\(\int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx\) [1282]

Optimal result

Integrand size = 28, antiderivative size = 184 \[ \int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) \sqrt {a+b x}}+\frac {2 \sqrt {f} \sqrt {-b e+a f} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {a+b x}}{\sqrt {-b e+a f}}\right )|\frac {d (b e-a f)}{(b c-a d) f}\right )}{b (b c-a d) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x}} \] Output:

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

Mathematica [A] (verified)

Time = 12.99 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {a-\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )}{b \sqrt {a-\frac {b c}{d}} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{f (a+b x)}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {110, 27, 124, 123}

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.

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

\(\Big \downarrow \) 110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

Input:

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

Output:

(-2*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*Sqrt[a + b*x]) + (2*Sqrt[f]* 
Sqrt[-(b*e) + a*f]*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticE 
[ArcSin[(Sqrt[f]*Sqrt[a + b*x])/Sqrt[-(b*e) + a*f]], (d*(b*e - a*f))/((b*c 
 - a*d)*f)])/(b*(b*c - a*d)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*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 110

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 + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, 
 m + n])
 

rule 123

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

rule 124

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(636\) vs. \(2(161)=322\).

Time = 4.16 (sec) , antiderivative size = 637, normalized size of antiderivative = 3.46

method	result	size
elliptic	\(\frac {\sqrt {\left (f x +e \right ) \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 b d f \,x^{2}+2 b c f x +2 b d e x +2 b c e}{\left (a d -b c \right ) b \sqrt {\left (x +\frac {a}{b}\right ) \left (b d f \,x^{2}+b c f x +b d e x +b c e \right )}}+\frac {2 \left (\frac {f}{b}-\frac {a d f -b c f -b d e}{b \left (a d -b c \right )}-\frac {b c f +b d e}{\left (a d -b c \right ) b}\right ) \left (\frac {c}{d}-\frac {a}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}-\frac {2 d f \left (\frac {c}{d}-\frac {a}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \left (\left (-\frac {c}{d}+\frac {e}{f}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )-\frac {e \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )}{f}\right )}{\left (a d -b c \right ) \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\right )}{\sqrt {f x +e}\, \sqrt {b x +a}\, \sqrt {x d +c}}\)	\(637\)
default	\(\text {Expression too large to display}\)	\(1014\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 764 vs. \(2 (161) = 322\).

Time = 0.09 (sec) , antiderivative size = 764, normalized size of antiderivative = 4.15 \[ \int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:

integrate((f*x+e)^(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(f*x + e)*b^2*d*f + (a*b*d*e - (2* 
a*b*c - a^2*d)*f + (b^2*d*e - (2*b^2*c - a*b*d)*f)*x)*sqrt(b*d*f)*weierstr 
assPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c* 
d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b 
^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c 
^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3*(3 
*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f)) + 3*(b^2*d*f*x + a*b*d*f)*sqrt( 
b*d*f)*weierstrassZeta(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c 
^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3* 
c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 
 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f 
^3), weierstrassPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2 
*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^ 
3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f 
^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3 
*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f))))/((b^4*c*d - a*b^ 
3*d^2)*f*x + (a*b^3*c*d - a^2*b^2*d^2)*f)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*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)