\(\int \sqrt {d+e x} \sqrt {b x+c x^2} \, dx\) [186]

Optimal result

Integrand size = 23, antiderivative size = 333 \[ \int \sqrt {d+e x} \sqrt {b x+c x^2} \, dx=\frac {4 \left (3 b c d e-(c d+b e)^2\right ) x \sqrt {d+e x}}{15 c e^2 \sqrt {b x+c x^2}}+\frac {2 (c d+b e) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 c e}+\frac {2}{5} x \sqrt {d+e x} \sqrt {b x+c x^2}-\frac {4 \sqrt {b} \left (3 b c d e-(c d+b e)^2\right ) \sqrt {x} \sqrt {d+e x} E\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|1-\frac {b e}{c d}\right )}{15 c^{3/2} e^2 \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}}-\frac {2 b^{3/2} (c d+b e) \sqrt {x} \sqrt {d+e x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),1-\frac {b e}{c d}\right )}{15 c^{3/2} e \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}} \] Output:

4/15*(3*b*c*d*e-(b*e+c*d)^2)*x*(e*x+d)^(1/2)/c/e^2/(c*x^2+b*x)^(1/2)+2/15* 
(b*e+c*d)*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/c/e+2/5*x*(e*x+d)^(1/2)*(c*x^2+b 
*x)^(1/2)-4/15*b^(1/2)*(3*b*c*d*e-(b*e+c*d)^2)*x^(1/2)*(e*x+d)^(1/2)*Ellip 
ticE(c^(1/2)*x^(1/2)/b^(1/2)/(1+c*x/b)^(1/2),(1-b*e/c/d)^(1/2))/c^(3/2)/e^ 
2/(b*(e*x+d)/d/(c*x+b))^(1/2)/(c*x^2+b*x)^(1/2)-2/15*b^(3/2)*(b*e+c*d)*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^(3/2)/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 = 12.18 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.88 \[ \int \sqrt {d+e x} \sqrt {b x+c x^2} \, dx=\frac {2 \left (b e x (b+c x) (d+e x) (b e+c (d+3 e x))+\sqrt {\frac {b}{c}} \left (-2 \sqrt {\frac {b}{c}} \left (c^2 d^2-b c d e+b^2 e^2\right ) (b+c x) (d+e x)-2 i b e \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e \left (c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{15 b c e^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \] Input:

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

Output:

(2*(b*e*x*(b + c*x)*(d + e*x)*(b*e + c*(d + 3*e*x)) + Sqrt[b/c]*(-2*Sqrt[b 
/c]*(c^2*d^2 - b*c*d*e + b^2*e^2)*(b + c*x)*(d + e*x) - (2*I)*b*e*(c^2*d^2 
 - b*c*d*e + b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Elliptic 
E[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(c^2*d^2 - 3*b*c*d*e 
+ 2*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSi 
nh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(15*b*c*e^2*Sqrt[x*(b + c*x)]*Sqrt[ 
d + e*x])
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1162, 1236, 27, 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.

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

\(\Big \downarrow \) 1162

\(\displaystyle \frac {2 \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 e}-\frac {\int \frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{\sqrt {c x^2+b x}}dx}{5 e}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2 \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 e}-\frac {\frac {2 \int \frac {b d (c d+b e)+2 \left (c^2 d^2-b c e d+b^2 e^2\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (2 c d-b e)}{3 c}}{5 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 e}-\frac {\frac {\int \frac {b d (c d+b e)+2 \left (c^2 d^2-b c e d+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (2 c d-b e)}{3 c}}{5 e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 e}-\frac {\frac {\frac {2 \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{3 c}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (2 c d-b e)}{3 c}}{5 e}\)

\(\Big \downarrow \) 1169

\(\displaystyle \frac {2 \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 e}-\frac {\frac {\frac {2 \sqrt {x} \sqrt {b+c x} \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 c}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (2 c d-b e)}{3 c}}{5 e}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {2 \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 e}-\frac {\frac {\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 c}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (2 c d-b e)}{3 c}}{5 e}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {2 \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 e}-\frac {\frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 c}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (2 c d-b e)}{3 c}}{5 e}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {2 \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 e}-\frac {\frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 c}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (2 c d-b e)}{3 c}}{5 e}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {2 \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 e}-\frac {\frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 c}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (2 c d-b e)}{3 c}}{5 e}\)

Input:

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

Output:

(2*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2])/(5*e) - ((2*(2*c*d - b*e)*Sqrt[d + e 
*x]*Sqrt[b*x + c*x^2])/(3*c) + ((4*Sqrt[-b]*(c^2*d^2 - b*c*d*e + b^2*e^2)* 
Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x]) 
/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) 
- (2*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 
 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(S 
qrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2]))/(3*c))/(5*e)
 

Defintions of rubi rules used

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 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 1236

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

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 = 0.79 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.33

method	result	size
elliptic	\(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (c x +b \right ) x \left (e x +d \right )}\, \left (\frac {2 x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5}+\frac {2 \left (\frac {b e}{5}+\frac {c d}{5}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}-\frac {2 \left (\frac {b e}{5}+\frac {c d}{5}\right ) b \,d^{2} \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{3 c \,e^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {2 b d}{5}-\frac {2 \left (\frac {b e}{5}+\frac {c d}{5}\right ) \left (b e +c d \right )}{3 c e}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )-\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\)	\(444\)
default	\(\frac {2 \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}\, \left (2 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{3} d \,e^{3}-3 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} c \,d^{2} e^{2}+\sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b \,c^{2} d^{3} e -2 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{3} d \,e^{3}+4 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} c \,d^{2} e^{2}-4 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b \,c^{2} d^{3} e +2 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) c^{3} d^{4}+3 c^{3} e^{4} x^{4}+4 b \,c^{2} e^{4} x^{3}+4 c^{3} d \,e^{3} x^{3}+b^{2} c \,e^{4} x^{2}+5 b \,c^{2} d \,e^{3} x^{2}+c^{3} d^{2} e^{2} x^{2}+b^{2} c d \,e^{3} x +b \,c^{2} d^{2} e^{2} x \right )}{15 x \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{2} e^{3}}\)	\(681\)

Input:

int((e*x+d)^(1/2)*(c*x^2+b*x)^(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)*(2/5 
*x*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/3*(1/5*b*e+1/5*c*d)/c/e*(c*e*x^ 
3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)-2/3*(1/5*b*e+1/5*c*d)/c/e^2*b*d^2*((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* 
(2/5*b*d-2/3*(1/5*b*e+1/5*c*d)/c/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*E 
llipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e+b/c))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.19 \[ \int \sqrt {d+e x} \sqrt {b x+c x^2} \, dx=\frac {2 \, {\left ({\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 6 \, {\left (c^{3} d^{2} e - b c^{2} d e^{2} + b^{2} c e^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (3 \, c^{3} e^{3} x + c^{3} d e^{2} + b c^{2} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{45 \, c^{3} e^{3}} \] Input:

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

Output:

2/45*((2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)*sqrt(c*e)*we 
ierstrassPInverse(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)) + 6*(c^3*d^2*e - b*c^2*d*e^2 + b^2*c*e^3)*sqrt(c*e)*w 
eierstrassZeta(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), weierstrassPInv 
erse(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*(3*c^3*e^3*x + c^3*d*e^2 + b*c^2*e^3)*sqrt(c*x^2 + b*x)*sqrt( 
e*x + d))/(c^3*e^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int \sqrt {d+e x} \sqrt {b x+c x^2} \, dx=\frac {2 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b d +2 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, b e x +2 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, c d x +\left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{b c \,e^{2} x^{2}+c^{2} d e \,x^{2}+b^{2} e^{2} x +2 b c d e x +c^{2} d^{2} x +b^{2} d e +b c \,d^{2}}d x \right ) b^{3} e^{3}+\left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{b c \,e^{2} x^{2}+c^{2} d e \,x^{2}+b^{2} e^{2} x +2 b c d e x +c^{2} d^{2} x +b^{2} d e +b c \,d^{2}}d x \right ) c^{3} d^{3}-\left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}}{b c \,e^{2} x^{3}+c^{2} d e \,x^{3}+b^{2} e^{2} x^{2}+2 b c d e \,x^{2}+c^{2} d^{2} x^{2}+b^{2} d e x +b c \,d^{2} x}d x \right ) b^{3} d^{2} e -\left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}}{b c \,e^{2} x^{3}+c^{2} d e \,x^{3}+b^{2} e^{2} x^{2}+2 b c d e \,x^{2}+c^{2} d^{2} x^{2}+b^{2} d e x +b c \,d^{2} x}d x \right ) b^{2} c \,d^{3}}{5 b e +5 c d} \] Input:

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

Output:

(2*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*d + 2*sqrt(x)*sqrt(d + e*x)*sqrt( 
b + c*x)*b*e*x + 2*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c*d*x + 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**3*e**3 + 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**3 
 - 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**2*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**3)/(5*(b*e + c*d))