\(\int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx\) [410]

Optimal result

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

[Out]

2*EllipticF(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)*(1+e*x/d)^(1/2)/c^(
1/2)/(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, 118, 117} \[ \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}} \]

[In]

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

[Out]

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

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> 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 118

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[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] /; FreeQ[{b, c, d, e, f}, x] &&  !(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 {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{\sqrt {b x+c x^2}} \\ & = \frac {\left (\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{\sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \frac {2 \sqrt {-b} \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.46 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {\frac {c+\frac {b}{x}}{c}} \sqrt {\frac {e+\frac {d}{x}}{e}} x^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{\sqrt {-\frac {b}{c}} \sqrt {x (b+c x)} \sqrt {d+e x}} \]

[In]

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

[Out]

(-2*Sqrt[(c + b/x)/c]*Sqrt[(e + d/x)/e]*x^(3/2)*EllipticF[ArcSin[Sqrt[-(b/c)]/Sqrt[x]], (c*d)/(b*e)])/(Sqrt[-(
b/c)]*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

Maple [A] (verified)

Time = 1.96 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {2 F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {-\frac {c x}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {\frac {c x +b}{b}}\, b \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}}{c x \left (c e \,x^{2}+b e x +c d x +b d \right )}\)	\(113\)
elliptic	\(\frac {2 \sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}\, c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\)	\(146\)

[In]

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

[Out]

2*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*(-c*x/b)^(1/2)*(-c*(e*x+d)/(b*e-c*d))^(1/2)*((c*x+b)/b)^(
1/2)*b*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)/c/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 = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx=\frac {2 \, \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )}{c e} \]

[In]

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

[Out]

2*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*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x} \sqrt {e x + d}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x} \sqrt {e x + d}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx=\int \frac {1}{\sqrt {c\,x^2+b\,x}\,\sqrt {d+e\,x}} \,d x \]

[In]

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

[Out]

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