\(\int \frac {\sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx\) [187]

Optimal result

Integrand size = 23, antiderivative size = 268 \[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx=-\frac {2 (2 c d-b e) x \sqrt {d+e x}}{3 e^2 \sqrt {b x+c x^2}}+\frac {2 \sqrt {d+e x} \sqrt {b x+c x^2}}{3 e}+\frac {2 \sqrt {b} (2 c d-b e) \sqrt {x} \sqrt {d+e x} E\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|1-\frac {b e}{c d}\right )}{3 \sqrt {c} e^2 \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}}-\frac {2 b^{3/2} \sqrt {x} \sqrt {d+e x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),1-\frac {b e}{c d}\right )}{3 \sqrt {c} e \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}} \] Output:

-2/3*(-b*e+2*c*d)*x*(e*x+d)^(1/2)/e^2/(c*x^2+b*x)^(1/2)+2/3*(e*x+d)^(1/2)* 
(c*x^2+b*x)^(1/2)/e+2/3*b^(1/2)*(-b*e+2*c*d)*x^(1/2)*(e*x+d)^(1/2)*Ellipti 
cE(c^(1/2)*x^(1/2)/b^(1/2)/(1+c*x/b)^(1/2),(1-b*e/c/d)^(1/2))/c^(1/2)/e^2/ 
(b*(e*x+d)/d/(c*x+b))^(1/2)/(c*x^2+b*x)^(1/2)-2/3*b^(3/2)*x^(1/2)*(e*x+d)^ 
(1/2)*InverseJacobiAM(arctan(c^(1/2)*x^(1/2)/b^(1/2)),(1-b*e/c/d)^(1/2))/c 
^(1/2)/e/(b*(e*x+d)/d/(c*x+b))^(1/2)/(c*x^2+b*x)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.24 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx=\frac {2 \left ((b+c x) (d+e x) (-2 c d+b e+c e x)+i \sqrt {\frac {b}{c}} c e (-2 c d+b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i \sqrt {\frac {b}{c}} c e (-c d+b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{3 c e^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \] Input:

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

Output:

(2*((b + c*x)*(d + e*x)*(-2*c*d + b*e + c*e*x) + I*Sqrt[b/c]*c*e*(-2*c*d + 
 b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt 
[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*Sqrt[b/c]*c*e*(-(c*d) + b*e)*Sqrt[1 + b/( 
c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c 
*d)/(b*e)]))/(3*c*e^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1162, 1269, 1169, 122, 120, 127, 126}

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.

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

\(\Big \downarrow \) 1162

\(\displaystyle \frac {2 \sqrt {b x+c x^2} \sqrt {d+e x}}{3 e}-\frac {\int \frac {b d+(2 c d-b e) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 \sqrt {b x+c x^2} \sqrt {d+e x}}{3 e}-\frac {\frac {(2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {2 d (c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{3 e}\)

\(\Big \downarrow \) 1169

\(\displaystyle \frac {2 \sqrt {b x+c x^2} \sqrt {d+e x}}{3 e}-\frac {\frac {\sqrt {x} \sqrt {b+c x} (2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {2 d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {2 \sqrt {b x+c x^2} \sqrt {d+e x}}{3 e}-\frac {\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {2 \sqrt {b x+c x^2} \sqrt {d+e x}}{3 e}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {2 \sqrt {b x+c x^2} \sqrt {d+e x}}{3 e}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 e}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {2 \sqrt {b x+c x^2} \sqrt {d+e x}}{3 e}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {4 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 e}\)

Input:

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

Output:

(2*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*e) - ((2*Sqrt[-b]*(2*c*d - b*e)*Sqr 
t[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sq 
rt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - ( 
4*Sqrt[-b]*d*(c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ellip 
ticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + 
 e*x]*Sqrt[b*x + c*x^2]))/(3*e)
 

Defintions of rubi rules used

rule 120

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- 
b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt 
Q[e, 0] &&  !LtQ[-b/d, 0]
 

rule 122

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) 
)   Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b 
, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 126

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* 
Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & 
& GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
 

rule 127

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x 
]))   Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free 
Q[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

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 1169

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> 
 Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2])   Int[(d + e*x)^m/(Sqrt[x]* 
Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq 
Q[m^2, 1/4]
 

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 = 0.76 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.41

method	result	size
elliptic	\(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (c x +b \right ) x \left (e x +d \right )}\, \left (\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 e}-\frac {2 d^{2} b \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{3 e^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (b -\frac {2 \left (b e +c d \right )}{3 e}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )-\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\)	\(377\)
default	\(-\frac {2 \sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}\, \left (\sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} d \,e^{2}-d^{2} b \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) e c -\sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} d \,e^{2}+3 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b c \,d^{2} e -2 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) c^{2} d^{3}-e^{3} x^{3} c^{2}-e^{3} x^{2} c b -d \,e^{2} c^{2} x^{2}-d \,e^{2} c b x \right )}{3 x \left (c e \,x^{2}+b e x +c d x +b d \right ) e^{3} c}\)	\(467\)

Input:

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

Output:

1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*((c*x+b)*x*(e*x+d))^(1/2)/x/(c*x+b)*(2/3 
/e*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)-2/3*d^2/e^2*b*((x+d/e)/d*e)^(1/2) 
*((b/c+x)/(-d/e+b/c))^(1/2)*(-e*x/d)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x) 
^(1/2)*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e+b/c))^(1/2))+2*(b-2/3/e*( 
b*e+c*d))*d/e*((x+d/e)/d*e)^(1/2)*((b/c+x)/(-d/e+b/c))^(1/2)*(-e*x/d)^(1/2 
)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-d/e+b/c)*EllipticE(((x+d/e)/d*e 
)^(1/2),(-d/e/(-d/e+b/c))^(1/2))-b/c*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/( 
-d/e+b/c))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {c x^{2} + b x} \sqrt {e x + d} c^{2} e^{2} + {\left (2 \, c^{2} d^{2} - 2 \, b c d e - b^{2} e^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right )\right )}}{9 \, c^{2} e^{3}} \] Input:

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

Output:

2/9*(3*sqrt(c*x^2 + b*x)*sqrt(e*x + d)*c^2*e^2 + (2*c^2*d^2 - 2*b*c*d*e - 
b^2*e^2)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/( 
c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c 
^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*(2*c^2*d*e - b*c*e^2)*sqrt(c 
*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2* 
c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstras 
sPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 
3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + 
 b*e)/(c*e))))/(c^2*e^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b -\left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{b c \,e^{2} x^{2}+c^{2} d e \,x^{2}+b^{2} e^{2} x +2 b c d e x +c^{2} d^{2} x +b^{2} d e +b c \,d^{2}}d x \right ) b^{2} c \,e^{2}+\left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{b c \,e^{2} x^{2}+c^{2} d e \,x^{2}+b^{2} e^{2} x +2 b c d e x +c^{2} d^{2} x +b^{2} d e +b c \,d^{2}}d x \right ) b \,c^{2} d e +2 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{b c \,e^{2} x^{2}+c^{2} d e \,x^{2}+b^{2} e^{2} x +2 b c d e x +c^{2} d^{2} x +b^{2} d e +b c \,d^{2}}d x \right ) c^{3} d^{2}-\left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}}{b c \,e^{2} x^{3}+c^{2} d e \,x^{3}+b^{2} e^{2} x^{2}+2 b c d e \,x^{2}+c^{2} d^{2} x^{2}+b^{2} d e x +b c \,d^{2} x}d x \right ) b^{3} d e -\left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}}{b c \,e^{2} x^{3}+c^{2} d e \,x^{3}+b^{2} e^{2} x^{2}+2 b c d e \,x^{2}+c^{2} d^{2} x^{2}+b^{2} d e x +b c \,d^{2} x}d x \right ) b^{2} c \,d^{2}}{2 b e +2 c d} \] Input:

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

Output:

(2*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b - int((sqrt(x)*sqrt(d + e*x)*sqrt 
(b + c*x)*x)/(b**2*d*e + b**2*e**2*x + b*c*d**2 + 2*b*c*d*e*x + b*c*e**2*x 
**2 + c**2*d**2*x + c**2*d*e*x**2),x)*b**2*c*e**2 + int((sqrt(x)*sqrt(d + 
e*x)*sqrt(b + c*x)*x)/(b**2*d*e + b**2*e**2*x + b*c*d**2 + 2*b*c*d*e*x + b 
*c*e**2*x**2 + c**2*d**2*x + c**2*d*e*x**2),x)*b*c**2*d*e + 2*int((sqrt(x) 
*sqrt(d + e*x)*sqrt(b + c*x)*x)/(b**2*d*e + b**2*e**2*x + b*c*d**2 + 2*b*c 
*d*e*x + b*c*e**2*x**2 + c**2*d**2*x + c**2*d*e*x**2),x)*c**3*d**2 - int(( 
sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x))/(b**2*d*e*x + b**2*e**2*x**2 + b*c*d* 
*2*x + 2*b*c*d*e*x**2 + b*c*e**2*x**3 + c**2*d**2*x**2 + c**2*d*e*x**3),x) 
*b**3*d*e - int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x))/(b**2*d*e*x + b**2*e 
**2*x**2 + b*c*d**2*x + 2*b*c*d*e*x**2 + b*c*e**2*x**3 + c**2*d**2*x**2 + 
c**2*d*e*x**3),x)*b**2*c*d**2)/(2*(b*e + c*d))