### 3.271 $$\int \frac{A+B x+C x^2}{(d+e x)^{7/2} \sqrt{a+b x+c x^2}} \, dx$$

Optimal. Leaf size=944 $-\frac{2 \sqrt{c x^2+b x+a} \left (C d^2-e (B d-A e)\right )}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (c^2 \left (2 C d^2+e (3 B d-23 A e)\right ) d^2-e^2 \left (\left (3 C d^2+2 B e d+8 A e^2\right ) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right )-c e \left (b d \left (7 C d^2-7 B e d-23 A e^2\right )-a e \left (19 C d^2-29 B e d+9 A e^2\right )\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b e d+a e^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c d \left (2 C d^2+e (3 B d-8 A e)\right )+e \left (5 a e (2 C d-B e)-b \left (6 C d^2-B e d-4 A e^2\right )\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}+\frac{2 \left (c^2 \left (2 C d^2+e (3 B d-23 A e)\right ) d^2-e^2 \left (\left (3 C d^2+2 B e d+8 A e^2\right ) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right )-c e \left (b d \left (7 C d^2-7 B e d-23 A e^2\right )-a e \left (19 C d^2-29 B e d+9 A e^2\right )\right )\right ) \sqrt{c x^2+b x+a}}{15 e \left (c d^2-b e d+a e^2\right )^3 \sqrt{d+e x}}+\frac{2 \left (c d \left (2 C d^2+e (3 B d-8 A e)\right )+e \left (5 a e (2 C d-B e)-b \left (6 C d^2-B e d-4 A e^2\right )\right )\right ) \sqrt{c x^2+b x+a}}{15 e \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}}$

[Out]

(-2*(C*d^2 - e*(B*d - A*e))*Sqrt[a + b*x + c*x^2])/(5*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) + (2*(c*d*(2*
C*d^2 + e*(3*B*d - 8*A*e)) + e*(5*a*e*(2*C*d - B*e) - b*(6*C*d^2 - B*d*e - 4*A*e^2)))*Sqrt[a + b*x + c*x^2])/(
15*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) + (2*(c^2*d^2*(2*C*d^2 + e*(3*B*d - 23*A*e)) - e^2*(15*a^2*C*e
^2 - 10*a*b*e*(C*d + B*e) + b^2*(3*C*d^2 + 2*B*d*e + 8*A*e^2)) - c*e*(b*d*(7*C*d^2 - 7*B*d*e - 23*A*e^2) - a*e
*(19*C*d^2 - 29*B*d*e + 9*A*e^2)))*Sqrt[a + b*x + c*x^2])/(15*e*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) - (Sq
rt[2]*Sqrt[b^2 - 4*a*c]*(c^2*d^2*(2*C*d^2 + e*(3*B*d - 23*A*e)) - e^2*(15*a^2*C*e^2 - 10*a*b*e*(C*d + B*e) + b
^2*(3*C*d^2 + 2*B*d*e + 8*A*e^2)) - c*e*(b*d*(7*C*d^2 - 7*B*d*e - 23*A*e^2) - a*e*(19*C*d^2 - 29*B*d*e + 9*A*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)])/(15*e^2*(
c*d^2 - b*d*e + a*e^2)^3*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*S
qrt[2]*Sqrt[b^2 - 4*a*c]*(c*d*(2*C*d^2 + e*(3*B*d - 8*A*e)) + e*(5*a*e*(2*C*d - B*e) - b*(6*C*d^2 - B*d*e - 4*
A*e^2)))*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]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2
*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(15*e^2*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 2.25575, antiderivative size = 942, 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 = {1650, 834, 843, 718, 424, 419} $-\frac{2 \sqrt{c x^2+b x+a} \left (C d^2-e (B d-A e)\right )}{5 e \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (\left (2 C d^4+e (3 B d-23 A e) d^2\right ) c^2-e \left (b d \left (7 C d^2-7 B e d-23 A e^2\right )-a e \left (19 C d^2-29 B e d+9 A e^2\right )\right ) c-e^2 \left (\left (3 C d^2+2 B e d+8 A e^2\right ) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b e d+a e^2\right )^3 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (2 c C d^3+c e (3 B d-8 A e) d+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt{d+e x} \sqrt{c x^2+b x+a}}+\frac{2 \left (\left (2 C d^4+e (3 B d-23 A e) d^2\right ) c^2-e \left (b d \left (7 C d^2-7 B e d-23 A e^2\right )-a e \left (19 C d^2-29 B e d+9 A e^2\right )\right ) c-e^2 \left (\left (3 C d^2+2 B e d+8 A e^2\right ) b^2-10 a e (C d+B e) b+15 a^2 C e^2\right )\right ) \sqrt{c x^2+b x+a}}{15 e \left (c d^2-b e d+a e^2\right )^3 \sqrt{d+e x}}+\frac{2 \left (2 c C d^3+c e (3 B d-8 A e) d+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{c x^2+b x+a}}{15 e \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(A + B*x + C*x^2)/((d + e*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(-2*(C*d^2 - e*(B*d - A*e))*Sqrt[a + b*x + c*x^2])/(5*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) + (2*(2*c*C*d
^3 + c*d*e*(3*B*d - 8*A*e) + 5*a*e^2*(2*C*d - B*e) - b*e*(6*C*d^2 - e*(B*d + 4*A*e)))*Sqrt[a + b*x + c*x^2])/(
15*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) + (2*(c^2*(2*C*d^4 + d^2*e*(3*B*d - 23*A*e)) - e^2*(15*a^2*C*e
^2 - 10*a*b*e*(C*d + B*e) + b^2*(3*C*d^2 + 2*B*d*e + 8*A*e^2)) - c*e*(b*d*(7*C*d^2 - 7*B*d*e - 23*A*e^2) - a*e
*(19*C*d^2 - 29*B*d*e + 9*A*e^2)))*Sqrt[a + b*x + c*x^2])/(15*e*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) - (Sq
rt[2]*Sqrt[b^2 - 4*a*c]*(c^2*(2*C*d^4 + d^2*e*(3*B*d - 23*A*e)) - e^2*(15*a^2*C*e^2 - 10*a*b*e*(C*d + B*e) + b
^2*(3*C*d^2 + 2*B*d*e + 8*A*e^2)) - c*e*(b*d*(7*C*d^2 - 7*B*d*e - 23*A*e^2) - a*e*(19*C*d^2 - 29*B*d*e + 9*A*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)])/(15*e^2*(
c*d^2 - b*d*e + a*e^2)^3*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*S
qrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*C*d^3 + c*d*e*(3*B*d - 8*A*e) + 5*a*e^2*(2*C*d - B*e) - b*e*(6*C*d^2 - e*(B*d +
4*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]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2
*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(15*e^2*(c*d^2 - b*d*e + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(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*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

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 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+C x^2}{(d+e x)^{7/2} \sqrt{a+b x+c x^2}} \, dx &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \int \frac{-\frac{b C d^2-b e (B d+4 A e)+5 e (A c d-a C d+a B e)}{2 e}-\frac{1}{2} \left (3 B c d-5 b C d+\frac{2 c C d^2}{e}-3 A c e+5 a C e\right ) x}{(d+e x)^{5/2} \sqrt{a+b x+c x^2}} \, dx}{5 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{4 \int \frac{\frac{b^2 e \left (3 C d^2+2 e (B d+4 A e)\right )+b \left (c C d^3-c d e (6 B d+19 A e)-10 a e^2 (C d+B e)\right )+3 e \left (A c \left (5 c d^2-3 a e^2\right )+a \left (5 a C e^2-c d (3 C d-8 B e)\right )\right )}{4 e}+\frac{c \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) x}{4 e}}{(d+e x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{8 \int \frac{-\frac{c \left (b^2 d e \left (9 C d^2+e (B d+4 A e)\right )-b \left (c C d^4+26 a C d^2 e^2+4 a e^3 (B d-A e)+c d^2 e (9 B d+11 A e)\right )+e \left (A c d \left (15 c d^2-17 a e^2\right )-a \left (c d^2 (7 C d-27 B e)-5 a e^2 (5 C d-B e)\right )\right )\right )}{8 e}+\frac{c \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) x}{8 e}}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}+\frac{\left (c \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )^2}-\frac{\left (c \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{15 e^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 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 )}{15 e^2 \left (c d^2-b d e+a e^2\right )^3 \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 (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 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 )}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \left (C d^2-e (B d-A e)\right ) \sqrt{a+b x+c x^2}}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}+\frac{2 \left (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 A e)\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 A e^2\right )\right )\right ) \sqrt{a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (c^2 \left (2 C d^4+d^2 e (3 B d-23 A e)\right )-e^2 \left (15 a^2 C e^2-10 a b e (C d+B e)+b^2 \left (3 C d^2+2 B d e+8 A e^2\right )\right )-c e \left (b d \left (7 C d^2-7 B d e-23 A e^2\right )-a e \left (19 C d^2-29 B d e+9 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 )}{15 e^2 \left (c d^2-b d e+a e^2\right )^3 \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 (2 c C d^3+c d e (3 B d-8 A e)+5 a e^2 (2 C d-B e)-b e \left (6 C d^2-e (B d+4 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 )}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 15.2817, size = 12295, normalized size = 13.02 $\text{Result too large to show}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(A + B*x + C*x^2)/((d + e*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]

[Out]

Result too large to show

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Maple [B]  time = 0.628, size = 46695, normalized size = 49.5 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/(e*x+d)^(7/2)/(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{C x^{2} + B x + A}{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C x^{2} + B x + A\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}}{c e^{4} x^{6} +{\left (4 \, c d e^{3} + b e^{4}\right )} x^{5} + a d^{4} +{\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{4} + 2 \,{\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{3} +{\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{2} +{\left (b d^{4} + 4 \, a d^{3} e\right )} x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)/(c*e^4*x^6 + (4*c*d*e^3 + b*e^4)*x^5 + a*d^4 +
(6*c*d^2*e^2 + 4*b*d*e^3 + a*e^4)*x^4 + 2*(2*c*d^3*e + 3*b*d^2*e^2 + 2*a*d*e^3)*x^3 + (c*d^4 + 4*b*d^3*e + 6*a
*d^2*e^2)*x^2 + (b*d^4 + 4*a*d^3*e)*x), 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((C*x**2+B*x+A)/(e*x+d)**(7/2)/(c*x**2+b*x+a)**(1/2),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((C*x^2+B*x+A)/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

Timed out