3.1266 \(\int \frac{(A+B x) \sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=254 \[ -\frac{2 \sqrt{-b} B d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 c^{3/2} e \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (3 A c e-2 b B e+B c d) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 B \sqrt{b x+c x^2} \sqrt{d+e x}}{3 c} \]

[Out]

(2*B*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*c) + (2*Sqrt[-b]*(B*c*d - 2*b*B*e + 3*A*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b
]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*c^(3/2)*e*Sqrt[1 + (e*x)/d]*Sqr
t[b*x + c*x^2]) - (2*Sqrt[-b]*B*d*(c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sq
rt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*c^(3/2)*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.283494, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {832, 843, 715, 112, 110, 117, 116} \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (3 A c e-2 b B e+B c d) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{-b} B d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} e \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 B \sqrt{b x+c x^2} \sqrt{d+e x}}{3 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*c) + (2*Sqrt[-b]*(B*c*d - 2*b*B*e + 3*A*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b
]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*c^(3/2)*e*Sqrt[1 + (e*x)/d]*Sqr
t[b*x + c*x^2]) - (2*Sqrt[-b]*B*d*(c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sq
rt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*c^(3/2)*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 832

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

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

Mathematica [C]  time = 1.17227, size = 263, normalized size = 1.04 \[ \frac{2 x \left (\frac{i \sqrt{x} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (2 b B-3 A c) (b e-c d) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )}{b}+\frac{(b+c x) (d+e x) (3 A c e-2 b B e+B c d)}{c e x}+i \sqrt{x} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (3 A c e-2 b B e+B c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+B (b+c x) (d+e x)\right )}{3 c \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(B*(b + c*x)*(d + e*x) + ((B*c*d - 2*b*B*e + 3*A*c*e)*(b + c*x)*(d + e*x))/(c*e*x) + I*Sqrt[b/c]*(B*c*d -
2*b*B*e + 3*A*c*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*Sqrt[x]*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/
(b*e)] + (I*Sqrt[b/c]*(2*b*B - 3*A*c)*(-(c*d) + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*Sqrt[x]*EllipticF[I*A
rcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/b))/(3*c*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.019, size = 629, normalized size = 2.5 \begin{align*} -{\frac{2}{3\,x \left ( ce{x}^{2}+bex+cdx+bd \right ){c}^{3}e}\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}c{e}^{2}\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}-3\,A{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}de\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}-B{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}cde\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}+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{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}-2\,B{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{3}{e}^{2}\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}+3\,B{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}cde\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}-B{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}{d}^{2}\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}-B{x}^{3}{c}^{3}{e}^{2}-B{x}^{2}b{c}^{2}{e}^{2}-B{x}^{2}{c}^{3}de-Bxb{c}^{2}de \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(3*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c*e^2*((c*x+b
)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-3*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*
b*c^2*d*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*
e-c*d))^(1/2))*b^2*c*d*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+B*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)-2*B*Elli
pticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)
^(1/2)+3*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c*d*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d
))^(1/2)*(-c*x/b)^(1/2)-B*EllipticE(((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)-B*x^3*c^3*e^2-B*x^2*b*c^2*e^2-B*x^2*c^3*d*e-B*x*b*c^2*d*e)/x/(c*e*x^2+b
*e*x+c*d*x+b*d)/c^3/e

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*x + A)*sqrt(e*x + d)/sqrt(c*x^2 + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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