3.1273 \(\int \frac{(A+B x) (d+e x)^{3/2}}{(b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=295 \[ \frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (b B-2 A c) (c d-b e) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]

[Out]

(-2*Sqrt[d + e*x]*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2*c*Sqrt[b*x + c*x^2]) + (2*(2*A*c
^2*d + 2*b^2*B*e - b*c*(B*d + A*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)*c^(3/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*(b*B - 2*A*c)*d*(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)*c^(3/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.311367, antiderivative size = 295, 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 = {818, 843, 715, 112, 110, 117, 116} \[ -\frac{2 \sqrt{d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (b B-2 A c) (c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt{b x+c x^2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[d + e*x]*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2*c*Sqrt[b*x + c*x^2]) + (2*(2*A*c
^2*d + 2*b^2*B*e - b*c*(B*d + A*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)*c^(3/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*(b*B - 2*A*c)*d*(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)*c^(3/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 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) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 \sqrt{d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 \int \frac{\frac{1}{2} b (b B+A c) d e+\frac{1}{2} e \left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{b^2 c}\\ &=-\frac{2 \sqrt{d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{((b B-2 A c) d (c d-b e)) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{b^2 c}+\frac{\left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{b^2 c}\\ &=-\frac{2 \sqrt{d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{\left ((b B-2 A c) 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}{b^2 c \sqrt{b x+c x^2}}+\frac{\left (\left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{b^2 c \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{\left (\left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) \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 c \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left ((b B-2 A c) 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}{b^2 c \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 \left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) \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} c^{3/2} \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{2 (b B-2 A c) 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 )}{(-b)^{3/2} c^{3/2} \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 1.67401, size = 302, normalized size = 1.02 \[ \frac{2 \left (b (d+e x) (x (b B-A c) (c d-b e)-A c d (b+c x))+\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (A c-2 b B) (c d-b e) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right )\right )\right )}{b^3 c \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*(d + e*x)*((b*B - A*c)*(c*d - b*e)*x - A*c*d*(b + c*x)) + Sqrt[b/c]*(Sqrt[b/c]*(2*A*c^2*d + 2*b^2*B*e -
b*c*(B*d + A*e))*(b + c*x)*(d + e*x) + I*b*e*(2*A*c^2*d + 2*b^2*B*e - b*c*(B*d + A*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)] - I*b*(-2*b*B + A*c)*e*(c*d - b*e)*S
qrt[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^3*c*Sqrt
[x*(b + c*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.032, size = 898, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(A*((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*c*e^2-3*A*((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^2*d*e+2*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipti
cE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^2+2*A*((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^2*c^2*d*e-2*A*((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^3*d^2-2*B*((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^4*e^2+3*B*(
(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*c*d*e-B*((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^2*d^2-B*((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^3*c*d*e+B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*El
lipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^2+A*x^2*b*c^3*e^2-2*A*x^2*c^4*d*e-B*x^2*b^2*c^2*e^2
+B*x^2*b*c^3*d*e-2*A*x*c^4*d^2-B*x*b^2*c^2*d*e+B*x*b*c^3*d^2-A*b*c^3*d^2)/x*(x*(c*x+b))^(1/2)/(c*x+b)/c^3/b^2/
(e*x+d)^(1/2)

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B e x^{2} + A d +{\left (B d + A e\right )} x\right )} \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((B*x+A)*(e*x+d)^(3/2)/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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