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

Optimal. Leaf size=225 \[ \frac {b^{5/2} (2 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{7/2}}+\frac {d \left (-2 a^2 d^2+6 a b c d+b^2 c^2\right )}{2 a c^2 \sqrt {c+d x^2} (b c-a d)^3}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 c^{5/2}}+\frac {b}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {d (2 a d+3 b c)}{6 a c \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

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Rubi [A]  time = 0.33, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 103, 152, 156, 63, 208} \begin {gather*} \frac {d \left (-2 a^2 d^2+6 a b c d+b^2 c^2\right )}{2 a c^2 \sqrt {c+d x^2} (b c-a d)^3}+\frac {b^{5/2} (2 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{7/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 c^{5/2}}+\frac {b}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {d (2 a d+3 b c)}{6 a c \left (c+d x^2\right )^{3/2} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 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]]

Rubi steps

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

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Mathematica [C]  time = 0.12, size = 114, normalized size = 0.51 \begin {gather*} \frac {-\frac {b (2 b c-7 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)^2}+\frac {3 a b}{\left (a+b x^2\right ) (b c-a d)}+\frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {d x^2}{c}+1\right )}{c}}{6 a^2 \left (c+d x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a*b)/((b*c - a*d)*(a + b*x^2)) - (b*(2*b*c - 7*a*d)*Hypergeometric2F1[-3/2, 1, -1/2, (b*(c + d*x^2))/(b*c
- a*d)])/(b*c - a*d)^2 + (2*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (d*x^2)/c])/c)/(6*a^2*(c + d*x^2)^(3/2))

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IntegrateAlgebraic [A]  time = 0.65, size = 276, normalized size = 1.23 \begin {gather*} \frac {\left (2 b^{7/2} c-7 a b^{5/2} d\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2} \sqrt {a d-b c}}{b c-a d}\right )}{2 a^2 (b c-a d)^3 \sqrt {a d-b c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 c^{5/2}}+\frac {8 a^3 c d^3+6 a^3 d^4 x^2-20 a^2 b c^2 d^2-10 a^2 b c d^3 x^2+6 a^2 b d^4 x^4-20 a b^2 c^2 d^2 x^2-18 a b^2 c d^3 x^4-3 b^3 c^4-6 b^3 c^3 d x^2-3 b^3 c^2 d^2 x^4}{6 a c^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (a d-b c)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b^3*c^4 - 20*a^2*b*c^2*d^2 + 8*a^3*c*d^3 - 6*b^3*c^3*d*x^2 - 20*a*b^2*c^2*d^2*x^2 - 10*a^2*b*c*d^3*x^2 + 6
*a^3*d^4*x^2 - 3*b^3*c^2*d^2*x^4 - 18*a*b^2*c*d^3*x^4 + 6*a^2*b*d^4*x^4)/(6*a*c^2*(-(b*c) + a*d)^3*(a + b*x^2)
*(c + d*x^2)^(3/2)) + ((2*b^(7/2)*c - 7*a*b^(5/2)*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c
- a*d)])/(2*a^2*(b*c - a*d)^3*Sqrt[-(b*c) + a*d]) - ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]/(a^2*c^(5/2))

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fricas [B]  time = 15.72, size = 3403, normalized size = 15.12

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*(3*(2*a*b^3*c^6 - 7*a^2*b^2*c^5*d + (2*b^4*c^4*d^2 - 7*a*b^3*c^3*d^3)*x^6 + (4*b^4*c^5*d - 12*a*b^3*c^4*
d^2 - 7*a^2*b^2*c^3*d^3)*x^4 + (2*b^4*c^6 - 3*a*b^3*c^5*d - 14*a^2*b^2*c^4*d^2)*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)) + 12*(a*b^3*
c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3
*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b
^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqr
t(c) + 2*c)/x^2) + 4*(3*a*b^3*c^5 + 20*a^3*b*c^3*d^2 - 8*a^4*c^2*d^3 + 3*(a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 -
2*a^3*b*c*d^4)*x^4 + 2*(3*a*b^3*c^4*d + 10*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 3*a^4*c*d^4)*x^2)*sqrt(d*x^2 +
c))/(a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3 + (a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*
a^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^6 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^
4 - a^6*c^3*d^5)*x^4 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2
), 1/24*(24*(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*
a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x
^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x^2)*sqrt(-c)*arctan(sqrt(-c)
/sqrt(d*x^2 + c)) + 3*(2*a*b^3*c^6 - 7*a^2*b^2*c^5*d + (2*b^4*c^4*d^2 - 7*a*b^3*c^3*d^3)*x^6 + (4*b^4*c^5*d -
12*a*b^3*c^4*d^2 - 7*a^2*b^2*c^3*d^3)*x^4 + (2*b^4*c^6 - 3*a*b^3*c^5*d - 14*a^2*b^2*c^4*d^2)*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*b^3*c^5 + 20*a^3*b*c^3*d^2 - 8*a^4*c^2*d^3 + 3*(a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 2*a^3*b*c*d^4)*x
^4 + 2*(3*a*b^3*c^4*d + 10*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 3*a^4*c*d^4)*x^2)*sqrt(d*x^2 + c))/(a^3*b^3*c^8
- 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3 + (a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4
- a^5*b*c^3*d^5)*x^6 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5)
*x^4 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2), -1/12*(3*(2*a
*b^3*c^6 - 7*a^2*b^2*c^5*d + (2*b^4*c^4*d^2 - 7*a*b^3*c^3*d^3)*x^6 + (4*b^4*c^5*d - 12*a*b^3*c^4*d^2 - 7*a^2*b
^2*c^3*d^3)*x^4 + (2*b^4*c^6 - 3*a*b^3*c^5*d - 14*a^2*b^2*c^4*d^2)*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*(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a
^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d
- 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^
2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*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*b
^3*c^5 + 20*a^3*b*c^3*d^2 - 8*a^4*c^2*d^3 + 3*(a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 2*a^3*b*c*d^4)*x^4 + 2*(3*a
*b^3*c^4*d + 10*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 3*a^4*c*d^4)*x^2)*sqrt(d*x^2 + c))/(a^3*b^3*c^8 - 3*a^4*b^
2*c^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3 + (a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3
*d^5)*x^6 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5)*x^4 + (a^2
*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2), -1/12*(3*(2*a*b^3*c^6 -
7*a^2*b^2*c^5*d + (2*b^4*c^4*d^2 - 7*a*b^3*c^3*d^3)*x^6 + (4*b^4*c^5*d - 12*a*b^3*c^4*d^2 - 7*a^2*b^2*c^3*d^3)
*x^4 + (2*b^4*c^6 - 3*a*b^3*c^5*d - 14*a^2*b^2*c^4*d^2)*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)) - 12*(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d
^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*
c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*
b*c^2*d^3 - 2*a^4*c*d^4)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 + c)) - 2*(3*a*b^3*c^5 + 20*a^3*b*c^3*d^2 -
8*a^4*c^2*d^3 + 3*(a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 2*a^3*b*c*d^4)*x^4 + 2*(3*a*b^3*c^4*d + 10*a^2*b^2*c^3*
d^2 + 5*a^3*b*c^2*d^3 - 3*a^4*c*d^4)*x^2)*sqrt(d*x^2 + c))/(a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d^2 -
a^6*c^5*d^3 + (a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4 - a^5*b*c^3*d^5)*x^6 + (2*a^2*b^4*c^7*d
- 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^6*c^3*d^5)*x^4 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3
*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^2)]

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giac [A]  time = 0.42, size = 298, normalized size = 1.32 \begin {gather*} \frac {\sqrt {d x^{2} + c} b^{3} d}{2 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} - \frac {{\left (2 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\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} - 3 \, {\left (d x^{2} + c\right )} a d^{3} - a c d^{3}}{3 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {\arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(d*x^2 + c)*b^3*d/((a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*((d*x^2 + c)*b - b*c + a*d)
) - 1/2*(2*b^4*c - 7*a*b^3*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/((a^2*b^3*c^3 - 3*a^3*b^2*c^2*d +
3*a^4*b*c*d^2 - a^5*d^3)*sqrt(-b^2*c + a*b*d)) + 1/3*(9*(d*x^2 + c)*b*c*d^2 + b*c^2*d^2 - 3*(d*x^2 + c)*a*d^3
- a*c*d^3)/((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*(d*x^2 + c)^(3/2)) + arctan(sqrt(d*x^2
+ c)/sqrt(-c))/(a^2*sqrt(-c)*c^2)

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maple [B]  time = 0.02, size = 2837, normalized size = 12.61 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(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), x)

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mupad [B]  time = 5.14, size = 8467, normalized size = 37.63

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d^2*(c + d*x^2)*(3*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*(b^3*c^2
- 2*a^2*b*d^2 + 6*a*b^2*c*d))/(2*a*c*(b*c^2 - a*c*d)*(a*d - b*c)^2))/(b*(c + d*x^2)^(5/2) + (c + d*x^2)^(3/2)
*(a*d - b*c)) - atanh((560*a^3*b^16*c^19*d^4*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4
*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^
11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b
^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*
d^17 + 64*a^17*b^2*c^3*d^18)) - (7280*a^4*b^15*c^18*d^5*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4
- 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 59
3440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 27
8768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^
16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (42560*a^5*b^14*c^17*d^6*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b
^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c
^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c
^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^
16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) - (149184*a^6*b^13*c^16*d^7*(c + d*x^2)^(1/2))/((c^5)^(1/
2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 3519
04*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 5050
08*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^
15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (351904*a^7*b^12*c^15*d^8*(c + d*x^2)^(1/2
))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^
14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^1
0*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d
^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) - (593440*a^8*b^11*c^14*d^9*(c
+ d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 14918
4*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840
*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a
^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (741120*a^9*b^10
*c^13*d^10*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^
15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11
*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*
d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) - (6
99840*a^10*b^9*c^12*d^11*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 425
60*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120
*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480
*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c
^3*d^18)) + (505008*a^11*b^8*c^11*d^12*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*
c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^1
2*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8
*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 +
64*a^17*b^2*c^3*d^18)) - (278768*a^12*b^7*c^10*d^13*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 -
7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 59344
0*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 27876
8*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*
b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (116480*a^13*b^6*c^9*d^14*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^1
6*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^1
3*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9
*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16
- 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) - (35840*a^14*b^5*c^8*d^15*(c + d*x^2)^(1/2))/((c^5)^(1/2)*
(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^7 + 351904*
a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^11 + 505008*
a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 + 7680*a^15*
b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (7680*a^15*b^4*c^7*d^16*(c + d*x^2)^(1/2))/((
c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b^13*c^14*d^
7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b^9*c^10*d^1
1 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5*c^6*d^15 +
7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) - (1024*a^16*b^3*c^6*d^17*(c + d*x^2
)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149184*a^6*b
^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 699840*a^10*b
^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840*a^14*b^5
*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)) + (64*a^17*b^2*c^5*d^18*(
c + d*x^2)^(1/2))/((c^5)^(1/2)*(560*a^3*b^16*c^17*d^4 - 7280*a^4*b^15*c^16*d^5 + 42560*a^5*b^14*c^15*d^6 - 149
184*a^6*b^13*c^14*d^7 + 351904*a^7*b^12*c^13*d^8 - 593440*a^8*b^11*c^12*d^9 + 741120*a^9*b^10*c^11*d^10 - 6998
40*a^10*b^9*c^10*d^11 + 505008*a^11*b^8*c^9*d^12 - 278768*a^12*b^7*c^8*d^13 + 116480*a^13*b^6*c^7*d^14 - 35840
*a^14*b^5*c^6*d^15 + 7680*a^15*b^4*c^5*d^16 - 1024*a^16*b^3*c^4*d^17 + 64*a^17*b^2*c^3*d^18)))/(a^2*(c^5)^(1/2
)) + (atan((((-b^5*(a*d - b*c)^7)^(1/2)*((c + d*x^2)^(1/2)*(128*a^3*b^18*c^21*d^2 - 1984*a^4*b^17*c^20*d^3 + 1
3840*a^5*b^16*c^19*d^4 - 57680*a^6*b^15*c^18*d^5 + 161280*a^7*b^14*c^17*d^6 - 322560*a^8*b^13*c^16*d^7 + 48092
8*a^9*b^12*c^15*d^8 - 550560*a^10*b^11*c^14*d^9 + 494400*a^11*b^10*c^13*d^10 - 352640*a^12*b^9*c^12*d^11 + 199
696*a^13*b^8*c^11*d^12 - 88144*a^14*b^7*c^10*d^13 + 29120*a^15*b^6*c^9*d^14 - 6720*a^16*b^5*c^8*d^15 + 960*a^1
7*b^4*c^7*d^16 - 64*a^18*b^3*c^6*d^17) - ((-b^5*(a*d - b*c)^7)^(1/2)*(7*a*d - 2*b*c)*(1536*a^7*b^16*c^22*d^4 -
64*a^6*b^17*c^23*d^3 - 13952*a^8*b^15*c^21*d^5 + 71040*a^9*b^14*c^20*d^6 - 235968*a^10*b^13*c^19*d^7 + 551936
*a^11*b^12*c^18*d^8 - 948992*a^12*b^11*c^17*d^9 + 1229184*a^13*b^10*c^16*d^10 - 1214400*a^14*b^9*c^15*d^11 + 9
18016*a^15*b^8*c^14*d^12 - 528000*a^16*b^7*c^13*d^13 + 227456*a^17*b^6*c^12*d^14 - 71232*a^18*b^5*c^11*d^15 +
15360*a^19*b^4*c^10*d^16 - 2048*a^20*b^3*c^9*d^17 + 128*a^21*b^2*c^8*d^18 + ((-b^5*(a*d - b*c)^7)^(1/2)*(c + d
*x^2)^(1/2)*(7*a*d - 2*b*c)*(512*a^7*b^18*c^26*d^2 - 7936*a^8*b^17*c^25*d^3 + 57600*a^9*b^16*c^24*d^4 - 259840
*a^10*b^15*c^23*d^5 + 815360*a^11*b^14*c^22*d^6 - 1886976*a^12*b^13*c^21*d^7 + 3331328*a^13*b^12*c^20*d^8 - 45
76000*a^14*b^11*c^19*d^9 + 4942080*a^15*b^10*c^18*d^10 - 4209920*a^16*b^9*c^17*d^11 + 2818816*a^17*b^8*c^16*d^
12 - 1467648*a^18*b^7*c^15*d^13 + 582400*a^19*b^6*c^14*d^14 - 170240*a^20*b^5*c^13*d^15 + 34560*a^21*b^4*c^12*
d^16 - 4352*a^22*b^3*c^11*d^17 + 256*a^23*b^2*c^10*d^18))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4
*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6))))/(4*(a^9*d^7 -
a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*
d^5 - 7*a^8*b*c*d^6)))*(7*a*d - 2*b*c)*1i)/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 +
35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6)) + ((-b^5*(a*d - b*c)^7)^(1/2)*(
(c + d*x^2)^(1/2)*(128*a^3*b^18*c^21*d^2 - 1984*a^4*b^17*c^20*d^3 + 13840*a^5*b^16*c^19*d^4 - 57680*a^6*b^15*c
^18*d^5 + 161280*a^7*b^14*c^17*d^6 - 322560*a^8*b^13*c^16*d^7 + 480928*a^9*b^12*c^15*d^8 - 550560*a^10*b^11*c^
14*d^9 + 494400*a^11*b^10*c^13*d^10 - 352640*a^12*b^9*c^12*d^11 + 199696*a^13*b^8*c^11*d^12 - 88144*a^14*b^7*c
^10*d^13 + 29120*a^15*b^6*c^9*d^14 - 6720*a^16*b^5*c^8*d^15 + 960*a^17*b^4*c^7*d^16 - 64*a^18*b^3*c^6*d^17) -
((-b^5*(a*d - b*c)^7)^(1/2)*(7*a*d - 2*b*c)*(64*a^6*b^17*c^23*d^3 - 1536*a^7*b^16*c^22*d^4 + 13952*a^8*b^15*c^
21*d^5 - 71040*a^9*b^14*c^20*d^6 + 235968*a^10*b^13*c^19*d^7 - 551936*a^11*b^12*c^18*d^8 + 948992*a^12*b^11*c^
17*d^9 - 1229184*a^13*b^10*c^16*d^10 + 1214400*a^14*b^9*c^15*d^11 - 918016*a^15*b^8*c^14*d^12 + 528000*a^16*b^
7*c^13*d^13 - 227456*a^17*b^6*c^12*d^14 + 71232*a^18*b^5*c^11*d^15 - 15360*a^19*b^4*c^10*d^16 + 2048*a^20*b^3*
c^9*d^17 - 128*a^21*b^2*c^8*d^18 + ((-b^5*(a*d - b*c)^7)^(1/2)*(c + d*x^2)^(1/2)*(7*a*d - 2*b*c)*(512*a^7*b^18
*c^26*d^2 - 7936*a^8*b^17*c^25*d^3 + 57600*a^9*b^16*c^24*d^4 - 259840*a^10*b^15*c^23*d^5 + 815360*a^11*b^14*c^
22*d^6 - 1886976*a^12*b^13*c^21*d^7 + 3331328*a^13*b^12*c^20*d^8 - 4576000*a^14*b^11*c^19*d^9 + 4942080*a^15*b
^10*c^18*d^10 - 4209920*a^16*b^9*c^17*d^11 + 2818816*a^17*b^8*c^16*d^12 - 1467648*a^18*b^7*c^15*d^13 + 582400*
a^19*b^6*c^14*d^14 - 170240*a^20*b^5*c^13*d^15 + 34560*a^21*b^4*c^12*d^16 - 4352*a^22*b^3*c^11*d^17 + 256*a^23
*b^2*c^10*d^18))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^
6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6))))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^
5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6)))*(7*a*d - 2*b*c)*1i
)/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 +
21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6)))/(208*a^3*b^16*c^17*d^4 - 32*a^2*b^17*c^18*d^3 + 304*a^4*b^15*c^16*d^5 -
7040*a^5*b^14*c^15*d^6 + 31200*a^6*b^13*c^14*d^7 - 75936*a^7*b^12*c^13*d^8 + 118944*a^8*b^11*c^12*d^9 - 12652
8*a^9*b^10*c^11*d^10 + 92640*a^10*b^9*c^10*d^11 - 46000*a^11*b^8*c^9*d^12 + 14768*a^12*b^7*c^8*d^13 - 2752*a^1
3*b^6*c^7*d^14 + 224*a^14*b^5*c^6*d^15 + ((-b^5*(a*d - b*c)^7)^(1/2)*((c + d*x^2)^(1/2)*(128*a^3*b^18*c^21*d^2
- 1984*a^4*b^17*c^20*d^3 + 13840*a^5*b^16*c^19*d^4 - 57680*a^6*b^15*c^18*d^5 + 161280*a^7*b^14*c^17*d^6 - 322
560*a^8*b^13*c^16*d^7 + 480928*a^9*b^12*c^15*d^8 - 550560*a^10*b^11*c^14*d^9 + 494400*a^11*b^10*c^13*d^10 - 35
2640*a^12*b^9*c^12*d^11 + 199696*a^13*b^8*c^11*d^12 - 88144*a^14*b^7*c^10*d^13 + 29120*a^15*b^6*c^9*d^14 - 672
0*a^16*b^5*c^8*d^15 + 960*a^17*b^4*c^7*d^16 - 64*a^18*b^3*c^6*d^17) - ((-b^5*(a*d - b*c)^7)^(1/2)*(7*a*d - 2*b
*c)*(1536*a^7*b^16*c^22*d^4 - 64*a^6*b^17*c^23*d^3 - 13952*a^8*b^15*c^21*d^5 + 71040*a^9*b^14*c^20*d^6 - 23596
8*a^10*b^13*c^19*d^7 + 551936*a^11*b^12*c^18*d^8 - 948992*a^12*b^11*c^17*d^9 + 1229184*a^13*b^10*c^16*d^10 - 1
214400*a^14*b^9*c^15*d^11 + 918016*a^15*b^8*c^14*d^12 - 528000*a^16*b^7*c^13*d^13 + 227456*a^17*b^6*c^12*d^14
- 71232*a^18*b^5*c^11*d^15 + 15360*a^19*b^4*c^10*d^16 - 2048*a^20*b^3*c^9*d^17 + 128*a^21*b^2*c^8*d^18 + ((-b^
5*(a*d - b*c)^7)^(1/2)*(c + d*x^2)^(1/2)*(7*a*d - 2*b*c)*(512*a^7*b^18*c^26*d^2 - 7936*a^8*b^17*c^25*d^3 + 576
00*a^9*b^16*c^24*d^4 - 259840*a^10*b^15*c^23*d^5 + 815360*a^11*b^14*c^22*d^6 - 1886976*a^12*b^13*c^21*d^7 + 33
31328*a^13*b^12*c^20*d^8 - 4576000*a^14*b^11*c^19*d^9 + 4942080*a^15*b^10*c^18*d^10 - 4209920*a^16*b^9*c^17*d^
11 + 2818816*a^17*b^8*c^16*d^12 - 1467648*a^18*b^7*c^15*d^13 + 582400*a^19*b^6*c^14*d^14 - 170240*a^20*b^5*c^1
3*d^15 + 34560*a^21*b^4*c^12*d^16 - 4352*a^22*b^3*c^11*d^17 + 256*a^23*b^2*c^10*d^18))/(4*(a^9*d^7 - a^2*b^7*c
^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a
^8*b*c*d^6))))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*
b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6)))*(7*a*d - 2*b*c))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6
*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6)) - ((-
b^5*(a*d - b*c)^7)^(1/2)*((c + d*x^2)^(1/2)*(128*a^3*b^18*c^21*d^2 - 1984*a^4*b^17*c^20*d^3 + 13840*a^5*b^16*c
^19*d^4 - 57680*a^6*b^15*c^18*d^5 + 161280*a^7*b^14*c^17*d^6 - 322560*a^8*b^13*c^16*d^7 + 480928*a^9*b^12*c^15
*d^8 - 550560*a^10*b^11*c^14*d^9 + 494400*a^11*b^10*c^13*d^10 - 352640*a^12*b^9*c^12*d^11 + 199696*a^13*b^8*c^
11*d^12 - 88144*a^14*b^7*c^10*d^13 + 29120*a^15*b^6*c^9*d^14 - 6720*a^16*b^5*c^8*d^15 + 960*a^17*b^4*c^7*d^16
- 64*a^18*b^3*c^6*d^17) - ((-b^5*(a*d - b*c)^7)^(1/2)*(7*a*d - 2*b*c)*(64*a^6*b^17*c^23*d^3 - 1536*a^7*b^16*c^
22*d^4 + 13952*a^8*b^15*c^21*d^5 - 71040*a^9*b^14*c^20*d^6 + 235968*a^10*b^13*c^19*d^7 - 551936*a^11*b^12*c^18
*d^8 + 948992*a^12*b^11*c^17*d^9 - 1229184*a^13*b^10*c^16*d^10 + 1214400*a^14*b^9*c^15*d^11 - 918016*a^15*b^8*
c^14*d^12 + 528000*a^16*b^7*c^13*d^13 - 227456*a^17*b^6*c^12*d^14 + 71232*a^18*b^5*c^11*d^15 - 15360*a^19*b^4*
c^10*d^16 + 2048*a^20*b^3*c^9*d^17 - 128*a^21*b^2*c^8*d^18 + ((-b^5*(a*d - b*c)^7)^(1/2)*(c + d*x^2)^(1/2)*(7*
a*d - 2*b*c)*(512*a^7*b^18*c^26*d^2 - 7936*a^8*b^17*c^25*d^3 + 57600*a^9*b^16*c^24*d^4 - 259840*a^10*b^15*c^23
*d^5 + 815360*a^11*b^14*c^22*d^6 - 1886976*a^12*b^13*c^21*d^7 + 3331328*a^13*b^12*c^20*d^8 - 4576000*a^14*b^11
*c^19*d^9 + 4942080*a^15*b^10*c^18*d^10 - 4209920*a^16*b^9*c^17*d^11 + 2818816*a^17*b^8*c^16*d^12 - 1467648*a^
18*b^7*c^15*d^13 + 582400*a^19*b^6*c^14*d^14 - 170240*a^20*b^5*c^13*d^15 + 34560*a^21*b^4*c^12*d^16 - 4352*a^2
2*b^3*c^11*d^17 + 256*a^23*b^2*c^10*d^18))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 +
35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6))))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7
*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c
*d^6)))*(7*a*d - 2*b*c))/(4*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3
- 35*a^6*b^3*c^3*d^4 + 21*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6))))*(-b^5*(a*d - b*c)^7)^(1/2)*(7*a*d - 2*b*c)*1i)/
(2*(a^9*d^7 - a^2*b^7*c^7 + 7*a^3*b^6*c^6*d - 21*a^4*b^5*c^5*d^2 + 35*a^5*b^4*c^4*d^3 - 35*a^6*b^3*c^3*d^4 + 2
1*a^7*b^2*c^2*d^5 - 7*a^8*b*c*d^6))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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