### 3.411 $$\int \frac{1}{(d+e x)^{3/2} \sqrt{b x+c x^2}} \, dx$$

Optimal. Leaf size=146 $\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 e \sqrt{b x+c x^2}}{d \sqrt{d+e x} (c d-b e)}$

[Out]

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

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Rubi [A]  time = 0.0806936, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.217, Rules used = {744, 21, 715, 112, 110} $\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 e \sqrt{b x+c x^2}}{d \sqrt{d+e x} (c d-b e)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 744

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

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 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]

Rubi steps

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

Mathematica [A]  time = 0.161222, size = 127, normalized size = 0.87 $\frac{2 \sqrt{x (b+c x)} \left (e \sqrt{x} \sqrt{-\frac{d}{e}} \sqrt{\frac{d}{e x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right )|\frac{b e}{c d}\right )+d \sqrt{\frac{b}{c x}+1}\right )}{d x \sqrt{\frac{b}{c x}+1} \sqrt{d+e x} (c d-b e)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.291, size = 216, normalized size = 1.5 \begin{align*} 2\,{\frac{\sqrt{x \left ( cx+b \right ) }\sqrt{ex+d}}{ \left ( be-cd \right ) cdx \left ( ce{x}^{2}+bxe+cdx+bd \right ) } \left ({\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}}}-{\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}}}+{x}^{2}{c}^{2}e+xbce \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(c*x^2 + b*x)*(e*x + d)^(3/2)), 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 e^{2} x^{4} + b d^{2} x +{\left (2 \, c d e + b e^{2}\right )} x^{3} +{\left (c d^{2} + 2 \, b d e\right )} x^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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