3.2635 \(\int \frac{(A+B x) (d+e x)^{5/2}}{(a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=678 \[ -\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b c \left (-29 a B e^2+6 A c d e+3 B c d^2\right )-2 c^2 \left (-9 a A e^2-20 a B d e+3 A c d^2\right )-b^2 c e (6 A e+13 B d)+8 b^3 B e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right )}{3 c^2 \left (b^2-4 a c\right )}+\frac{2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

[Out]

(2*(d + e*x)^(3/2)*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*
x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (2*e*(4*b^2*B*e - 3*b*c*(B*d + A*e) + 2*c*(3*A*c*d - 5*a*B*e))*S
qrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c^2*(b^2 - 4*a*c)) - (Sqrt[2]*(8*b^3*B*e^2 - b^2*c*e*(13*B*d + 6*A*e) -
2*c^2*(3*A*c*d^2 - 20*a*B*d*e - 9*a*A*e^2) + b*c*(3*B*c*d^2 + 6*A*c*d*e - 29*a*B*e^2))*Sqrt[d + e*x]*Sqrt[-((
c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/
Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*c^3*Sqrt[b^2 - 4*a*c]*Sqrt[(c*(d +
e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*(4*b^2
*B*e - 3*b*c*(B*d + A*e) + 2*c*(3*A*c*d - 5*a*B*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sq
rt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*
a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*c^3*Sqrt[b^2 - 4*a*c]*Sqrt[d
+ e*x]*Sqrt[a + b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.03134, antiderivative size = 678, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {818, 832, 843, 718, 424, 419} \[ -\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b c \left (-29 a B e^2+6 A c d e+3 B c d^2\right )-2 c^2 \left (-9 a A e^2-20 a B d e+3 A c d^2\right )-b^2 c e (6 A e+13 B d)+8 b^3 B e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right )}{3 c^2 \left (b^2-4 a c\right )}+\frac{2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*
x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (2*e*(4*b^2*B*e - 3*b*c*(B*d + A*e) + 2*c*(3*A*c*d - 5*a*B*e))*S
qrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c^2*(b^2 - 4*a*c)) - (Sqrt[2]*(8*b^3*B*e^2 - b^2*c*e*(13*B*d + 6*A*e) -
2*c^2*(3*A*c*d^2 - 20*a*B*d*e - 9*a*A*e^2) + b*c*(3*B*c*d^2 + 6*A*c*d*e - 29*a*B*e^2))*Sqrt[d + e*x]*Sqrt[-((
c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/
Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*c^3*Sqrt[b^2 - 4*a*c]*Sqrt[(c*(d +
e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*(4*b^2
*B*e - 3*b*c*(B*d + A*e) + 2*c*(3*A*c*d - 5*a*B*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sq
rt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*
a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*c^3*Sqrt[b^2 - 4*a*c]*Sqrt[d
+ e*x]*Sqrt[a + 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 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 718

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*Rt[b^2 - 4*a*c, 2]
*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d -
b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c,
2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, 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] && EqQ[m^2, 1/4]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac{2 (d+e x)^{3/2} \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 \int \frac{\sqrt{d+e x} \left (\frac{1}{2} e \left (b^2 B d+3 A b c d-10 a B c d+3 a b B e-6 a A c e\right )+\frac{1}{2} e \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac{2 (d+e x)^{3/2} \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 e \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}+\frac{4 \int \frac{-\frac{1}{4} e \left (4 b^3 B d e-b^2 \left (6 B c d^2+3 A c d e-4 a B e^2\right )+2 a c \left (12 A c d e+5 B \left (3 c d^2-a e^2\right )\right )-b c \left (22 a B d e+3 A \left (c d^2+a e^2\right )\right )\right )-\frac{1}{4} e \left (8 b^3 B e^2-b^2 c e (13 B d+6 A e)-2 c^2 \left (3 A c d^2-20 a B d e-9 a A e^2\right )+b c \left (3 B c d^2+6 A c d e-29 a B e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )}\\ &=\frac{2 (d+e x)^{3/2} \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 e \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}-\frac{\left (\left (c d^2-b d e+a e^2\right ) \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )}-\frac{\left (8 b^3 B e^2-b^2 c e (13 B d+6 A e)-2 c^2 \left (3 A c d^2-20 a B d e-9 a A e^2\right )+b c \left (3 B c d^2+6 A c d e-29 a B e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{3 c^2 \left (b^2-4 a c\right )}\\ &=\frac{2 (d+e x)^{3/2} \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 e \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}-\frac{\left (\sqrt{2} \left (8 b^3 B e^2-b^2 c e (13 B d+6 A e)-2 c^2 \left (3 A c d^2-20 a B d e-9 a A e^2\right )+b c \left (3 B c d^2+6 A c d e-29 a B e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{a+b x+c x^2}}-\frac{\left (2 \sqrt{2} \left (c d^2-b d e+a e^2\right ) \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 (d+e x)^{3/2} \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 e \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}-\frac{\sqrt{2} \left (8 b^3 B e^2-b^2 c e (13 B d+6 A e)-2 c^2 \left (3 A c d^2-20 a B d e-9 a A e^2\right )+b c \left (3 B c d^2+6 A c d e-29 a B e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{2} \left (c d^2-b d e+a e^2\right ) \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 13.0522, size = 1287, normalized size = 1.9 \[ \frac{\sqrt{d+e x} \left (\frac{2 B e^2}{3 c^2}+\frac{2 \left (-B e^2 x b^3-a B e^2 b^2+A c e^2 x b^2+2 B c d e x b^2+A c^2 d^2 b+a A c e^2 b+2 a B c d e b-B c^2 d^2 x b+3 a B c e^2 x b-2 A c^2 d e x b-2 a B c^2 d^2+2 a^2 B c e^2-4 a A c^2 d e+2 A c^3 d^2 x-2 a A c^2 e^2 x-4 a B c^2 d e x\right )}{c^2 \left (4 a c-b^2\right ) \left (c x^2+b x+a\right )}\right ) \left (c x^2+b x+a\right )^2}{(a+x (b+c x))^{3/2}}+\frac{2 (d+e x)^{3/2} \left (\left (8 B e^2 b^3-c e (13 B d+6 A e) b^2+c \left (3 B c d^2+6 A c e d-29 a B e^2\right ) b+2 c^2 \left (-3 A c d^2+20 a B e d+9 a A e^2\right )\right ) \left (c \left (\frac{d}{d+e x}-1\right )^2+\frac{e \left (-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right )}{d+e x}\right )-\frac{i \sqrt{1-\frac{2 \left (c d^2+e (a e-b d)\right )}{\left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt{\frac{2 \left (c d^2+e (a e-b d)\right )}{\left (-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}+1} \left (\left (2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}\right ) \left (8 B e^2 b^3-c e (13 B d+6 A e) b^2+c \left (3 B c d^2+6 A c e d-29 a B e^2\right ) b+2 c^2 \left (-3 A c d^2+20 a B e d+9 a A e^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )+\left (8 B e^3 b^4-e^2 \left (21 c d B+8 \sqrt{\left (b^2-4 a c\right ) e^2} B+6 A c e\right ) b^3+c e \left (6 A e \left (2 c d+\sqrt{\left (b^2-4 a c\right ) e^2}\right )+B \left (15 c d^2+13 \sqrt{\left (b^2-4 a c\right ) e^2} d-37 a e^2\right )\right ) b^2+c \left (a e^2 \left (84 c d B+29 \sqrt{\left (b^2-4 a c\right ) e^2} B+24 A c e\right )-3 c d \sqrt{\left (b^2-4 a c\right ) e^2} (B d+2 A e)\right ) b+2 c^2 \left (10 a^2 B e^3-a \left (10 B d \left (3 c d+2 \sqrt{\left (b^2-4 a c\right ) e^2}\right )+3 A e \left (8 c d+3 \sqrt{\left (b^2-4 a c\right ) e^2}\right )\right ) e+3 A c d^2 \sqrt{\left (b^2-4 a c\right ) e^2}\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right ),-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt{2} \sqrt{\frac{c d^2+e (a e-b d)}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}} \sqrt{d+e x}}\right ) \left (c x^2+b x+a\right )^{3/2}}{3 c^3 \left (4 a c-b^2\right ) e (a+x (b+c x))^{3/2} \sqrt{\frac{(d+e x)^2 \left (c \left (\frac{d}{d+e x}-1\right )^2+\frac{e \left (-\frac{d b}{d+e x}+b+\frac{a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[d + e*x]*(a + b*x + c*x^2)^2*((2*B*e^2)/(3*c^2) + (2*(A*b*c^2*d^2 - 2*a*B*c^2*d^2 + 2*a*b*B*c*d*e - 4*a*
A*c^2*d*e - a*b^2*B*e^2 + a*A*b*c*e^2 + 2*a^2*B*c*e^2 - b*B*c^2*d^2*x + 2*A*c^3*d^2*x + 2*b^2*B*c*d*e*x - 2*A*
b*c^2*d*e*x - 4*a*B*c^2*d*e*x - b^3*B*e^2*x + A*b^2*c*e^2*x + 3*a*b*B*c*e^2*x - 2*a*A*c^2*e^2*x))/(c^2*(-b^2 +
4*a*c)*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(3/2) + (2*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)*((8*b^3*B*e^
2 - b^2*c*e*(13*B*d + 6*A*e) + 2*c^2*(-3*A*c*d^2 + 20*a*B*d*e + 9*a*A*e^2) + b*c*(3*B*c*d^2 + 6*A*c*d*e - 29*a
*B*e^2))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) - ((I/2)*Sqrt[1 - (2
*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*
d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(8*
b^3*B*e^2 - b^2*c*e*(13*B*d + 6*A*e) + 2*c^2*(-3*A*c*d^2 + 20*a*B*d*e + 9*a*A*e^2) + b*c*(3*B*c*d^2 + 6*A*c*d*
e - 29*a*B*e^2))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*
e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] +
(8*b^4*B*e^3 - b^3*e^2*(21*B*c*d + 6*A*c*e + 8*B*Sqrt[(b^2 - 4*a*c)*e^2]) + b*c*(-3*c*d*Sqrt[(b^2 - 4*a*c)*e^
2]*(B*d + 2*A*e) + a*e^2*(84*B*c*d + 24*A*c*e + 29*B*Sqrt[(b^2 - 4*a*c)*e^2])) + b^2*c*e*(6*A*e*(2*c*d + Sqrt[
(b^2 - 4*a*c)*e^2]) + B*(15*c*d^2 - 37*a*e^2 + 13*d*Sqrt[(b^2 - 4*a*c)*e^2])) + 2*c^2*(10*a^2*B*e^3 + 3*A*c*d^
2*Sqrt[(b^2 - 4*a*c)*e^2] - a*e*(10*B*d*(3*c*d + 2*Sqrt[(b^2 - 4*a*c)*e^2]) + 3*A*e*(8*c*d + 3*Sqrt[(b^2 - 4*a
*c)*e^2]))))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2]
)])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sq
rt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[d + e*x])))/(3*c^3*(-b^2
+ 4*a*c)*e*(a + x*(b + c*x))^(3/2)*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)
/(d + e*x)))/(d + e*x)))/e^2])

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Maple [B]  time = 0.103, size = 10385, normalized size = 15.3 \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+a)^(3/2),x)

[Out]

result too large to display

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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{5}{2}}}{{\left (c x^{2} + b x + a\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)^(5/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 + b*x + a)^(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^{2} x^{3} + A d^{2} +{\left (2 \, B d e + A e^{2}\right )} x^{2} +{\left (B d^{2} + 2 \, A d e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, 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+a)^(3/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [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+a)^(3/2),x, algorithm="giac")

[Out]

Timed out