\(\int \frac {\sqrt {x} \sqrt {c+d x}}{(a+b x)^{3/2}} \, dx\) [466]

Optimal result

Integrand size = 24, antiderivative size = 191 \[ \int \frac {\sqrt {x} \sqrt {c+d x}}{(a+b x)^{3/2}} \, dx=\frac {2 \sqrt {x} \sqrt {c+d x}}{b \sqrt {a+b x}}-\frac {4 \sqrt {a} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{b^{3/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}+\frac {2 \sqrt {a} \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{b^{3/2} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

2*x^(1/2)*(d*x+c)^(1/2)/b/(b*x+a)^(1/2)-4*a^(1/2)*(d*x+c)^(1/2)*EllipticE( 
b^(1/2)*x^(1/2)/a^(1/2)/(1+b*x/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(3/2)/(b*x+a) 
^(1/2)/(a*(d*x+c)/c/(b*x+a))^(1/2)+2*a^(1/2)*(d*x+c)^(1/2)*InverseJacobiAM 
(arctan(b^(1/2)*x^(1/2)/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(3/2)/(b*x+a)^(1/2)/ 
(a*(d*x+c)/c/(b*x+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {x} \sqrt {c+d x}}{(a+b x)^{3/2}} \, dx=\frac {2 a (2 a+b x) (c+d x)+4 i a \sqrt {\frac {a}{b}} b d \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )-2 i \sqrt {\frac {a}{b}} b (-b c+2 a d) \sqrt {1+\frac {a}{b x}} \sqrt {1+\frac {c}{d x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{b}}}{\sqrt {x}}\right ),\frac {b c}{a d}\right )}{a b^2 \sqrt {x} \sqrt {a+b x} \sqrt {c+d x}} \] Input:

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

Output:

(2*a*(2*a + b*x)*(c + d*x) + (4*I)*a*Sqrt[a/b]*b*d*Sqrt[1 + a/(b*x)]*Sqrt[ 
1 + c/(d*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)] 
- (2*I)*Sqrt[a/b]*b*(-(b*c) + 2*a*d)*Sqrt[1 + a/(b*x)]*Sqrt[1 + c/(d*x)]*x 
^(3/2)*EllipticF[I*ArcSinh[Sqrt[a/b]/Sqrt[x]], (b*c)/(a*d)])/(a*b^2*Sqrt[x 
]*Sqrt[a + b*x]*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {108, 27, 176, 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.

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

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

Output:

(-2*Sqrt[x]*Sqrt[c + d*x])/(b*Sqrt[a + b*x]) + ((4*Sqrt[-a]*Sqrt[1 + (b*x) 
/a]*Sqrt[c + d*x]*EllipticE[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], (a*d)/(b*c 
)])/(Sqrt[b]*Sqrt[a + b*x]*Sqrt[1 + (d*x)/c]) - (2*Sqrt[-a]*c*Sqrt[1 + (b* 
x)/a]*Sqrt[1 + (d*x)/c]*EllipticF[ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[-a]], (a*d 
)/(b*c)])/(Sqrt[b]*Sqrt[a + b*x]*Sqrt[c + d*x]))/b
 

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 108

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 
*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

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 176

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* 
Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f   Int[Sqrt[e + f*x]/(Sqrt[a + b*x 
]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f   Int[1/(Sqrt[a + b*x]*Sqrt[c 
 + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim 
plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(349\) vs. \(2(166)=332\).

Time = 0.54 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.83

method	result	size
default	\(-\frac {2 \sqrt {x d +c}\, \sqrt {b x +a}\, \left (2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d -\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b \,c^{2}-2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d +2 \sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b \,c^{2}+b \,d^{2} x^{2}+b c d x \right )}{\sqrt {x}\, \left (b d \,x^{2}+a d x +b c x +a c \right ) b^{2} d}\)	\(350\)
elliptic	\(\frac {\sqrt {x \left (x d +c \right ) \left (b x +a \right )}\, \left (-\frac {2 \left (b d \,x^{2}+b c x \right )}{b^{2} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d \,x^{2}+b c x \right )}}+\frac {2 c^{2} \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b d \sqrt {b d \,x^{3}+a d \,x^{2}+x^{2} b c +a c x}}+\frac {4 c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{b \sqrt {b d \,x^{3}+a d \,x^{2}+x^{2} b c +a c x}}\right )}{\sqrt {x}\, \sqrt {x d +c}\, \sqrt {b x +a}}\)	\(363\)

Input:

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

Output:

-2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(2*((d*x+c)/c)^(1/2)*(d*(b*x+a)/(a*d-b*c))^ 
(1/2)*(-1/c*x*d)^(1/2)*EllipticF(((d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^(1/2)) 
*a*c*d-((d*x+c)/c)^(1/2)*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-1/c*x*d)^(1/2)*Elli 
pticF(((d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^(1/2))*b*c^2-2*((d*x+c)/c)^(1/2)* 
(d*(b*x+a)/(a*d-b*c))^(1/2)*(-1/c*x*d)^(1/2)*EllipticE(((d*x+c)/c)^(1/2),( 
-b*c/(a*d-b*c))^(1/2))*a*c*d+2*((d*x+c)/c)^(1/2)*(d*(b*x+a)/(a*d-b*c))^(1/ 
2)*(-1/c*x*d)^(1/2)*EllipticE(((d*x+c)/c)^(1/2),(-b*c/(a*d-b*c))^(1/2))*b* 
c^2+b*d^2*x^2+b*c*d*x)/x^(1/2)/(b*d*x^2+a*d*x+b*c*x+a*c)/b^2/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (166) = 332\).

Time = 0.13 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {x} \sqrt {c+d x}}{(a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b x + a} \sqrt {d x + c} b^{2} d \sqrt {x} - {\left (a b c - 2 \, a^{2} d + {\left (b^{2} c - 2 \, a b d\right )} x\right )} \sqrt {b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 6 \, {\left (b^{2} d x + a b d\right )} \sqrt {b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{3 \, {\left (b^{4} d x + a b^{3} d\right )}} \] Input:

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

Output:

-2/3*(3*sqrt(b*x + a)*sqrt(d*x + c)*b^2*d*sqrt(x) - (a*b*c - 2*a^2*d + (b^ 
2*c - 2*a*b*d)*x)*sqrt(b*d)*weierstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a 
^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^ 
3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + b*c + a*d)/(b*d)) + 6*(b^2*d*x + a*b*d)*s 
qrt(b*d)*weierstrassZeta(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/2 
7*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), weier 
strassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2)/(b^2*d^2), -4/27*(2*b^3*c 
^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)/(b^3*d^3), 1/3*(3*b*d*x + 
b*c + a*d)/(b*d))))/(b^4*d*x + a*b^3*d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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