### 3.266 $$\int \frac{(d+e x)^{3/2} (A+B x+C x^2)}{\sqrt{a+b x+c x^2}} \, dx$$

Optimal. Leaf size=724 $-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (-c e (25 a C e+28 b B e+15 b C d)+c^2 \left (-\left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{105 c^4 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (25 a C e+28 b B e+15 b C d)+c^2 \left (-\left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right )}{105 c^3 e}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e \left (a e (63 B e+82 C d)+b \left (70 A e^2+91 B d e+12 C d^2\right )\right )-8 b c e^2 (13 a C e+7 b B e+9 b C d)+c^3 d \left (6 C d^2-7 e (20 A e+3 B d)\right )+48 b^3 C e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 (d+e x)^{3/2} \sqrt{a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}$

[Out]

(2*(24*b^2*C*e^2 - c*e*(15*b*C*d + 28*b*B*e + 25*a*C*e) - c^2*(6*C*d^2 - 7*e*(3*B*d + 5*A*e)))*Sqrt[d + e*x]*S
qrt[a + b*x + c*x^2])/(105*c^3*e) - (2*(2*c*C*d - 7*B*c*e + 6*b*C*e)*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(3
5*c^2*e) + (2*C*(d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7*c*e) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(48*b^3*C*e^3 - 8*
b*c*e^2*(9*b*C*d + 7*b*B*e + 13*a*C*e) + c^3*d*(6*C*d^2 - 7*e*(3*B*d + 20*A*e)) + c^2*e*(a*e*(82*C*d + 63*B*e)
+ b*(12*C*d^2 + 91*B*d*e + 70*A*e^2)))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[A
rcSin[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)])/(105*c^4*e^2*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]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(24*b^2*C*e^2 - c*e*(15*b*C*d + 28*b*B*e + 25*a*
C*e) - c^2*(6*C*d^2 - 7*e*(3*B*d + 5*A*e)))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((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]]/Sq
rt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(105*c^4*e^2*Sqrt[d + e*x]*Sqrt[a + b*x
+ c*x^2])

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Rubi [A]  time = 1.77854, antiderivative size = 724, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 34, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.176, Rules used = {1653, 832, 843, 718, 424, 419} $\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (25 a C e+28 b B e+15 b C d)+c^2 \left (-\left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right )}{105 c^3 e}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \left (-c e (25 a C e+28 b B e+15 b C d)+c^2 \left (-\left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e \left (a e (63 B e+82 C d)+b \left (70 A e^2+91 B d e+12 C d^2\right )\right )-8 b c e^2 (13 a C e+7 b B e+9 b C d)+c^3 \left (6 C d^3-7 d e (20 A e+3 B d)\right )+48 b^3 C e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 (d+e x)^{3/2} \sqrt{a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(24*b^2*C*e^2 - c*e*(15*b*C*d + 28*b*B*e + 25*a*C*e) - c^2*(6*C*d^2 - 7*e*(3*B*d + 5*A*e)))*Sqrt[d + e*x]*S
qrt[a + b*x + c*x^2])/(105*c^3*e) - (2*(2*c*C*d - 7*B*c*e + 6*b*C*e)*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(3
5*c^2*e) + (2*C*(d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7*c*e) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(48*b^3*C*e^3 - 8*
b*c*e^2*(9*b*C*d + 7*b*B*e + 13*a*C*e) + c^3*(6*C*d^3 - 7*d*e*(3*B*d + 20*A*e)) + c^2*e*(a*e*(82*C*d + 63*B*e)
+ b*(12*C*d^2 + 91*B*d*e + 70*A*e^2)))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[A
rcSin[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)])/(105*c^4*e^2*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]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(24*b^2*C*e^2 - c*e*(15*b*C*d + 28*b*B*e + 25*a*
C*e) - c^2*(6*C*d^2 - 7*e*(3*B*d + 5*A*e)))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((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]]/Sq
rt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(105*c^4*e^2*Sqrt[d + e*x]*Sqrt[a + b*x
+ c*x^2])

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 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{(d+e x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt{a+b x+c x^2}} \, dx &=\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}+\frac{2 \int \frac{(d+e x)^{3/2} \left (-\frac{1}{2} e (b C d-7 A c e+5 a C e)-\frac{1}{2} e (2 c C d-7 B c e+6 b C e) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{7 c e^2}\\ &=-\frac{2 (2 c C d-7 B c e+6 b C e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}+\frac{4 \int \frac{\sqrt{d+e x} \left (\frac{1}{4} e \left (6 b^2 C d e+18 a b C e^2-b c d (3 C d+7 B e)+c e (35 A c d-19 a C d-21 a B e)\right )+\frac{1}{4} e \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{35 c^2 e^2}\\ &=\frac{2 \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3 e}-\frac{2 (2 c C d-7 B c e+6 b C e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}+\frac{8 \int \frac{-\frac{1}{8} e \left (24 b^3 C d e^2+b c \left (3 c C d^3+7 c d e (6 B d+5 A e)-2 a e^2 (47 C d+14 B e)\right )+b^2 \left (24 a C e^3-c d e (33 C d+28 B e)\right )-c e \left (35 A c \left (3 c d^2-a e^2\right )-a \left (51 c C d^2+84 B c d e-25 a C e^2\right )\right )\right )-\frac{1}{8} e \left (48 b^3 C e^3-8 b c e^2 (9 b C d+7 b B e+13 a C e)+c^3 \left (6 C d^3-7 d e (3 B d+20 A e)\right )+c^2 e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B d e+70 A e^2\right )\right )\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3 e^2}\\ &=\frac{2 \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3 e}-\frac{2 (2 c C d-7 B c e+6 b C e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}-\frac{\left (\left (c d^2-b d e+a e^2\right ) \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{105 c^3 e^2}-\frac{\left (48 b^3 C e^3-8 b c e^2 (9 b C d+7 b B e+13 a C e)+c^3 \left (6 C d^3-7 d e (3 B d+20 A e)\right )+c^2 e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B d e+70 A e^2\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{105 c^3 e^2}\\ &=\frac{2 \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3 e}-\frac{2 (2 c C d-7 B c e+6 b C e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (48 b^3 C e^3-8 b c e^2 (9 b C d+7 b B e+13 a C e)+c^3 \left (6 C d^3-7 d e (3 B d+20 A e)\right )+c^2 e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B d e+70 A e^2\right )\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 )}{105 c^4 e^2 \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} \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\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 )}{105 c^4 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}{105 c^3 e}-\frac{2 (2 c C d-7 B c e+6 b C e) (d+e x)^{3/2} \sqrt{a+b x+c x^2}}{35 c^2 e}+\frac{2 C (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c e}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (48 b^3 C e^3-8 b c e^2 (9 b C d+7 b B e+13 a C e)+c^3 \left (6 C d^3-7 d e (3 B d+20 A e)\right )+c^2 e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B d e+70 A e^2\right )\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 )}{105 c^4 e^2 \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} \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\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 )}{105 c^4 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 14.5728, size = 9972, normalized size = 13.77 $\text{Result too large to show}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.438, size = 14084, normalized size = 19.5 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(C*x^2+B*x+A)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(C*x^2+B*x+A)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A + B x + C x^{2}\right )}{\sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(C*x**2+B*x+A)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((d + e*x)**(3/2)*(A + B*x + C*x**2)/sqrt(a + b*x + c*x**2), x)

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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((e*x+d)^(3/2)*(C*x^2+B*x+A)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

Timed out