\(\int \frac {1}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}} \, dx\) [471]

Optimal result

Integrand size = 26, antiderivative size = 211 \[ \int \frac {1}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\frac {2 \sqrt {b} \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) \sqrt {e} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}}-\frac {2 \sqrt {a} d \sqrt {c+d x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c (b c-a d) \sqrt {e} \sqrt {a+b x} \sqrt {\frac {a (c+d x)}{c (a+b x)}}} \] Output:

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

Mathematica [A] (verified)

Time = 4.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\frac {2 \sqrt {c+d x} \left (\sqrt {-\frac {a}{b}} \sqrt {1+\frac {c}{d x}}-\sqrt {1+\frac {a}{b x}} \sqrt {x} E\left (\arcsin \left (\frac {\sqrt {-\frac {a}{b}}}{\sqrt {x}}\right )|\frac {b c}{a d}\right )\right )}{\sqrt {-\frac {a}{b}} (-b c+a d) \sqrt {1+\frac {c}{d x}} \sqrt {e x} \sqrt {a+b x}} \] Input:

Integrate[1/(Sqrt[e*x]*(a + b*x)^(3/2)*Sqrt[c + d*x]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.73, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {115, 27, 122, 120}

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.

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

\(\Big \downarrow \) 115

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

Input:

Int[1/(Sqrt[e*x]*(a + b*x)^(3/2)*Sqrt[c + d*x]),x]
 

Output:

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

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 115

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

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])
 

Maple [A] (verified)

Time = 2.90 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {2 \left (\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) a c d -\sqrt {\frac {x d +c}{c}}\, \sqrt {\frac {d \left (b x +a \right )}{a d -b c}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +c}{c}}, \sqrt {-\frac {b c}{a d -b c}}\right ) b \,c^{2}-b \,d^{2} x^{2}-b c d x \right ) \sqrt {x d +c}\, \sqrt {b x +a}}{a d \left (a d -b c \right ) \left (b d \,x^{2}+a d x +b c x +a c \right ) \sqrt {e x}}\)	\(218\)
elliptic	\(\frac {\sqrt {e x \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \left (b d e \,x^{2}+b c e x \right )}{\left (a d -b c \right ) a e \sqrt {\left (x +\frac {a}{b}\right ) \left (b d e \,x^{2}+b c e x \right )}}+\frac {2 \left (\frac {1}{a}+\frac {b c}{\left (a d -b c \right ) a}\right ) c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{d \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}+\frac {2 b c \sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \sqrt {-\frac {x d}{c}}\, \left (\left (-\frac {c}{d}+\frac {a}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {c}{d}\right ) d}{c}}, \sqrt {-\frac {c}{d \left (-\frac {c}{d}+\frac {a}{b}\right )}}\right )}{b}\right )}{\left (a d -b c \right ) a \sqrt {b d e \,x^{3}+a d e \,x^{2}+b c e \,x^{2}+a c e x}}\right )}{\sqrt {e x}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(417\)

Input:

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

Output:

2*(((d*x+c)/c)^(1/2)*(d*(b*x+a)/(a*d-b*c))^(1/2)*(-1/c*x*d)^(1/2)*Elliptic 
E(((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)*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)*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a/d/ 
(a*d-b*c)/(b*d*x^2+a*d*x+b*c*x+a*c)/(e*x)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (184) = 368\).

Time = 0.11 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.84 \[ \int \frac {1}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {b x + a} \sqrt {d x + c} \sqrt {e x} b^{2} d + {\left (a b c - 2 \, a^{2} d + {\left (b^{2} c - 2 \, a b d\right )} x\right )} \sqrt {b d e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right ) + 3 \, {\left (b^{2} d x + a b d\right )} \sqrt {b d e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}}{27 \, b^{3} d^{3}}, \frac {3 \, b d x + b c + a d}{3 \, b d}\right )\right )\right )}}{3 \, {\left ({\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} e x + {\left (a^{2} b^{2} c d - a^{3} b d^{2}\right )} e\right )}} \] Input:

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

Output:

2/3*(3*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(e*x)*b^2*d + (a*b*c - 2*a^2*d + (b 
^2*c - 2*a*b*d)*x)*sqrt(b*d*e)*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)) + 3*(b^2*d*x + a*b*d 
)*sqrt(b*d*e)*weierstrassZeta(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), 
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))))/((a*b^3*c*d - a^2*b^2*d^2)*e*x + (a^2*b^2*c*d - a 
^3*b*d^2)*e)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int \frac {1}{\sqrt {e x} \left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \] Input:

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

Output:

Integral(1/(sqrt(e*x)*(a + b*x)**(3/2)*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {e x}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {e x}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int \frac {1}{\sqrt {e\,x}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {e x} (a+b x)^{3/2} \sqrt {c+d x}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {b x +a}}{\sqrt {x}\, a^{2} c +\sqrt {x}\, a^{2} d x +2 \sqrt {x}\, a b c x +2 \sqrt {x}\, a b d \,x^{2}+\sqrt {x}\, b^{2} c \,x^{2}+\sqrt {x}\, b^{2} d \,x^{3}}d x \right )}{e} \] Input:

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

Output:

(sqrt(e)*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 + sqr 
t(x)*b**2*d*x**3),x))/e