3.1277 \(\int \frac{A+B x}{(d+e x)^{5/2} (b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=570 \[ -\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^2 (-e) (B d-4 A e)-3 b c d (2 A e+B d)+6 A c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 e \sqrt{b x+c x^2} \left (-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{3 b^2 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 e \sqrt{b x+c x^2} \left (b^2 (-e) (B d-4 A e)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^{3/2} (c d-b e)} \]

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2]) -
(2*e*(6*A*c^2*d^2 - b^2*e*(B*d - 4*A*e) - 3*b*c*d*(B*d + 2*A*e))*Sqrt[b*x + c*x^2])/(3*b^2*d^2*(c*d - b*e)^2*
(d + e*x)^(3/2)) - (2*e*(6*A*c^3*d^3 - b^2*c*d*e*(7*B*d - 19*A*e) + 2*b^3*e^2*(B*d - 4*A*e) - 3*b*c^2*d^2*(B*d
+ 3*A*e))*Sqrt[b*x + c*x^2])/(3*b^2*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) + (2*Sqrt[c]*(6*A*c^3*d^3 - b^2*c*d*e*(7
*B*d - 19*A*e) + 2*b^3*e^2*(B*d - 4*A*e) - 3*b*c^2*d^2*(B*d + 3*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)^(3/2)*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*
Sqrt[b*x + c*x^2]) - (2*Sqrt[c]*(6*A*c^2*d^2 - b^2*e*(B*d - 4*A*e) - 3*b*c*d*(B*d + 2*A*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)])/(3*(-b)^(3/2)*d^2*(c*d -
b*e)^2*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.987087, antiderivative size = 570, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {822, 834, 843, 715, 112, 110, 117, 116} \[ -\frac{2 e \sqrt{b x+c x^2} \left (-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{3 b^2 d^3 \sqrt{d+e x} (c d-b e)^3}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^3 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)^3}-\frac{2 e \sqrt{b x+c x^2} \left (b^2 (-e) (B d-4 A e)-3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 b^2 d^2 (d+e x)^{3/2} (c d-b e)^2}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^2 (-e) (B d-4 A e)-3 b c d (2 A e+B d)+6 A c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{3/2} d^2 \sqrt{b x+c x^2} \sqrt{d+e x} (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^{3/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(d + e*x)^(3/2)*Sqrt[b*x + c*x^2]) -
(2*e*(6*A*c^2*d^2 - b^2*e*(B*d - 4*A*e) - 3*b*c*d*(B*d + 2*A*e))*Sqrt[b*x + c*x^2])/(3*b^2*d^2*(c*d - b*e)^2*
(d + e*x)^(3/2)) - (2*e*(6*A*c^3*d^3 - b^2*c*d*e*(7*B*d - 19*A*e) + 2*b^3*e^2*(B*d - 4*A*e) - 3*b*c^2*d^2*(B*d
+ 3*A*e))*Sqrt[b*x + c*x^2])/(3*b^2*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) + (2*Sqrt[c]*(6*A*c^3*d^3 - b^2*c*d*e*(7
*B*d - 19*A*e) + 2*b^3*e^2*(B*d - 4*A*e) - 3*b*c^2*d^2*(B*d + 3*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)^(3/2)*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*
Sqrt[b*x + c*x^2]) - (2*Sqrt[c]*(6*A*c^2*d^2 - b^2*e*(B*d - 4*A*e) - 3*b*c*d*(B*d + 2*A*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)])/(3*(-b)^(3/2)*d^2*(c*d -
b*e)^2*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 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(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)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -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}{(d+e x)^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} b e (b B d+3 A c d-4 A b e)-\frac{3}{2} c e (b B d-2 A c d+A b e) x}{(d+e x)^{5/2} \sqrt{b x+c x^2}} \, dx}{b^2 d (c d-b e)}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{2 e \left (6 A c^2 d^2-b^2 e (B d-4 A e)-3 b c d (B d+2 A e)\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac{4 \int \frac{-\frac{1}{4} b e \left (3 A c^2 d^2+3 b c d (2 B d-5 A e)-2 b^2 e (B d-4 A e)\right )-\frac{1}{4} c e \left (6 A c^2 d^2-b^2 e (B d-4 A e)-3 b c d (B d+2 A e)\right ) x}{(d+e x)^{3/2} \sqrt{b x+c x^2}} \, dx}{3 b^2 d^2 (c d-b e)^2}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{2 e \left (6 A c^2 d^2-b^2 e (B d-4 A e)-3 b c d (B d+2 A e)\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 e \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (B d+3 A e)\right ) \sqrt{b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{8 \int \frac{-\frac{1}{8} b c d e \left (3 A c^2 d^2+b^2 e (B d-4 A e)-9 b c d (B d-A e)\right )-\frac{1}{8} c e \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (B d+3 A e)\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{3 b^2 d^3 (c d-b e)^3}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{2 e \left (6 A c^2 d^2-b^2 e (B d-4 A e)-3 b c d (B d+2 A e)\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 e \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (B d+3 A e)\right ) \sqrt{b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{\left (c \left (6 A c^2 d^2-b^2 e (B d-4 A e)-3 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^2 d^2 (c d-b e)^2}+\frac{\left (c \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (B d+3 A e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{3 b^2 d^3 (c d-b e)^3}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{2 e \left (6 A c^2 d^2-b^2 e (B d-4 A e)-3 b c d (B d+2 A e)\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 e \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (B d+3 A e)\right ) \sqrt{b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt{d+e x}}-\frac{\left (c \left (6 A c^2 d^2-b^2 e (B d-4 A e)-3 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^2 d^2 (c d-b e)^2 \sqrt{b x+c x^2}}+\frac{\left (c \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (B d+3 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^2 d^3 (c d-b e)^3 \sqrt{b x+c x^2}}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{2 e \left (6 A c^2 d^2-b^2 e (B d-4 A e)-3 b c d (B d+2 A e)\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 e \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (B d+3 A e)\right ) \sqrt{b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt{d+e x}}+\frac{\left (c \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (B d+3 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^2 d^3 (c d-b e)^3 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (c \left (6 A c^2 d^2-b^2 e (B d-4 A e)-3 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^2 d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}-\frac{2 e \left (6 A c^2 d^2-b^2 e (B d-4 A e)-3 b c d (B d+2 A e)\right ) \sqrt{b x+c x^2}}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac{2 e \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (B d+3 A e)\right ) \sqrt{b x+c x^2}}{3 b^2 d^3 (c d-b e)^3 \sqrt{d+e x}}+\frac{2 \sqrt{c} \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (B d+3 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)^{3/2} d^3 (c d-b e)^3 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 \sqrt{c} \left (6 A c^2 d^2-b^2 e (B d-4 A e)-3 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)^{3/2} d^2 (c d-b e)^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 4.3909, size = 506, normalized size = 0.89 \[ \frac{2 \left (b \left (b^2 d e^2 x (b+c x) (B d-A e) (c d-b e)+b^2 e^2 x (b+c x) (d+e x) (5 A e (b e-2 c d)+B d (7 c d-2 b e))+3 c^3 d^3 x (d+e x)^2 (b B-A c)-3 A (b+c x) (d+e x)^2 (c d-b e)^3\right )+c \sqrt{\frac{b}{c}} (d+e x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (2 b^2 e (4 A e-B d)+3 b c d (2 B d-5 A e)+3 A c^2 d^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^2 c d e (19 A e-7 B d)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^2 c d e (19 A e-7 B d)+2 b^3 e^2 (B d-4 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )\right )\right )}{3 b^3 d^3 \sqrt{x (b+c x)} (d+e x)^{3/2} (c d-b e)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

(2*(b*(b^2*d*e^2*(B*d - A*e)*(c*d - b*e)*x*(b + c*x) + b^2*e^2*(B*d*(7*c*d - 2*b*e) + 5*A*e*(-2*c*d + b*e))*x*
(b + c*x)*(d + e*x) + 3*c^3*(b*B - A*c)*d^3*x*(d + e*x)^2 - 3*A*(c*d - b*e)^3*(b + c*x)*(d + e*x)^2) + Sqrt[b/
c]*c*(d + e*x)*(Sqrt[b/c]*(6*A*c^3*d^3 + 2*b^3*e^2*(B*d - 4*A*e) - 3*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(-7*B
*d + 19*A*e))*(b + c*x)*(d + e*x) + I*b*e*(6*A*c^3*d^3 + 2*b^3*e^2*(B*d - 4*A*e) - 3*b*c^2*d^2*(B*d + 3*A*e) +
b^2*c*d*e*(-7*B*d + 19*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*e*(c*d - b*e)*(3*A*c^2*d^2 + 3*b*c*d*(2*B*d - 5*A*e) + 2*b^2*e*(-(B*d) + 4*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^3*d^3*(c*d
- b*e)^3*Sqrt[x*(b + c*x)]*(d + e*x)^(3/2))

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Maple [B]  time = 0.054, size = 3024, normalized size = 5.3 \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)/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x)

[Out]

-2/3*(x*(c*x+b))^(1/2)/x*(3*B*x*b*c^4*d^5+4*A*x*((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^4*c*d*e^4-10*A*x*((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^2*d^2*e^3+12*A*x*((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^3*d^3
*e^2-6*A*x*((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^4*d^4*e-27*A*x*((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*c*d*e^4+28*A*x*((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^2*d^2*e^3-15*A*x*((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^3*d^3*e^2+6*A*x*((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^4*d^4*e-B*x*((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^4*c*d^2*e^3-2*B*x*((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^2*d^3*e^2+3*B*x*((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^3*d^4*e+9*B*x*((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*c*d^2*e^3-4*B*x*(
(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^2*d^3*e^2-3*B*x*((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^3*d^4*e-3*A*b*c^4*d^5-2*B*x^2*b^4*c*d*e^4+6*B*x^2*b*c^4*d^4*e+8*B*x*b^3*c^2*d
^3*e^2+9*A*x*b^2*c^3*d^3*e^2+3*A*x*b*c^4*d^4*e-19*A*x^3*b^2*c^3*d*e^4+9*A*x^3*b*c^4*d^2*e^3+15*A*x^2*b*c^4*d^3
*e^2+4*B*x^2*b^3*c^2*d^2*e^3+12*A*x*b^4*c*d*e^4-26*A*x*b^3*c^2*d^2*e^3-3*B*x*b^4*c*d^2*e^3-7*A*x^2*b^3*c^2*d*e
^4-20*A*x^2*b^2*c^3*d^2*e^3+8*B*x^2*b^2*c^3*d^3*e^2-2*B*x^3*b^3*c^2*d*e^4+3*B*x^3*b*c^4*d^3*e^2-6*A*x*c^5*d^5+
7*B*x^3*b^2*c^3*d^2*e^3-3*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^2*c^3*d^5+8*A*x^3*b^3*c^2*e^5-6*A*x^3*c^5*d^3*e^2+8*A*x^2*b^4*c*e^5-12*A*x^2*
c^5*d^4*e+6*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*c^4*d^5+3*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^2*c^3*d^5-2*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^5*d^2*e^3+8*A*x*((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^5*e^5-6*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^4*d^5+8*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^5*
d*e^4+3*A*b^4*c*d^2*e^3-9*A*b^3*c^2*d^3*e^2+9*A*b^2*c^3*d^4*e-2*B*x*((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^5*d*e^4+4*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^4*c*d^2*e^3-10*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^3*c^2*d^3*e^2+12*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^3*d^4*e-27*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellip
ticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^2*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))*b^3*c^2*d^3*e^2-15*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^3*d^4*e-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^4*c*d^3*e^2-2*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^2*d^4*e+9*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elliptic
E(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d^3*e^2-4*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^2*d^4*e)/b^2/d^3/c/(e*x+d)^(3/2)/(b*e-c*
d)^3/(c*x+b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(3/2)*(e*x + d)^(5/2)), x)