\(\int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx\) [133]

Optimal result

Integrand size = 21, antiderivative size = 113 \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\frac {\sqrt {b x+c x^2}}{e}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{\sqrt {c} e^2}+\frac {2 \sqrt {d} \sqrt {c d-b e} \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{e^2} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.43 (sec) , antiderivative size = 402, normalized size of antiderivative = 3.56 \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (c \sqrt {d} e \sqrt {x} \sqrt {b+c x}+2 \left (c d-b e-i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (\sqrt {b}-\sqrt {b+c x}\right )}\right )+2 \left (c d-b e+i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (\sqrt {b}-\sqrt {b+c x}\right )}\right )+2 \sqrt {c} \sqrt {d} (2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )\right )}{c \sqrt {d} e^2 \sqrt {x (b+c x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.22, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1162, 1269, 1091, 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.

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

\(\Big \downarrow \) 1162

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1091

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

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 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{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 1162

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, 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]
 

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(-\frac {-e \sqrt {x \left (c x +b \right )}-\frac {\left (b e -2 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{\sqrt {c}}-\frac {2 \left (b e -c d \right ) d \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{\sqrt {d \left (b e -c d \right )}}}{e^{2}}\)	\(99\)
risch	\(\frac {x \left (c x +b \right )}{e \sqrt {x \left (c x +b \right )}}+\frac {\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e \sqrt {c}}+\frac {2 d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{2 e}\)	\(206\)
default	\(\frac {\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{e}\)	\(284\)

Input:

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

Output:

-1/e^2*(-e*(x*(c*x+b))^(1/2)-(b*e-2*c*d)/c^(1/2)*arctanh((x*(c*x+b))^(1/2) 
/x/c^(1/2))-2*(b*e-c*d)*d/(d*(b*e-c*d))^(1/2)*arctan((x*(c*x+b))^(1/2)/x*d 
/(d*(b*e-c*d))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 482, normalized size of antiderivative = 4.27 \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\left [\frac {2 \, \sqrt {c x^{2} + b x} c e - {\left (2 \, c d - b e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, \sqrt {c d^{2} - b d e} c \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right )}{2 \, c e^{2}}, \frac {2 \, \sqrt {c x^{2} + b x} c e - 4 \, \sqrt {-c d^{2} + b d e} c \arctan \left (\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x + b d}\right ) - {\left (2 \, c d - b e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c e^{2}}, \frac {\sqrt {c x^{2} + b x} c e + {\left (2 \, c d - b e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x + b}\right ) + \sqrt {c d^{2} - b d e} c \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right )}{c e^{2}}, \frac {\sqrt {c x^{2} + b x} c e - 2 \, \sqrt {-c d^{2} + b d e} c \arctan \left (\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x + b d}\right ) + {\left (2 \, c d - b e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x + b}\right )}{c e^{2}}\right ] \] Input:

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

Output:

[1/2*(2*sqrt(c*x^2 + b*x)*c*e - (2*c*d - b*e)*sqrt(c)*log(2*c*x + b + 2*sq 
rt(c*x^2 + b*x)*sqrt(c)) + 2*sqrt(c*d^2 - b*d*e)*c*log((b*d + (2*c*d - b*e 
)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)))/(c*e^2), 1/2*(2 
*sqrt(c*x^2 + b*x)*c*e - 4*sqrt(-c*d^2 + b*d*e)*c*arctan(sqrt(-c*d^2 + b*d 
*e)*sqrt(c*x^2 + b*x)/(c*d*x + b*d)) - (2*c*d - b*e)*sqrt(c)*log(2*c*x + b 
 + 2*sqrt(c*x^2 + b*x)*sqrt(c)))/(c*e^2), (sqrt(c*x^2 + b*x)*c*e + (2*c*d 
- b*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x + b)) + sqrt(c*d^2 
- b*d*e)*c*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + 
 b*x))/(e*x + d)))/(c*e^2), (sqrt(c*x^2 + b*x)*c*e - 2*sqrt(-c*d^2 + b*d*e 
)*c*arctan(sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x + b*d)) + (2*c*d 
- b*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x + b)))/(c*e^2)]
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((c*x^2+b*x)^(1/2)/(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(b*e-c*d>0)', see `assume?` for m 
ore detail
 

Giac [F(-2)]

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

integrate((c*x^2+b*x)^(1/2)/(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. \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}}{d+e\,x} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\frac {2 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}-\sqrt {e}\, \sqrt {c x +b}-\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c +2 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}+\sqrt {e}\, \sqrt {c x +b}+\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c +\sqrt {x}\, \sqrt {c x +b}\, c e +\sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b e -2 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) c d}{c \,e^{2}} \] Input:

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

Output:

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