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

Optimal. Leaf size=145 \[ \frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a (b c-a d)^{5/2}}-\frac {d (2 b c-a d)}{c^2 \sqrt {c+d x^2} (b c-a d)^2}-\frac {d}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a c^{5/2}} \]

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Rubi [A]  time = 0.18, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 85, 152, 156, 63, 208} \begin {gather*} \frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{a (b c-a d)^{5/2}}-\frac {d (2 b c-a d)}{c^2 \sqrt {c+d x^2} (b c-a d)^2}-\frac {d}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 85

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

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

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Mathematica [C]  time = 0.03, size = 90, normalized size = 0.62 \begin {gather*} \frac {(b c-a d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {d x^2}{c}+1\right )-b c \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b \left (d x^2+c\right )}{b c-a d}\right )}{3 a c \left (c+d x^2\right )^{3/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.33, size = 152, normalized size = 1.05 \begin {gather*} \frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2} \sqrt {a d-b c}}{b c-a d}\right )}{a (a d-b c)^{5/2}}+\frac {4 a c d^2+3 a d^3 x^2-7 b c^2 d-6 b c d^2 x^2}{3 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 3.20, size = 1711, normalized size = 11.80

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.34, size = 176, normalized size = 1.21 \begin {gather*} -\frac {b^{3} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {-b^{2} c + a b d}} - \frac {6 \, {\left (d x^{2} + c\right )} b c d + b c^{2} d - 3 \, {\left (d x^{2} + c\right )} a d^{2} - a c d^{2}}{3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {\arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a \sqrt {-c} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.02, size = 1186, normalized size = 8.18 \begin {gather*} \frac {b^{2} \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \left (a d -b c \right )^{2} \sqrt {-\frac {a d -b c}{b}}\, a}+\frac {b^{2} \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \left (a d -b c \right )^{2} \sqrt {-\frac {a d -b c}{b}}\, a}-\frac {b^{2}}{2 \left (a d -b c \right )^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a}-\frac {b^{2}}{2 \left (a d -b c \right )^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a}-\frac {\sqrt {-a b}\, b d x}{2 \left (a d -b c \right )^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a c}+\frac {\sqrt {-a b}\, b d x}{2 \left (a d -b c \right )^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a c}+\frac {b}{6 \left (a d -b c \right ) \left (\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}} a}+\frac {b}{6 \left (a d -b c \right ) \left (\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}} a}+\frac {\sqrt {-a b}\, d x}{6 \left (a d -b c \right ) \left (\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}} a c}-\frac {\sqrt {-a b}\, d x}{6 \left (a d -b c \right ) \left (\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}} a c}+\frac {\sqrt {-a b}\, d x}{3 \left (a d -b c \right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a \,c^{2}}-\frac {\sqrt {-a b}\, d x}{3 \left (a d -b c \right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a \,c^{2}}+\frac {1}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a c}-\frac {\ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{a \,c^{\frac {5}{2}}}+\frac {1}{\sqrt {d \,x^{2}+c}\, a \,c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} {\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)/(d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.48, size = 4558, normalized size = 31.43

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((-b^5*(a*d - b*c)^5)^(1/2)*(((c + d*x^2)^(1/2)*(4*b^13*c^16*d^2 - 32*a*b^12*c^15*d^3 + 120*a^2*b^11*c^
14*d^4 - 280*a^3*b^10*c^13*d^5 + 450*a^4*b^9*c^12*d^6 - 516*a^5*b^8*c^11*d^7 + 422*a^6*b^7*c^10*d^8 - 240*a^7*
b^6*c^9*d^9 + 90*a^8*b^5*c^8*d^10 - 20*a^9*b^4*c^7*d^11 + 2*a^10*b^3*c^6*d^12))/2 + ((-b^5*(a*d - b*c)^5)^(1/2
)*(6*a^2*b^12*c^18*d^3 - 54*a^3*b^11*c^17*d^4 + 218*a^4*b^10*c^16*d^5 - 520*a^5*b^9*c^15*d^6 + 812*a^6*b^8*c^1
4*d^7 - 868*a^7*b^7*c^13*d^8 + 644*a^8*b^6*c^12*d^9 - 328*a^9*b^5*c^11*d^10 + 110*a^10*b^4*c^10*d^11 - 22*a^11
*b^3*c^9*d^12 + 2*a^12*b^2*c^8*d^13 - ((-b^5*(a*d - b*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(16*a^2*b^13*c^21*d^2 - 16
8*a^3*b^12*c^20*d^3 + 800*a^4*b^11*c^19*d^4 - 2280*a^5*b^10*c^18*d^5 + 4320*a^6*b^9*c^17*d^6 - 5712*a^7*b^8*c^
16*d^7 + 5376*a^8*b^7*c^15*d^8 - 3600*a^9*b^6*c^14*d^9 + 1680*a^10*b^5*c^13*d^10 - 520*a^11*b^4*c^12*d^11 + 96
*a^12*b^3*c^11*d^12 - 8*a^13*b^2*c^10*d^13))/(4*a*(a*d - b*c)^5)))/(2*a*(a*d - b*c)^5))*1i)/(a*(a*d - b*c)^5)
+ ((-b^5*(a*d - b*c)^5)^(1/2)*(((c + d*x^2)^(1/2)*(4*b^13*c^16*d^2 - 32*a*b^12*c^15*d^3 + 120*a^2*b^11*c^14*d^
4 - 280*a^3*b^10*c^13*d^5 + 450*a^4*b^9*c^12*d^6 - 516*a^5*b^8*c^11*d^7 + 422*a^6*b^7*c^10*d^8 - 240*a^7*b^6*c
^9*d^9 + 90*a^8*b^5*c^8*d^10 - 20*a^9*b^4*c^7*d^11 + 2*a^10*b^3*c^6*d^12))/2 - ((-b^5*(a*d - b*c)^5)^(1/2)*(6*
a^2*b^12*c^18*d^3 - 54*a^3*b^11*c^17*d^4 + 218*a^4*b^10*c^16*d^5 - 520*a^5*b^9*c^15*d^6 + 812*a^6*b^8*c^14*d^7
- 868*a^7*b^7*c^13*d^8 + 644*a^8*b^6*c^12*d^9 - 328*a^9*b^5*c^11*d^10 + 110*a^10*b^4*c^10*d^11 - 22*a^11*b^3*
c^9*d^12 + 2*a^12*b^2*c^8*d^13 + ((-b^5*(a*d - b*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(16*a^2*b^13*c^21*d^2 - 168*a^3
*b^12*c^20*d^3 + 800*a^4*b^11*c^19*d^4 - 2280*a^5*b^10*c^18*d^5 + 4320*a^6*b^9*c^17*d^6 - 5712*a^7*b^8*c^16*d^
7 + 5376*a^8*b^7*c^15*d^8 - 3600*a^9*b^6*c^14*d^9 + 1680*a^10*b^5*c^13*d^10 - 520*a^11*b^4*c^12*d^11 + 96*a^12
*b^3*c^11*d^12 - 8*a^13*b^2*c^10*d^13))/(4*a*(a*d - b*c)^5)))/(2*a*(a*d - b*c)^5))*1i)/(a*(a*d - b*c)^5))/(4*b
^12*c^13*d^3 - 26*a*b^11*c^12*d^4 + 72*a^2*b^10*c^11*d^5 - 110*a^3*b^9*c^10*d^6 + 100*a^4*b^8*c^9*d^7 - 54*a^5
*b^7*c^8*d^8 + 16*a^6*b^6*c^7*d^9 - 2*a^7*b^5*c^6*d^10 + ((-b^5*(a*d - b*c)^5)^(1/2)*(((c + d*x^2)^(1/2)*(4*b^
13*c^16*d^2 - 32*a*b^12*c^15*d^3 + 120*a^2*b^11*c^14*d^4 - 280*a^3*b^10*c^13*d^5 + 450*a^4*b^9*c^12*d^6 - 516*
a^5*b^8*c^11*d^7 + 422*a^6*b^7*c^10*d^8 - 240*a^7*b^6*c^9*d^9 + 90*a^8*b^5*c^8*d^10 - 20*a^9*b^4*c^7*d^11 + 2*
a^10*b^3*c^6*d^12))/2 + ((-b^5*(a*d - b*c)^5)^(1/2)*(6*a^2*b^12*c^18*d^3 - 54*a^3*b^11*c^17*d^4 + 218*a^4*b^10
*c^16*d^5 - 520*a^5*b^9*c^15*d^6 + 812*a^6*b^8*c^14*d^7 - 868*a^7*b^7*c^13*d^8 + 644*a^8*b^6*c^12*d^9 - 328*a^
9*b^5*c^11*d^10 + 110*a^10*b^4*c^10*d^11 - 22*a^11*b^3*c^9*d^12 + 2*a^12*b^2*c^8*d^13 - ((-b^5*(a*d - b*c)^5)^
(1/2)*(c + d*x^2)^(1/2)*(16*a^2*b^13*c^21*d^2 - 168*a^3*b^12*c^20*d^3 + 800*a^4*b^11*c^19*d^4 - 2280*a^5*b^10*
c^18*d^5 + 4320*a^6*b^9*c^17*d^6 - 5712*a^7*b^8*c^16*d^7 + 5376*a^8*b^7*c^15*d^8 - 3600*a^9*b^6*c^14*d^9 + 168
0*a^10*b^5*c^13*d^10 - 520*a^11*b^4*c^12*d^11 + 96*a^12*b^3*c^11*d^12 - 8*a^13*b^2*c^10*d^13))/(4*a*(a*d - b*c
)^5)))/(2*a*(a*d - b*c)^5)))/(a*(a*d - b*c)^5) - ((-b^5*(a*d - b*c)^5)^(1/2)*(((c + d*x^2)^(1/2)*(4*b^13*c^16*
d^2 - 32*a*b^12*c^15*d^3 + 120*a^2*b^11*c^14*d^4 - 280*a^3*b^10*c^13*d^5 + 450*a^4*b^9*c^12*d^6 - 516*a^5*b^8*
c^11*d^7 + 422*a^6*b^7*c^10*d^8 - 240*a^7*b^6*c^9*d^9 + 90*a^8*b^5*c^8*d^10 - 20*a^9*b^4*c^7*d^11 + 2*a^10*b^3
*c^6*d^12))/2 - ((-b^5*(a*d - b*c)^5)^(1/2)*(6*a^2*b^12*c^18*d^3 - 54*a^3*b^11*c^17*d^4 + 218*a^4*b^10*c^16*d^
5 - 520*a^5*b^9*c^15*d^6 + 812*a^6*b^8*c^14*d^7 - 868*a^7*b^7*c^13*d^8 + 644*a^8*b^6*c^12*d^9 - 328*a^9*b^5*c^
11*d^10 + 110*a^10*b^4*c^10*d^11 - 22*a^11*b^3*c^9*d^12 + 2*a^12*b^2*c^8*d^13 + ((-b^5*(a*d - b*c)^5)^(1/2)*(c
+ d*x^2)^(1/2)*(16*a^2*b^13*c^21*d^2 - 168*a^3*b^12*c^20*d^3 + 800*a^4*b^11*c^19*d^4 - 2280*a^5*b^10*c^18*d^5
+ 4320*a^6*b^9*c^17*d^6 - 5712*a^7*b^8*c^16*d^7 + 5376*a^8*b^7*c^15*d^8 - 3600*a^9*b^6*c^14*d^9 + 1680*a^10*b
^5*c^13*d^10 - 520*a^11*b^4*c^12*d^11 + 96*a^12*b^3*c^11*d^12 - 8*a^13*b^2*c^10*d^13))/(4*a*(a*d - b*c)^5)))/(
2*a*(a*d - b*c)^5)))/(a*(a*d - b*c)^5)))*(-b^5*(a*d - b*c)^5)^(1/2)*1i)/(a*(a*d - b*c)^5) - atanh((10*b^12*c^1
5*d^3*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(10*b^12*c^13*d^3 - 80*a*b^11*c^12*d^4 + 290*a^2*b^10*c^11*d^5 - 630*a^3
*b^9*c^10*d^6 + 912*a^4*b^8*c^9*d^7 - 922*a^5*b^7*c^8*d^8 + 660*a^6*b^6*c^7*d^9 - 330*a^7*b^5*c^6*d^10 + 110*a
^8*b^4*c^5*d^11 - 22*a^9*b^3*c^4*d^12 + 2*a^10*b^2*c^3*d^13)) + (290*a^2*b^10*c^13*d^5*(c + d*x^2)^(1/2))/((c^
5)^(1/2)*(10*b^12*c^13*d^3 - 80*a*b^11*c^12*d^4 + 290*a^2*b^10*c^11*d^5 - 630*a^3*b^9*c^10*d^6 + 912*a^4*b^8*c
^9*d^7 - 922*a^5*b^7*c^8*d^8 + 660*a^6*b^6*c^7*d^9 - 330*a^7*b^5*c^6*d^10 + 110*a^8*b^4*c^5*d^11 - 22*a^9*b^3*
c^4*d^12 + 2*a^10*b^2*c^3*d^13)) - (630*a^3*b^9*c^12*d^6*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(10*b^12*c^13*d^3 - 8
0*a*b^11*c^12*d^4 + 290*a^2*b^10*c^11*d^5 - 630*a^3*b^9*c^10*d^6 + 912*a^4*b^8*c^9*d^7 - 922*a^5*b^7*c^8*d^8 +
660*a^6*b^6*c^7*d^9 - 330*a^7*b^5*c^6*d^10 + 110*a^8*b^4*c^5*d^11 - 22*a^9*b^3*c^4*d^12 + 2*a^10*b^2*c^3*d^13
)) + (912*a^4*b^8*c^11*d^7*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(10*b^12*c^13*d^3 - 80*a*b^11*c^12*d^4 + 290*a^2*b^
10*c^11*d^5 - 630*a^3*b^9*c^10*d^6 + 912*a^4*b^8*c^9*d^7 - 922*a^5*b^7*c^8*d^8 + 660*a^6*b^6*c^7*d^9 - 330*a^7
*b^5*c^6*d^10 + 110*a^8*b^4*c^5*d^11 - 22*a^9*b^3*c^4*d^12 + 2*a^10*b^2*c^3*d^13)) - (922*a^5*b^7*c^10*d^8*(c
+ d*x^2)^(1/2))/((c^5)^(1/2)*(10*b^12*c^13*d^3 - 80*a*b^11*c^12*d^4 + 290*a^2*b^10*c^11*d^5 - 630*a^3*b^9*c^10
*d^6 + 912*a^4*b^8*c^9*d^7 - 922*a^5*b^7*c^8*d^8 + 660*a^6*b^6*c^7*d^9 - 330*a^7*b^5*c^6*d^10 + 110*a^8*b^4*c^
5*d^11 - 22*a^9*b^3*c^4*d^12 + 2*a^10*b^2*c^3*d^13)) + (660*a^6*b^6*c^9*d^9*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(1
0*b^12*c^13*d^3 - 80*a*b^11*c^12*d^4 + 290*a^2*b^10*c^11*d^5 - 630*a^3*b^9*c^10*d^6 + 912*a^4*b^8*c^9*d^7 - 92
2*a^5*b^7*c^8*d^8 + 660*a^6*b^6*c^7*d^9 - 330*a^7*b^5*c^6*d^10 + 110*a^8*b^4*c^5*d^11 - 22*a^9*b^3*c^4*d^12 +
2*a^10*b^2*c^3*d^13)) - (330*a^7*b^5*c^8*d^10*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(10*b^12*c^13*d^3 - 80*a*b^11*c^
12*d^4 + 290*a^2*b^10*c^11*d^5 - 630*a^3*b^9*c^10*d^6 + 912*a^4*b^8*c^9*d^7 - 922*a^5*b^7*c^8*d^8 + 660*a^6*b^
6*c^7*d^9 - 330*a^7*b^5*c^6*d^10 + 110*a^8*b^4*c^5*d^11 - 22*a^9*b^3*c^4*d^12 + 2*a^10*b^2*c^3*d^13)) + (110*a
^8*b^4*c^7*d^11*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(10*b^12*c^13*d^3 - 80*a*b^11*c^12*d^4 + 290*a^2*b^10*c^11*d^5
- 630*a^3*b^9*c^10*d^6 + 912*a^4*b^8*c^9*d^7 - 922*a^5*b^7*c^8*d^8 + 660*a^6*b^6*c^7*d^9 - 330*a^7*b^5*c^6*d^
10 + 110*a^8*b^4*c^5*d^11 - 22*a^9*b^3*c^4*d^12 + 2*a^10*b^2*c^3*d^13)) - (22*a^9*b^3*c^6*d^12*(c + d*x^2)^(1/
2))/((c^5)^(1/2)*(10*b^12*c^13*d^3 - 80*a*b^11*c^12*d^4 + 290*a^2*b^10*c^11*d^5 - 630*a^3*b^9*c^10*d^6 + 912*a
^4*b^8*c^9*d^7 - 922*a^5*b^7*c^8*d^8 + 660*a^6*b^6*c^7*d^9 - 330*a^7*b^5*c^6*d^10 + 110*a^8*b^4*c^5*d^11 - 22*
a^9*b^3*c^4*d^12 + 2*a^10*b^2*c^3*d^13)) + (2*a^10*b^2*c^5*d^13*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(10*b^12*c^13*
d^3 - 80*a*b^11*c^12*d^4 + 290*a^2*b^10*c^11*d^5 - 630*a^3*b^9*c^10*d^6 + 912*a^4*b^8*c^9*d^7 - 922*a^5*b^7*c^
8*d^8 + 660*a^6*b^6*c^7*d^9 - 330*a^7*b^5*c^6*d^10 + 110*a^8*b^4*c^5*d^11 - 22*a^9*b^3*c^4*d^12 + 2*a^10*b^2*c
^3*d^13)) - (80*a*b^11*c^14*d^4*(c + d*x^2)^(1/2))/((c^5)^(1/2)*(10*b^12*c^13*d^3 - 80*a*b^11*c^12*d^4 + 290*a
^2*b^10*c^11*d^5 - 630*a^3*b^9*c^10*d^6 + 912*a^4*b^8*c^9*d^7 - 922*a^5*b^7*c^8*d^8 + 660*a^6*b^6*c^7*d^9 - 33
0*a^7*b^5*c^6*d^10 + 110*a^8*b^4*c^5*d^11 - 22*a^9*b^3*c^4*d^12 + 2*a^10*b^2*c^3*d^13)))/(a*(c^5)^(1/2)) - (d/
(3*(b*c^2 - a*c*d)) - (d*(c + d*x^2)*(a*d - 2*b*c))/(b*c^2 - a*c*d)^2)/(c + d*x^2)^(3/2)

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sympy [A]  time = 26.34, size = 133, normalized size = 0.92 \begin {gather*} \frac {d}{3 c \left (c + d x^{2}\right )^{\frac {3}{2}} \left (a d - b c\right )} + \frac {d \left (a d - 2 b c\right )}{c^{2} \sqrt {c + d x^{2}} \left (a d - b c\right )^{2}} - \frac {b^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{a \sqrt {\frac {a d - b c}{b}} \left (a d - b c\right )^{2}} + \frac {\operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {- c}} \right )}}{a c^{2} \sqrt {- c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d/(3*c*(c + d*x**2)**(3/2)*(a*d - b*c)) + d*(a*d - 2*b*c)/(c**2*sqrt(c + d*x**2)*(a*d - b*c)**2) - b**2*atan(s
qrt(c + d*x**2)/sqrt((a*d - b*c)/b))/(a*sqrt((a*d - b*c)/b)*(a*d - b*c)**2) + atan(sqrt(c + d*x**2)/sqrt(-c))/
(a*c**2*sqrt(-c))

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