3.2459 \(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=731 \[ \frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 e^6 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 c \sqrt {a+b x+c x^2} \left (e x \left (-4 c e (8 b d-5 a e)+3 b^2 e^2+32 c^2 d^2\right )-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+128 c^2 d^3\right )}{21 e^5 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{21 e^3 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}} \]

[Out]

-2/21*(16*c^2*d^3+3*a*b*e^3-c*d*e*(-4*a*e+13*b*d)+e*(22*c^2*d^2+3*b^2*e^2-2*c*e*(-5*a*e+11*b*d))*x)*(c*x^2+b*x
+a)^(3/2)/e^3/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(5/2)-2/7*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^(7/2)+2/21*c*(128*c^2*d^3-4*
c*d*e*(-29*a*e+44*b*d)+3*b*e^2*(-16*a*e+17*b*d)+e*(32*c^2*d^2+3*b^2*e^2-4*c*e*(-5*a*e+8*b*d))*x)*(c*x^2+b*x+a)
^(1/2)/e^5/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)-1/21*(-b*e+2*c*d)*(128*c^2*d^2+3*b^2*e^2-4*c*e*(-29*a*e+32*b*d))*
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^6/(a*e^2-b*d*e+c*d^2)/(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)+4/21*(128*c^
2*d^2+27*b^2*e^2-4*c*e*(-5*a*e+32*b*d))*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^6/(e*x+d)^(1/2)/(c*x^2+b*x+a
)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.89, antiderivative size = 731, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {732, 810, 812, 843, 718, 424, 419} \[ -\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{21 e^3 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}+\frac {2 c \sqrt {a+b x+c x^2} \left (e x \left (-4 c e (8 b d-5 a e)+3 b^2 e^2+32 c^2 d^2\right )-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+128 c^2 d^3\right )}{21 e^5 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 e^6 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(2*c*(128*c^2*d^3 - 4*c*d*e*(44*b*d - 29*a*e) + 3*b*e^2*(17*b*d - 16*a*e) + e*(32*c^2*d^2 + 3*b^2*e^2 - 4*c*e*
(8*b*d - 5*a*e))*x)*Sqrt[a + b*x + c*x^2])/(21*e^5*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) - (2*(16*c^2*d^3 + 3
*a*b*e^3 - c*d*e*(13*b*d - 4*a*e) + e*(22*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(11*b*d - 5*a*e))*x)*(a + b*x + c*x^2)^(
3/2))/(21*e^3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) - (2*(a + b*x + c*x^2)^(5/2))/(7*e*(d + e*x)^(7/2)) - (
Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(32*b*d - 29*a*e))*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)])/(21*e^6*(c*d^2 - b*d*e + a*e^2)*S
qrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(
128*c^2*d^2 + 27*b^2*e^2 - 4*c*e*(32*b*d - 5*a*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqr
t[-((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)])/(21*e^6*Sqrt[d + e*x]*Sqrt[a + b
*x + c*x^2])

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)])

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 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 732

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^
(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 810

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*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*
c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 812

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*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] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx &=-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac {5 \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx}{7 e}\\ &=-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {2 \int \frac {\left (-\frac {1}{2} c \left (16 b c d^2-13 b^2 d e-12 a c d e+16 a b e^2\right )-\frac {1}{2} c \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^{3/2}} \, dx}{7 e^3 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{21 e^5 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac {4 \int \frac {-\frac {1}{4} c \left (51 b^3 d e^2-8 a c e \left (8 c d^2+5 a e^2\right )+4 b c d \left (32 c d^2+45 a e^2\right )-2 b^2 \left (88 c d^2 e+27 a e^3\right )\right )-\frac {1}{4} c (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{21 e^5 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{21 e^5 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{21 e^6 \left (c d^2-b d e+a e^2\right )}+\frac {\left (2 c \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{21 e^6}\\ &=\frac {2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{21 e^5 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\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 )}{21 e^6 \left (c d^2-b d e+a e^2\right ) \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 (4 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a 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 )}{21 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{21 e^5 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\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 )}{21 e^6 \left (c d^2-b d e+a e^2\right ) \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 {4 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a 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 )}{21 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 13.20, size = 5482, normalized size = 7.50 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

Result too large to show

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fricas [F]  time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}}{e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)/(e^5*x^
5 + 5*d*e^4*x^4 + 10*d^2*e^3*x^3 + 10*d^3*e^2*x^2 + 5*d^4*e*x + d^5), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(9/2), x)

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maple [B]  time = 0.24, size = 25728, normalized size = 35.20 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(9/2),x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(9/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(9/2),x)

[Out]

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(9/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

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