
 

3.15.69.3 Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {654, 27, 1478, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 654

\(\displaystyle 2 \int \frac {3 (1-3 x)}{(3 x+4)^2-8 (3 x+4)+25}d\sqrt {3 x+4}\)

\(\Big \downarrow \) 27

\(\displaystyle 6 \int \frac {1-3 x}{(3 x+4)^2-8 (3 x+4)+25}d\sqrt {3 x+4}\)

\(\Big \downarrow \) 1478

\(\displaystyle 6 \left (-\frac {\int -\frac {3 \sqrt {2}-2 \sqrt {3 x+4}}{3 x-3 \sqrt {2} \sqrt {3 x+4}+9}d\sqrt {3 x+4}}{6 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {3 x+4}+3\right )}{3 x+3 \sqrt {2} \sqrt {3 x+4}+9}d\sqrt {3 x+4}}{6 \sqrt {2}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 6 \left (\frac {\int \frac {3 \sqrt {2}-2 \sqrt {3 x+4}}{3 x-3 \sqrt {2} \sqrt {3 x+4}+9}d\sqrt {3 x+4}}{6 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {3 x+4}+3\right )}{3 x+3 \sqrt {2} \sqrt {3 x+4}+9}d\sqrt {3 x+4}}{6 \sqrt {2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 6 \left (\frac {\int \frac {3 \sqrt {2}-2 \sqrt {3 x+4}}{3 x-3 \sqrt {2} \sqrt {3 x+4}+9}d\sqrt {3 x+4}}{6 \sqrt {2}}+\frac {1}{6} \int \frac {\sqrt {2} \sqrt {3 x+4}+3}{3 x+3 \sqrt {2} \sqrt {3 x+4}+9}d\sqrt {3 x+4}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle 6 \left (\frac {\log \left (3 x+3 \sqrt {2} \sqrt {3 x+4}+9\right )}{6 \sqrt {2}}-\frac {\log \left (3 x-3 \sqrt {2} \sqrt {3 x+4}+9\right )}{6 \sqrt {2}}\right )\)

input

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

output

6*(-1/6*Log[9 + 3*x - 3*Sqrt[2]*Sqrt[4 + 3*x]]/Sqrt[2] + Log[9 + 3*x + 3*S 
qrt[2]*Sqrt[4 + 3*x]]/(6*Sqrt[2]))
 

3.15.69.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 654

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), 
x_Symbol] :> Simp[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]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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]
 

rule 1478

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[-2*(d/e) - b/c, 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e 
 + q*x - x^2, x], x], x] + Simp[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]
 

3.15.69.4 Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.75

method	result	size
pseudoelliptic	\(-\frac {\left (\ln \left (3+x -\sqrt {2}\, \sqrt {4+3 x}\right )-\ln \left (3+x +\sqrt {2}\, \sqrt {4+3 x}\right )\right ) \sqrt {2}}{2}\)	\(40\)
derivativedivides	\(-\frac {\sqrt {2}\, \ln \left (9+3 x -3 \sqrt {2}\, \sqrt {4+3 x}\right )}{2}+\frac {\sqrt {2}\, \ln \left (9+3 x +3 \sqrt {2}\, \sqrt {4+3 x}\right )}{2}\)	\(48\)
default	\(-\frac {\sqrt {2}\, \ln \left (9+3 x -3 \sqrt {2}\, \sqrt {4+3 x}\right )}{2}+\frac {\sqrt {2}\, \ln \left (9+3 x +3 \sqrt {2}\, \sqrt {4+3 x}\right )}{2}\)	\(48\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {4+3 x}\, x -12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +12 \sqrt {4+3 x}-17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{2}+1}\right )}{2}\)	\(66\)

input

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

output

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

3.15.69.5 Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {1-3 x}{\sqrt {4+3 x} \left (1+x^2\right )} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, \sqrt {2} \sqrt {3 \, x + 4} {\left (x + 3\right )} + x^{2} + 12 \, x + 17}{x^{2} + 1}\right ) \]

input

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

output

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

3.15.69.6 Sympy [F]

\[ \int \frac {1-3 x}{\sqrt {4+3 x} \left (1+x^2\right )} \, dx=- \int \frac {3 x}{x^{2} \sqrt {3 x + 4} + \sqrt {3 x + 4}}\, dx - \int \left (- \frac {1}{x^{2} \sqrt {3 x + 4} + \sqrt {3 x + 4}}\right )\, dx \]

input

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

output

-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)
 

3.15.69.7 Maxima [F]

\[ \int \frac {1-3 x}{\sqrt {4+3 x} \left (1+x^2\right )} \, dx=\int { -\frac {3 \, x - 1}{{\left (x^{2} + 1\right )} \sqrt {3 \, x + 4}} \,d x } \]

input

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

output

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

3.15.69.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {1-3 x}{\sqrt {4+3 x} \left (1+x^2\right )} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {3}{5} \cdot 25^{\frac {1}{4}} \sqrt {10} \sqrt {3 \, x + 4} + 3 \, x + 9\right ) - \frac {1}{2} \, \sqrt {2} \log \left (-\frac {3}{5} \cdot 25^{\frac {1}{4}} \sqrt {10} \sqrt {3 \, x + 4} + 3 \, x + 9\right ) \]

input

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

output

1/2*sqrt(2)*log(3/5*25^(1/4)*sqrt(10)*sqrt(3*x + 4) + 3*x + 9) - 1/2*sqrt( 
2)*log(-3/5*25^(1/4)*sqrt(10)*sqrt(3*x + 4) + 3*x + 9)
 

3.15.69.9 Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.40 \[ \int \frac {1-3 x}{\sqrt {4+3 x} \left (1+x^2\right )} \, dx=\sqrt {2}\,\mathrm {atanh}\left (\frac {24\,\sqrt {6\,x+8}}{24\,x+72}\right ) \]

input

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

output

2^(1/2)*atanh((24*(6*x + 8)^(1/2))/(24*x + 72))