3.6.77 \(\int \frac {1}{x^9 (a+b x^8)^2 \sqrt {c+d x^8}} \, dx\)

Optimal. Leaf size=185 \[ -\frac {b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 a^3 (b c-a d)^{3/2}}+\frac {(a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{8 a^3 c^{3/2}}-\frac {b \sqrt {c+d x^8} (2 b c-a d)}{8 a^2 c \left (a+b x^8\right ) (b c-a d)}-\frac {\sqrt {c+d x^8}}{8 a c x^8 \left (a+b x^8\right )} \]

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Rubi [A]  time = 0.23, antiderivative size = 185, 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, 103, 151, 156, 63, 208} \begin {gather*} -\frac {b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 a^3 (b c-a d)^{3/2}}+\frac {(a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{8 a^3 c^{3/2}}-\frac {b \sqrt {c+d x^8} (2 b c-a d)}{8 a^2 c \left (a+b x^8\right ) (b c-a d)}-\frac {\sqrt {c+d x^8}}{8 a c x^8 \left (a+b x^8\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

-(b*(2*b*c - a*d)*Sqrt[c + d*x^8])/(8*a^2*c*(b*c - a*d)*(a + b*x^8)) - Sqrt[c + d*x^8]/(8*a*c*x^8*(a + b*x^8))
+ ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^8]/Sqrt[c]])/(8*a^3*c^(3/2)) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTanh[(Sqrt[b
]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]])/(8*a^3*(b*c - a*d)^(3/2))

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 151

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

Rule 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^9 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx,x,x^8\right )\\ &=-\frac {\sqrt {c+d x^8}}{8 a c x^8 \left (a+b x^8\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (4 b c+a d)+\frac {3 b d x}{2}}{x (a+b x)^2 \sqrt {c+d x}} \, dx,x,x^8\right )}{8 a c}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^8}}{8 a^2 c (b c-a d) \left (a+b x^8\right )}-\frac {\sqrt {c+d x^8}}{8 a c x^8 \left (a+b x^8\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (b c-a d) (4 b c+a d)+\frac {1}{2} b d (2 b c-a d) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )}{8 a^2 c (b c-a d)}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^8}}{8 a^2 c (b c-a d) \left (a+b x^8\right )}-\frac {\sqrt {c+d x^8}}{8 a c x^8 \left (a+b x^8\right )}+\frac {\left (b^2 (4 b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^8\right )}{16 a^3 (b c-a d)}-\frac {(4 b c+a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^8\right )}{16 a^3 c}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^8}}{8 a^2 c (b c-a d) \left (a+b x^8\right )}-\frac {\sqrt {c+d x^8}}{8 a c x^8 \left (a+b x^8\right )}+\frac {\left (b^2 (4 b c-5 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^8}\right )}{8 a^3 d (b c-a d)}-\frac {(4 b c+a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^8}\right )}{8 a^3 c d}\\ &=-\frac {b (2 b c-a d) \sqrt {c+d x^8}}{8 a^2 c (b c-a d) \left (a+b x^8\right )}-\frac {\sqrt {c+d x^8}}{8 a c x^8 \left (a+b x^8\right )}+\frac {(4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{8 a^3 c^{3/2}}-\frac {b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{8 a^3 (b c-a d)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.60, size = 163, normalized size = 0.88 \begin {gather*} \frac {\frac {a \sqrt {c+d x^8} \left (a^2 d+a b \left (d x^8-c\right )-2 b^2 c x^8\right )}{x^8 \left (a+b x^8\right ) (b c-a d)}+\frac {b^{3/2} c (5 a d-4 b c) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}+\frac {(a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{\sqrt {c}}}{8 a^3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

((a*Sqrt[c + d*x^8]*(a^2*d - 2*b^2*c*x^8 + a*b*(-c + d*x^8)))/((b*c - a*d)*x^8*(a + b*x^8)) + ((4*b*c + a*d)*A
rcTanh[Sqrt[c + d*x^8]/Sqrt[c]])/Sqrt[c] + (b^(3/2)*c*(-4*b*c + 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[
b*c - a*d]])/(b*c - a*d)^(3/2))/(8*a^3*c)

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IntegrateAlgebraic [A]  time = 0.65, size = 187, normalized size = 1.01 \begin {gather*} \frac {\left (4 b^{5/2} c-5 a b^{3/2} d\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^8} \sqrt {a d-b c}}{b c-a d}\right )}{8 a^3 (a d-b c)^{3/2}}+\frac {(a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^8}}{\sqrt {c}}\right )}{8 a^3 c^{3/2}}+\frac {\sqrt {c+d x^8} \left (-a^2 d+a b c-a b d x^8+2 b^2 c x^8\right )}{8 a^2 c x^8 \left (a+b x^8\right ) (a d-b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^9*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]

[Out]

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

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fricas [A]  time = 0.56, size = 1236, normalized size = 6.68

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="fricas")

[Out]

[1/16*(((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^16 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x^8)*sqrt(b/(b*c - a*d))*log((b*d*x^8
+ 2*b*c - a*d - 2*sqrt(d*x^8 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^8 + a)) + ((4*b^3*c^2 - 3*a*b^2*c*d -
a^2*b*d^2)*x^16 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x^8)*sqrt(c)*log((d*x^8 + 2*sqrt(d*x^8 + c)*sqrt(c) +
2*c)/x^8) - 2*((2*a*b^2*c^2 - a^2*b*c*d)*x^8 + a^2*b*c^2 - a^3*c*d)*sqrt(d*x^8 + c))/((a^3*b^2*c^3 - a^4*b*c^
2*d)*x^16 + (a^4*b*c^3 - a^5*c^2*d)*x^8), -1/16*(2*((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^16 + (4*a*b^2*c^3 - 5*a^2*b*
c^2*d)*x^8)*sqrt(-b/(b*c - a*d))*arctan(-sqrt(d*x^8 + c)*(b*c - a*d)*sqrt(-b/(b*c - a*d))/(b*d*x^8 + b*c)) - (
(4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^16 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x^8)*sqrt(c)*log((d*x^8 + 2
*sqrt(d*x^8 + c)*sqrt(c) + 2*c)/x^8) + 2*((2*a*b^2*c^2 - a^2*b*c*d)*x^8 + a^2*b*c^2 - a^3*c*d)*sqrt(d*x^8 + c)
)/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^16 + (a^4*b*c^3 - a^5*c^2*d)*x^8), -1/16*(2*((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b
*d^2)*x^16 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x^8)*sqrt(-c)*arctan(sqrt(d*x^8 + c)*sqrt(-c)/c) - ((4*b^3*
c^3 - 5*a*b^2*c^2*d)*x^16 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x^8)*sqrt(b/(b*c - a*d))*log((b*d*x^8 + 2*b*c - a*d
- 2*sqrt(d*x^8 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^8 + a)) + 2*((2*a*b^2*c^2 - a^2*b*c*d)*x^8 + a^2*b*c
^2 - a^3*c*d)*sqrt(d*x^8 + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^16 + (a^4*b*c^3 - a^5*c^2*d)*x^8), -1/8*(((4*b^3
*c^3 - 5*a*b^2*c^2*d)*x^16 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x^8)*sqrt(-b/(b*c - a*d))*arctan(-sqrt(d*x^8 + c)*(
b*c - a*d)*sqrt(-b/(b*c - a*d))/(b*d*x^8 + b*c)) + ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^16 + (4*a*b^2*c^2
- 3*a^2*b*c*d - a^3*d^2)*x^8)*sqrt(-c)*arctan(sqrt(d*x^8 + c)*sqrt(-c)/c) + ((2*a*b^2*c^2 - a^2*b*c*d)*x^8 + a
^2*b*c^2 - a^3*c*d)*sqrt(d*x^8 + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^16 + (a^4*b*c^3 - a^5*c^2*d)*x^8)]

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giac [A]  time = 0.17, size = 257, normalized size = 1.39 \begin {gather*} \frac {{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{8} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (a^{3} b c - a^{4} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x^{8} + c\right )}^{\frac {3}{2}} b^{2} c d - 2 \, \sqrt {d x^{8} + c} b^{2} c^{2} d - {\left (d x^{8} + c\right )}^{\frac {3}{2}} a b d^{2} + 2 \, \sqrt {d x^{8} + c} a b c d^{2} - \sqrt {d x^{8} + c} a^{2} d^{3}}{8 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} {\left ({\left (d x^{8} + c\right )}^{2} b - 2 \, {\left (d x^{8} + c\right )} b c + b c^{2} + {\left (d x^{8} + c\right )} a d - a c d\right )}} - \frac {{\left (4 \, b c + a d\right )} \arctan \left (\frac {\sqrt {d x^{8} + c}}{\sqrt {-c}}\right )}{8 \, a^{3} \sqrt {-c} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="giac")

[Out]

1/8*(4*b^3*c - 5*a*b^2*d)*arctan(sqrt(d*x^8 + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b*c - a^4*d)*sqrt(-b^2*c + a*b*
d)) - 1/8*(2*(d*x^8 + c)^(3/2)*b^2*c*d - 2*sqrt(d*x^8 + c)*b^2*c^2*d - (d*x^8 + c)^(3/2)*a*b*d^2 + 2*sqrt(d*x^
8 + c)*a*b*c*d^2 - sqrt(d*x^8 + c)*a^2*d^3)/((a^2*b*c^2 - a^3*c*d)*((d*x^8 + c)^2*b - 2*(d*x^8 + c)*b*c + b*c^
2 + (d*x^8 + c)*a*d - a*c*d)) - 1/8*(4*b*c + a*d)*arctan(sqrt(d*x^8 + c)/sqrt(-c))/(a^3*sqrt(-c)*c)

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maple [F]  time = 0.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}\, x^{9}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)

[Out]

int(1/x^9/(b*x^8+a)^2/(d*x^8+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^{8} + a\right )}^{2} \sqrt {d x^{8} + c} x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)*x^9), x)

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mupad [B]  time = 7.85, size = 3832, normalized size = 20.71

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^9*(a + b*x^8)^2*(c + d*x^8)^(1/2)),x)

[Out]

(((c + d*x^8)^(1/2)*(a^2*d^3 + 2*b^2*c^2*d - 2*a*b*c*d^2))/(2*a^2*(b*c^2 - a*c*d)) + (b*(c + d*x^8)^(3/2)*(a*d
^2 - 2*b*c*d))/(2*a^2*(b*c^2 - a*c*d)))/((c + d*x^8)*(4*a*d - 8*b*c) + 4*b*(c + d*x^8)^2 + 4*b*c^2 - 4*a*c*d)
+ (atan((((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(((c + d*x^8)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*
b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(32*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)) + ((-b^3
*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(((a^9*b^2*c*d^6)/2 + a^6*b^5*c^4*d^3 - 2*a^7*b^4*c^3*d^4 + (a^8*b^3*c^2
*d^5)/2)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) - ((-b^3*(a*d - b*c)^3)^(1/2)*(c + d*x^8)^(1/2)*(5*a*d -
4*b*c)*(512*a^6*b^5*c^5*d^2 - 1280*a^7*b^4*c^4*d^3 + 1024*a^8*b^3*c^3*d^4 - 256*a^9*b^2*c^2*d^5))/(512*(a^4*b^
2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2))))/(16*(a^6*d^3
- a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))*1i)/(16*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b
*c*d^2)) + ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(((c + d*x^8)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*
a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(32*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)) - ((-b
^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(((a^9*b^2*c*d^6)/2 + a^6*b^5*c^4*d^3 - 2*a^7*b^4*c^3*d^4 + (a^8*b^3*c
^2*d^5)/2)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) + ((-b^3*(a*d - b*c)^3)^(1/2)*(c + d*x^8)^(1/2)*(5*a*d
- 4*b*c)*(512*a^6*b^5*c^5*d^2 - 1280*a^7*b^4*c^4*d^3 + 1024*a^8*b^3*c^3*d^4 - 256*a^9*b^2*c^2*d^5))/(512*(a^4*
b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2))))/(16*(a^6*d
^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))*1i)/(16*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5
*b*c*d^2)))/(((5*a^3*b^4*d^6)/256 + (b^7*c^3*d^3)/8 - (3*a*b^6*c^2*d^4)/16 + (3*a^2*b^5*c*d^5)/128)/(a^6*b^2*c
^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) - ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(((c + d*x^8)^(1/2)*(a^4*b^3*d
^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(32*(a^4*b^2*c^4 + a^6*c^2*d^2
- 2*a^5*b*c^3*d)) + ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(((a^9*b^2*c*d^6)/2 + a^6*b^5*c^4*d^3 - 2*a^7
*b^4*c^3*d^4 + (a^8*b^3*c^2*d^5)/2)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) - ((-b^3*(a*d - b*c)^3)^(1/2)*
(c + d*x^8)^(1/2)*(5*a*d - 4*b*c)*(512*a^6*b^5*c^5*d^2 - 1280*a^7*b^4*c^4*d^3 + 1024*a^8*b^3*c^3*d^4 - 256*a^9
*b^2*c^2*d^5))/(512*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a
^5*b*c*d^2))))/(16*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2))))/(16*(a^6*d^3 - a^3*b^3*c^3 + 3
*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)) + ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(((c + d*x^8)^(1/2)*(a^4*b^3*d^
6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(32*(a^4*b^2*c^4 + a^6*c^2*d^2
- 2*a^5*b*c^3*d)) - ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(((a^9*b^2*c*d^6)/2 + a^6*b^5*c^4*d^3 - 2*a^7*
b^4*c^3*d^4 + (a^8*b^3*c^2*d^5)/2)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) + ((-b^3*(a*d - b*c)^3)^(1/2)*(
c + d*x^8)^(1/2)*(5*a*d - 4*b*c)*(512*a^6*b^5*c^5*d^2 - 1280*a^7*b^4*c^4*d^3 + 1024*a^8*b^3*c^3*d^4 - 256*a^9*
b^2*c^2*d^5))/(512*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^
5*b*c*d^2))))/(16*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2))))/(16*(a^6*d^3 - a^3*b^3*c^3 + 3*
a^4*b^2*c^2*d - 3*a^5*b*c*d^2))))*(-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*1i)/(8*(a^6*d^3 - a^3*b^3*c^3 + 3
*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)) + (atan((((((c + d*x^8)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^
3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(32*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)) + ((((a^9*b^2*c*d^
6)/2 + a^6*b^5*c^4*d^3 - 2*a^7*b^4*c^3*d^4 + (a^8*b^3*c^2*d^5)/2)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d)
- ((c + d*x^8)^(1/2)*(a*d + 4*b*c)*(512*a^6*b^5*c^5*d^2 - 1280*a^7*b^4*c^4*d^3 + 1024*a^8*b^3*c^3*d^4 - 256*a^
9*b^2*c^2*d^5))/(512*a^3*(c^3)^(1/2)*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)))*(a*d + 4*b*c))/(16*a^3*(c^3
)^(1/2)))*(a*d + 4*b*c)*1i)/(16*a^3*(c^3)^(1/2)) + ((((c + d*x^8)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b
^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(32*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)) - ((((a^9
*b^2*c*d^6)/2 + a^6*b^5*c^4*d^3 - 2*a^7*b^4*c^3*d^4 + (a^8*b^3*c^2*d^5)/2)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*
b*c^3*d) + ((c + d*x^8)^(1/2)*(a*d + 4*b*c)*(512*a^6*b^5*c^5*d^2 - 1280*a^7*b^4*c^4*d^3 + 1024*a^8*b^3*c^3*d^4
- 256*a^9*b^2*c^2*d^5))/(512*a^3*(c^3)^(1/2)*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)))*(a*d + 4*b*c))/(16
*a^3*(c^3)^(1/2)))*(a*d + 4*b*c)*1i)/(16*a^3*(c^3)^(1/2)))/(((5*a^3*b^4*d^6)/256 + (b^7*c^3*d^3)/8 - (3*a*b^6*
c^2*d^4)/16 + (3*a^2*b^5*c*d^5)/128)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) - ((((c + d*x^8)^(1/2)*(a^4*b
^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(32*(a^4*b^2*c^4 + a^6*c^2
*d^2 - 2*a^5*b*c^3*d)) + ((((a^9*b^2*c*d^6)/2 + a^6*b^5*c^4*d^3 - 2*a^7*b^4*c^3*d^4 + (a^8*b^3*c^2*d^5)/2)/(a^
6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) - ((c + d*x^8)^(1/2)*(a*d + 4*b*c)*(512*a^6*b^5*c^5*d^2 - 1280*a^7*b^
4*c^4*d^3 + 1024*a^8*b^3*c^3*d^4 - 256*a^9*b^2*c^2*d^5))/(512*a^3*(c^3)^(1/2)*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a
^5*b*c^3*d)))*(a*d + 4*b*c))/(16*a^3*(c^3)^(1/2)))*(a*d + 4*b*c))/(16*a^3*(c^3)^(1/2)) + ((((c + d*x^8)^(1/2)*
(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(32*(a^4*b^2*c^4 + a
^6*c^2*d^2 - 2*a^5*b*c^3*d)) - ((((a^9*b^2*c*d^6)/2 + a^6*b^5*c^4*d^3 - 2*a^7*b^4*c^3*d^4 + (a^8*b^3*c^2*d^5)/
2)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) + ((c + d*x^8)^(1/2)*(a*d + 4*b*c)*(512*a^6*b^5*c^5*d^2 - 1280*
a^7*b^4*c^4*d^3 + 1024*a^8*b^3*c^3*d^4 - 256*a^9*b^2*c^2*d^5))/(512*a^3*(c^3)^(1/2)*(a^4*b^2*c^4 + a^6*c^2*d^2
- 2*a^5*b*c^3*d)))*(a*d + 4*b*c))/(16*a^3*(c^3)^(1/2)))*(a*d + 4*b*c))/(16*a^3*(c^3)^(1/2))))*(a*d + 4*b*c)*1
i)/(8*a^3*(c^3)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)

[Out]

Timed out

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