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

Optimal. Leaf size=668 \[ \frac{\left (a^2 f \left (e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (\sqrt{e^2-4 d f}+e\right )+c^2 d^3 \left (e-\sqrt{e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\left (a^2 f \left (-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (e-\sqrt{e^2-4 d f}\right )+c^2 d^3 \left (\sqrt{e^2-4 d f}+e\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{a^{3/2} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^3}+\frac{a \sqrt{a+c x^2} \left (e^2-d f\right )}{d^3}-\frac{\sqrt{a+c x^2} \left (2 \left (a \left (e^2-d f\right )+c d^2\right )-c d e x\right )}{2 d^3}+\frac{e \left (a+c x^2\right )^{3/2}}{d^2 x}-\frac{3 c e x \sqrt{a+c x^2}}{2 d^2}-\frac{\left (a+c x^2\right )^{3/2}}{2 d x^2}+\frac{3 c \sqrt{a+c x^2}}{2 d}-\frac{3 \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 d} \]

[Out]

(3*c*Sqrt[a + c*x^2])/(2*d) + (a*(e^2 - d*f)*Sqrt[a + c*x^2])/d^3 - (3*c*e*x*Sqrt[a + c*x^2])/(2*d^2) - ((2*(c
*d^2 + a*(e^2 - d*f)) - c*d*e*x)*Sqrt[a + c*x^2])/(2*d^3) - (a + c*x^2)^(3/2)/(2*d*x^2) + (e*(a + c*x^2)^(3/2)
)/(d^2*x) + ((c^2*d^3*(e - Sqrt[e^2 - 4*d*f]) + 2*a*c*d^2*f*(e + Sqrt[e^2 - 4*d*f]) + a^2*f*(e^3 - 3*d*e*f + e
^2*Sqrt[e^2 - 4*d*f] - d*f*Sqrt[e^2 - 4*d*f]))*ArcTanh[(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])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 +
c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - ((2*a*c*d^2*f*(e - Sqrt[e^2 - 4*d*f]) + c^2*d^3*(e + Sqrt[e^2 - 4*d
*f]) + a^2*f*(e^3 - 3*d*e*f - e^2*Sqrt[e^2 - 4*d*f] + d*f*Sqrt[e^2 - 4*d*f]))*ArcTanh[(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])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^3*S
qrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) - (3*Sqrt[a]*c*ArcTanh[Sqrt[a + c*x^2]
/Sqrt[a]])/(2*d) - (a^(3/2)*(e^2 - d*f)*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^3

________________________________________________________________________________________

Rubi [A]  time = 3.46473, antiderivative size = 668, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 15, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {6728, 266, 47, 50, 63, 208, 277, 195, 217, 206, 1020, 1068, 1080, 1034, 725} \[ \frac{\left (a^2 f \left (e^2 \sqrt{e^2-4 d f}-d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (\sqrt{e^2-4 d f}+e\right )+c^2 d^3 \left (e-\sqrt{e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (e-\sqrt{e^2-4 d f}\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{\left (a^2 f \left (-e^2 \sqrt{e^2-4 d f}+d f \sqrt{e^2-4 d f}-3 d e f+e^3\right )+2 a c d^2 f \left (e-\sqrt{e^2-4 d f}\right )+c^2 d^3 \left (\sqrt{e^2-4 d f}+e\right )\right ) \tanh ^{-1}\left (\frac{2 a f-c x \left (\sqrt{e^2-4 d f}+e\right )}{\sqrt{2} \sqrt{a+c x^2} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt{2} d^3 \sqrt{e^2-4 d f} \sqrt{2 a f^2+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}-\frac{a^{3/2} \left (e^2-d f\right ) \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^3}+\frac{a \sqrt{a+c x^2} \left (e^2-d f\right )}{d^3}-\frac{\sqrt{a+c x^2} \left (2 \left (a \left (e^2-d f\right )+c d^2\right )-c d e x\right )}{2 d^3}+\frac{e \left (a+c x^2\right )^{3/2}}{d^2 x}-\frac{3 c e x \sqrt{a+c x^2}}{2 d^2}-\frac{\left (a+c x^2\right )^{3/2}}{2 d x^2}+\frac{3 c \sqrt{a+c x^2}}{2 d}-\frac{3 \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*c*Sqrt[a + c*x^2])/(2*d) + (a*(e^2 - d*f)*Sqrt[a + c*x^2])/d^3 - (3*c*e*x*Sqrt[a + c*x^2])/(2*d^2) - ((2*(c
*d^2 + a*(e^2 - d*f)) - c*d*e*x)*Sqrt[a + c*x^2])/(2*d^3) - (a + c*x^2)^(3/2)/(2*d*x^2) + (e*(a + c*x^2)^(3/2)
)/(d^2*x) + ((c^2*d^3*(e - Sqrt[e^2 - 4*d*f]) + 2*a*c*d^2*f*(e + Sqrt[e^2 - 4*d*f]) + a^2*f*(e^3 - 3*d*e*f + e
^2*Sqrt[e^2 - 4*d*f] - d*f*Sqrt[e^2 - 4*d*f]))*ArcTanh[(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])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^3*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 +
c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) - ((2*a*c*d^2*f*(e - Sqrt[e^2 - 4*d*f]) + c^2*d^3*(e + Sqrt[e^2 - 4*d
*f]) + a^2*f*(e^3 - 3*d*e*f - e^2*Sqrt[e^2 - 4*d*f] + d*f*Sqrt[e^2 - 4*d*f]))*ArcTanh[(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])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^3*S
qrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]) - (3*Sqrt[a]*c*ArcTanh[Sqrt[a + c*x^2]
/Sqrt[a]])/(2*d) - (a^(3/2)*(e^2 - d*f)*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^3

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

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 206

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

Rule 1020

Int[((g_.) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[(h*(a + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*f*(p + q + 1)), x] + Dist[1/(2*f*(p + q + 1)), Int[(a + c*x^2)
^(p - 1)*(d + e*x + f*x^2)^q*Simp[a*h*e*p - a*(h*e - 2*g*f)*(p + q + 1) - 2*h*p*(c*d - a*f)*x - (h*c*e*p + c*(
h*e - 2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, h, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[
p, 0] && NeQ[p + q + 1, 0]

Rule 1068

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 1080

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]

Rule 1034

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 725

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]

Rubi steps

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

Mathematica [C]  time = 3.54652, size = 904, normalized size = 1.35 \[ \frac{\frac{6 c d^2 \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{c x^2}{a}+1\right ) \left (c x^2+a\right )^{5/2}}{a^2}-5 \left (e^2-\frac{\left (e^2-3 d f\right ) e}{\sqrt{e^2-4 d f}}-d f\right ) \left (c x^2+a\right )^{3/2}-5 \left (e^2+\frac{\left (e^2-3 d f\right ) e}{\sqrt{e^2-4 d f}}-d f\right ) \left (c x^2+a\right )^{3/2}+\frac{30 a d e \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{c x^2}{a}\right ) \sqrt{c x^2+a}}{x \sqrt{\frac{c x^2}{a}+1}}+\frac{15 \left (-e^2-\frac{\left (e^2-3 d f\right ) e}{\sqrt{e^2-4 d f}}+d f\right ) \left (\frac{2 \sqrt{c} \left (\sqrt{e^2-4 d f}-e\right ) \sqrt{c x^2+a} \left (\sqrt{c} \sqrt{\frac{c x^2}{a}+1} x+\sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )\right )}{\sqrt{\frac{c x^2}{a}+1}}+\frac{2 \left (2 a f^2+c \left (e^2-\sqrt{e^2-4 d f} e-2 d f\right )\right ) \left (2 \sqrt{c x^2+a} f+\sqrt{c} \left (\sqrt{e^2-4 d f}-e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2+a}}\right )-\sqrt{2 c e^2-2 c \sqrt{e^2-4 d f} e+4 a f^2-4 c d f} \tanh ^{-1}\left (\frac{2 a f+c \left (\sqrt{e^2-4 d f}-e\right ) x}{\sqrt{4 a f^2-2 c \left (-e^2+\sqrt{e^2-4 d f} e+2 d f\right )} \sqrt{c x^2+a}}\right )\right )}{f^2}\right )}{8 f}-\frac{15 \left (-e^2+\frac{\left (e^2-3 d f\right ) e}{\sqrt{e^2-4 d f}}+d f\right ) \left (\frac{2 \sqrt{c} \left (e+\sqrt{e^2-4 d f}\right ) \sqrt{c x^2+a} \left (\sqrt{c} \sqrt{\frac{c x^2}{a}+1} x+\sqrt{a} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )\right )}{\sqrt{\frac{c x^2}{a}+1}}+\frac{2 \left (2 a f^2+c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )\right ) \left (-2 \sqrt{c x^2+a} f+\sqrt{c} \left (e+\sqrt{e^2-4 d f}\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2+a}}\right )+\sqrt{4 a f^2+2 c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )} \tanh ^{-1}\left (\frac{2 a f-c \left (e+\sqrt{e^2-4 d f}\right ) x}{\sqrt{4 a f^2+2 c \left (e^2+\sqrt{e^2-4 d f} e-2 d f\right )} \sqrt{c x^2+a}}\right )\right )}{f^2}\right )}{8 f}+10 \left (e^2-d f\right ) \left (\sqrt{c x^2+a} \left (c x^2+4 a\right )-3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c x^2+a}}{\sqrt{a}}\right )\right )}{30 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(e^2 - d*f - (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*(a + c*x^2)^(3/2) - 5*(e^2 - d*f + (e*(e^2 - 3*d*f))/Sqr
t[e^2 - 4*d*f])*(a + c*x^2)^(3/2) + (15*(-e^2 + d*f - (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*((2*Sqrt[c]*(-e + S
qrt[e^2 - 4*d*f])*Sqrt[a + c*x^2]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/Sqrt
[1 + (c*x^2)/a] + (2*(2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]))*(2*f*Sqrt[a + c*x^2] + Sqrt[c]*(-e + Sq
rt[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - Sqrt[2*c*e^2 - 4*c*d*f + 4*a*f^2 - 2*c*e*Sqrt[e^2 - 4*
d*f]]*ArcTanh[(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 - 2*c*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f])]
*Sqrt[a + c*x^2])]))/f^2))/(8*f) - (15*(-e^2 + d*f + (e*(e^2 - 3*d*f))/Sqrt[e^2 - 4*d*f])*((2*Sqrt[c]*(e + Sqr
t[e^2 - 4*d*f])*Sqrt[a + c*x^2]*(Sqrt[c]*x*Sqrt[1 + (c*x^2)/a] + Sqrt[a]*ArcSinh[(Sqrt[c]*x)/Sqrt[a]]))/Sqrt[1
+ (c*x^2)/a] + (2*(2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*(-2*f*Sqrt[a + c*x^2] + Sqrt[c]*(e + Sqrt
[e^2 - 4*d*f])*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] + Sqrt[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*
ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d*f])*x)/(Sqrt[4*a*f^2 + 2*c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt[a
+ c*x^2])]))/f^2))/(8*f) + 10*(e^2 - d*f)*(Sqrt[a + c*x^2]*(4*a + c*x^2) - 3*a^(3/2)*ArcTanh[Sqrt[a + c*x^2]/S
qrt[a]]) + (30*a*d*e*Sqrt[a + c*x^2]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((c*x^2)/a)])/(x*Sqrt[1 + (c*x^2)/a])
+ (6*c*d^2*(a + c*x^2)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (c*x^2)/a])/a^2)/(30*d^3)

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Maple [B]  time = 0.312, size = 10298, normalized size = 15.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}{{\left (f x^{2} + e x + d\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage2