3.8.27 \(\int \frac {1}{x (a+b x^2) (c+d x^2)^{5/2}} \, dx\) [727]

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

[Out]

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

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Rubi [A]
time = 0.13, 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 = {457, 87, 157, 162, 65, 214} \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]

-1/3*d/(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[Sqr
t[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 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 87

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 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*} \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx &=\frac {1}{2} \text {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 {\text {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 {\text {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 {\text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a c^2}-\frac {b^3 \text {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 {\text {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 \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 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 [A]
time = 0.42, size = 138, normalized size = 0.95 \begin {gather*} \frac {d \left (a d \left (4 c+3 d x^2\right )-b c \left (7 c+6 d x^2\right )\right )}{3 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{a (-b c+a d)^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(a*d*(4*c + 3*d*x^2) - b*c*(7*c + 6*d*x^2)))/(3*c^2*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - (b^(5/2)*ArcTan[(Sqr
t[b]*Sqrt[c + d*x^2])/Sqrt[-(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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1454\) vs. \(2(123)=246\).
time = 0.10, size = 1455, normalized size = 10.03

method	result	size
default	\(\text {Expression too large to display}\)	\(1455\)

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,method=_RETURNVERBOSE)

[Out]

-1/2/a*(-1/3/(a*d-b*c)*b/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+
d*(-a*b)^(1/2)/(a*d-b*c)*(2/3*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x
-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+16/3*d/(-4*d*(a*d-b*c)/b+4*d^2
*a/b)^2*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*
b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/(a*d-b*c)*b*(-1/(a*d-b*c)*b/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/
b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+2*d*(-a*b)^(1/2)/(a*d-b*c)*(2*d*(x-1/b*(-a*b)^(1/2))+2*d*(-a*b)^(1/2)/b)/(-
4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x-1/b*(-a*b)^(1/2))^2+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2
)+1/(a*d-b*c)*b/(-(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)*(d*(x-1/b*(-a*b)^(1/2))^2+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*(-1/3/(a*d-b*c)*b/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b
)^(3/2)-d*(-a*b)^(1/2)/(a*d-b*c)*(2/3*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/
b)/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+16/3*d/(-4*d*(a*d-b*c)
/b+4*d^2*a/b)^2*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/2)/b)/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+
1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/(a*d-b*c)*b*(-1/(a*d-b*c)*b/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)
/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-2*d*(-a*b)^(1/2)/(a*d-b*c)*(2*d*(x+1/b*(-a*b)^(1/2))-2*d*(-a*b)^(1/
2)/b)/(-4*d*(a*d-b*c)/b+4*d^2*a/b)/(d*(x+1/b*(-a*b)^(1/2))^2-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)
/b)^(1/2)+1/(a*d-b*c)*b/(-(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)*(d*(x+1/b*(-a*b)^(1/2))^2-2*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/a*(1/3/c/(d*x^2+c)^(3/2)+1/c*(1/c/(d*x^2+c)^(1/2)-1/c^(3/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/
2))/x)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (123) = 246\).
time = 2.39, size = 1711, normalized size = 11.80 \begin {gather*} \left [\frac {3 \, {\left (b^{2} c^{3} d^{2} x^{4} + 2 \, b^{2} c^{4} d x^{2} + b^{2} c^{5}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 6 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 4 \, {\left (7 \, a b c^{3} d - 4 \, a^{2} c^{2} d^{2} + 3 \, {\left (2 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2} + {\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{4} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{2}\right )}}, \frac {12 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + 3 \, {\left (b^{2} c^{3} d^{2} x^{4} + 2 \, b^{2} c^{4} d x^{2} + b^{2} c^{5}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (7 \, a b c^{3} d - 4 \, a^{2} c^{2} d^{2} + 3 \, {\left (2 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2} + {\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{4} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{2}\right )}}, -\frac {3 \, {\left (b^{2} c^{3} d^{2} x^{4} + 2 \, b^{2} c^{4} d x^{2} + b^{2} c^{5}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) - 3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (7 \, a b c^{3} d - 4 \, a^{2} c^{2} d^{2} + 3 \, {\left (2 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2} + {\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{4} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{2}\right )}}, -\frac {3 \, {\left (b^{2} c^{3} d^{2} x^{4} + 2 \, b^{2} c^{4} d x^{2} + b^{2} c^{5}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) - 6 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + 2 \, {\left (7 \, a b c^{3} d - 4 \, a^{2} c^{2} d^{2} + 3 \, {\left (2 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left (a b^{2} c^{7} - 2 \, a^{2} b c^{6} d + a^{3} c^{5} d^{2} + {\left (a b^{2} c^{5} d^{2} - 2 \, a^{2} b c^{4} d^{3} + a^{3} c^{3} d^{4}\right )} x^{4} + 2 \, {\left (a b^{2} c^{6} d - 2 \, a^{2} b c^{5} d^{2} + a^{3} c^{4} d^{3}\right )} x^{2}\right )}}\right ] \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="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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Sympy [A]
time = 9.86, 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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Giac [A]
time = 0.55, 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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Mupad [B]
time = 1.44, size = 2500, normalized size = 17.24 \begin {gather*} \text {Too large to display} \end {gather*}

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 + 1...

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