3.1.58 \(\int \frac {x^2 (a+c x^2)^{3/2}}{d+e x+f x^2} \, dx\) [58]

Optimal. Leaf size=795 \[ -\frac {\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt {a+c x^2}}{8 f^4}-\frac {(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac {\left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c} f^5}-\frac {\left (a^2 f^4 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+2 a c f^2 \left (e^4-4 d e^2 f+2 d^2 f^2-e^3 \sqrt {e^2-4 d f}+2 d e f \sqrt {e^2-4 d f}\right )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3-e^5 \sqrt {e^2-4 d f}+4 d e^3 f \sqrt {e^2-4 d f}-3 d^2 e f^2 \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^5 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (a^2 f^4 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+2 a c f^2 \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3+e^5 \sqrt {e^2-4 d f}-4 d e^3 f \sqrt {e^2-4 d f}+3 d^2 e f^2 \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^5 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \]

[Out]

-1/12*(-3*f*x+4*e)*(c*x^2+a)^(3/2)/f^2+1/8*(3*a^2*f^4+12*a*c*f^2*(-d*f+e^2)+8*c^2*(d^2*f^2-3*d*e^2*f+e^4))*arc
tanh(x*c^(1/2)/(c*x^2+a)^(1/2))/f^5/c^(1/2)-1/8*(8*e*(a*f^2+c*(-2*d*f+e^2))-f*(3*a*f^2+4*c*(-d*f+e^2))*x)*(c*x
^2+a)^(1/2)/f^4-1/2*arctanh(1/2*(2*a*f-c*x*(e-(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+a)^(1/2)/(2*a*f^2+c*(e^2-2*d
*f-e*(-4*d*f+e^2)^(1/2)))^(1/2))*(a^2*f^4*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2))+2*a*c*f^2*(e^4-4*d*e^2*f+2*d^2*f^2-
e^3*(-4*d*f+e^2)^(1/2)+2*d*e*f*(-4*d*f+e^2)^(1/2))+c^2*(e^6-6*d*e^4*f+9*d^2*e^2*f^2-2*d^3*f^3-e^5*(-4*d*f+e^2)
^(1/2)+4*d*e^3*f*(-4*d*f+e^2)^(1/2)-3*d^2*e*f^2*(-4*d*f+e^2)^(1/2)))/f^5*2^(1/2)/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c
*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2)+1/2*arctanh(1/2*(2*a*f-c*x*(e+(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+a)^
(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2))*(a^2*f^4*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2))+2*a*c*f^2*
(e^4-4*d*e^2*f+2*d^2*f^2+e^3*(-4*d*f+e^2)^(1/2)-2*d*e*f*(-4*d*f+e^2)^(1/2))+c^2*(e^6-6*d*e^4*f+9*d^2*e^2*f^2-2
*d^3*f^3+e^5*(-4*d*f+e^2)^(1/2)-4*d*e^3*f*(-4*d*f+e^2)^(1/2)+3*d^2*e*f^2*(-4*d*f+e^2)^(1/2)))/f^5*2^(1/2)/(-4*
d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 2.70, antiderivative size = 795, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1083, 1082, 1094, 223, 212, 1048, 739} \begin {gather*} -\frac {(4 e-3 f x) \left (c x^2+a\right )^{3/2}}{12 f^2}-\frac {\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt {c x^2+a}}{8 f^4}+\frac {\left (3 a^2 f^4+12 a c \left (e^2-d f\right ) f^2+8 c^2 \left (e^4-3 d f e^2+d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {c x^2+a}}\right )}{8 \sqrt {c} f^5}-\frac {\left (a^2 \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right ) f^4+2 a c \left (e^4-\sqrt {e^2-4 d f} e^3-4 d f e^2+2 d f \sqrt {e^2-4 d f} e+2 d^2 f^2\right ) f^2+c^2 \left (e^6-\sqrt {e^2-4 d f} e^5-6 d f e^4+4 d f \sqrt {e^2-4 d f} e^3+9 d^2 f^2 e^2-3 d^2 f^2 \sqrt {e^2-4 d f} e-2 d^3 f^3\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )} \sqrt {c x^2+a}}\right )}{\sqrt {2} f^5 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )}}+\frac {\left (a^2 \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right ) f^4+2 a c \left (e^4+\sqrt {e^2-4 d f} e^3-4 d f e^2-2 d f \sqrt {e^2-4 d f} e+2 d^2 f^2\right ) f^2+c^2 \left (e^6+\sqrt {e^2-4 d f} e^5-6 d f e^4-4 d f \sqrt {e^2-4 d f} e^3+9 d^2 f^2 e^2+3 d^2 f^2 \sqrt {e^2-4 d f} e-2 d^3 f^3\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )} \sqrt {c x^2+a}}\right )}{\sqrt {2} f^5 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*((8*e*(a*f^2 + c*(e^2 - 2*d*f)) - f*(3*a*f^2 + 4*c*(e^2 - d*f))*x)*Sqrt[a + c*x^2])/f^4 - ((4*e - 3*f*x)*
(a + c*x^2)^(3/2))/(12*f^2) + ((3*a^2*f^4 + 12*a*c*f^2*(e^2 - d*f) + 8*c^2*(e^4 - 3*d*e^2*f + d^2*f^2))*ArcTan
h[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*Sqrt[c]*f^5) - ((a^2*f^4*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + 2*a*c*f^2*(e
^4 - 4*d*e^2*f + 2*d^2*f^2 - e^3*Sqrt[e^2 - 4*d*f] + 2*d*e*f*Sqrt[e^2 - 4*d*f]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2
*e^2*f^2 - 2*d^3*f^3 - e^5*Sqrt[e^2 - 4*d*f] + 4*d*e^3*f*Sqrt[e^2 - 4*d*f] - 3*d^2*e*f^2*Sqrt[e^2 - 4*d*f]))*A
rcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sq
rt[a + c*x^2])])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + ((a^2
*f^4*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + 2*a*c*f^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2 + e^3*Sqrt[e^2 - 4*d*f] - 2*
d*e*f*Sqrt[e^2 - 4*d*f]) + c^2*(e^6 - 6*d*e^4*f + 9*d^2*e^2*f^2 - 2*d^3*f^3 + e^5*Sqrt[e^2 - 4*d*f] - 4*d*e^3*
f*Sqrt[e^2 - 4*d*f] + 3*d^2*e*f^2*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*S
qrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*f^5*Sqrt[e^2 - 4*d*f]*Sqrt[2*
a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1048

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x], x] - Dist[(2*c
*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 1082

Int[((a_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_
Symbol] :> Simp[(B*c*f*(2*p + 2*q + 3) + C*((-c)*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + c*x^2)^p*((d +
e*x + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Dist[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)
), Int[(a + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Simp[p*((-a)*e)*(C*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3
)) + (p + q + 1)*(a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C
*(c*e)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(-4*a*c)))*x + (p*(c*e)*(C*(c*e)*(q + 1
) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(-4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2
*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, A, B, C, q}, x] && NeQ[e^2 - 4
*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] &&  !IGtQ[p, 0] &&  !IGtQ[q, 0]

Rule 1083

Int[((a_) + (c_.)*(x_)^2)^(p_)*((A_.) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Si
mp[(C*((-c)*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*c*f^2*(p + q
+ 1)*(2*p + 2*q + 3))), x] - Dist[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)), Int[(a + c*x^2)^(p - 1)*(d + e*x +
f*x^2)^q*Simp[p*((-a)*e)*(C*(c*e)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(a*c*(C*(2*d*f - e^2*(2*p
+ q + 2)) + f*(-2*A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p +
q + 1)*(C*e*f*p*(-4*a*c)))*x + (p*(c*e)*(C*(c*e)*(q + 1) - c*(C*e)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(-
4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x] /; FreeQ[{a
, c, d, e, f, A, C, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] &
&  !IGtQ[p, 0] &&  !IGtQ[q, 0]

Rule 1094

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (f_.)*(x_)^2]), x_Sym
bol] :> Dist[C/c, Int[1/Sqrt[d + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)
*Sqrt[d + f*x^2]), x], x] /; FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {x^2 \left (a+c x^2\right )^{3/2}}{d+e x+f x^2} \, dx &=-\frac {(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}-\frac {\int \frac {\sqrt {a+c x^2} \left (3 a c d f-3 c e (4 c d-a f) x-3 c \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x^2\right )}{d+e x+f x^2} \, dx}{12 c f^2}\\ &=-\frac {\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt {a+c x^2}}{8 f^4}-\frac {(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac {\int \frac {-3 a c^2 d f \left (5 a f^2+4 c \left (e^2-d f\right )\right )-3 c^2 e \left (5 a^2 f^3+4 a c f \left (e^2-5 d f\right )-8 c^2 d \left (e^2-2 d f\right )\right ) x+3 c^2 \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) x^2}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{24 c^2 f^4}\\ &=-\frac {\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt {a+c x^2}}{8 f^4}-\frac {(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac {\int \frac {-3 a c^2 d f^2 \left (5 a f^2+4 c \left (e^2-d f\right )\right )-3 c^2 d \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )+\left (-3 c^2 e f \left (5 a^2 f^3+4 a c f \left (e^2-5 d f\right )-8 c^2 d \left (e^2-2 d f\right )\right )-3 c^2 e \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right )\right ) x}{\sqrt {a+c x^2} \left (d+e x+f x^2\right )} \, dx}{24 c^2 f^5}+\frac {\left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{8 f^5}\\ &=-\frac {\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt {a+c x^2}}{8 f^4}-\frac {(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac {\left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{8 f^5}+\frac {\left (a^2 f^4 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+2 a c f^2 \left (e^4-4 d e^2 f+2 d^2 f^2-e^3 \sqrt {e^2-4 d f}+2 d e f \sqrt {e^2-4 d f}\right )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3-e^5 \sqrt {e^2-4 d f}+4 d e^3 f \sqrt {e^2-4 d f}-3 d^2 e f^2 \sqrt {e^2-4 d f}\right )\right ) \int \frac {1}{\left (e-\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{f^5 \sqrt {e^2-4 d f}}-\frac {\left (a^2 f^4 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+2 a c f^2 \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3+e^5 \sqrt {e^2-4 d f}-4 d e^3 f \sqrt {e^2-4 d f}+3 d^2 e f^2 \sqrt {e^2-4 d f}\right )\right ) \int \frac {1}{\left (e+\sqrt {e^2-4 d f}+2 f x\right ) \sqrt {a+c x^2}} \, dx}{f^5 \sqrt {e^2-4 d f}}\\ &=-\frac {\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt {a+c x^2}}{8 f^4}-\frac {(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac {\left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c} f^5}-\frac {\left (a^2 f^4 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+2 a c f^2 \left (e^4-4 d e^2 f+2 d^2 f^2-e^3 \sqrt {e^2-4 d f}+2 d e f \sqrt {e^2-4 d f}\right )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3-e^5 \sqrt {e^2-4 d f}+4 d e^3 f \sqrt {e^2-4 d f}-3 d^2 e f^2 \sqrt {e^2-4 d f}\right )\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e-\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{f^5 \sqrt {e^2-4 d f}}+\frac {\left (a^2 f^4 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+2 a c f^2 \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3+e^5 \sqrt {e^2-4 d f}-4 d e^3 f \sqrt {e^2-4 d f}+3 d^2 e f^2 \sqrt {e^2-4 d f}\right )\right ) \text {Subst}\left (\int \frac {1}{4 a f^2+c \left (e+\sqrt {e^2-4 d f}\right )^2-x^2} \, dx,x,\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {a+c x^2}}\right )}{f^5 \sqrt {e^2-4 d f}}\\ &=-\frac {\left (8 e \left (a f^2+c \left (e^2-2 d f\right )\right )-f \left (3 a f^2+4 c \left (e^2-d f\right )\right ) x\right ) \sqrt {a+c x^2}}{8 f^4}-\frac {(4 e-3 f x) \left (a+c x^2\right )^{3/2}}{12 f^2}+\frac {\left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c} f^5}-\frac {\left (a^2 f^4 \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )+2 a c f^2 \left (e^4-4 d e^2 f+2 d^2 f^2-e^3 \sqrt {e^2-4 d f}+2 d e f \sqrt {e^2-4 d f}\right )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3-e^5 \sqrt {e^2-4 d f}+4 d e^3 f \sqrt {e^2-4 d f}-3 d^2 e f^2 \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^5 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (a^2 f^4 \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )+2 a c f^2 \left (e^4-4 d e^2 f+2 d^2 f^2+e^3 \sqrt {e^2-4 d f}-2 d e f \sqrt {e^2-4 d f}\right )+c^2 \left (e^6-6 d e^4 f+9 d^2 e^2 f^2-2 d^3 f^3+e^5 \sqrt {e^2-4 d f}-4 d e^3 f \sqrt {e^2-4 d f}+3 d^2 e f^2 \sqrt {e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} f^5 \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.01, size = 942, normalized size = 1.18 \begin {gather*} \frac {f \sqrt {a+c x^2} \left (a f^2 (-32 e+15 f x)-2 c \left (12 e^3-6 e^2 f x+4 e f \left (-6 d+f x^2\right )-3 f^2 x \left (-2 d+f x^2\right )\right )\right )-\frac {3 \left (3 a^2 f^4+12 a c f^2 \left (e^2-d f\right )+8 c^2 \left (e^4-3 d e^2 f+d^2 f^2\right )\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}+24 \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {a c^2 e^5 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-4 a c^2 d e^3 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+3 a c^2 d^2 e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+2 a^2 c e^3 f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-4 a^2 c d e f^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+a^3 e f^4 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+2 c^{5/2} d e^4 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-6 c^{5/2} d^2 e^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 c^{5/2} d^3 f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a c^{3/2} d e^2 f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a c^{3/2} d^2 f^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a^2 \sqrt {c} d f^4 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-c^2 e^5 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+4 c^2 d e^3 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-3 c^2 d^2 e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-2 a c e^3 f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+4 a c d e f^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 e f^4 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{24 f^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f*Sqrt[a + c*x^2]*(a*f^2*(-32*e + 15*f*x) - 2*c*(12*e^3 - 6*e^2*f*x + 4*e*f*(-6*d + f*x^2) - 3*f^2*x*(-2*d +
f*x^2))) - (3*(3*a^2*f^4 + 12*a*c*f^2*(e^2 - d*f) + 8*c^2*(e^4 - 3*d*e^2*f + d^2*f^2))*Log[-(Sqrt[c]*x) + Sqrt
[a + c*x^2]])/Sqrt[c] + 24*RootSum[a^2*f + 2*a*Sqrt[c]*e*#1 + 4*c*d*#1^2 - 2*a*f*#1^2 - 2*Sqrt[c]*e*#1^3 + f*#
1^4 & , (a*c^2*e^5*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1] - 4*a*c^2*d*e^3*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^
2] - #1] + 3*a*c^2*d^2*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1] + 2*a^2*c*e^3*f^2*Log[-(Sqrt[c]*x) + Sqr
t[a + c*x^2] - #1] - 4*a^2*c*d*e*f^3*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1] + a^3*e*f^4*Log[-(Sqrt[c]*x) + S
qrt[a + c*x^2] - #1] + 2*c^(5/2)*d*e^4*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 - 6*c^(5/2)*d^2*e^2*f*Log[-
(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 + 2*c^(5/2)*d^3*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 + 4*a*c
^(3/2)*d*e^2*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 - 4*a*c^(3/2)*d^2*f^3*Log[-(Sqrt[c]*x) + Sqrt[a +
c*x^2] - #1]*#1 + 2*a^2*Sqrt[c]*d*f^4*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 - c^2*e^5*Log[-(Sqrt[c]*x)
+ Sqrt[a + c*x^2] - #1]*#1^2 + 4*c^2*d*e^3*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2 - 3*c^2*d^2*e*f^2*L
og[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2 - 2*a*c*e^3*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2 + 4
*a*c*d*e*f^3*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2 - a^2*e*f^4*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1
]*#1^2)/(a*Sqrt[c]*e + 4*c*d*#1 - 2*a*f*#1 - 3*Sqrt[c]*e*#1^2 + 2*f*#1^3) & ])/(24*f^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2366\) vs. \(2(722)=1444\).
time = 0.15, size = 2367, normalized size = 2.98

method	result	size
default	\(\text {Expression too large to display}\)	\(2367\)
risch	\(\text {Expression too large to display}\)	\(9198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(1/4*x*(c*x^2+a)^(3/2)+3/4*a*(1/2*x*(c*x^2+a)^(1/2)+1/2*a/c^(1/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))))+1/2*(-e*
(-4*d*f+e^2)^(1/2)+2*d*f-e^2)/f^2/(-4*d*f+e^2)^(1/2)*(1/3*((x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e+(-4*d*f+e
^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(3/2)-1/
2*c*(e+(-4*d*f+e^2)^(1/2))/f*(1/4*(2*c*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)-c*(e+(-4*d*f+e^2)^(1/2))/f)/c*((x+1/2*
(e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*((-4*d*f+e^2)^(1
/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)+1/8*(2*c*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c^2*(e+(
-4*d*f+e^2)^(1/2))^2/f^2)/c^(3/2)*ln((-1/2*c*(e+(-4*d*f+e^2)^(1/2))/f+c*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f))/c^(1
/2)+((x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*((-4
*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)))+1/2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2
*(1/2*(4*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-4*c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*
((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)-1/2*c^(1/2)*(e+(-4*d*f+e^2)^(1/2))/f*ln((-1/2*c*(e+(
-4*d*f+e^2)^(1/2))/f+c*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f))/c^(1/2)+((x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e+(-
4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(
1/2))-1/2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2*2^(1/2)/(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+
c*e^2)/f^2)^(1/2)*ln((((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+
(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x+1/2*(e+(-4
*d*f+e^2)^(1/2))/f)^2*c-4*c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*((-4*d*f+e^2)^(1/2)*c*
e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2))/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f))))+1/2*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2))/
f^2/(-4*d*f+e^2)^(1/2)*(1/3*((x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4
*d*f+e^2)^(1/2)))+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(3/2)-1/2*c*(e-(-4*d*f+e^2)^(1/2))/
f*(1/4*(2*c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))-c*(e-(-4*d*f+e^2)^(1/2))/f)/c*((x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))
^2*c-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f
+c*e^2)/f^2)^(1/2)+1/8*(2*c*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c^2*(e-(-4*d*f+e^2)^(1/2))^2/f
^2)/c^(3/2)*ln((-1/2*c*(e-(-4*d*f+e^2)^(1/2))/f+c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2))))/c^(1/2)+((x-1/2/f*(-e+(-4
*d*f+e^2)^(1/2)))^2*c-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*(-(-4*d*f+e^2)^(1/2)*c*
e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)))+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2*(1/2*(4*(x-1/2/f
*(-e+(-4*d*f+e^2)^(1/2)))^2*c-4*c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(-(-4*d*f+e^2)^
(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)-1/2*c^(1/2)*(e-(-4*d*f+e^2)^(1/2))/f*ln((-1/2*c*(e-(-4*d*f+e^2)^(1
/2))/f+c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2))))/c^(1/2)+((x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c-c*(e-(-4*d*f+e^2)^(
1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2))-1/2*
(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2*2^(1/2)/((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f
^2)^(1/2)*ln(((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*
d*f+e^2)^(1/2)))+1/2*2^(1/2)*((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x-1/2/f*(-e+(-4*d
*f+e^2)^(1/2)))^2*c-4*c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(-(-4*d*f+e^2)^(1/2)*c*e+
2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2))/(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2))))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*d*f-%e^2>0)', see `assume?`
for more det

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(c*x**2+a)**(3/2)/(f*x**2+e*x+d),x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(a + c*x^2)^(3/2))/(d + e*x + f*x^2),x)

[Out]

int((x^2*(a + c*x^2)^(3/2))/(d + e*x + f*x^2), x)

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