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

Optimal. Leaf size=918 \[ -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (5 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-4 b^2 e^2-c e (3 b d-14 a e)\right )-4 c (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 e \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {\sqrt {2} \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {8 \sqrt {2} (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[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]
time = 1.01, antiderivative size = 918, 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 = {754, 836, 848, 857, 732, 435, 430} \begin {gather*} -\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 (\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 ) \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}} 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 )}{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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 \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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 33.07, size = 7870, normalized size = 8.57 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)),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. \(27156\) vs. \(2(848)=1696\).
time = 0.87, size = 27157, normalized size = 29.58

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2305\)
default	\(\text {Expression too large to display}\)	\(27157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.36, size = 4986, normalized size = 5.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*((16*c^7*d^6*x^4 + 32*b*c^6*d^6*x^3 + 32*a*b*c^5*d^6*x + 16*a^2*c^5*d^6 + 16*(b^2*c^5 + 2*a*c^6)*d^6*x^2 -
((8*b^5*c^2 - 69*a*b^3*c^3 + 156*a^2*b*c^4)*x^5 + 2*(8*b^6*c - 69*a*b^4*c^2 + 156*a^2*b^2*c^3)*x^4 + (8*b^7 -
53*a*b^5*c + 18*a^2*b^3*c^2 + 312*a^3*b*c^3)*x^3 + 2*(8*a*b^6 - 69*a^2*b^4*c + 156*a^3*b^2*c^2)*x^2 + (8*a^2*
b^5 - 69*a^3*b^3*c + 156*a^4*b*c^2)*x)*e^6 + ((11*b^4*c^3 - 102*a*b^2*c^4 + 312*a^2*c^5)*d*x^5 + (14*b^5*c^2 -
135*a*b^3*c^3 + 468*a^2*b*c^4)*d*x^4 - (5*b^6*c - 58*a*b^4*c^2 + 204*a^2*b^2*c^3 - 624*a^3*c^4)*d*x^3 - (8*b^
7 - 75*a*b^5*c + 222*a^2*b^3*c^2 - 312*a^3*b*c^3)*d*x^2 - (16*a*b^6 - 149*a^2*b^4*c + 414*a^3*b^2*c^2 - 312*a^
4*c^3)*d*x - (8*a^2*b^5 - 69*a^3*b^3*c + 156*a^4*b*c^2)*d)*e^5 + ((7*b^3*c^4 - 108*a*b*c^5)*d^2*x^5 + (25*b^4*
c^3 - 318*a*b^2*c^4 + 312*a^2*c^5)*d^2*x^4 + (29*b^5*c^2 - 298*a*b^3*c^3 + 408*a^2*b*c^4)*d^2*x^3 + (11*b^6*c
- 66*a*b^4*c^2 - 108*a^2*b^2*c^3 + 624*a^3*c^4)*d^2*x^2 + (22*a*b^5*c - 197*a^2*b^3*c^2 + 516*a^3*b*c^3)*d^2*x
+ (11*a^2*b^4*c - 102*a^3*b^2*c^2 + 312*a^4*c^3)*d^2)*e^4 + (2*(11*b^2*c^5 + 36*a*c^6)*d^3*x^5 + 3*(17*b^3*c^
4 + 12*a*b*c^5)*d^3*x^4 + 4*(9*b^4*c^3 - 25*a*b^2*c^4 + 36*a^2*c^5)*d^3*x^3 + (7*b^5*c^2 - 50*a*b^3*c^3 - 72*a
^2*b*c^4)*d^3*x^2 + 2*(7*a*b^4*c^2 - 97*a^2*b^2*c^3 + 36*a^3*c^4)*d^3*x + (7*a^2*b^3*c^2 - 108*a^3*b*c^3)*d^3)
*e^3 - 2*(20*b*c^6*d^4*x^5 + (29*b^2*c^5 - 36*a*c^6)*d^4*x^4 - 2*(b^3*c^4 + 16*a*b*c^5)*d^4*x^3 - (11*b^4*c^3
+ 18*a*b^2*c^4 + 72*a^2*c^5)*d^4*x^2 - 2*(11*a*b^3*c^3 + 26*a^2*b*c^4)*d^4*x - (11*a^2*b^2*c^3 + 36*a^3*c^4)*d
^4)*e^2 + 8*(2*c^7*d^5*x^5 - b*c^6*d^5*x^4 - 5*a^2*b*c^4*d^5 - 4*(2*b^2*c^5 - a*c^6)*d^5*x^3 - (5*b^3*c^4 + 6*
a*b*c^5)*d^5*x^2 - 2*(5*a*b^2*c^4 - a^2*c^5)*d^5*x)*e)*sqrt(c)*e^(1/2)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*
d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9
*a*b*c)*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c) - 3*(((8*b^4*c^3 - 57*a*b^2*c^4 + 84*a^2*c^5)*x^5
+ 2*(8*b^5*c^2 - 57*a*b^3*c^3 + 84*a^2*b*c^4)*x^4 + (8*b^6*c - 41*a*b^4*c^2 - 30*a^2*b^2*c^3 + 168*a^3*c^4)*x
^3 + 2*(8*a*b^5*c - 57*a^2*b^3*c^2 + 84*a^3*b*c^3)*x^2 + (8*a^2*b^4*c - 57*a^3*b^2*c^2 + 84*a^4*c^3)*x)*e^6 -
((7*b^3*c^4 - 60*a*b*c^5)*d*x^5 + 3*(2*b^4*c^3 - 21*a*b^2*c^4 - 28*a^2*c^5)*d*x^4 - (9*b^5*c^2 - 68*a*b^3*c^3
+ 288*a^2*b*c^4)*d*x^3 - (8*b^6*c - 55*a*b^4*c^2 + 90*a^2*b^2*c^3 + 168*a^3*c^4)*d*x^2 - (16*a*b^5*c - 121*a^2
*b^3*c^2 + 228*a^3*b*c^3)*d*x - (8*a^2*b^4*c - 57*a^3*b^2*c^2 + 84*a^4*c^3)*d)*e^5 - (3*(3*b^2*c^5 + 20*a*c^6)
*d^2*x^5 + 5*(5*b^3*c^4 + 12*a*b*c^5)*d^2*x^4 + (23*b^4*c^3 - 42*a*b^2*c^4 + 120*a^2*c^5)*d^2*x^3 + 7*(b^5*c^2
- 4*a*b^3*c^3)*d^2*x^2 + (14*a*b^4*c^2 - 111*a^2*b^2*c^3 + 60*a^3*c^4)*d^2*x + (7*a^2*b^3*c^2 - 60*a^3*b*c^3)
*d^2)*e^4 + (32*b*c^6*d^3*x^5 + 5*(11*b^2*c^5 - 12*a*c^6)*d^3*x^4 + 14*(b^3*c^4 - 4*a*b*c^5)*d^3*x^3 - (9*b^4*
c^3 + 14*a*b^2*c^4 + 120*a^2*c^5)*d^3*x^2 - 2*(9*a*b^3*c^3 + 44*a^2*b*c^4)*d^3*x - 3*(3*a^2*b^2*c^3 + 20*a^3*c
^4)*d^3)*e^3 - 16*(c^7*d^4*x^5 - 2*a^2*b*c^4*d^4 - (3*b^2*c^5 - 2*a*c^6)*d^4*x^3 - 2*(b^3*c^4 + a*b*c^5)*d^4*x
^2 - (4*a*b^2*c^4 - a^2*c^5)*d^4*x)*e^2 - 16*(c^7*d^5*x^4 + 2*b*c^6*d^5*x^3 + 2*a*b*c^5*d^5*x + a^2*c^5*d^5 +
(b^2*c^5 + 2*a*c^6)*d^5*x^2)*e)*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, we
ierstrassPInverse(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*sqrt(c*
x^2 + b*x + a)*((3*a^2*b^4*c - 24*a^3*b^2*c^2 + 48*a^4*c^3 + (8*b^4*c^3 - 57*a*b^2*c^4 + 84*a^2*c^5)*x^4 + 2*(
8*b^5*c^2 - 59*a*b^3*c^3 + 96*a^2*b*c^4)*x^3 + (8*b^6*c - 49*a*b^4*c^2 + 21*a^2*b^2*c^3 + 140*a^3*c^4)*x^2 + 4
*(3*a*b^5*c - 23*a^2*b^3*c^2 + 42*a^3*b*c^3)*x)*e^6 - ((7*b^3*c^4 - 60*a*b*c^5)*d*x^4 + 2*(5*b^4*c^3 - 48*a*b^
2*c^4 + 24*a^2*c^5)*d*x^3 - (b^5*c^2 - 4*a*b^3*c^3 + 72*a^2*b*c^4)*d*x^2 - 2*(2*b^6*c - 17*a*b^4*c^2 + 45*a^2*
b^2*c^3 - 28*a^3*c^4)*d*x - 2*(3*a*b^5*c - 22*a^2*b^3*c^2 + 36*a^3*b*c^3)*d)*e^5 - (3*(3*b^2*c^5 + 20*a*c^6)*d
^2*x^4 + 2*(11*b^3*c^4 + 12*a*b*c^5)*d^2*x^3 + 6*(3*b^4*c^3 - 14*a*b^2*c^4 + 24*a^2*c^5)*d^2*x^2 + 6*(b^5*c^2
- 7*a*b^3*c^3 + 4*a^2*b*c^4)*d^2*x + (b^6*c + 3*a*b^4*c^2 - 69*a^2*b^2*c^3 + 100*a^3*c^4)*d^2)*e^4 + (32*b*c^6
*d^3*x^4 + 8*(5*b^2*c^5 - 8*a*c^6)*d^3*x^3 - 2*(b^3*c^4 + 20*a*b*c^5)*d^3*x^2 - 2*(3*b^4*c^3 + 6*a*b^2*c^4 + 4
0*a^2*c^5)*d^3*x + 3*(b^5*c^2 - 8*a*b^3*c^3)*d^3)*e^3 - (16*c^7*d^4*x^4 - 8*b*c^6*d^4*x^3 - (43*b^2*c^5 - 28*a
*c^6)*d^4*x^2 - 2*(7*b^3*c^4 + 16*a*b*c^5)*d^4*x + (3*b^4*c^3 - 33*a*b^2*c^4 + 20*a^2*c^5)*d^4)*e^2 - (16*c^7*
d^5*x^3 + 24*b*c^6*d^5*x^2 + 6*(b^2*c^5 + 4*a*c^6)*d^5*x - (b^3*c^4 - 12*a*b*c^5)*d^5)*e)*sqrt(x*e + d))/(((a^
3*b^4*c^3 - 8*a^4*b^2*c^4 + 16*a^5*c^5)*x^5 + 2*(a^3*b^5*c^2 - 8*a^4*b^3*c^3 + 16*a^5*b*c^4)*x^4 + (a^3*b^6*c
- 6*a^4*b^4*c^2 + 32*a^6*c^4)*x^3 + 2*(a^4*b^5*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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(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)*(x*e + d)^(3/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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