\(\int \frac {1}{x^3 (a+b x^2) (c+d x^2)^{5/2}} \, dx\) [729]

Optimal result

Integrand size = 24, antiderivative size = 211 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=-\frac {d (3 b c-5 a d)}{6 a c^2 (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {1}{2 a c x^2 \left (c+d x^2\right )^{3/2}}-\frac {d \left (b^2 c^2-8 a b c d+5 a^2 d^2\right )}{2 a c^3 (b c-a d)^2 \sqrt {c+d x^2}}+\frac {(2 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2 c^{7/2}}-\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 (b c-a d)^{5/2}} \]

[Out]

-1/6*d*(-5*a*d+3*b*c)/a/c^2/(-a*d+b*c)/(d*x^2+c)^(3/2)-1/2/a/c/x^2/(d*x^2+c)^(3/2)+1/2*(5*a*d+2*b*c)*arctanh((
d*x^2+c)^(1/2)/c^(1/2))/a^2/c^(7/2)-b^(7/2)*arctanh(b^(1/2)*(d*x^2+c)^(1/2)/(-a*d+b*c)^(1/2))/a^2/(-a*d+b*c)^(
5/2)-1/2*d*(5*a^2*d^2-8*a*b*c*d+b^2*c^2)/a/c^3/(-a*d+b*c)^2/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 105, 157, 162, 65, 214} \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=-\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a^2 (b c-a d)^{5/2}}+\frac {(5 a d+2 b c) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2 c^{7/2}}-\frac {d \left (5 a^2 d^2-8 a b c d+b^2 c^2\right )}{2 a c^3 \sqrt {c+d x^2} (b c-a d)^2}-\frac {d (3 b c-5 a d)}{6 a c^2 \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {1}{2 a c x^2 \left (c+d x^2\right )^{3/2}} \]

[In]

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

[Out]

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

Rule 65

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 105

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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 157

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 162

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 214

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

Rule 457

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]]

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\frac {-\frac {a \left (3 b^2 c^2 \left (c+d x^2\right )^2-2 a b c d \left (3 c^2+16 c d x^2+12 d^2 x^4\right )+a^2 d^2 \left (3 c^2+20 c d x^2+15 d^2 x^4\right )\right )}{c^3 (b c-a d)^2 x^2 \left (c+d x^2\right )^{3/2}}+\frac {6 b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{5/2}}+\frac {3 (2 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{c^{7/2}}}{6 a^2} \]

[In]

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

[Out]

(-((a*(3*b^2*c^2*(c + d*x^2)^2 - 2*a*b*c*d*(3*c^2 + 16*c*d*x^2 + 12*d^2*x^4) + a^2*d^2*(3*c^2 + 20*c*d*x^2 + 1
5*d^2*x^4)))/(c^3*(b*c - a*d)^2*x^2*(c + d*x^2)^(3/2))) + (6*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b
*c) + a*d]])/(-(b*c) + a*d)^(5/2) + (3*(2*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/c^(7/2))/(6*a^2)

Maple [A] (verified)

Time = 3.18 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(d^{2} \left (\frac {b^{4} \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{2} a^{2} d^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {-5 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right ) a d \,x^{2}-2 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right ) b c \,x^{2}+\sqrt {d \,x^{2}+c}\, a \sqrt {c}}{2 x^{2} c^{\frac {7}{2}} a^{2} d^{2}}-\frac {1}{3 \left (a d -b c \right ) c^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}-\frac {2 a d -3 b c}{\left (a d -b c \right )^{2} c^{3} \sqrt {d \,x^{2}+c}}\right )\)	\(190\)
risch	\(\text {Expression too large to display}\)	\(1338\)
default	\(\text {Expression too large to display}\)	\(1548\)

[In]

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

[Out]

d^2*(1/(a*d-b*c)^2*b^4/a^2/d^2/((a*d-b*c)*b)^(1/2)*arctan(b*(d*x^2+c)^(1/2)/((a*d-b*c)*b)^(1/2))-1/2*(-5*arcta
nh((d*x^2+c)^(1/2)/c^(1/2))*a*d*x^2-2*arctanh((d*x^2+c)^(1/2)/c^(1/2))*b*c*x^2+(d*x^2+c)^(1/2)*a*c^(1/2))/x^2/
c^(7/2)/a^2/d^2-1/3/(a*d-b*c)/c^2/(d*x^2+c)^(3/2)-(2*a*d-3*b*c)/(a*d-b*c)^2/c^3/(d*x^2+c)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (181) = 362\).

Time = 1.45 (sec) , antiderivative size = 2219, normalized size of antiderivative = 10.52 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/12*(3*(b^3*c^4*d^2*x^6 + 2*b^3*c^5*d*x^4 + b^3*c^6*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)) + 3*((2*b^3*c^3*d^2 + a*b^2*c^2*d^3 -
8*a^2*b*c*d^4 + 5*a^3*d^5)*x^6 + 2*(2*b^3*c^4*d + a*b^2*c^3*d^2 - 8*a^2*b*c^2*d^3 + 5*a^3*c*d^4)*x^4 + (2*b^3*
c^5 + a*b^2*c^4*d - 8*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)*x^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(c) + 2*
c)/x^2) - 2*(3*a*b^2*c^5 - 6*a^2*b*c^4*d + 3*a^3*c^3*d^2 + 3*(a*b^2*c^3*d^2 - 8*a^2*b*c^2*d^3 + 5*a^3*c*d^4)*x
^4 + 2*(3*a*b^2*c^4*d - 16*a^2*b*c^3*d^2 + 10*a^3*c^2*d^3)*x^2)*sqrt(d*x^2 + c))/((a^2*b^2*c^6*d^2 - 2*a^3*b*c
^5*d^3 + a^4*c^4*d^4)*x^6 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a^4*c^5*d^3)*x^4 + (a^2*b^2*c^8 - 2*a^3*b*c^7
*d + a^4*c^6*d^2)*x^2), -1/12*(6*((2*b^3*c^3*d^2 + a*b^2*c^2*d^3 - 8*a^2*b*c*d^4 + 5*a^3*d^5)*x^6 + 2*(2*b^3*c
^4*d + a*b^2*c^3*d^2 - 8*a^2*b*c^2*d^3 + 5*a^3*c*d^4)*x^4 + (2*b^3*c^5 + a*b^2*c^4*d - 8*a^2*b*c^3*d^2 + 5*a^3
*c^2*d^3)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 + c)) - 3*(b^3*c^4*d^2*x^6 + 2*b^3*c^5*d*x^4 + b^3*c^6*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)) + 2*(3*a*b^2*c^5 - 6*a^2*b*c^4*d + 3*a^3*c^3*d^2 + 3*(a*b^2*c^3*d^2 - 8*a^2*b*c^2*d^3 + 5*a^3*c*
d^4)*x^4 + 2*(3*a*b^2*c^4*d - 16*a^2*b*c^3*d^2 + 10*a^3*c^2*d^3)*x^2)*sqrt(d*x^2 + c))/((a^2*b^2*c^6*d^2 - 2*a
^3*b*c^5*d^3 + a^4*c^4*d^4)*x^6 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a^4*c^5*d^3)*x^4 + (a^2*b^2*c^8 - 2*a^3
*b*c^7*d + a^4*c^6*d^2)*x^2), 1/12*(6*(b^3*c^4*d^2*x^6 + 2*b^3*c^5*d*x^4 + b^3*c^6*x^2)*sqrt(-b/(b*c - a*d))*a
rctan(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*((2*b^3*c^3*d^2 +
a*b^2*c^2*d^3 - 8*a^2*b*c*d^4 + 5*a^3*d^5)*x^6 + 2*(2*b^3*c^4*d + a*b^2*c^3*d^2 - 8*a^2*b*c^2*d^3 + 5*a^3*c*d^
4)*x^4 + (2*b^3*c^5 + a*b^2*c^4*d - 8*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)*x^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 +
c)*sqrt(c) + 2*c)/x^2) - 2*(3*a*b^2*c^5 - 6*a^2*b*c^4*d + 3*a^3*c^3*d^2 + 3*(a*b^2*c^3*d^2 - 8*a^2*b*c^2*d^3
+ 5*a^3*c*d^4)*x^4 + 2*(3*a*b^2*c^4*d - 16*a^2*b*c^3*d^2 + 10*a^3*c^2*d^3)*x^2)*sqrt(d*x^2 + c))/((a^2*b^2*c^6
*d^2 - 2*a^3*b*c^5*d^3 + a^4*c^4*d^4)*x^6 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a^4*c^5*d^3)*x^4 + (a^2*b^2*c
^8 - 2*a^3*b*c^7*d + a^4*c^6*d^2)*x^2), 1/6*(3*(b^3*c^4*d^2*x^6 + 2*b^3*c^5*d*x^4 + b^3*c^6*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*((2*b^3*c
^3*d^2 + a*b^2*c^2*d^3 - 8*a^2*b*c*d^4 + 5*a^3*d^5)*x^6 + 2*(2*b^3*c^4*d + a*b^2*c^3*d^2 - 8*a^2*b*c^2*d^3 + 5
*a^3*c*d^4)*x^4 + (2*b^3*c^5 + a*b^2*c^4*d - 8*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sq
rt(d*x^2 + c)) - (3*a*b^2*c^5 - 6*a^2*b*c^4*d + 3*a^3*c^3*d^2 + 3*(a*b^2*c^3*d^2 - 8*a^2*b*c^2*d^3 + 5*a^3*c*d
^4)*x^4 + 2*(3*a*b^2*c^4*d - 16*a^2*b*c^3*d^2 + 10*a^3*c^2*d^3)*x^2)*sqrt(d*x^2 + c))/((a^2*b^2*c^6*d^2 - 2*a^
3*b*c^5*d^3 + a^4*c^4*d^4)*x^6 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a^4*c^5*d^3)*x^4 + (a^2*b^2*c^8 - 2*a^3*
b*c^7*d + a^4*c^6*d^2)*x^2)]

Sympy [F]

\[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}} x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\frac {b^{4} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {9 \, {\left (d x^{2} + c\right )} b c d^{2} + b c^{2} d^{2} - 6 \, {\left (d x^{2} + c\right )} a d^{3} - a c d^{3}}{3 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} - \frac {{\left (2 \, b c + 5 \, a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{2} \sqrt {-c} c^{3}} - \frac {\sqrt {d x^{2} + c}}{2 \, a c^{3} x^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.14 (sec) , antiderivative size = 5409, normalized size of antiderivative = 25.64 \[ \int \frac {1}{x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan(((((c + d*x^2)^(1/2)*(128*a^3*b^15*c^21*d^2 - 704*a^4*b^14*c^20*d^3 + 1040*a^5*b^13*c^19*d^4 + 1440*a^6*
b^12*c^18*d^5 - 6000*a^7*b^11*c^17*d^6 + 2688*a^8*b^10*c^16*d^7 + 16864*a^9*b^9*c^15*d^8 - 41280*a^10*b^8*c^14
*d^9 + 48480*a^11*b^7*c^13*d^10 - 34240*a^12*b^6*c^12*d^11 + 14864*a^13*b^5*c^11*d^12 - 3680*a^14*b^4*c^10*d^1
3 + 400*a^15*b^3*c^9*d^14) + ((5*a*d + 2*b*c)*(64*a^6*b^13*c^23*d^3 + 64*a^7*b^12*c^22*d^4 - 3648*a^8*b^11*c^2
1*d^5 + 19520*a^9*b^10*c^20*d^6 - 53632*a^10*b^9*c^19*d^7 + 92288*a^11*b^8*c^18*d^8 - 106624*a^12*b^7*c^17*d^9
+ 84608*a^13*b^6*c^16*d^10 - 45760*a^14*b^5*c^15*d^11 + 16192*a^15*b^4*c^14*d^12 - 3392*a^16*b^3*c^13*d^13 +
320*a^17*b^2*c^12*d^14 - ((c + d*x^2)^(1/2)*(5*a*d + 2*b*c)*(512*a^7*b^13*c^26*d^2 - 5376*a^8*b^12*c^25*d^3 +
25600*a^9*b^11*c^24*d^4 - 72960*a^10*b^10*c^23*d^5 + 138240*a^11*b^9*c^22*d^6 - 182784*a^12*b^8*c^21*d^7 + 172
032*a^13*b^7*c^20*d^8 - 115200*a^14*b^6*c^19*d^9 + 53760*a^15*b^5*c^18*d^10 - 16640*a^16*b^4*c^17*d^11 + 3072*
a^17*b^3*c^16*d^12 - 256*a^18*b^2*c^15*d^13))/(4*a^2*(c^7)^(1/2))))/(4*a^2*(c^7)^(1/2)))*(5*a*d + 2*b*c)*1i)/(
4*a^2*(c^7)^(1/2)) + (((c + d*x^2)^(1/2)*(128*a^3*b^15*c^21*d^2 - 704*a^4*b^14*c^20*d^3 + 1040*a^5*b^13*c^19*d
^4 + 1440*a^6*b^12*c^18*d^5 - 6000*a^7*b^11*c^17*d^6 + 2688*a^8*b^10*c^16*d^7 + 16864*a^9*b^9*c^15*d^8 - 41280
*a^10*b^8*c^14*d^9 + 48480*a^11*b^7*c^13*d^10 - 34240*a^12*b^6*c^12*d^11 + 14864*a^13*b^5*c^11*d^12 - 3680*a^1
4*b^4*c^10*d^13 + 400*a^15*b^3*c^9*d^14) - ((5*a*d + 2*b*c)*(64*a^6*b^13*c^23*d^3 + 64*a^7*b^12*c^22*d^4 - 364
8*a^8*b^11*c^21*d^5 + 19520*a^9*b^10*c^20*d^6 - 53632*a^10*b^9*c^19*d^7 + 92288*a^11*b^8*c^18*d^8 - 106624*a^1
2*b^7*c^17*d^9 + 84608*a^13*b^6*c^16*d^10 - 45760*a^14*b^5*c^15*d^11 + 16192*a^15*b^4*c^14*d^12 - 3392*a^16*b^
3*c^13*d^13 + 320*a^17*b^2*c^12*d^14 + ((c + d*x^2)^(1/2)*(5*a*d + 2*b*c)*(512*a^7*b^13*c^26*d^2 - 5376*a^8*b^
12*c^25*d^3 + 25600*a^9*b^11*c^24*d^4 - 72960*a^10*b^10*c^23*d^5 + 138240*a^11*b^9*c^22*d^6 - 182784*a^12*b^8*
c^21*d^7 + 172032*a^13*b^7*c^20*d^8 - 115200*a^14*b^6*c^19*d^9 + 53760*a^15*b^5*c^18*d^10 - 16640*a^16*b^4*c^1
7*d^11 + 3072*a^17*b^3*c^16*d^12 - 256*a^18*b^2*c^15*d^13))/(4*a^2*(c^7)^(1/2))))/(4*a^2*(c^7)^(1/2)))*(5*a*d
+ 2*b*c)*1i)/(4*a^2*(c^7)^(1/2)))/(32*a^2*b^15*c^18*d^3 - 368*a^3*b^14*c^17*d^4 + 1056*a^4*b^13*c^16*d^5 - 560
0*a^6*b^11*c^14*d^7 + 12768*a^7*b^10*c^13*d^8 - 14112*a^8*b^9*c^12*d^9 + 8704*a^9*b^8*c^11*d^10 - 2880*a^10*b^
7*c^10*d^11 + 400*a^11*b^6*c^9*d^12 - (((c + d*x^2)^(1/2)*(128*a^3*b^15*c^21*d^2 - 704*a^4*b^14*c^20*d^3 + 104
0*a^5*b^13*c^19*d^4 + 1440*a^6*b^12*c^18*d^5 - 6000*a^7*b^11*c^17*d^6 + 2688*a^8*b^10*c^16*d^7 + 16864*a^9*b^9
*c^15*d^8 - 41280*a^10*b^8*c^14*d^9 + 48480*a^11*b^7*c^13*d^10 - 34240*a^12*b^6*c^12*d^11 + 14864*a^13*b^5*c^1
1*d^12 - 3680*a^14*b^4*c^10*d^13 + 400*a^15*b^3*c^9*d^14) + ((5*a*d + 2*b*c)*(64*a^6*b^13*c^23*d^3 + 64*a^7*b^
12*c^22*d^4 - 3648*a^8*b^11*c^21*d^5 + 19520*a^9*b^10*c^20*d^6 - 53632*a^10*b^9*c^19*d^7 + 92288*a^11*b^8*c^18
*d^8 - 106624*a^12*b^7*c^17*d^9 + 84608*a^13*b^6*c^16*d^10 - 45760*a^14*b^5*c^15*d^11 + 16192*a^15*b^4*c^14*d^
12 - 3392*a^16*b^3*c^13*d^13 + 320*a^17*b^2*c^12*d^14 - ((c + d*x^2)^(1/2)*(5*a*d + 2*b*c)*(512*a^7*b^13*c^26*
d^2 - 5376*a^8*b^12*c^25*d^3 + 25600*a^9*b^11*c^24*d^4 - 72960*a^10*b^10*c^23*d^5 + 138240*a^11*b^9*c^22*d^6 -
182784*a^12*b^8*c^21*d^7 + 172032*a^13*b^7*c^20*d^8 - 115200*a^14*b^6*c^19*d^9 + 53760*a^15*b^5*c^18*d^10 - 1
6640*a^16*b^4*c^17*d^11 + 3072*a^17*b^3*c^16*d^12 - 256*a^18*b^2*c^15*d^13))/(4*a^2*(c^7)^(1/2))))/(4*a^2*(c^7
)^(1/2)))*(5*a*d + 2*b*c))/(4*a^2*(c^7)^(1/2)) + (((c + d*x^2)^(1/2)*(128*a^3*b^15*c^21*d^2 - 704*a^4*b^14*c^2
0*d^3 + 1040*a^5*b^13*c^19*d^4 + 1440*a^6*b^12*c^18*d^5 - 6000*a^7*b^11*c^17*d^6 + 2688*a^8*b^10*c^16*d^7 + 16
864*a^9*b^9*c^15*d^8 - 41280*a^10*b^8*c^14*d^9 + 48480*a^11*b^7*c^13*d^10 - 34240*a^12*b^6*c^12*d^11 + 14864*a
^13*b^5*c^11*d^12 - 3680*a^14*b^4*c^10*d^13 + 400*a^15*b^3*c^9*d^14) - ((5*a*d + 2*b*c)*(64*a^6*b^13*c^23*d^3
+ 64*a^7*b^12*c^22*d^4 - 3648*a^8*b^11*c^21*d^5 + 19520*a^9*b^10*c^20*d^6 - 53632*a^10*b^9*c^19*d^7 + 92288*a^
11*b^8*c^18*d^8 - 106624*a^12*b^7*c^17*d^9 + 84608*a^13*b^6*c^16*d^10 - 45760*a^14*b^5*c^15*d^11 + 16192*a^15*
b^4*c^14*d^12 - 3392*a^16*b^3*c^13*d^13 + 320*a^17*b^2*c^12*d^14 + ((c + d*x^2)^(1/2)*(5*a*d + 2*b*c)*(512*a^7
*b^13*c^26*d^2 - 5376*a^8*b^12*c^25*d^3 + 25600*a^9*b^11*c^24*d^4 - 72960*a^10*b^10*c^23*d^5 + 138240*a^11*b^9
*c^22*d^6 - 182784*a^12*b^8*c^21*d^7 + 172032*a^13*b^7*c^20*d^8 - 115200*a^14*b^6*c^19*d^9 + 53760*a^15*b^5*c^
18*d^10 - 16640*a^16*b^4*c^17*d^11 + 3072*a^17*b^3*c^16*d^12 - 256*a^18*b^2*c^15*d^13))/(4*a^2*(c^7)^(1/2))))/
(4*a^2*(c^7)^(1/2)))*(5*a*d + 2*b*c))/(4*a^2*(c^7)^(1/2))))*(5*a*d + 2*b*c)*1i)/(2*a^2*(c^7)^(1/2)) - ((d^2*(c
+ d*x^2)*(5*a*d - 8*b*c))/(3*(b*c^2 - a*c*d)^2) - d^2/(3*(b*c^2 - a*c*d)) + (d*(c + d*x^2)^2*(5*a^2*d^2 + b^2
*c^2 - 8*a*b*c*d))/(2*a*c^2*(b*c^2 - a*c*d)*(a*d - b*c)))/(c*(c + d*x^2)^(3/2) - (c + d*x^2)^(5/2)) + (atan(((
(-b^7*(a*d - b*c)^5)^(1/2)*(((c + d*x^2)^(1/2)*(128*a^3*b^15*c^21*d^2 - 704*a^4*b^14*c^20*d^3 + 1040*a^5*b^13*
c^19*d^4 + 1440*a^6*b^12*c^18*d^5 - 6000*a^7*b^11*c^17*d^6 + 2688*a^8*b^10*c^16*d^7 + 16864*a^9*b^9*c^15*d^8 -
41280*a^10*b^8*c^14*d^9 + 48480*a^11*b^7*c^13*d^10 - 34240*a^12*b^6*c^12*d^11 + 14864*a^13*b^5*c^11*d^12 - 36
80*a^14*b^4*c^10*d^13 + 400*a^15*b^3*c^9*d^14))/2 + ((-b^7*(a*d - b*c)^5)^(1/2)*(32*a^6*b^13*c^23*d^3 + 32*a^7
*b^12*c^22*d^4 - 1824*a^8*b^11*c^21*d^5 + 9760*a^9*b^10*c^20*d^6 - 26816*a^10*b^9*c^19*d^7 + 46144*a^11*b^8*c^
18*d^8 - 53312*a^12*b^7*c^17*d^9 + 42304*a^13*b^6*c^16*d^10 - 22880*a^14*b^5*c^15*d^11 + 8096*a^15*b^4*c^14*d^
12 - 1696*a^16*b^3*c^13*d^13 + 160*a^17*b^2*c^12*d^14 - ((-b^7*(a*d - b*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(512*a^7
*b^13*c^26*d^2 - 5376*a^8*b^12*c^25*d^3 + 25600*a^9*b^11*c^24*d^4 - 72960*a^10*b^10*c^23*d^5 + 138240*a^11*b^9
*c^22*d^6 - 182784*a^12*b^8*c^21*d^7 + 172032*a^13*b^7*c^20*d^8 - 115200*a^14*b^6*c^19*d^9 + 53760*a^15*b^5*c^
18*d^10 - 16640*a^16*b^4*c^17*d^11 + 3072*a^17*b^3*c^16*d^12 - 256*a^18*b^2*c^15*d^13))/(4*a^2*(a*d - b*c)^5))
)/(2*a^2*(a*d - b*c)^5))*1i)/(a^2*(a*d - b*c)^5) + ((-b^7*(a*d - b*c)^5)^(1/2)*(((c + d*x^2)^(1/2)*(128*a^3*b^
15*c^21*d^2 - 704*a^4*b^14*c^20*d^3 + 1040*a^5*b^13*c^19*d^4 + 1440*a^6*b^12*c^18*d^5 - 6000*a^7*b^11*c^17*d^6
+ 2688*a^8*b^10*c^16*d^7 + 16864*a^9*b^9*c^15*d^8 - 41280*a^10*b^8*c^14*d^9 + 48480*a^11*b^7*c^13*d^10 - 3424
0*a^12*b^6*c^12*d^11 + 14864*a^13*b^5*c^11*d^12 - 3680*a^14*b^4*c^10*d^13 + 400*a^15*b^3*c^9*d^14))/2 - ((-b^7
*(a*d - b*c)^5)^(1/2)*(32*a^6*b^13*c^23*d^3 + 32*a^7*b^12*c^22*d^4 - 1824*a^8*b^11*c^21*d^5 + 9760*a^9*b^10*c^
20*d^6 - 26816*a^10*b^9*c^19*d^7 + 46144*a^11*b^8*c^18*d^8 - 53312*a^12*b^7*c^17*d^9 + 42304*a^13*b^6*c^16*d^1
0 - 22880*a^14*b^5*c^15*d^11 + 8096*a^15*b^4*c^14*d^12 - 1696*a^16*b^3*c^13*d^13 + 160*a^17*b^2*c^12*d^14 + ((
-b^7*(a*d - b*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(512*a^7*b^13*c^26*d^2 - 5376*a^8*b^12*c^25*d^3 + 25600*a^9*b^11*c
^24*d^4 - 72960*a^10*b^10*c^23*d^5 + 138240*a^11*b^9*c^22*d^6 - 182784*a^12*b^8*c^21*d^7 + 172032*a^13*b^7*c^2
0*d^8 - 115200*a^14*b^6*c^19*d^9 + 53760*a^15*b^5*c^18*d^10 - 16640*a^16*b^4*c^17*d^11 + 3072*a^17*b^3*c^16*d^
12 - 256*a^18*b^2*c^15*d^13))/(4*a^2*(a*d - b*c)^5)))/(2*a^2*(a*d - b*c)^5))*1i)/(a^2*(a*d - b*c)^5))/(((-b^7*
(a*d - b*c)^5)^(1/2)*(((c + d*x^2)^(1/2)*(128*a^3*b^15*c^21*d^2 - 704*a^4*b^14*c^20*d^3 + 1040*a^5*b^13*c^19*d
^4 + 1440*a^6*b^12*c^18*d^5 - 6000*a^7*b^11*c^17*d^6 + 2688*a^8*b^10*c^16*d^7 + 16864*a^9*b^9*c^15*d^8 - 41280
*a^10*b^8*c^14*d^9 + 48480*a^11*b^7*c^13*d^10 - 34240*a^12*b^6*c^12*d^11 + 14864*a^13*b^5*c^11*d^12 - 3680*a^1
4*b^4*c^10*d^13 + 400*a^15*b^3*c^9*d^14))/2 - ((-b^7*(a*d - b*c)^5)^(1/2)*(32*a^6*b^13*c^23*d^3 + 32*a^7*b^12*
c^22*d^4 - 1824*a^8*b^11*c^21*d^5 + 9760*a^9*b^10*c^20*d^6 - 26816*a^10*b^9*c^19*d^7 + 46144*a^11*b^8*c^18*d^8
- 53312*a^12*b^7*c^17*d^9 + 42304*a^13*b^6*c^16*d^10 - 22880*a^14*b^5*c^15*d^11 + 8096*a^15*b^4*c^14*d^12 - 1
696*a^16*b^3*c^13*d^13 + 160*a^17*b^2*c^12*d^14 + ((-b^7*(a*d - b*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(512*a^7*b^13*
c^26*d^2 - 5376*a^8*b^12*c^25*d^3 + 25600*a^9*b^11*c^24*d^4 - 72960*a^10*b^10*c^23*d^5 + 138240*a^11*b^9*c^22*
d^6 - 182784*a^12*b^8*c^21*d^7 + 172032*a^13*b^7*c^20*d^8 - 115200*a^14*b^6*c^19*d^9 + 53760*a^15*b^5*c^18*d^1
0 - 16640*a^16*b^4*c^17*d^11 + 3072*a^17*b^3*c^16*d^12 - 256*a^18*b^2*c^15*d^13))/(4*a^2*(a*d - b*c)^5)))/(2*a
^2*(a*d - b*c)^5)))/(a^2*(a*d - b*c)^5) - ((-b^7*(a*d - b*c)^5)^(1/2)*(((c + d*x^2)^(1/2)*(128*a^3*b^15*c^21*d
^2 - 704*a^4*b^14*c^20*d^3 + 1040*a^5*b^13*c^19*d^4 + 1440*a^6*b^12*c^18*d^5 - 6000*a^7*b^11*c^17*d^6 + 2688*a
^8*b^10*c^16*d^7 + 16864*a^9*b^9*c^15*d^8 - 41280*a^10*b^8*c^14*d^9 + 48480*a^11*b^7*c^13*d^10 - 34240*a^12*b^
6*c^12*d^11 + 14864*a^13*b^5*c^11*d^12 - 3680*a^14*b^4*c^10*d^13 + 400*a^15*b^3*c^9*d^14))/2 + ((-b^7*(a*d - b
*c)^5)^(1/2)*(32*a^6*b^13*c^23*d^3 + 32*a^7*b^12*c^22*d^4 - 1824*a^8*b^11*c^21*d^5 + 9760*a^9*b^10*c^20*d^6 -
26816*a^10*b^9*c^19*d^7 + 46144*a^11*b^8*c^18*d^8 - 53312*a^12*b^7*c^17*d^9 + 42304*a^13*b^6*c^16*d^10 - 22880
*a^14*b^5*c^15*d^11 + 8096*a^15*b^4*c^14*d^12 - 1696*a^16*b^3*c^13*d^13 + 160*a^17*b^2*c^12*d^14 - ((-b^7*(a*d
- b*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(512*a^7*b^13*c^26*d^2 - 5376*a^8*b^12*c^25*d^3 + 25600*a^9*b^11*c^24*d^4 -
72960*a^10*b^10*c^23*d^5 + 138240*a^11*b^9*c^22*d^6 - 182784*a^12*b^8*c^21*d^7 + 172032*a^13*b^7*c^20*d^8 - 1
15200*a^14*b^6*c^19*d^9 + 53760*a^15*b^5*c^18*d^10 - 16640*a^16*b^4*c^17*d^11 + 3072*a^17*b^3*c^16*d^12 - 256*
a^18*b^2*c^15*d^13))/(4*a^2*(a*d - b*c)^5)))/(2*a^2*(a*d - b*c)^5)))/(a^2*(a*d - b*c)^5) + 32*a^2*b^15*c^18*d^
3 - 368*a^3*b^14*c^17*d^4 + 1056*a^4*b^13*c^16*d^5 - 5600*a^6*b^11*c^14*d^7 + 12768*a^7*b^10*c^13*d^8 - 14112*
a^8*b^9*c^12*d^9 + 8704*a^9*b^8*c^11*d^10 - 2880*a^10*b^7*c^10*d^11 + 400*a^11*b^6*c^9*d^12))*(-b^7*(a*d - b*c
)^5)^(1/2)*1i)/(a^2*(a*d - b*c)^5)