3.3.71 \(\int \frac {A+B x+C x^2}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx\) [271]

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

[Out]

-2/5*(C*d^2-e*(-A*e+B*d))*(c*x^2+b*x+a)^(1/2)/e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(5/2)+2/15*(c*d*(2*C*d^2+e*(-8*A*e
+3*B*d))+e*(5*a*e*(-B*e+2*C*d)-b*(-4*A*e^2-B*d*e+6*C*d^2)))*(c*x^2+b*x+a)^(1/2)/e/(a*e^2-b*d*e+c*d^2)^2/(e*x+d
)^(3/2)+2/15*(c^2*d^2*(2*C*d^2+e*(-23*A*e+3*B*d))-e^2*(15*a^2*C*e^2-10*a*b*e*(B*e+C*d)+b^2*(8*A*e^2+2*B*d*e+3*
C*d^2))-c*e*(b*d*(-23*A*e^2-7*B*d*e+7*C*d^2)-a*e*(9*A*e^2-29*B*d*e+19*C*d^2)))*(c*x^2+b*x+a)^(1/2)/e/(a*e^2-b*
d*e+c*d^2)^3/(e*x+d)^(1/2)-1/15*(c^2*d^2*(2*C*d^2+e*(-23*A*e+3*B*d))-e^2*(15*a^2*C*e^2-10*a*b*e*(B*e+C*d)+b^2*
(8*A*e^2+2*B*d*e+3*C*d^2))-c*e*(b*d*(-23*A*e^2-7*B*d*e+7*C*d^2)-a*e*(9*A*e^2-29*B*d*e+19*C*d^2)))*EllipticE(1/
2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c
+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/e^2/(a*e^
2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)+2/15*(c*d*(2*C*d^2+e*(
-8*A*e+3*B*d))+e*(5*a*e*(-B*e+2*C*d)-b*(-4*A*e^2-B*d*e+6*C*d^2)))*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/
(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-
4*a*c+b^2)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/e^2/
(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.34, antiderivative size = 942, normalized size of antiderivative = 1.00, 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 = {1664, 848, 857, 732, 435, 430} \begin {gather*} -\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 (\text {ArcSin}\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 (\text {ArcSin}\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}} \end {gather*}

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 430

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

Rule 435

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

Rule 732

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 848

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 857

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 1664

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]

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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 32.92, size = 1746, normalized size = 1.85 \begin {gather*} \frac {\sqrt {d+e x} \left (a+b x+c x^2\right ) \left (-\frac {2 \left (C d^2-B d e+A e^2\right )}{5 e \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {2 \left (-2 c C d^3-3 B c d^2 e+6 b C d^2 e-b B d e^2+8 A c d e^2-10 a C d e^2-4 A b e^3+5 a B e^3\right )}{15 e \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {2 \left (-2 c^2 C d^4-3 B c^2 d^3 e+7 b c C d^3 e-7 b B c d^2 e^2+23 A c^2 d^2 e^2+3 b^2 C d^2 e^2-19 a c C d^2 e^2+2 b^2 B d e^3-23 A b c d e^3+29 a B c d e^3-10 a b C d e^3+8 A b^2 e^4-10 a b B e^4-9 a A c e^4+15 a^2 C e^4\right )}{15 e \left (c d^2-b d e+a e^2\right )^3 (d+e x)}\right )}{\sqrt {a+x (b+c x)}}+\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (-\left (\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 (a e \left (19 C d^2-29 B d e+9 A e^2\right )+b d \left (-7 C d^2+7 B d e+23 A e^2\right )\right )\right ) \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )\right )-\frac {i \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \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 (a e \left (19 C d^2-29 B d e+9 A e^2\right )+b d \left (-7 C d^2+7 B d e+23 A e^2\right )\right )\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+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 (-30 A c^3 d^3 e^2+14 a c^2 C d^3 e^2-54 a B c^2 d^2 e^3+34 a A c^2 d e^4-50 a^2 c C d e^4+10 a^2 B c e^5+2 c^2 C d^4 \sqrt {\left (b^2-4 a c\right ) e^2}+3 B c^2 d^3 e \sqrt {\left (b^2-4 a c\right ) e^2}-23 A c^2 d^2 e^2 \sqrt {\left (b^2-4 a c\right ) e^2}+19 a c C d^2 e^2 \sqrt {\left (b^2-4 a c\right ) e^2}-29 a B c d e^3 \sqrt {\left (b^2-4 a c\right ) e^2}+9 a A c e^4 \sqrt {\left (b^2-4 a c\right ) e^2}-15 a^2 C e^4 \sqrt {\left (b^2-4 a c\right ) e^2}+b^3 e^3 \left (3 C d^2+2 e (B d+4 A e)\right )-b^2 e^2 \left (11 c C d^3+c d e (9 B d+31 A e)+10 a e^2 (C d+B e)+\sqrt {\left (b^2-4 a c\right ) e^2} \left (3 C d^2+2 e (B d+4 A e)\right )\right )+b \left (A c e^3 \left (45 c d^2-17 a e^2+23 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+C e \left (15 a^2 e^4-7 c d^3 \sqrt {\left (b^2-4 a c\right ) e^2}+a d e^2 \left (33 c d+10 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+B e^2 \left (15 c^2 d^3+10 a e^2 \sqrt {\left (b^2-4 a c\right ) e^2}+c d \left (37 a e^2+7 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right )\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+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 (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {d+e x}}\right )}{15 e^3 \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+x (b+c x)} \sqrt {\frac {(d+e x)^2 \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x]*(a + b*x + c*x^2)*((-2*(C*d^2 - B*d*e + A*e^2))/(5*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (2*
(-2*c*C*d^3 - 3*B*c*d^2*e + 6*b*C*d^2*e - b*B*d*e^2 + 8*A*c*d*e^2 - 10*a*C*d*e^2 - 4*A*b*e^3 + 5*a*B*e^3))/(15
*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (2*(-2*c^2*C*d^4 - 3*B*c^2*d^3*e + 7*b*c*C*d^3*e - 7*b*B*c*d^2*e^2
+ 23*A*c^2*d^2*e^2 + 3*b^2*C*d^2*e^2 - 19*a*c*C*d^2*e^2 + 2*b^2*B*d*e^3 - 23*A*b*c*d*e^3 + 29*a*B*c*d*e^3 - 1
0*a*b*C*d*e^3 + 8*A*b^2*e^4 - 10*a*b*B*e^4 - 9*a*A*c*e^4 + 15*a^2*C*e^4))/(15*e*(c*d^2 - b*d*e + a*e^2)^3*(d +
e*x))))/Sqrt[a + x*(b + c*x)] + (2*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^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*(a*e*(19*C*d^2 -
29*B*d*e + 9*A*e^2) + b*d*(-7*C*d^2 + 7*B*d*e + 23*A*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])*(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*(a*e*(19*C*d^2 - 29*B*d*e + 9*A*e^2) + b*d*
(-7*C*d^2 + 7*B*d*e + 23*A*e^2)))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sq
rt[(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]))] + (-30*A*c^3*d^3*e^2 + 14*a*c^2*C*d^3*e^2 - 54*a*B*c^2*d^2*e^3 + 34*a*A*c^2*d*e^4 - 50*a^2*c*C
*d*e^4 + 10*a^2*B*c*e^5 + 2*c^2*C*d^4*Sqrt[(b^2 - 4*a*c)*e^2] + 3*B*c^2*d^3*e*Sqrt[(b^2 - 4*a*c)*e^2] - 23*A*c
^2*d^2*e^2*Sqrt[(b^2 - 4*a*c)*e^2] + 19*a*c*C*d^2*e^2*Sqrt[(b^2 - 4*a*c)*e^2] - 29*a*B*c*d*e^3*Sqrt[(b^2 - 4*a
*c)*e^2] + 9*a*A*c*e^4*Sqrt[(b^2 - 4*a*c)*e^2] - 15*a^2*C*e^4*Sqrt[(b^2 - 4*a*c)*e^2] + b^3*e^3*(3*C*d^2 + 2*e
*(B*d + 4*A*e)) - b^2*e^2*(11*c*C*d^3 + c*d*e*(9*B*d + 31*A*e) + 10*a*e^2*(C*d + B*e) + Sqrt[(b^2 - 4*a*c)*e^2
]*(3*C*d^2 + 2*e*(B*d + 4*A*e))) + b*(A*c*e^3*(45*c*d^2 - 17*a*e^2 + 23*d*Sqrt[(b^2 - 4*a*c)*e^2]) + C*e*(15*a
^2*e^4 - 7*c*d^3*Sqrt[(b^2 - 4*a*c)*e^2] + a*d*e^2*(33*c*d + 10*Sqrt[(b^2 - 4*a*c)*e^2])) + B*e^2*(15*c^2*d^3
+ 10*a*e^2*Sqrt[(b^2 - 4*a*c)*e^2] + c*d*(37*a*e^2 + 7*d*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]))]))/(Sqrt[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])))/(15*e^3*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[a + x*(b + c
*x)]*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] Leaf count of result is larger than twice the leaf count of optimal. \(46696\) vs. \(2(874)=1748\).
time = 0.21, size = 46697, normalized size = 49.47

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1757\)
default	\(\text {Expression too large to display}\)	\(46697\)

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,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)*(x*e + d)^(7/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.20, size = 2611, normalized size = 2.77 \begin {gather*} \text {Too large to display} \end {gather*}

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]

2/45*((2*C*c^3*d^8 - (15*C*a^2*b - 10*B*a*b^2 + 8*A*b^3 + 3*(5*B*a^2 - 7*A*a*b)*c)*x^3*e^8 + ((10*C*a*b^2 - 2*
B*b^3 - 42*A*a*c^2 + (60*C*a^2 - 31*B*a*b + 27*A*b^2)*c)*d*x^3 - 3*(15*C*a^2*b - 10*B*a*b^2 + 8*A*b^3 + 3*(5*B
*a^2 - 7*A*a*b)*c)*d*x^2)*e^7 - ((3*C*b^3 - (52*B*a - 33*A*b)*c^2 + (49*C*a*b - 8*B*b^2)*c)*d^2*x^3 - 3*(10*C*
a*b^2 - 2*B*b^3 - 42*A*a*c^2 + (60*C*a^2 - 31*B*a*b + 27*A*b^2)*c)*d^2*x^2 + 3*(15*C*a^2*b - 10*B*a*b^2 + 8*A*
b^3 + 3*(5*B*a^2 - 7*A*a*b)*c)*d^2*x)*e^6 + ((17*C*b^2*c + 22*A*c^3 - (2*C*a + 17*B*b)*c^2)*d^3*x^3 - 3*(3*C*b
^3 - (52*B*a - 33*A*b)*c^2 + (49*C*a*b - 8*B*b^2)*c)*d^3*x^2 + 3*(10*C*a*b^2 - 2*B*b^3 - 42*A*a*c^2 + (60*C*a^
2 - 31*B*a*b + 27*A*b^2)*c)*d^3*x - (15*C*a^2*b - 10*B*a*b^2 + 8*A*b^3 + 3*(5*B*a^2 - 7*A*a*b)*c)*d^3)*e^5 - (
(8*C*b*c^2 - 3*B*c^3)*d^4*x^3 - 3*(17*C*b^2*c + 22*A*c^3 - (2*C*a + 17*B*b)*c^2)*d^4*x^2 + 3*(3*C*b^3 - (52*B*
a - 33*A*b)*c^2 + (49*C*a*b - 8*B*b^2)*c)*d^4*x - (10*C*a*b^2 - 2*B*b^3 - 42*A*a*c^2 + (60*C*a^2 - 31*B*a*b +
27*A*b^2)*c)*d^4)*e^4 + (2*C*c^3*d^5*x^3 - 3*(8*C*b*c^2 - 3*B*c^3)*d^5*x^2 + 3*(17*C*b^2*c + 22*A*c^3 - (2*C*a
+ 17*B*b)*c^2)*d^5*x - (3*C*b^3 - (52*B*a - 33*A*b)*c^2 + (49*C*a*b - 8*B*b^2)*c)*d^5)*e^3 + (6*C*c^3*d^6*x^2
- 3*(8*C*b*c^2 - 3*B*c^3)*d^6*x + (17*C*b^2*c + 22*A*c^3 - (2*C*a + 17*B*b)*c^2)*d^6)*e^2 + (6*C*c^3*d^7*x -
(8*C*b*c^2 - 3*B*c^3)*d^7)*e)*sqrt(c)*e^(1/2)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*
e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/c^3,
1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c) + 3*(2*C*c^3*d^7*e + (9*A*a*c^2 - (15*C*a^2 - 10*B*a*b + 8*A*b^2)*c)*x^3*e
^8 - (((29*B*a - 23*A*b)*c^2 - 2*(5*C*a*b - B*b^2)*c)*d*x^3 - 3*(9*A*a*c^2 - (15*C*a^2 - 10*B*a*b + 8*A*b^2)*c
)*d*x^2)*e^7 - ((3*C*b^2*c + 23*A*c^3 - (19*C*a + 7*B*b)*c^2)*d^2*x^3 + 3*((29*B*a - 23*A*b)*c^2 - 2*(5*C*a*b
- B*b^2)*c)*d^2*x^2 - 3*(9*A*a*c^2 - (15*C*a^2 - 10*B*a*b + 8*A*b^2)*c)*d^2*x)*e^6 - ((7*C*b*c^2 - 3*B*c^3)*d^
3*x^3 + 3*(3*C*b^2*c + 23*A*c^3 - (19*C*a + 7*B*b)*c^2)*d^3*x^2 + 3*((29*B*a - 23*A*b)*c^2 - 2*(5*C*a*b - B*b^
2)*c)*d^3*x - (9*A*a*c^2 - (15*C*a^2 - 10*B*a*b + 8*A*b^2)*c)*d^3)*e^5 + (2*C*c^3*d^4*x^3 - 3*(7*C*b*c^2 - 3*B
*c^3)*d^4*x^2 - 3*(3*C*b^2*c + 23*A*c^3 - (19*C*a + 7*B*b)*c^2)*d^4*x - ((29*B*a - 23*A*b)*c^2 - 2*(5*C*a*b -
B*b^2)*c)*d^4)*e^4 + (6*C*c^3*d^5*x^2 - 3*(7*C*b*c^2 - 3*B*c^3)*d^5*x - (3*C*b^2*c + 23*A*c^3 - (19*C*a + 7*B*
b)*c^2)*d^5)*e^3 + (6*C*c^3*d^6*x - (7*C*b*c^2 - 3*B*c^3)*d^6)*e^2)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c^2*d
^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (
2*b^3 - 9*a*b*c)*e^3)*e^(-3)/c^3, weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2,
-4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/c^3, 1/3*(c*d + (
3*c*x + b)*e)*e^(-1)/c)) + 3*(C*c^3*d^6*e^2 - (3*A*a^2*c + (5*B*a^2 - 4*A*a*b)*c*x - (9*A*a*c^2 - (15*C*a^2 -
10*B*a*b + 8*A*b^2)*c)*x^2)*e^8 - (((29*B*a - 23*A*b)*c^2 - 2*(5*C*a*b - B*b^2)*c)*d*x^2 + 2*(B*a^2 - 5*A*a*b)
*c*d - 2*(5*A*a*c^2 - (10*C*a^2 - 13*B*a*b + 10*A*b^2)*c)*d*x)*e^7 - ((3*C*b^2*c + 23*A*c^3 - (19*C*a + 7*B*b)
*c^2)*d^2*x^2 + (2*(30*B*a - 29*A*b)*c^2 - (4*C*a*b - 5*B*b^2)*c)*d^2*x + (5*A*a*c^2 + (8*C*a^2 - 10*B*a*b + 1
5*A*b^2)*c)*d^2)*e^6 - ((25*B*a - 41*A*b)*c^2*d^3 + (7*C*b*c^2 - 3*B*c^3)*d^3*x^2 + 2*(27*A*c^3 - (25*C*a + 6*
B*b)*c^2)*d^3*x)*e^5 + (2*C*c^3*d^4*x^2 - (22*C*b*c^2 - 9*B*c^3)*d^4*x - (34*A*c^3 - (25*C*a - B*b)*c^2)*d^4)*
e^4 + 3*(2*C*c^3*d^5*x - 3*(C*b*c^2 - B*c^3)*d^5)*e^3)*sqrt(c*x^2 + b*x + a)*sqrt(x*e + d))/(c^4*d^9*e^3 + a^3
*c*x^3*e^12 - 3*(a^2*b*c*d*x^3 - a^3*c*d*x^2)*e^11 - 3*(3*a^2*b*c*d^2*x^2 - a^3*c*d^2*x - (a*b^2*c + a^2*c^2)*
d^2*x^3)*e^10 - (9*a^2*b*c*d^3*x - a^3*c*d^3 + (b^3*c + 6*a*b*c^2)*d^3*x^3 - 9*(a*b^2*c + a^2*c^2)*d^3*x^2)*e^
9 - 3*(a^2*b*c*d^4 - (b^2*c^2 + a*c^3)*d^4*x^3 + (b^3*c + 6*a*b*c^2)*d^4*x^2 - 3*(a*b^2*c + a^2*c^2)*d^4*x)*e^
8 - 3*(b*c^3*d^5*x^3 - 3*(b^2*c^2 + a*c^3)*d^5*x^2 + (b^3*c + 6*a*b*c^2)*d^5*x - (a*b^2*c + a^2*c^2)*d^5)*e^7
+ (c^4*d^6*x^3 - 9*b*c^3*d^6*x^2 + 9*(b^2*c^2 + a*c^3)*d^6*x - (b^3*c + 6*a*b*c^2)*d^6)*e^6 + 3*(c^4*d^7*x^2 -
3*b*c^3*d^7*x + (b^2*c^2 + a*c^3)*d^7)*e^5 + 3*(c^4*d^8*x - b*c^3*d^8)*e^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x + C x^{2}}{\left (d + e x\right )^{\frac {7}{2}} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {C\,x^2+B\,x+A}{{\left (d+e\,x\right )}^{7/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2)/((d + e*x)^(7/2)*(a + b*x + c*x^2)^(1/2)),x)

[Out]

int((A + B*x + C*x^2)/((d + e*x)^(7/2)*(a + b*x + c*x^2)^(1/2)), x)

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