\(\int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx\) [409]

Optimal result

Integrand size = 23, antiderivative size = 94 \[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {-b} \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {729, 113, 111} \[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}} \]

[In]

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

[Out]

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

Rule 111

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

Rule 113

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

Rule 729

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[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] &
& NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

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

Mathematica [A] (verified)

Time = 5.59 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=-\frac {2 \left (c+\frac {b}{x}\right ) \sqrt {x} \sqrt {d+e x} \left (-\sqrt {x}+\frac {d \sqrt {1+\frac {d}{e x}} E\left (\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{\sqrt {x}}\right )|\frac {b e}{c d}\right )}{\sqrt {-\frac {d}{e}} \sqrt {1+\frac {b}{c x}} \left (e+\frac {d}{x}\right )}\right )}{c \sqrt {x (b+c x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.94 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.29

method	result	size
default	\(-\frac {2 \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \left (b e -c d \right )}{c^{2} x \left (c e \,x^{2}+b e x +c d x +b d \right )}\)	\(121\)
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 d b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 e b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(322\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.12 (sec) , antiderivative size = 299, normalized size of antiderivative = 3.18 \[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {c e} c e {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - {\left (2 \, c d - b e\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right )}}{3 \, c^{2} e} \]

[In]

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

[Out]

-2/3*(3*sqrt(c*e)*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), 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))) -
(2*c*d - b*e)*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)))/(c^2*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\sqrt {d+e\,x}}{\sqrt {c\,x^2+b\,x}} \,d x \]

[In]

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

[Out]

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