3.418 \(\int \frac{1}{\sqrt{d+e x} (b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=274 \[ -\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{(-b)^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (c d-b e)}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)} \]

[Out]

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

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Rubi [A]  time = 0.219433, antiderivative size = 274, 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 = {740, 843, 715, 112, 110, 117, 116} \[ -\frac{2 \sqrt{d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (c d-b e)}-\frac{4 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*(b*x + c*x^2)^(3/2)),x]

[Out]

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

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && 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{1}{\sqrt{d+e x} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 \sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt{b x+c x^2}}-\frac{2 \int \frac{-\frac{1}{2} b c d e-\frac{1}{2} c e (2 c d-b e) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{b^2 d (c d-b e)}\\ &=-\frac{2 \sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt{b x+c x^2}}-\frac{(2 c) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{b^2}+\frac{(c (2 c d-b e)) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{b^2 d (c d-b e)}\\ &=-\frac{2 \sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt{b x+c x^2}}-\frac{\left (2 c \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{\left (c (2 c d-b e) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{b^2 d (c d-b e) \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt{b x+c x^2}}+\frac{\left (c (2 c d-b e) \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 d (c d-b e) \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (2 c \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 \sqrt{d+e x} (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt{b x+c x^2}}+\frac{2 \sqrt{c} (2 c d-b e) \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} d (c d-b e) \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{4 \sqrt{c} \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{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 0.591614, size = 220, normalized size = 0.8 \[ \frac{-2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-c d) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+2 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 b c d (d+e x)}{b^2 d \sqrt{x (b+c x)} \sqrt{d+e x} (b e-c d)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*(b*x + c*x^2)^(3/2)),x]

[Out]

(-2*b*c*d*(d + e*x) + (2*I)*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE
[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (2*I)*Sqrt[b/c]*c*e*(-(c*d) + b*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^2*d*(-(c*d) + b*e)*Sqrt[x*(b + c*x)]*S
qrt[d + e*x])

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Maple [B]  time = 0.317, size = 480, normalized size = 1.8 \begin{align*} -2\,{\frac{\sqrt{x \left ( cx+b \right ) }}{x \left ( cx+b \right ) \left ( be-cd \right ) c{b}^{2}d\sqrt{ex+d}} \left ( 2\,{b}^{2}d\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) ec-2\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}{d}^{2}+\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{3}{e}^{2}-3\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}cde+2\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}{d}^{2}+{x}^{2}b{c}^{2}{e}^{2}-2\,{x}^{2}{c}^{3}de+x{b}^{2}c{e}^{2}-2\,x{c}^{3}{d}^{2}+{b}^{2}cde-b{c}^{2}{d}^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} \sqrt{e x + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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} e x^{5} + b^{2} d x^{2} +{\left (c^{2} d + 2 \, b c e\right )} x^{4} +{\left (2 \, b c d + b^{2} e\right )} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2 + b*x)*sqrt(e*x + d)/(c^2*e*x^5 + b^2*d*x^2 + (c^2*d + 2*b*c*e)*x^4 + (2*b*c*d + b^2*e)*x^
3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \sqrt{d + e x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} \sqrt{e x + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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