3.25.60 \(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\) [2460]

Optimal. Leaf size=923 \[ -\frac {2 \left (128 c^4 d^5-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)-b c e^3 \left (b^2 d^2+9 a b d e-24 a^2 e^2\right )+3 c^2 d e^2 \left (37 b^2 d^2-52 a b d e+12 a^2 e^2\right )+e \left (160 c^4 d^4-2 b^4 e^4-4 c^3 d^2 e (80 b d-69 a e)-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b d e+28 a^2 e^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{63 e^5 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2/63*(16*c^2*d^3-b*e^2*(-5*a*e+2*b*d)-c*d*e*(-4*a*e+11*b*d)+e*(26*c^2*d^2+3*b^2*e^2-2*c*e*(-7*a*e+13*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)^(7/2)-2/9*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^(9/2)-2/63*(128*c
^4*d^5-2*a*b^3*e^5-4*c^3*d^3*e*(-49*a*e+60*b*d)-b*c*e^3*(-24*a^2*e^2+9*a*b*d*e+b^2*d^2)+3*c^2*d*e^2*(12*a^2*e^
2-52*a*b*d*e+37*b^2*d^2)+e*(160*c^4*d^4-2*b^4*e^4-4*c^3*d^2*e*(-69*a*e+80*b*d)-b^2*c*e^3*(-27*a*e+11*b*d)+3*c^
2*e^2*(28*a^2*e^2-92*a*b*d*e+57*b^2*d^2))*x)*(c*x^2+b*x+a)^(1/2)/e^5/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(3/2)+2/63*
(128*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(-57*a*e+64*b*d)-b^2*c*e^3*(-15*a*e+7*b*d)+3*c^2*e^2*(28*a^2*e^2-76*a*b*d*e+4
5*b^2*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^6/(a*e^2-b*d*e+c*d^2)^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
)-2/63*(-b*e+2*c*d)*(128*c^2*d^2-b^2*e^2-4*c*e*(-33*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/(
a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.80, antiderivative size = 923, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {746, 824, 857, 732, 435, 430} \begin {gather*} -\frac {2 \left (c x^2+b x+a\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {2 \left (16 c^2 d^3-c e (11 b d-4 a e) d-b e^2 (2 b d-5 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{63 e^3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (128 c^4 d^5-4 c^3 e (60 b d-49 a e) d^3+3 c^2 e^2 \left (37 b^2 d^2-52 a b e d+12 a^2 e^2\right ) d-2 a b^3 e^5-b c e^3 \left (b^2 d^2+9 a b e d-24 a^2 e^2\right )+e \left (160 c^4 d^4-4 c^3 e (80 b d-69 a e) d^2-2 b^4 e^4-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b e d+28 a^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{63 e^5 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 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 )}{63 e^6 \left (c d^2-b e d+a e^2\right )^2 \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} (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 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 (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 )}{63 e^6 \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x} \sqrt {c x^2+b x+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-2*(128*c^4*d^5 - 2*a*b^3*e^5 - 4*c^3*d^3*e*(60*b*d - 49*a*e) - b*c*e^3*(b^2*d^2 + 9*a*b*d*e - 24*a^2*e^2) +
3*c^2*d*e^2*(37*b^2*d^2 - 52*a*b*d*e + 12*a^2*e^2) + e*(160*c^4*d^4 - 2*b^4*e^4 - 4*c^3*d^2*e*(80*b*d - 69*a*e
) - b^2*c*e^3*(11*b*d - 27*a*e) + 3*c^2*e^2*(57*b^2*d^2 - 92*a*b*d*e + 28*a^2*e^2))*x)*Sqrt[a + b*x + c*x^2])/
(63*e^5*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) - (2*(16*c^2*d^3 - b*e^2*(2*b*d - 5*a*e) - c*d*e*(11*b*d -
4*a*e) + e*(26*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(13*b*d - 7*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(63*e^3*(c*d^2 - b*d*
e + a*e^2)*(d + e*x)^(7/2)) - (2*(a + b*x + c*x^2)^(5/2))/(9*e*(d + e*x)^(9/2)) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]
*(128*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 57*a*e) - b^2*c*e^3*(7*b*d - 15*a*e) + 3*c^2*e^2*(45*b^2*d^2 -
76*a*b*d*e + 28*a^2*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)])/(63*e^6*(c*d^2 - b*d*e + a*e^2)^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]*(2*c*d - b*e)*(128*c^2*d^2 - b^2*e^2 - 4*c*e*(32*b*d - 33*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)])/(63*e^6*(c*d^2 - b*d*e + a*e^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 746

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 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((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), 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 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]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx &=-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {5 \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx}{9 e}\\ &=-\frac {2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {2 \int \frac {\left (\frac {1}{2} \left (11 b^2 c d e+20 a c^2 d e+2 b^3 e^2-8 b c \left (2 c d^2+3 a e^2\right )\right )-\frac {1}{2} c \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx}{21 e^3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (128 c^4 d^5-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)-b c e^3 \left (b^2 d^2+9 a b d e-24 a^2 e^2\right )+3 c^2 d e^2 \left (37 b^2 d^2-52 a b d e+12 a^2 e^2\right )+e \left (160 c^4 d^4-2 b^4 e^4-4 c^3 d^2 e (80 b d-69 a e)-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b d e+28 a^2 e^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{63 e^5 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {4 \int \frac {-\frac {1}{4} c \left (b^4 d e^3+32 a c^2 d e \left (2 c d^2+3 a e^2\right )+12 b^2 c d e \left (20 c d^2+19 a e^2\right )-b^3 \left (111 c d^2 e^2-a e^4\right )-4 b c \left (32 c^2 d^4+81 a c d^2 e^2+33 a^2 e^4\right )\right )+\frac {1}{2} c \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{63 e^5 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (128 c^4 d^5-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)-b c e^3 \left (b^2 d^2+9 a b d e-24 a^2 e^2\right )+3 c^2 d e^2 \left (37 b^2 d^2-52 a b d e+12 a^2 e^2\right )+e \left (160 c^4 d^4-2 b^4 e^4-4 c^3 d^2 e (80 b d-69 a e)-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b d e+28 a^2 e^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{63 e^5 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac {\left (c (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{63 e^6 \left (c d^2-b d e+a e^2\right )}+\frac {\left (2 c \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{63 e^6 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (128 c^4 d^5-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)-b c e^3 \left (b^2 d^2+9 a b d e-24 a^2 e^2\right )+3 c^2 d e^2 \left (37 b^2 d^2-52 a b d e+12 a^2 e^2\right )+e \left (160 c^4 d^4-2 b^4 e^4-4 c^3 d^2 e (80 b d-69 a e)-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b d e+28 a^2 e^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{63 e^5 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 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 ) \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 )}{63 e^6 \left (c d^2-b d e+a e^2\right )^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} (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 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 ) \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 )}{63 e^6 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 \left (128 c^4 d^5-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)-b c e^3 \left (b^2 d^2+9 a b d e-24 a^2 e^2\right )+3 c^2 d e^2 \left (37 b^2 d^2-52 a b d e+12 a^2 e^2\right )+e \left (160 c^4 d^4-2 b^4 e^4-4 c^3 d^2 e (80 b d-69 a e)-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b d e+28 a^2 e^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{63 e^5 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 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 )}{63 e^6 \left (c d^2-b d e+a e^2\right )^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} (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 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 )}{63 e^6 \left (c d^2-b d e+a e^2\right ) \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 = 33.12, size = 8108, normalized size = 8.78 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(44993\) vs. \(2(853)=1706\).
time = 0.96, size = 44994, normalized size = 48.75

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1744\)
default	\(\text {Expression too large to display}\)	\(44994\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(11/2),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)^(5/2)/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(x*e + d)^(11/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.56, size = 2755, normalized size = 2.98 \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)^(5/2)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

-2/189*((256*c^5*d^10 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*x^5*e^10 - ((13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c
^3)*d*x^5 + 5*(2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d*x^4)*e^9 - ((77*b^3*c^2 + 972*a*b*c^3)*d^2*x^5 + 5*(13*b^
4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d^2*x^4 + 10*(2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^2*x^3)*e^8 + (2*(239*b^
2*c^3 + 324*a*c^4)*d^3*x^5 - 5*(77*b^3*c^2 + 972*a*b*c^3)*d^3*x^4 - 10*(13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3
)*d^3*x^3 - 10*(2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^3*x^2)*e^7 - 5*(128*b*c^4*d^4*x^5 - 2*(239*b^2*c^3 + 324
*a*c^4)*d^4*x^4 + 2*(77*b^3*c^2 + 972*a*b*c^3)*d^4*x^3 + 2*(13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d^4*x^2 +
(2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^4*x)*e^6 + (256*c^5*d^5*x^5 - 3200*b*c^4*d^5*x^4 + 20*(239*b^2*c^3 + 32
4*a*c^4)*d^5*x^3 - 10*(77*b^3*c^2 + 972*a*b*c^3)*d^5*x^2 - 5*(13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d^5*x -
(2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^5)*e^5 + (1280*c^5*d^6*x^4 - 6400*b*c^4*d^6*x^3 + 20*(239*b^2*c^3 + 324
*a*c^4)*d^6*x^2 - 5*(77*b^3*c^2 + 972*a*b*c^3)*d^6*x - (13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d^6)*e^4 + (25
60*c^5*d^7*x^3 - 6400*b*c^4*d^7*x^2 + 10*(239*b^2*c^3 + 324*a*c^4)*d^7*x - (77*b^3*c^2 + 972*a*b*c^3)*d^7)*e^3
+ 2*(1280*c^5*d^8*x^2 - 1600*b*c^4*d^8*x + (239*b^2*c^3 + 324*a*c^4)*d^8)*e^2 + 640*(2*c^5*d^9*x - b*c^4*d^9)
*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) + 6*(128*c^5*d^9*e - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*x^5*e^10 - ((7*b^3*c^2 + 228*a*b*c^3)*d*x^5
+ 5*(b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*d*x^4)*e^9 + (3*(45*b^2*c^3 + 76*a*c^4)*d^2*x^5 - 5*(7*b^3*c^2 + 228*
a*b*c^3)*d^2*x^4 - 10*(b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*d^2*x^3)*e^8 - (256*b*c^4*d^3*x^5 - 15*(45*b^2*c^3 +
76*a*c^4)*d^3*x^4 + 10*(7*b^3*c^2 + 228*a*b*c^3)*d^3*x^3 + 10*(b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*d^3*x^2)*e^
7 + (128*c^5*d^4*x^5 - 1280*b*c^4*d^4*x^4 + 30*(45*b^2*c^3 + 76*a*c^4)*d^4*x^3 - 10*(7*b^3*c^2 + 228*a*b*c^3)*
d^4*x^2 - 5*(b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*d^4*x)*e^6 + (640*c^5*d^5*x^4 - 2560*b*c^4*d^5*x^3 + 30*(45*b^
2*c^3 + 76*a*c^4)*d^5*x^2 - 5*(7*b^3*c^2 + 228*a*b*c^3)*d^5*x - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*d^5)*e^5 +
(1280*c^5*d^6*x^3 - 2560*b*c^4*d^6*x^2 + 15*(45*b^2*c^3 + 76*a*c^4)*d^6*x - (7*b^3*c^2 + 228*a*b*c^3)*d^6)*e^
4 + (1280*c^5*d^7*x^2 - 1280*b*c^4*d^7*x + 3*(45*b^2*c^3 + 76*a*c^4)*d^7)*e^3 + 128*(5*c^5*d^8*x - 2*b*c^4*d^8
)*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*(128*c^5*d^8*e^2 + (19*a^3*b*c*x
+ 7*a^4*c - (2*b^4*c - 30*a*b^2*c^2 - 105*a^2*c^3)*x^4 + (a*b^3*c + 57*a^2*b*c^2)*x^3 + (15*a^2*b^2*c + 28*a^
3*c^2)*x^2)*e^10 - (9*a^3*b*c*d + 2*(7*b^3*c^2 + 165*a*b*c^3)*d*x^4 + 9*(b^4*c - 34*a^2*c^3)*d*x^3 + 27*(a*b^3
*c - a^2*b*c^2)*d*x^2 + 9*(3*a^2*b^2*c - 2*a^3*c^2)*d*x)*e^9 + (54*a^2*b*c^2*d^2*x + 18*a^3*c^2*d^2 + 3*(69*b^
2*c^3 + 110*a*c^4)*d^2*x^4 + 2*(4*b^3*c^2 - 483*a*b*c^3)*d^2*x^3 + 54*(a*b^2*c^2 + 8*a^2*c^3)*d^2*x^2)*e^8 - 2
*(193*b*c^4*d^3*x^4 - 126*a^2*c^3*d^3*x - (291*b^2*c^3 + 542*a*c^4)*d^3*x^3 + (5*b^3*c^2 + 663*a*b*c^3)*d^3*x^
2)*e^7 + (193*c^5*d^4*x^4 - 1239*b*c^4*d^4*x^3 + 63*a^2*c^3*d^4 + 9*(87*b^2*c^3 + 164*a*c^4)*d^4*x^2 - (5*b^3*
c^2 + 789*a*b*c^3)*d^4*x)*e^6 + (650*c^5*d^5*x^3 - 1665*b*c^4*d^5*x^2 + 3*(159*b^2*c^3 + 302*a*c^4)*d^5*x - (b
^3*c^2 + 183*a*b*c^3)*d^5)*e^5 + (880*c^5*d^6*x^2 - 1024*b*c^4*d^6*x + (111*b^2*c^3 + 212*a*c^4)*d^6)*e^4 + 16
*(34*c^5*d^7*x - 15*b*c^4*d^7)*e^3)*sqrt(c*x^2 + b*x + a)*sqrt(x*e + d))/(c^3*d^9*e^7 + a^2*c*x^5*e^16 - (2*a*
b*c*d*x^5 - 5*a^2*c*d*x^4)*e^15 - (10*a*b*c*d^2*x^4 - 10*a^2*c*d^2*x^3 - (b^2*c + 2*a*c^2)*d^2*x^5)*e^14 - (2*
b*c^2*d^3*x^5 + 20*a*b*c*d^3*x^3 - 10*a^2*c*d^3*x^2 - 5*(b^2*c + 2*a*c^2)*d^3*x^4)*e^13 + (c^3*d^4*x^5 - 10*b*
c^2*d^4*x^4 - 20*a*b*c*d^4*x^2 + 5*a^2*c*d^4*x + 10*(b^2*c + 2*a*c^2)*d^4*x^3)*e^12 + (5*c^3*d^5*x^4 - 20*b*c^
2*d^5*x^3 - 10*a*b*c*d^5*x + a^2*c*d^5 + 10*(b^2*c + 2*a*c^2)*d^5*x^2)*e^11 + (10*c^3*d^6*x^3 - 20*b*c^2*d^6*x
^2 - 2*a*b*c*d^6 + 5*(b^2*c + 2*a*c^2)*d^6*x)*e^10 + (10*c^3*d^7*x^2 - 10*b*c^2*d^7*x + (b^2*c + 2*a*c^2)*d^7)
*e^9 + (5*c^3*d^8*x - 2*b*c^2*d^8)*e^8)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Integral((a + b*x + c*x**2)**(5/2)/(d + e*x)**(11/2), 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)^(5/2)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(x*e + d)^(11/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x)

[Out]

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(11/2), x)

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