\(\int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [68]

Optimal result

Integrand size = 36, antiderivative size = 137 \[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \sqrt {-f g+e h} i \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}} \]

[Out]

2*i*EllipticE(h^(1/2)*(f*x+e)^(1/2)/(e*h-f*g)^(1/2),(-d*(-e*h+f*g)/(-c*f+d*e)/h)^(1/2))*(e*h-f*g)^(1/2)*(d*x+c
)^(1/2)*(f*(h*x+g)/(-e*h+f*g))^(1/2)/f/h^(1/2)/(-f*(d*x+c)/(-c*f+d*e))^(1/2)/(h*x+g)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {21, 115, 114} \[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 i \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{f \sqrt {h} \sqrt {g+h x} \sqrt {-\frac {f (c+d x)}{d e-c f}}} \]

[In]

Int[(c*i + d*i*x)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*Sqrt[-(f*g) + e*h]*i*Sqrt[c + d*x]*Sqrt[(f*(g + h*x))/(f*g - e*h)]*EllipticE[ArcSin[(Sqrt[h]*Sqrt[e + f*x])
/Sqrt[-(f*g) + e*h]], -((d*(f*g - e*h))/((d*e - c*f)*h))])/(f*Sqrt[h]*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*Sqrt[
g + h*x])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> 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] &&  !LtQ[-(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 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[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] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]

Rubi steps \begin{align*} \text {integral}& = i \int \frac {\sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx \\ & = \frac {\left (i \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}}\right ) \int \frac {\sqrt {\frac {c f}{-d e+c f}+\frac {d f x}{-d e+c f}}}{\sqrt {e+f x} \sqrt {\frac {f g}{f g-e h}+\frac {f h x}{f g-e h}}} \, dx}{\sqrt {\frac {f (c+d x)}{-d e+c f}} \sqrt {g+h x}} \\ & = \frac {2 \sqrt {-f g+e h} i \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.74 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.31 \[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 i i \sqrt {c+d x} \sqrt {g+h x} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {f (c+d x)}{d e-c f}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {f (c+d x)}{d e-c f}}\right ),\frac {d e h-c f h}{d f g-c f h}\right )\right )}{h \sqrt {\frac {f (c+d x)}{d (e+f x)}} \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}} \]

[In]

Integrate[(c*i + d*i*x)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

((-2*I)*i*Sqrt[c + d*x]*Sqrt[g + h*x]*(EllipticE[I*ArcSinh[Sqrt[(f*(c + d*x))/(d*e - c*f)]], (d*e*h - c*f*h)/(
d*f*g - c*f*h)] - EllipticF[I*ArcSinh[Sqrt[(f*(c + d*x))/(d*e - c*f)]], (d*e*h - c*f*h)/(d*f*g - c*f*h)]))/(h*
Sqrt[(f*(c + d*x))/(d*(e + f*x))]*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)])

Maple [A] (verified)

Time = 1.63 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.53

method	result	size
default	\(-\frac {2 i \left (c e \,h^{2}-c f g h -d e g h +d f \,g^{2}\right ) E\left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) \sqrt {\frac {\left (f x +e \right ) h}{e h -f g}}\, \sqrt {\frac {\left (d x +c \right ) h}{c h -d g}}\, \sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}{h^{2} f \left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right )}\)	\(210\)
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 c i \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 d i \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \left (\left (-\frac {g}{h}+\frac {c}{d}\right ) E\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )-\frac {c F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{d}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\)	\(500\)

[In]

int((d*i*x+c*i)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*i*(c*e*h^2-c*f*g*h-d*e*g*h+d*f*g^2)*EllipticE((-(h*x+g)*f/(e*h-f*g))^(1/2),((e*h-f*g)*d/f/(c*h-d*g))^(1/2))
*((f*x+e)*h/(e*h-f*g))^(1/2)*((d*x+c)*h/(c*h-d*g))^(1/2)*(-(h*x+g)*f/(e*h-f*g))^(1/2)/h^2/f*(d*x+c)^(1/2)*(f*x
+e)^(1/2)*(h*x+g)^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 664, normalized size of antiderivative = 4.85 \[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {d f h} d f h i {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + {\left (d e + c f\right )} h}{3 \, d f h}\right )\right ) + {\left (d f g + {\left (d e - 2 \, c f\right )} h\right )} \sqrt {d f h} i {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + {\left (d e + c f\right )} h}{3 \, d f h}\right )\right )}}{3 \, d f^{2} h^{2}} \]

[In]

integrate((d*i*x+c*i)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")

[Out]

-2/3*(3*sqrt(d*f*h)*d*f*h*i*weierstrassZeta(4/3*(d^2*f^2*g^2 - (d^2*e*f + c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f +
c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2*f - 4*c*d^2*e
*f^2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + 2*c^3*f^3)*h^3)/(d^3*f^3*h^3), weierstr
assPInverse(4/3*(d^2*f^2*g^2 - (d^2*e*f + c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/
27*(2*d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2*f - 4*c*d^2*e*f^2 + c^2*d*f^3)*g*h^2 + (2*d^3
*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + 2*c^3*f^3)*h^3)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + (d*e + c*f)*h)/
(d*f*h))) + (d*f*g + (d*e - 2*c*f)*h)*sqrt(d*f*h)*i*weierstrassPInverse(4/3*(d^2*f^2*g^2 - (d^2*e*f + c*d*f^2)
*g*h + (d^2*e^2 - c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*
h - 3*(d^3*e^2*f - 4*c*d^2*e*f^2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + 2*c^3*f^3)*
h^3)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + (d*e + c*f)*h)/(d*f*h)))/(d*f^2*h^2)

Sympy [F]

\[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=i \int \frac {\sqrt {c + d x}}{\sqrt {e + f x} \sqrt {g + h x}}\, dx \]

[In]

integrate((d*i*x+c*i)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

i*Integral(sqrt(c + d*x)/(sqrt(e + f*x)*sqrt(g + h*x)), x)

Maxima [F]

\[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {d i x + c i}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

integrate((d*i*x+c*i)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")

[Out]

integrate((d*i*x + c*i)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)

Giac [F]

\[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {d i x + c i}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

integrate((d*i*x+c*i)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

integrate((d*i*x + c*i)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {c\,i+d\,i\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]

[In]

int((c*i + d*i*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(c + d*x)^(1/2)),x)

[Out]

int((c*i + d*i*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(c + d*x)^(1/2)), x)