3.25.75 \(\int \frac {1}{(d+e x)^{5/2} (a+b x+c x^2)^{3/2}} \, dx\) [2475]

Optimal. Leaf size=744 \[ -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}-\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}+\frac {\sqrt {2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2*(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)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2)-4
/3*e*(3*c^2*d^2+2*b^2*e^2-c*e*(5*a*e+3*b*d))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(3
/2)-2/3*e*(-b*e+2*c*d)*(3*c^2*d^2+8*b^2*e^2-c*e*(29*a*e+3*b*d))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+
c*d^2)^3/(e*x+d)^(1/2)+1/3*(-b*e+2*c*d)*(3*c^2*d^2+8*b^2*e^2-c*e*(29*a*e+3*b*d))*EllipticE(1/2*((b+2*c*x+(-4*a
*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1
/2))*2^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/(a*e^2-b*d*e+c*d^2)^3/(-4*a*c+b^2)^(1/2)/(c*x
^2+b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)-4/3*(3*c^2*d^2+2*b^2*e^2-c*e*(5*a*e+3*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)/(a*e^2-b*d*e+c*d^2)^2/(-4*a*c+b^2)^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.63, antiderivative size = 744, 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 = {754, 848, 857, 732, 435, 430} \begin {gather*} -\frac {4 \sqrt {2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\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 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac {\sqrt {2} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) 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 )}{3 \sqrt {b^2-4 a c} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 e \sqrt {a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 e \sqrt {a+b x+c x^2} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right )}{3 \left (b^2-4 a c\right ) \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^3}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(3/2)*Sqrt
[a + b*x + c*x^2]) - (4*e*(3*c^2*d^2 + 2*b^2*e^2 - c*e*(3*b*d + 5*a*e))*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c
)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) - (2*e*(2*c*d - b*e)*(3*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*a*e
))*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*Sqrt[d + e*x]) + (Sqrt[2]*(2*c*d - b*e)*(
3*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellip
ticE[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*Sqrt[b^2 - 4*a*c]*(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]) - (4*Sqrt[2]*(3*c^2*d^2 + 2*b^2*e^2 - c*e*(3*b*d + 5*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[Arc
Sin[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 + S
qrt[b^2 - 4*a*c])*e)])/(3*Sqrt[b^2 - 4*a*c]*(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 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)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} e \left (3 b c d-4 b^2 e+10 a c e\right )+\frac {3}{2} c e (2 c d-b e) x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx}{\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 )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}-\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}+\frac {4 \int \frac {\frac {1}{4} e \left (15 b^2 c d e-48 a c^2 d e-8 b^3 e^2-b c \left (3 c d^2-29 a e^2\right )\right )-\frac {1}{2} c e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 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 ) \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 )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}-\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \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^3 d e^2+2 a c e \left (27 c d^2-5 a e^2\right )-b c d \left (3 c d^2+25 a e^2\right )-b^2 \left (9 c d^2 e-4 a e^3\right )\right )-\frac {1}{8} c e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right ) \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 )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}-\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (2 c \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 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 ) \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (c (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right ) \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 )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}-\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}+\frac {\left (\sqrt {2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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 \sqrt {b^2-4 a c} \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 (4 \sqrt {2} \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \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 \sqrt {b^2-4 a c} \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 )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}-\frac {4 e \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 e (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}+\frac {\sqrt {2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {b^2-4 a c} \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 {4 \sqrt {2} \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {b^2-4 a c} \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] Result contains complex when optimal does not.
time = 32.73, size = 5565, normalized size = 7.48 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(12894\) vs. \(2(676)=1352\).
time = 0.99, size = 12895, normalized size = 17.33

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1964\)
default	\(\text {Expression too large to display}\)	\(12895\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.61, size = 3001, normalized size = 4.03 \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)^(5/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

-2/9*((6*c^5*d^6*x^2 + 6*b*c^4*d^6*x + 6*a*c^4*d^6 - ((8*b^4*c - 41*a*b^2*c^2 + 30*a^2*c^3)*x^4 + (8*b^5 - 41*
a*b^3*c + 30*a^2*b*c^2)*x^3 + (8*a*b^4 - 41*a^2*b^2*c + 30*a^3*c^2)*x^2)*e^6 + ((23*b^3*c^2 - 104*a*b*c^3)*d*x
^4 + (7*b^4*c - 22*a*b^2*c^2 - 60*a^2*c^3)*d*x^3 - (16*b^5 - 105*a*b^3*c + 164*a^2*b*c^2)*d*x^2 - 2*(8*a*b^4 -
41*a^2*b^2*c + 30*a^3*c^2)*d*x)*e^5 - ((17*b^2*c^3 - 104*a*c^4)*d^2*x^4 - (29*b^3*c^2 - 104*a*b*c^3)*d^2*x^3
- 2*(19*b^4*c - 92*a*b^2*c^2 + 37*a^2*c^3)*d^2*x^2 + (8*b^5 - 87*a*b^3*c + 238*a^2*b*c^2)*d^2*x + (8*a*b^4 - 4
1*a^2*b^2*c + 30*a^3*c^2)*d^2)*e^4 - (12*b*c^4*d^3*x^4 + 2*(23*b^2*c^3 - 104*a*c^4)*d^3*x^3 + (11*b^3*c^2 - 92
*a*b*c^3)*d^3*x^2 - (23*b^4*c - 138*a*b^2*c^2 + 208*a^2*c^3)*d^3*x - (23*a*b^3*c - 104*a^2*b*c^2)*d^3)*e^3 + (
6*c^5*d^4*x^4 - 18*b*c^4*d^4*x^3 - (41*b^2*c^3 - 110*a*c^4)*d^4*x^2 - (17*b^3*c^2 - 80*a*b*c^3)*d^4*x - (17*a*
b^2*c^2 - 104*a^2*c^3)*d^4)*e^2 + 12*(c^5*d^5*x^3 - a*b*c^3*d^5 - (b^2*c^3 - a*c^4)*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
^3*c^2 - 29*a*b*c^3)*x^4 + (8*b^4*c - 29*a*b^2*c^2)*x^3 + (8*a*b^3*c - 29*a^2*b*c^2)*x^2)*e^6 - (3*b^3*c^2*d*x
^3 + (19*b^2*c^3 - 58*a*c^4)*d*x^4 - (16*b^4*c - 77*a*b^2*c^2 + 58*a^2*c^3)*d*x^2 - 2*(8*a*b^3*c - 29*a^2*b*c^
2)*d*x)*e^5 + (9*b*c^4*d^2*x^4 - 29*(b^2*c^3 - 4*a*c^4)*d^2*x^3 - 6*(5*b^3*c^2 - 16*a*b*c^3)*d^2*x^2 + (8*b^4*
c - 67*a*b^2*c^2 + 116*a^2*c^3)*d^2*x + (8*a*b^3*c - 29*a^2*b*c^2)*d^2)*e^4 - (6*c^5*d^3*x^4 - 12*b*c^4*d^3*x^
3 + (b^2*c^3 - 52*a*c^4)*d^3*x^2 + 19*(b^3*c^2 - 4*a*b*c^3)*d^3*x + (19*a*b^2*c^2 - 58*a^2*c^3)*d^3)*e^3 - 3*(
4*c^5*d^4*x^3 + b*c^4*d^4*x^2 - 3*a*b*c^3*d^4 - (3*b^2*c^3 - 4*a*c^4)*d^4*x)*e^2 - 6*(c^5*d^5*x^2 + b*c^4*d^5*
x + a*c^4*d^5)*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, -4/2
7*(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, weierstrassPInve
rse(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)
*((a^2*b^2*c - 4*a^3*c^2 - (8*b^3*c^2 - 29*a*b*c^3)*x^3 - (8*b^4*c - 33*a*b^2*c^2 + 10*a^2*c^3)*x^2 - 4*(a*b^3
*c - 4*a^2*b*c^2)*x)*e^6 + ((19*b^2*c^3 - 58*a*c^4)*d*x^3 + (7*b^3*c^2 - 25*a*b*c^3)*d*x^2 - 2*(6*b^4*c - 29*a
*b^2*c^2 + 26*a^2*c^3)*d*x - 6*(a*b^3*c - 4*a^2*b*c^2)*d)*e^5 - (9*b*c^4*d^2*x^3 - 2*(10*b^2*c^3 - 31*a*c^4)*d
^2*x^2 - (26*b^3*c^2 - 89*a*b*c^3)*d^2*x + (3*b^4*c - 23*a*b^2*c^2 + 50*a^2*c^3)*d^2)*e^4 + 3*(2*c^5*d^3*x^3 -
5*b*c^4*d^3*x^2 - 3*(b^2*c^3 - 2*a*c^4)*d^3*x + 3*(b^3*c^2 - 3*a*b*c^3)*d^3)*e^3 + 3*(4*c^5*d^4*x^2 - b*c^4*d
^4*x - 3*(b^2*c^3 - 2*a*c^4)*d^4)*e^2 + 3*(2*c^5*d^5*x + b*c^4*d^5)*e)*sqrt(x*e + d))/(((a^3*b^2*c^2 - 4*a^4*c
^3)*x^4 + (a^3*b^3*c - 4*a^4*b*c^2)*x^3 + (a^4*b^2*c - 4*a^5*c^2)*x^2)*e^9 - (3*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d*
x^4 + (3*a^2*b^4*c - 14*a^3*b^2*c^2 + 8*a^4*c^3)*d*x^3 + (a^3*b^3*c - 4*a^4*b*c^2)*d*x^2 - 2*(a^4*b^2*c - 4*a^
5*c^2)*d*x)*e^8 + (3*(a*b^4*c^2 - 3*a^2*b^2*c^3 - 4*a^3*c^4)*d^2*x^4 + 3*(a*b^5*c - 5*a^2*b^3*c^2 + 4*a^3*b*c^
3)*d^2*x^3 - (3*a^2*b^4*c - 16*a^3*b^2*c^2 + 16*a^4*c^3)*d^2*x^2 - 5*(a^3*b^3*c - 4*a^4*b*c^2)*d^2*x + (a^4*b^
2*c - 4*a^5*c^2)*d^2)*e^7 - ((b^5*c^2 + 2*a*b^3*c^3 - 24*a^2*b*c^4)*d^3*x^4 + (b^6*c - 4*a*b^4*c^2 - 6*a^2*b^2
*c^3 + 24*a^3*c^4)*d^3*x^3 - (5*a*b^5*c - 23*a^2*b^3*c^2 + 12*a^3*b*c^3)*d^3*x^2 - 3*(a^2*b^4*c - 2*a^3*b^2*c^
2 - 8*a^4*c^3)*d^3*x + 3*(a^3*b^3*c - 4*a^4*b*c^2)*d^3)*e^6 + (3*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*d^4*x^4 +
(b^5*c^2 - 13*a*b^3*c^3 + 36*a^2*b*c^4)*d^4*x^3 - 2*(b^6*c - a*b^4*c^2 - 15*a^2*b^2*c^3 + 12*a^3*c^4)*d^4*x^2
+ (a*b^5*c - 13*a^2*b^3*c^2 + 36*a^3*b*c^3)*d^4*x + 3*(a^2*b^4*c - 3*a^3*b^2*c^2 - 4*a^4*c^3)*d^4)*e^5 - (3*(
b^3*c^4 - 4*a*b*c^5)*d^5*x^4 - 3*(b^4*c^3 - 2*a*b^2*c^4 - 8*a^2*c^5)*d^5*x^3 - (5*b^5*c^2 - 23*a*b^3*c^3 + 12*
a^2*b*c^4)*d^5*x^2 + (b^6*c - 4*a*b^4*c^2 - 6*a^2*b^2*c^3 + 24*a^3*c^4)*d^5*x + (a*b^5*c + 2*a^2*b^3*c^2 - 24*
a^3*b*c^3)*d^5)*e^4 + ((b^2*c^5 - 4*a*c^6)*d^6*x^4 - 5*(b^3*c^4 - 4*a*b*c^5)*d^6*x^3 - (3*b^4*c^3 - 16*a*b^2*c
^4 + 16*a^2*c^5)*d^6*x^2 + 3*(b^5*c^2 - 5*a*b^3*c^3 + 4*a^2*b*c^4)*d^6*x + 3*(a*b^4*c^2 - 3*a^2*b^2*c^3 - 4*a^
3*c^4)*d^6)*e^3 + (2*(b^2*c^5 - 4*a*c^6)*d^7*x^3 - (b^3*c^4 - 4*a*b*c^5)*d^7*x^2 - (3*b^4*c^3 - 14*a*b^2*c^4 +
8*a^2*c^5)*d^7*x - 3*(a*b^3*c^3 - 4*a^2*b*c^4)*d^7)*e^2 + ((b^2*c^5 - 4*a*c^6)*d^8*x^2 + (b^3*c^4 - 4*a*b*c^5
)*d^8*x + (a*b^2*c^4 - 4*a^2*c^5)*d^8)*e)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________