\(\int \frac {1}{(d+e x)^{3/2} (a+b x+c x^2)^{5/2}} \, dx\) [2481]

Optimal result

Integrand size = 24, antiderivative size = 918 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {\sqrt {2} \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {8 \sqrt {2} (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2/3*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^(3/2)/(e*x+d)^(1/2)
-2/3*(5*a*c*e*(-b*e+2*c*d)^2-(2*a*c*e-b^2*e+b*c*d)*(8*c^2*d^2-4*b^2*e^2-c*e*(-14*a*e+3*b*d))-4*c*(-b*e+2*c*d)*
(2*c^2*d^2-b^2*e^2-2*c*e*(-3*a*e+b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/
2)+2/3*e*(16*c^4*d^4-8*b^4*e^4-4*c^3*d^2*e*(-15*a*e+8*b*d)+b^2*c*e^3*(57*a*e+7*b*d)+3*c^2*e^2*(-28*a^2*e^2-20*
a*b*d*e+3*b^2*d^2))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^(1/2)-1/3*(16*c^4*d^4-8*b
^4*e^4-4*c^3*d^2*e*(-15*a*e+8*b*d)+b^2*c*e^3*(57*a*e+7*b*d)+3*c^2*e^2*(-28*a^2*e^2-20*a*b*d*e+3*b^2*d^2))*Elli
pticE(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)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/(-4*a*c+b^2)^(3/2)/(
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)+8/3*(-b*e+2*c*d)*(
2*c^2*d^2-b^2*e^2-2*c*e*(-3*a*e+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)*(-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)/(-4*a*c+b^2)^(3/2)/(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] (verified)

Time = 1.00 (sec) , antiderivative size = 918, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {754, 836, 848, 857, 732, 435, 430} \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}+\frac {2 e \sqrt {c x^2+b x+a} \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^3 \sqrt {d+e x}}+\frac {8 \sqrt {2} (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b e d+a e^2\right )^2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {2 \left (5 a c e (2 c d-b e)^2-4 c \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {2 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x} \left (c x^2+b x+a\right )^{3/2}} \]

[In]

Int[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*(a +
b*x + c*x^2)^(3/2)) - (2*(5*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 4*b^2*e^2 - c*e*(3
*b*d - 14*a*e)) - 4*c*(2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*x))/(3*(b^2 - 4*a*c)^2*(c*d^2
- b*d*e + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]) + (2*e*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d -
15*a*e) + b^2*c*e^3*(7*b*d + 57*a*e) + 3*c^2*e^2*(3*b^2*d^2 - 20*a*b*d*e - 28*a^2*e^2))*Sqrt[a + b*x + c*x^2])
/(3*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) - (Sqrt[2]*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*
(8*b*d - 15*a*e) + b^2*c*e^3*(7*b*d + 57*a*e) + 3*c^2*e^2*(3*b^2*d^2 - 20*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)])/(3*(b^2 - 4*a*c)^(3/2)*(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]) + (8*Sqrt[
2]*(2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*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)])/(3*(b^2 - 4*a*c)^(3/2)
*(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 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, 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] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-3 b c d e-4 b^2 e^2+14 a c e^2\right )+\frac {5}{2} c e (2 c d-b e) x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^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 {4 \int \frac {-\frac {1}{4} e \left (3 b^3 c d e^2-8 b^4 e^3+12 a c^2 e \left (c d^2-7 a e^2\right )-4 b c^2 d \left (2 c d^2+9 a e^2\right )+3 b^2 c e \left (3 c d^2+19 a e^2\right )\right )+c e (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^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 e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {8 \int \frac {-\frac {1}{8} c e \left (4 b^4 d e^3+3 b^2 c d e \left (5 c d^2-11 a e^2\right )+4 a c^2 d e \left (c d^2+33 a e^2\right )-b^3 \left (3 c d^2 e^2-4 a e^4\right )-4 b c \left (2 c^2 d^4+9 a c d^2 e^2+6 a^2 e^4\right )\right )+\frac {1}{8} c e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^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 e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}+\frac {\left (4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (c \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 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}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^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 e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (\sqrt {2} \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 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 )}{3 \left (b^2-4 a c\right )^{3/2} \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 (8 \sqrt {2} (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 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 )}{3 \left (b^2-4 a c\right )^{3/2} \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 (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^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 e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {\sqrt {2} \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 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 )}{3 \left (b^2-4 a c\right )^{3/2} \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 {8 \sqrt {2} (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 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 )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 34.97 (sec) , antiderivative size = 7870, normalized size of antiderivative = 8.57 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

Result too large to show

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2304\) vs. \(2(848)=1696\).

Time = 5.63 (sec) , antiderivative size = 2305, normalized size of antiderivative = 2.51

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2305\)
default	\(\text {Expression too large to display}\)	\(27156\)

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(5/2),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*(c*e*x^2+b*e*x+a*e)*e^4/(a*e^2-b*d*e+c*d^2
)^3/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+(-2/3*(2*a*c*e^2-b^2*e^2+2*b*c*d*e-2*c^2*d^2)/c/(4*a^2*c*e^2-a*b^2*e^2
-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)/(a*e^2-b*d*e+c*d^2)*x-2/3*(3*a*b*c*e^2-4*a*c^2*d*e-b^3*e^2+2*b^2*c
*d*e-b*c^2*d^2)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)/c^2/(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)/(a/c+b/c*x+x^2)^2-2*(c*e*x+c*d)*(1/3*(36*a^2*c^2*e^4-33*a*b^2*c*e
^4+60*a*b*c^2*d*e^3-60*a*c^3*d^2*e^2+5*b^4*e^4-7*b^3*c*d*e^3-9*b^2*c^2*d^2*e^2+32*b*c^3*d^3*e-16*c^4*d^4)/(4*a
^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)^2/(a*e^2-b*d*e+c*d^2)*x+1/3*(60*a^2*b*c^2*e^4-84
*a^2*c^3*d*e^3-37*a*b^3*c*e^4+69*a*b^2*c^2*d*e^3-24*a*b*c^3*d^2*e^2-4*a*c^4*d^3*e+5*b^5*e^4-8*b^4*c*d*e^3-6*b^
3*c^2*d^2*e^2+17*b^2*c^3*d^3*e-8*b*c^4*d^4)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)^
2/c/(a*e^2-b*d*e+c*d^2))/((a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+2*(-e^4*(b*e-c*d)/(a*e^2-b*d*e+c*d^2)^3+b*e^5/(a*
e^2-b*d*e+c*d^2)^3-1/3*(84*a^2*b*c^2*e^5-96*a^2*c^3*d*e^4-41*a*b^3*c*e^5+12*a*b^2*c^2*d*e^4+132*a*b*c^3*d^2*e^
3-128*a*c^4*d^3*e^2+5*b^5*e^5+b^4*c*d*e^4-17*b^3*c^2*d^2*e^3-16*b^2*c^3*d^3*e^2+64*b*c^4*d^4*e-32*c^5*d^5)/(4*
a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)^2/(a*e^2-b*d*e+c*d^2)+1/3*e*(60*a^2*b*c^2*e^4-8
4*a^2*c^3*d*e^3-37*a*b^3*c*e^4+69*a*b^2*c^2*d*e^3-24*a*b*c^3*d^2*e^2-4*a*c^4*d^3*e+5*b^5*e^4-8*b^4*c*d*e^3-6*b
^3*c^2*d^2*e^2+17*b^2*c^3*d^3*e-8*b*c^4*d^4)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)
^2/(a*e^2-b*d*e+c*d^2)+2/3*c*d*(36*a^2*c^2*e^4-33*a*b^2*c*e^4+60*a*b*c^2*d*e^3-60*a*c^3*d^2*e^2+5*b^4*e^4-7*b^
3*c*d*e^3-9*b^2*c^2*d^2*e^2+32*b*c^3*d^3*e-16*c^4*d^4)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-
b^2*c*d^2)^2/(a*e^2-b*d*e+c*d^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)*Ellipt
icF(((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*(e^5*c/(a*e^2-b*d*e+c*d^2)^3+1/3*c*e*(36*a^2*c^2*e^4-33*a*b^2*c*e^4+60*a*b*c^2*d*
e^3-60*a*c^3*d^2*e^2+5*b^4*e^4-7*b^3*c*d*e^3-9*b^2*c^2*d^2*e^2+32*b*c^3*d^3*e-16*c^4*d^4)/(4*a^2*c*e^2-a*b^2*e
^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)^2/(a*e^2-b*d*e+c*d^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 = 0.48 (sec) , antiderivative size = 5060, normalized size of antiderivative = 5.51 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/((d + e*x)**(3/2)*(a + b*x + c*x**2)**(5/2)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)^(3/2)), x)

Giac [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]

[In]

int(1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)),x)

[Out]

int(1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)), x)