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

Optimal. Leaf size=304 \[ -\frac{d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{2 a^2 c^3 \sqrt{c+d x^2} (b c-a d)^3}-\frac{d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{6 a^2 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 (b c-a d)^{7/2}}+\frac{(5 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3 c^{7/2}}-\frac{b (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \]

[Out]

-(d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(6*a^2*c^2*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - (b*(2*b*c - a*d))/(2*a^
2*c*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) - 1/(2*a*c*x^2*(a + b*x^2)*(c + d*x^2)^(3/2)) - (d*(2*b*c - a*d
)*(b^2*c^2 - a*b*c*d + 5*a^2*d^2))/(2*a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x^2]) + ((4*b*c + 5*a*d)*ArcTanh[Sqrt[c
+ d*x^2]/Sqrt[c]])/(2*a^3*c^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*
d]])/(2*a^3*(b*c - a*d)^(7/2))

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Rubi [A]  time = 0.484342, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {446, 103, 151, 152, 156, 63, 208} \[ -\frac{d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{2 a^2 c^3 \sqrt{c+d x^2} (b c-a d)^3}-\frac{d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{6 a^2 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 (b c-a d)^{7/2}}+\frac{(5 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3 c^{7/2}}-\frac{b (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(6*a^2*c^2*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - (b*(2*b*c - a*d))/(2*a^
2*c*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) - 1/(2*a*c*x^2*(a + b*x^2)*(c + d*x^2)^(3/2)) - (d*(2*b*c - a*d
)*(b^2*c^2 - a*b*c*d + 5*a^2*d^2))/(2*a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x^2]) + ((4*b*c + 5*a*d)*ArcTanh[Sqrt[c
+ d*x^2]/Sqrt[c]])/(2*a^3*c^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*
d]])/(2*a^3*(b*c - a*d)^(7/2))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 103

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx,x,x^2\right )\\ &=-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (4 b c+5 a d)+\frac{7 b d x}{2}}{x (a+b x)^2 (c+d x)^{5/2}} \, dx,x,x^2\right )}{2 a c}\\ &=-\frac{b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (b c-a d) (4 b c+5 a d)+\frac{5}{2} b d (2 b c-a d) x}{x (a+b x) (c+d x)^{5/2}} \, dx,x,x^2\right )}{2 a^2 c (b c-a d)}\\ &=-\frac{d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{6 a^2 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac{b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{4} (b c-a d)^2 (4 b c+5 a d)-\frac{3}{4} b d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{x (a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{3 a^2 c^2 (b c-a d)^2}\\ &=-\frac{d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{6 a^2 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac{b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{2 a^2 c^3 (b c-a d)^3 \sqrt{c+d x^2}}-\frac{2 \operatorname{Subst}\left (\int \frac{\frac{3}{8} (b c-a d)^3 (4 b c+5 a d)+\frac{3}{8} b d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right ) x}{x (a+b x) \sqrt{c+d x}} \, dx,x,x^2\right )}{3 a^2 c^3 (b c-a d)^3}\\ &=-\frac{d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{6 a^2 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac{b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{2 a^2 c^3 (b c-a d)^3 \sqrt{c+d x^2}}+\frac{\left (b^4 (4 b c-9 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^2\right )}{4 a^3 (b c-a d)^3}-\frac{(4 b c+5 a d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^2\right )}{4 a^3 c^3}\\ &=-\frac{d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{6 a^2 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac{b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{2 a^2 c^3 (b c-a d)^3 \sqrt{c+d x^2}}+\frac{\left (b^4 (4 b c-9 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{2 a^3 d (b c-a d)^3}-\frac{(4 b c+5 a d) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{2 a^3 c^3 d}\\ &=-\frac{d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{6 a^2 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac{b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac{d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{2 a^2 c^3 (b c-a d)^3 \sqrt{c+d x^2}}+\frac{(4 b c+5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3 c^{7/2}}-\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 (b c-a d)^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.11999, size = 190, normalized size = 0.62 \[ \frac{b^2 c^2 x^2 \left (a+b x^2\right ) (4 b c-9 a d) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b \left (d x^2+c\right )}{b c-a d}\right )-(b c-a d) \left (x^2 \left (a+b x^2\right ) \left (-5 a^2 d^2+a b c d+4 b^2 c^2\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{d x^2}{c}+1\right )-3 a c \left (a^2 d+a b \left (d x^2-c\right )-2 b^2 c x^2\right )\right )}{6 a^3 c^2 x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*c^2*(4*b*c - 9*a*d)*x^2*(a + b*x^2)*Hypergeometric2F1[-3/2, 1, -1/2, (b*(c + d*x^2))/(b*c - a*d)] - (b*c
- a*d)*(-3*a*c*(a^2*d - 2*b^2*c*x^2 + a*b*(-c + d*x^2)) + (4*b^2*c^2 + a*b*c*d - 5*a^2*d^2)*x^2*(a + b*x^2)*Hy
pergeometric2F1[-3/2, 1, -1/2, 1 + (d*x^2)/c]))/(6*a^3*c^2*(b*c - a*d)^2*x^2*(a + b*x^2)*(c + d*x^2)^(3/2))

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Maple [B]  time = 0.016, size = 2980, normalized size = 9.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/6*b^2/a/(-a*b)^(1/2)*d^2/(a*d-b*c)^2/c^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(
a*d-b*c)/b)^(1/2)*x-5/12*b^2/a/(-a*b)^(1/2)*d^2/(a*d-b*c)^2/c/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-
1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x-5/6*b^2/a/(-a*b)^(1/2)*d^2/(a*d-b*c)^2/c^2/((x-1/b*(-a*b)^(1/2))^2*d+2*
d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+5/4*b^3/a/(-a*b)^(1/2)*d^2/(a*d-b*c)^3/c/((x-1/b*(-
a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+1/3*b^2/a^2/(-a*b)^(1/2)*d/(a*d-b
*c)/c/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x+2/3*b^2/a^2/(-a*b
)^(1/2)*d/(a*d-b*c)/c^2/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x
-b^3/a^3/(a*d-b*c)^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-
b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a
*b)^(1/2)))-5/12*b^2/a^2*d/(a*d-b*c)^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-
b*c)/b)^(3/2)+5/4*b^3/a^2*d/(a*d-b*c)^3/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d
-b*c)/b)^(1/2)-b^3/a^3/(a*d-b*c)^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1
/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/
2))/(x+1/b*(-a*b)^(1/2)))-1/2/a^2/c/x^2/(d*x^2+c)^(3/2)-5/6/a^2*d/c^2/(d*x^2+c)^(3/2)-5/2/a^2*d/c^3/(d*x^2+c)^
(1/2)+5/2/a^2*d/c^(7/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-1/3*b^2/a^3/(a*d-b*c)/((x-1/b*(-a*b)^(1/2))^2*d+
2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+b^3/a^3/(a*d-b*c)^2/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(
-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-2/3*b/a^3/c/(d*x^2+c)^(3/2)-2*b/a^3/c^2/(d*x^2+c)^(1/2)+
2*b/a^3/c^(5/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-1/3*b^2/a^3/(a*d-b*c)/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*
b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+b^3/a^3/(a*d-b*c)^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1
/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1/4*b^2/a^2/(-a*b)^(1/2)/(a*d-b*c)/(x+1/b*(-a*b)^(1/2))/((x+1/b*
(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+b^2/a^3/(a*d-b*c)^2*(-a*b)^(1/2)/
c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d-5/12*b^2/a^2*d/(a*d
-b*c)^2/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+5/4*b^3/a^2*d/(a*
d-b*c)^3/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-5/4*b^3/a^2*d/(a
*d-b*c)^3/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/
2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))
+1/4*b^2/a^2/(-a*b)^(1/2)/(a*d-b*c)/(x-1/b*(-a*b)^(1/2))/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(
-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)-5/4*b^3/a^2*d/(a*d-b*c)^3/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(
1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^
(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))-b^2/a^3/(a*d-b*c)^2*(-a*b)^(1/2)/c/((x-1/b*(-a*b)^(1/2))^2*d+
2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d-1/3*b/a^3*d*(-a*b)^(1/2)/(a*d-b*c)/c/((x+1/b*(-
a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x-2/3*b/a^3*d*(-a*b)^(1/2)/(a*d-b*c
)/c^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+1/3*b/a^3*d*(-a*b
)^(1/2)/(a*d-b*c)/c/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x-5/4
*b^3/a/(-a*b)^(1/2)*d^2/(a*d-b*c)^3/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b
*c)/b)^(1/2)*x-1/3*b^2/a^2/(-a*b)^(1/2)*d/(a*d-b*c)/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*
b)^(1/2))-(a*d-b*c)/b)^(3/2)*x-2/3*b^2/a^2/(-a*b)^(1/2)*d/(a*d-b*c)/c^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(
1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+5/12*b^2/a/(-a*b)^(1/2)*d^2/(a*d-b*c)^2/c/((x+1/b*(-a*b)^(1/2
))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x+2/3*b/a^3*d*(-a*b)^(1/2)/(a*d-b*c)/c^2/((x
-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 69.8787, size = 8246, normalized size = 27.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*(3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^8 + (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^6 + (
4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^4 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*x^2)*sqrt(b/(b*c - a*d))*l
og((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(2*b^2*c^2 - 3*a*b*c*d +
a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 6*((4*
b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^8 + (8*b^5*c^5*d - 10*a*
b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6 + a*b^4*c^5*
d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^4 + (4*a*b^4*c^6 - 7*a^2*b^3*c
^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt
(c) + 2*c)/x^2) - 4*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 -
3*a^2*b^3*c^3*d^3 + 11*a^3*b^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^6 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^
2*c^3*d^3 + 13*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^4 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4
*b*c^3*d^3 - 20*a^5*c^2*d^4)*x^2)*sqrt(d*x^2 + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 -
a^6*b*c^4*d^5)*x^8 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*
x^6 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^4 + (a^4*b^3*c^9 -
3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^2), -1/24*(12*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^
3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^8 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^
3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d
^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^4 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*
d^3 - 5*a^5*c^2*d^4)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 + c)) - 3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^8
+ (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^6 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x
^4 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*x^2)*sqrt(b/(b*c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d
^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2
+ c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2
- 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3 + 11*a^3*b^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^6 + (12*a*b^4
*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 + 13*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^4 + (6*a*b^4*c^6 - 3*a^2
*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3 - 20*a^5*c^2*d^4)*x^2)*sqrt(d*x^2 + c))/((a^3*b^4*c^7*d^2 -
3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^8 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*
c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^6 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^
3 - 2*a^7*c^5*d^4)*x^4 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^2), 1/12*(3*((4*b^5
*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^8 + (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^6 + (4*b^5*c^7 - a*b^
4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^4 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*x^2)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*
x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b*c)) + 3*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3
- 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^8 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*
d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^
3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^4 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11
*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*(3*a^2*b^
3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3 + 11*a^3*b^
2*c^2*d^4 - 5*a^4*b*c*d^5)*x^6 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 + 13*a^4*b*c^2*d^4
- 15*a^5*c*d^5)*x^4 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3 - 20*a^5*c^2*d^4)*
x^2)*sqrt(d*x^2 + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^8 + (2*a^3*
b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^6 + (a^3*b^4*c^9 - a^4*b^3*
c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^4 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^
7*d^2 - a^7*c^6*d^3)*x^2), 1/12*(3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^8 + (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 -
9*a^2*b^3*c^4*d^3)*x^6 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^4 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*
x^2)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b
*c)) - 6*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^8 + (8*b^5*
c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6
+ a*b^4*c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^4 + (4*a*b^4*c^6
- 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d
*x^2 + c)) - 2*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2
*b^3*c^3*d^3 + 11*a^3*b^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^6 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3
*d^3 + 13*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^4 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^
3*d^3 - 20*a^5*c^2*d^4)*x^2)*sqrt(d*x^2 + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*
b*c^4*d^5)*x^8 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^6 +
(a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^4 + (a^4*b^3*c^9 - 3*a^
5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.21261, size = 684, normalized size = 2.25 \begin{align*} \frac{1}{6} \, d^{3}{\left (\frac{3 \,{\left (4 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{3} c^{3} d^{3} - 3 \, a^{4} b^{2} c^{2} d^{4} + 3 \, a^{5} b c d^{5} - a^{6} d^{6}\right )} \sqrt{-b^{2} c + a b d}} - \frac{3 \,{\left (2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{4} c^{3} - 2 \, \sqrt{d x^{2} + c} b^{4} c^{4} - 3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b^{3} c^{2} d + 4 \, \sqrt{d x^{2} + c} a b^{3} c^{3} d + 3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} b^{2} c d^{2} - 6 \, \sqrt{d x^{2} + c} a^{2} b^{2} c^{2} d^{2} -{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{3} b d^{3} + 4 \, \sqrt{d x^{2} + c} a^{3} b c d^{3} - \sqrt{d x^{2} + c} a^{4} d^{4}\right )}}{{\left (a^{2} b^{3} c^{6} d^{2} - 3 \, a^{3} b^{2} c^{5} d^{3} + 3 \, a^{4} b c^{4} d^{4} - a^{5} c^{3} d^{5}\right )}{\left ({\left (d x^{2} + c\right )}^{2} b - 2 \,{\left (d x^{2} + c\right )} b c + b c^{2} +{\left (d x^{2} + c\right )} a d - a c d\right )}} - \frac{2 \,{\left (12 \,{\left (d x^{2} + c\right )} b c + b c^{2} - 6 \,{\left (d x^{2} + c\right )} a d - a c d\right )}}{{\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}}} - \frac{3 \,{\left (4 \, b c + 5 \, a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c^{3} d^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*d^3*(3*(4*b^5*c - 9*a*b^4*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b^3*c^3*d^3 - 3*a^4*b^2*
c^2*d^4 + 3*a^5*b*c*d^5 - a^6*d^6)*sqrt(-b^2*c + a*b*d)) - 3*(2*(d*x^2 + c)^(3/2)*b^4*c^3 - 2*sqrt(d*x^2 + c)*
b^4*c^4 - 3*(d*x^2 + c)^(3/2)*a*b^3*c^2*d + 4*sqrt(d*x^2 + c)*a*b^3*c^3*d + 3*(d*x^2 + c)^(3/2)*a^2*b^2*c*d^2
- 6*sqrt(d*x^2 + c)*a^2*b^2*c^2*d^2 - (d*x^2 + c)^(3/2)*a^3*b*d^3 + 4*sqrt(d*x^2 + c)*a^3*b*c*d^3 - sqrt(d*x^2
+ c)*a^4*d^4)/((a^2*b^3*c^6*d^2 - 3*a^3*b^2*c^5*d^3 + 3*a^4*b*c^4*d^4 - a^5*c^3*d^5)*((d*x^2 + c)^2*b - 2*(d*
x^2 + c)*b*c + b*c^2 + (d*x^2 + c)*a*d - a*c*d)) - 2*(12*(d*x^2 + c)*b*c + b*c^2 - 6*(d*x^2 + c)*a*d - a*c*d)/
((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*(d*x^2 + c)^(3/2)) - 3*(4*b*c + 5*a*d)*arctan(sqrt(
d*x^2 + c)/sqrt(-c))/(a^3*sqrt(-c)*c^3*d^3))