3.1282 \(\int \frac{A+B x}{\sqrt{d+e x} (b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=543 \[ \frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^2 e (9 B d-A e)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}+\frac{2 \sqrt{d+e x} \left (c x \left (b^2 c d e (4 A e+13 B d)+b^3 \left (-e^2\right ) (3 B d-2 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (3 B d-2 A e)-b c d (5 A e+4 B d)+8 A c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 c d e (4 A e+13 B d)+b^3 \left (-e^2\right ) (3 B d-2 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}-\frac{2 \sqrt{d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)} \]

[Out]

(-2*Sqrt[d + e*x]*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(3*b^2*d*(c*d - b*e)*(b*x + c*x^2)^(3/2))
+ (2*Sqrt[d + e*x]*(b*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d - 2*A*e) - b*c*d*(4*B*d + 5*A*e)) + c*(16*A*c^3
*d^3 - b^3*e^2*(3*B*d - 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(13*B*d + 4*A*e))*x))/(3*b^4*d^2*(c*d -
b*e)^2*Sqrt[b*x + c*x^2]) - (2*Sqrt[c]*(16*A*c^3*d^3 - b^3*e^2*(3*B*d - 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) +
b^2*c*d*e*(13*B*d + 4*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)])/(3*(-b)^(7/2)*d^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[c]*(16*A*c^2*
d^2 + b^2*e*(9*B*d - A*e) - 8*b*c*d*(B*d + 2*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSi
n[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.755695, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {822, 843, 715, 112, 110, 117, 116} \[ \frac{2 \sqrt{d+e x} \left (c x \left (b^2 c d e (4 A e+13 B d)+b^3 \left (-e^2\right ) (3 B d-2 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (3 B d-2 A e)-b c d (5 A e+4 B d)+8 A c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (b^2 c d e (4 A e+13 B d)+b^3 \left (-e^2\right ) (3 B d-2 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^2}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^2 e (9 B d-A e)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)}-\frac{2 \sqrt{d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)^(5/2)),x]

[Out]

(-2*Sqrt[d + e*x]*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(3*b^2*d*(c*d - b*e)*(b*x + c*x^2)^(3/2))
+ (2*Sqrt[d + e*x]*(b*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d - 2*A*e) - b*c*d*(4*B*d + 5*A*e)) + c*(16*A*c^3
*d^3 - b^3*e^2*(3*B*d - 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(13*B*d + 4*A*e))*x))/(3*b^4*d^2*(c*d -
b*e)^2*Sqrt[b*x + c*x^2]) - (2*Sqrt[c]*(16*A*c^3*d^3 - b^3*e^2*(3*B*d - 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) +
b^2*c*d*e*(13*B*d + 4*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)])/(3*(-b)^(7/2)*d^2*(c*d - b*e)^2*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[c]*(16*A*c^2*
d^2 + b^2*e*(9*B*d - A*e) - 8*b*c*d*(B*d + 2*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSi
n[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d*(c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 822

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

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} \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 \sqrt{d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )-\frac{3}{2} c e (b B d-2 A c d+A b e) x}{\sqrt{d+e x} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)}\\ &=-\frac{2 \sqrt{d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}+\frac{4 \int \frac{-\frac{1}{4} b c d e \left (8 A c^2 d^2+b^2 e (6 B d+A e)-b c d (4 B d+11 A e)\right )-\frac{1}{4} c e \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}\\ &=-\frac{2 \sqrt{d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}+\frac{\left (c \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 A e)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 b^4 d (c d-b e)}-\frac{\left (c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}\\ &=-\frac{2 \sqrt{d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}+\frac{\left (c \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 A e)\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{3 b^4 d (c d-b e) \sqrt{b x+c x^2}}-\frac{\left (c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}-\frac{\left (c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 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}{3 b^4 d^2 (c d-b e)^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (c \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 A e)\right ) \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 b^4 d (c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 \sqrt{d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac{2 \sqrt{d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}-\frac{2 \sqrt{c} \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 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 )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{2 \sqrt{c} \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 A e)\right ) \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 (-b)^{7/2} d (c d-b e) \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 4.21695, size = 514, normalized size = 0.95 \[ -\frac{2 \left (b (d+e x) \left (c^2 d^2 x^2 (b+c x) \left (5 b c (2 A e+B d)-8 A c^2 d-7 b^2 B e\right )+b c^2 d^2 x^2 (b B-A c) (c d-b e)+x (b+c x)^2 (c d-b e)^2 (-2 A b e-8 A c d+3 b B d)+A b d (b+c x)^2 (c d-b e)^2\right )+c x \sqrt{\frac{b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (b^2 e (3 B d-2 A e)-b c d (5 A e+4 B d)+8 A c^2 d^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 c d e (4 A e+13 B d)+b^3 e^2 (2 A e-3 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^2 c d e (4 A e+13 B d)+b^3 e^2 (2 A e-3 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )\right )}{3 b^5 d^2 (x (b+c x))^{3/2} \sqrt{d+e x} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*(d + e*x)*(b*c^2*(b*B - A*c)*d^2*(c*d - b*e)*x^2 + c^2*d^2*(-8*A*c^2*d - 7*b^2*B*e + 5*b*c*(B*d + 2*A*e
))*x^2*(b + c*x) + A*b*d*(c*d - b*e)^2*(b + c*x)^2 + (c*d - b*e)^2*(3*b*B*d - 8*A*c*d - 2*A*b*e)*x*(b + c*x)^2
) + Sqrt[b/c]*c*x*(b + c*x)*(Sqrt[b/c]*(16*A*c^3*d^3 + b^3*e^2*(-3*B*d + 2*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e) +
b^2*c*d*e*(13*B*d + 4*A*e))*(b + c*x)*(d + e*x) + I*b*e*(16*A*c^3*d^3 + b^3*e^2*(-3*B*d + 2*A*e) - 8*b*c^2*d^2
*(B*d + 3*A*e) + b^2*c*d*e*(13*B*d + 4*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[S
qrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d - 2*A*e) - b*c*d*(4*B*d + 5*A*
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)])))/(3*b^5
*d^2*(c*d - b*e)^2*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])

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Maple [B]  time = 0.053, size = 3140, normalized size = 5.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x)

[Out]

2/3*(24*A*x^2*b*c^5*d^4+4*A*x^4*b^2*c^4*d*e^3-3*B*x^4*b^3*c^3*d*e^3+B*x^3*b^2*c^4*d^3*e+6*A*x^2*b^4*c^2*d*e^3-
3*B*x^2*b^5*c*d*e^3-3*B*x*b^5*c*d^2*e^2+6*B*x*b^4*c^2*d^3*e-11*A*x*b^3*c^3*d^3*e-31*A*x^2*b^2*c^4*d^3*e+A*x*b^
5*c*d*e^3+4*A*x*b^4*c^2*d^2*e^2-33*A*x^3*b^2*c^4*d^2*e^2-6*B*x^3*b^4*c^2*d*e^3+17*B*x^3*b^3*c^3*d^2*e^2-3*A*x^
2*b^3*c^3*d^2*e^2-8*B*x^4*b*c^5*d^3*e+9*A*x^3*b^3*c^3*d*e^3-24*A*x^4*b*c^5*d^2*e^2+13*B*x^4*b^2*c^4*d^2*e^2-A*
b^3*c^3*d^4+16*A*x^3*c^6*d^4+17*B*x^2*b^3*c^3*d^3*e+2*A*x^2*b^5*c*e^4+6*A*x*b^2*c^4*d^4-3*B*x*b^3*c^3*d^4-12*B
*x^2*b^2*c^4*d^4+2*A*x^4*b^3*c^3*e^4+16*A*x^4*c^6*d^3*e-8*B*x^3*b*c^5*d^4+4*A*x^3*b^4*c^2*e^4+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))*x^2*b^5*c
*e^4-16*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))*x^2*b*c^5*d^4+16*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))*x^2*b*c^5*d^4+8*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))*x^2*b^2*c^4*d^4-8*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))*x^2*b^2*c^4*d^4-16*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))*x*b^2*c
^4*d^4+16*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))*x*b^2*c^4*d^4-3*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))*x*b^6*d*e^3+8*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))*x*b^3*c^3*d^4-8*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))*x*b^3*c^3*d^4+15*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))*x^2*b^3*c^3*
d^2*e^2-32*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))*x^2*b^2*c^4*d^3*e-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))*x^2*b^5*c*d*e^3+16*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))*x^2*b^4*c^2*d^2*e^2-21*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))*x^2*b^3*c^3*d^3*e
-9*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))*x^2*b^4*c^2*d^2*e^2+17*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))*x^2*b^3*c^3*d^3*e+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))*x*b^5*c*d*e^3-28*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))*x*b^4*c^2*d^2*e^2+40*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))*x
*b^3*c^3*d^3*e+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))*x*b^5*c*d*e^3+15*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))*x*b^4*c^2*d^2*e^2-32*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))*x*b^3*c^3*d^3*e+16*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))*x*b^5*c*d^2*e^2-21
*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))*x*b^4*c^2*d^3*e-9*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))*x*b^5*c*d^2*e^2+17*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))*x*b^4*c^2*d^3*e+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))*x^2*b^4*c^2*d*e^3-28*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))*x^2*b^3*
c^3*d^2*e^2+40*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))*x^2*b^2*c^4*d^3*e+A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elliptic
F(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^4*c^2*d*e^3+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))*x*b^6*e^4-A*b^5*c*d^2*e^2+2*A*b^4*c^2*d^3*
e)/x^2*(x*(c*x+b))^(1/2)/b^4/d^2/c/(c*x+b)^2/(b*e-c*d)^2/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}} \sqrt{e x + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*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}{\left (B x + A\right )} \sqrt{e x + d}}{c^{3} e x^{7} + b^{3} d x^{3} +{\left (c^{3} d + 3 \, b c^{2} e\right )} x^{6} + 3 \,{\left (b c^{2} d + b^{2} c e\right )} x^{5} +{\left (3 \, b^{2} c d + b^{3} e\right )} x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^2 + b*x)*(B*x + A)*sqrt(e*x + d)/(c^3*e*x^7 + b^3*d*x^3 + (c^3*d + 3*b*c^2*e)*x^6 + 3*(b*c^2
*d + b^2*c*e)*x^5 + (3*b^2*c*d + b^3*e)*x^4), 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)/(c*x**2+b*x)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}} \sqrt{e x + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+b*x)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*sqrt(e*x + d)), x)