\(\int \frac {\sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx\) [188]

Optimal result

Integrand size = 23, antiderivative size = 208 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {b x+c x^2}}{e \sqrt {d+e x}}-\frac {4 \sqrt {d} \sqrt {b x+c x^2} E\left (\arctan \left (\frac {\sqrt {e} \sqrt {x}}{\sqrt {d}}\right )|1-\frac {c d}{b e}\right )}{e^{3/2} \sqrt {x} \sqrt {\frac {d (b+c x)}{b (d+e x)}} \sqrt {d+e x}}+\frac {2 \sqrt {d} \sqrt {b x+c x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} \sqrt {x}}{\sqrt {d}}\right ),1-\frac {c d}{b e}\right )}{e^{3/2} \sqrt {x} \sqrt {\frac {d (b+c x)}{b (d+e x)}} \sqrt {d+e x}} \] Output:

2*(c*x^2+b*x)^(1/2)/e/(e*x+d)^(1/2)-4*d^(1/2)*(c*x^2+b*x)^(1/2)*EllipticE( 
e^(1/2)*x^(1/2)/d^(1/2)/(1+e*x/d)^(1/2),(1-c*d/b/e)^(1/2))/e^(3/2)/x^(1/2) 
/(d*(c*x+b)/b/(e*x+d))^(1/2)/(e*x+d)^(1/2)+2*d^(1/2)*(c*x^2+b*x)^(1/2)*Inv 
erseJacobiAM(arctan(e^(1/2)*x^(1/2)/d^(1/2)),(1-c*d/b/e)^(1/2))/e^(3/2)/x^ 
(1/2)/(d*(c*x+b)/b/(e*x+d))^(1/2)/(e*x+d)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.70 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \left (\sqrt {\frac {b}{c}} (b+c x) (2 d+e x)+2 i b e \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{\sqrt {\frac {b}{c}} e^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \] Input:

Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^(3/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1161, 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.

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1169

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

Int[Sqrt[b*x + c*x^2]/(d + e*x)^(3/2),x]
 

Output:

(-2*Sqrt[b*x + c*x^2])/(e*Sqrt[d + e*x]) + ((4*Sqrt[-b]*Sqrt[c]*Sqrt[x]*Sq 
rt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]] 
, (b*e)/(c*d)])/(e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*(2*c 
*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sq 
rt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x 
+ c*x^2]))/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 1161

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 + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 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 = 0.79 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.33

method	result	size
default	\(-\frac {2 \sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}\, \left (\sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b d e -2 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b d e +2 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) c \,d^{2}+x^{2} c \,e^{2}+x b \,e^{2}\right )}{x \left (c e \,x^{2}+b e x +c d x +b d \right ) e^{3}}\)	\(276\)
elliptic	\(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (c x +b \right ) x \left (e x +d \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x \right )}{e^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 b d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{e^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {4 c d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )-\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{c}\right )}{e^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\)	\(369\)

Input:

int((c*x^2+b*x)^(1/2)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.72 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {c x^{2} + b x} \sqrt {e x + d} c e^{2} + {\left (2 \, c d^{2} - b d e + {\left (2 \, c d e - b e^{2}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 6 \, {\left (c e^{2} x + c d e\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right )\right )}}{3 \, {\left (c e^{4} x + c d e^{3}\right )}} \] Input:

integrate((c*x^2+b*x)^(1/2)/(e*x+d)^(3/2),x, algorithm="fricas")
 

Output:

-2/3*(3*sqrt(c*x^2 + b*x)*sqrt(e*x + d)*c*e^2 + (2*c*d^2 - b*d*e + (2*c*d* 
e - b*e^2)*x)*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)) + 6*(c*e^2*x + c*d*e)*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), weierstra 
ssPInverse(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*e^4*x + c*d*e^3)
 

Sympy [F]

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

integrate((c*x**2+b*x)**(1/2)/(e*x+d)**(3/2),x)
 

Output:

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

Maxima [F]

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

integrate((c*x^2+b*x)^(1/2)/(e*x+d)^(3/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x^2 + b*x)/(e*x + d)^(3/2), x)
 

Giac [F]

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

integrate((c*x^2+b*x)^(1/2)/(e*x+d)^(3/2),x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^2 + b*x)/(e*x + d)^(3/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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