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

Optimal. Leaf size=231 \[ -\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{(-b)^{3/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 (b+2 c x) \sqrt{d+e x}}{b^2 \sqrt{b x+c x^2}}+\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]

[Out]

(-2*(b + 2*c*x)*Sqrt[d + e*x])/(b^2*Sqrt[b*x + c*x^2]) + (4*Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*El
lipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/((-b)^(3/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (
2*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e
)/(c*d)])/((-b)^(3/2)*Sqrt[c]*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.154094, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {736, 843, 715, 112, 110, 117, 116} \[ -\frac{2 (b+2 c x) \sqrt{d+e x}}{b^2 \sqrt{b x+c x^2}}-\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b + 2*c*x)*Sqrt[d + e*x])/(b^2*Sqrt[b*x + c*x^2]) + (4*Sqrt[c]*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*El
lipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/((-b)^(3/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (
2*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e
)/(c*d)])/((-b)^(3/2)*Sqrt[c]*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 736

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*(b + 2*
c*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m
- 1)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d
, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m
, 0] && (LtQ[m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 715

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]

Rule 112

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 110

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Sqrt[e]*Rt[-(b/d)
, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/b, 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 117

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 116

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (PosQ[-(b/d)] || NegQ[-(b/f)])

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (b+2 c x) \sqrt{d+e x}}{b^2 \sqrt{b x+c x^2}}+\frac{2 \int \frac{\frac{b e}{2}+c e x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{b^2}\\ &=-\frac{2 (b+2 c x) \sqrt{d+e x}}{b^2 \sqrt{b x+c x^2}}+\frac{(2 c) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{b^2}-\frac{(2 c d-b e) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{b^2}\\ &=-\frac{2 (b+2 c x) \sqrt{d+e x}}{b^2 \sqrt{b x+c x^2}}+\frac{\left (2 c \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{b^2 \sqrt{b x+c x^2}}-\frac{\left ((2 c d-b e) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{b^2 \sqrt{b x+c x^2}}\\ &=-\frac{2 (b+2 c x) \sqrt{d+e x}}{b^2 \sqrt{b x+c x^2}}+\frac{\left (2 c \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}{b^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left ((2 c d-b e) \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}{b^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 (b+2 c x) \sqrt{d+e x}}{b^2 \sqrt{b x+c x^2}}+\frac{4 \sqrt{c} \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 )}{(-b)^{3/2} \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 (2 c d-b e) \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 )}{(-b)^{3/2} \sqrt{c} \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 0.41625, size = 186, normalized size = 0.81 \[ \frac{-2 i e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+4 i e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+2 \sqrt{\frac{b}{c}} (d+e x)}{b \sqrt{\frac{b}{c}} \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b/c]*(d + e*x) + (4*I)*e*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqr
t[x]], (c*d)/(b*e)] - (2*I)*e*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x
]], (c*d)/(b*e)])/(b*Sqrt[b/c]*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

________________________________________________________________________________________

Maple [A]  time = 0.286, size = 352, normalized size = 1.5 \begin{align*} 2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{x \left ( cx+b \right ){b}^{2}c\sqrt{ex+d}} \left ({\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}e\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}-2\,{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) bcd\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}-2\,{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}e\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}+2\,{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) bcd\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}-2\,{x}^{2}{c}^{2}e-xbce-2\,x{c}^{2}d-bcd \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + b x} \sqrt{e x + d}}{c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2 + b*x)*sqrt(e*x + d)/(c^2*x^4 + 2*b*c*x^3 + b^2*x^2), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d + e x}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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