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

Optimal result

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

[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] (verified)

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

[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*} \text {integral}& = -\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*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 36.14 (sec) , antiderivative size = 8108, normalized size of antiderivative = 8.78 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1743\) vs. \(2(853)=1706\).

Time = 2.61 (sec) , antiderivative size = 1744, normalized size of antiderivative = 1.89

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

[In]

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

[Out]

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/9*(a^2*e^4-2*a*b*d*e^3+2*a*c*d^2*e^2+b^2*d
^2*e^2-2*b*c*d^3*e+c^2*d^4)/e^10*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^5-38/63*(a*b*e^3-2*a*
c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/e^9*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^4-2/63*(2
8*a*c*e^2+15*b^2*e^2-88*b*c*d*e+88*c^2*d^2)/e^8*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^3-2/63
*(57*a*b*c*e^3-114*a*c^2*d*e^2+b^3*e^3-63*b^2*c*d*e^2+183*b*c^2*d^2*e-122*c^3*d^3)/e^7/(a*e^2-b*d*e+c*d^2)*(c*
e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^2-2/63*(c*e*x^2+b*e*x+a*e)/(a*e^2-b*d*e+c*d^2)^2/e^6*(105
*a^2*c^2*e^4+30*a*b^2*c*e^4-330*a*b*c^2*d*e^3+330*a*c^3*d^2*e^2-2*b^4*e^4-14*b^3*c*d*e^3+207*b^2*c^2*d^2*e^2-3
86*b*c^3*d^3*e+193*c^4*d^4)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2*(c^2*(3*b*e-5*c*d)/e^6-1/63*c*(57*a*b*c*e^3-
114*a*c^2*d*e^2+b^3*e^3-63*b^2*c*d*e^2+183*b*c^2*d^2*e-122*c^3*d^3)/e^6/(a*e^2-b*d*e+c*d^2)-1/63/e^6*(b*e-c*d)
*(105*a^2*c^2*e^4+30*a*b^2*c*e^4-330*a*b*c^2*d*e^3+330*a*c^3*d^2*e^2-2*b^4*e^4-14*b^3*c*d*e^3+207*b^2*c^2*d^2*
e^2-386*b*c^3*d^3*e+193*c^4*d^4)/(a*e^2-b*d*e+c*d^2)^2+1/63*b/e^5/(a*e^2-b*d*e+c*d^2)^2*(105*a^2*c^2*e^4+30*a*
b^2*c*e^4-330*a*b*c^2*d*e^3+330*a*c^3*d^2*e^2-2*b^4*e^4-14*b^3*c*d*e^3+207*b^2*c^2*d^2*e^2-386*b*c^3*d^3*e+193
*c^4*d^4))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b
+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*
(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/2*(
b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/
2))+2*(c^3/e^5+1/63/e^5*c*(105*a^2*c^2*e^4+30*a*b^2*c*e^4-330*a*b*c^2*d*e^3+330*a*c^3*d^2*e^2-2*b^4*e^4-14*b^3
*c*d*e^3+207*b^2*c^2*d^2*e^2-386*b*c^3*d^3*e+193*c^4*d^4)/(a*e^2-b*d*e+c*d^2)^2)*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2
))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-
4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^
3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+d/e)/(d/e-1/2*(b+
(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)
)+1/2/c*(-b+(-4*a*c+b^2)^(1/2))*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4
*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.36 (sec) , antiderivative size = 2808, normalized size of antiderivative = 3.04 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Too large to display} \]

[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 - 640*b*c^4*d^9*e + 2*(239*b^2*c^3 + 324*a*c^4)*d^8*e^2 - (77*b^3*c^2 + 972*a*b*c^3)*d^7
*e^3 - (13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d^6*e^4 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^5*e^5 + (256*
c^5*d^5*e^5 - 640*b*c^4*d^4*e^6 + 2*(239*b^2*c^3 + 324*a*c^4)*d^3*e^7 - (77*b^3*c^2 + 972*a*b*c^3)*d^2*e^8 - (
13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d*e^9 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*e^10)*x^5 + 5*(256*c^5*d^
6*e^4 - 640*b*c^4*d^5*e^5 + 2*(239*b^2*c^3 + 324*a*c^4)*d^4*e^6 - (77*b^3*c^2 + 972*a*b*c^3)*d^3*e^7 - (13*b^4
*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d^2*e^8 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d*e^9)*x^4 + 10*(256*c^5*d^7*
e^3 - 640*b*c^4*d^6*e^4 + 2*(239*b^2*c^3 + 324*a*c^4)*d^5*e^5 - (77*b^3*c^2 + 972*a*b*c^3)*d^4*e^6 - (13*b^4*c
- 258*a*b^2*c^2 - 456*a^2*c^3)*d^3*e^7 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^2*e^8)*x^3 + 10*(256*c^5*d^8*
e^2 - 640*b*c^4*d^7*e^3 + 2*(239*b^2*c^3 + 324*a*c^4)*d^6*e^4 - (77*b^3*c^2 + 972*a*b*c^3)*d^5*e^5 - (13*b^4*c
- 258*a*b^2*c^2 - 456*a^2*c^3)*d^4*e^6 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^3*e^7)*x^2 + 5*(256*c^5*d^9*e
- 640*b*c^4*d^8*e^2 + 2*(239*b^2*c^3 + 324*a*c^4)*d^7*e^3 - (77*b^3*c^2 + 972*a*b*c^3)*d^6*e^4 - (13*b^4*c -
258*a*b^2*c^2 - 456*a^2*c^3)*d^5*e^5 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^4*e^6)*x)*sqrt(c*e)*weierstrassP
Inverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^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)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 6*(128*c^5*d^9*e - 256*b*
c^4*d^8*e^2 + 3*(45*b^2*c^3 + 76*a*c^4)*d^7*e^3 - (7*b^3*c^2 + 228*a*b*c^3)*d^6*e^4 - (b^4*c - 15*a*b^2*c^2 -
84*a^2*c^3)*d^5*e^5 + (128*c^5*d^4*e^6 - 256*b*c^4*d^3*e^7 + 3*(45*b^2*c^3 + 76*a*c^4)*d^2*e^8 - (7*b^3*c^2 +
228*a*b*c^3)*d*e^9 - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*e^10)*x^5 + 5*(128*c^5*d^5*e^5 - 256*b*c^4*d^4*e^6 +
3*(45*b^2*c^3 + 76*a*c^4)*d^3*e^7 - (7*b^3*c^2 + 228*a*b*c^3)*d^2*e^8 - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*d*
e^9)*x^4 + 10*(128*c^5*d^6*e^4 - 256*b*c^4*d^5*e^5 + 3*(45*b^2*c^3 + 76*a*c^4)*d^4*e^6 - (7*b^3*c^2 + 228*a*b*
c^3)*d^3*e^7 - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*d^2*e^8)*x^3 + 10*(128*c^5*d^7*e^3 - 256*b*c^4*d^6*e^4 + 3*
(45*b^2*c^3 + 76*a*c^4)*d^5*e^5 - (7*b^3*c^2 + 228*a*b*c^3)*d^4*e^6 - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*d^3*
e^7)*x^2 + 5*(128*c^5*d^8*e^2 - 256*b*c^4*d^7*e^3 + 3*(45*b^2*c^3 + 76*a*c^4)*d^6*e^4 - (7*b^3*c^2 + 228*a*b*c
^3)*d^5*e^5 - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*d^4*e^6)*x)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e
+ (b^2 - 3*a*c)*e^2)/(c^2*e^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)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^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)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c
*e))) + 3*(128*c^5*d^8*e^2 - 240*b*c^4*d^7*e^3 + 63*a^2*c^3*d^4*e^6 + 18*a^3*c^2*d^2*e^8 - 9*a^3*b*c*d*e^9 + 7
*a^4*c*e^10 + (111*b^2*c^3 + 212*a*c^4)*d^6*e^4 - (b^3*c^2 + 183*a*b*c^3)*d^5*e^5 + (193*c^5*d^4*e^6 - 386*b*c
^4*d^3*e^7 + 3*(69*b^2*c^3 + 110*a*c^4)*d^2*e^8 - 2*(7*b^3*c^2 + 165*a*b*c^3)*d*e^9 - (2*b^4*c - 30*a*b^2*c^2
- 105*a^2*c^3)*e^10)*x^4 + (650*c^5*d^5*e^5 - 1239*b*c^4*d^4*e^6 + 2*(291*b^2*c^3 + 542*a*c^4)*d^3*e^7 + 2*(4*
b^3*c^2 - 483*a*b*c^3)*d^2*e^8 - 9*(b^4*c - 34*a^2*c^3)*d*e^9 + (a*b^3*c + 57*a^2*b*c^2)*e^10)*x^3 + (880*c^5*
d^6*e^4 - 1665*b*c^4*d^5*e^5 + 9*(87*b^2*c^3 + 164*a*c^4)*d^4*e^6 - 2*(5*b^3*c^2 + 663*a*b*c^3)*d^3*e^7 + 54*(
a*b^2*c^2 + 8*a^2*c^3)*d^2*e^8 - 27*(a*b^3*c - a^2*b*c^2)*d*e^9 + (15*a^2*b^2*c + 28*a^3*c^2)*e^10)*x^2 + (544
*c^5*d^7*e^3 - 1024*b*c^4*d^6*e^4 + 252*a^2*c^3*d^3*e^7 + 54*a^2*b*c^2*d^2*e^8 + 19*a^3*b*c*e^10 + 3*(159*b^2*
c^3 + 302*a*c^4)*d^5*e^5 - (5*b^3*c^2 + 789*a*b*c^3)*d^4*e^6 - 9*(3*a^2*b^2*c - 2*a^3*c^2)*d*e^9)*x)*sqrt(c*x^
2 + b*x + a)*sqrt(e*x + d))/(c^3*d^9*e^7 - 2*b*c^2*d^8*e^8 - 2*a*b*c*d^6*e^10 + a^2*c*d^5*e^11 + (b^2*c + 2*a*
c^2)*d^7*e^9 + (c^3*d^4*e^12 - 2*b*c^2*d^3*e^13 - 2*a*b*c*d*e^15 + a^2*c*e^16 + (b^2*c + 2*a*c^2)*d^2*e^14)*x^
5 + 5*(c^3*d^5*e^11 - 2*b*c^2*d^4*e^12 - 2*a*b*c*d^2*e^14 + a^2*c*d*e^15 + (b^2*c + 2*a*c^2)*d^3*e^13)*x^4 + 1
0*(c^3*d^6*e^10 - 2*b*c^2*d^5*e^11 - 2*a*b*c*d^3*e^13 + a^2*c*d^2*e^14 + (b^2*c + 2*a*c^2)*d^4*e^12)*x^3 + 10*
(c^3*d^7*e^9 - 2*b*c^2*d^6*e^10 - 2*a*b*c*d^4*e^12 + a^2*c*d^3*e^13 + (b^2*c + 2*a*c^2)*d^5*e^11)*x^2 + 5*(c^3
*d^8*e^8 - 2*b*c^2*d^7*e^9 - 2*a*b*c*d^5*e^11 + a^2*c*d^4*e^12 + (b^2*c + 2*a*c^2)*d^6*e^10)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[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)/(e*x + d)^(11/2), x)

Giac [F]

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

[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)/(e*x + d)^(11/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]

[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)