\(\int \frac {\sqrt {a+b x+c x^2}}{x (d+e x)} \, dx\) [16]

Optimal result

Integrand size = 25, antiderivative size = 167 \[ \int \frac {\sqrt {a+b x+c x^2}}{x (d+e x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d}+\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e}-\frac {\sqrt {c d^2-b d e+a e^2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d e} \] Output:

-a^(1/2)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/d+c^(1/2)*arct 
anh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e-(a*e^2-b*d*e+c*d^2)^(1/2) 
*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b 
*x+a)^(1/2))/d/e
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b x+c x^2}}{x (d+e x)} \, dx=-\frac {2 \sqrt {-c d^2+b d e-a e^2} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+b d e-a e^2}}\right )-2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+\sqrt {c} d \log \left (e \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{d e} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {1270, 25, 1154, 219, 1269, 1092, 219, 1154, 219}

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.

\(\displaystyle \int \frac {\sqrt {a+b x+c x^2}}{x (d+e x)} \, dx\)

\(\Big \downarrow \) 1270

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\int \frac {b d+c x d-a e}{(d+e x) \sqrt {c x^2+b x+a}}dx}{d}-\frac {2 a \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\int \frac {b d+c x d-a e}{(d+e x) \sqrt {c x^2+b x+a}}dx}{d}-\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {c d \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{d}-\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {2 c d \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {\left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{d}-\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\sqrt {c} d \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e}-\frac {\left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{d}-\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {2 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {\sqrt {c} d \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e}}{d}-\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\sqrt {c} d \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e}-\frac {\sqrt {a e^2-b d e+c d^2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{d}-\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 219

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

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1270

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/(((d_.) + (e_.)*(x_))*((f_.) + 
 (g_.)*(x_))), x_Symbol] :> Simp[(c*d^2 - b*d*e + a*e^2)/(e*(e*f - d*g)) 
Int[(a + b*x + c*x^2)^(p - 1)/(d + e*x), x], x] - Simp[1/(e*(e*f - d*g)) 
Int[Simp[c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x, x]*((a + b*x + c*x^2)^(p 
- 1)/(f + g*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[p] 
&& GtQ[p, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(412\) vs. \(2(143)=286\).

Time = 1.40 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.47

method	result	size
default	\(\frac {\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d}-\frac {\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{d}\)	\(413\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 39.79 (sec) , antiderivative size = 2218, normalized size of antiderivative = 13.28 \[ \int \frac {\sqrt {a+b x+c x^2}}{x (d+e x)} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(sqrt(c)*d*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*( 
2*c*x + b)*sqrt(c) - 4*a*c) + sqrt(a)*e*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 
- 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) + sqrt(c*d^2 - 
 b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^ 
2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqr 
t(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b* 
e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)))/(d*e), -1/2*(2*s 
qrt(-c)*d*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + 
 b*c*x + a*c)) - sqrt(a)*e*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^ 
2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - sqrt(c*d^2 - b*d*e + a*e^ 
2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e 
 + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x 
 + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 
+ 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)))/(d*e), 1/2*(sqrt(c)*d*log(-8* 
c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4* 
a*c) + sqrt(a)*e*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + 
a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 2*sqrt(-c*d^2 + b*d*e - a*e^2)*arct 
an(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + 
(2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e 
^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)))/(d*e), -1/2*(2*sqrt(-c)*d*...
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{x (d+e x)} \, dx=\int \frac {\sqrt {a + b x + c x^{2}}}{x \left (d + e x\right )}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{x (d+e x)} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{x (d+e x)} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{x\,\left (d+e\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {a+b x+c x^2}}{x (d+e x)} \, dx=\frac {\sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (-2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right )-\sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right )+\sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) e -\sqrt {a}\, \mathrm {log}\left (x \right ) e +\sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) d}{d e} \] Input:

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

Output:

(sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 
 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x) - sqrt(a*e**2 - b*d*e 
+ c*d**2)*log(d + e*x) + sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2* 
a - b*x)*e - sqrt(a)*log(x)*e + sqrt(c)*log( - 2*sqrt(c)*sqrt(a + b*x + c* 
x**2) - b - 2*c*x)*d)/(d*e)