3.2.23 \(\int \frac {1}{x^8 (a+b x^3) (c+d x^3)} \, dx\)

Optimal. Leaf size=352 \[ -\frac {b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}+\frac {b^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{10/3} (b c-a d)}+\frac {a d+b c}{4 a^2 c^2 x^4}-\frac {a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}+\frac {d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}-\frac {d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac {d^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{10/3} (b c-a d)}-\frac {1}{7 a c x^7} \]

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Rubi [A]  time = 0.50, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {480, 583, 584, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {a^2 d^2+a b c d+b^2 c^2}{a^3 c^3 x}-\frac {b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}+\frac {b^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{10/3} (b c-a d)}+\frac {a d+b c}{4 a^2 c^2 x^4}+\frac {d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}-\frac {d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac {d^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{10/3} (b c-a d)}-\frac {1}{7 a c x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(a + b*x^3)*(c + d*x^3)),x]

[Out]

-1/(7*a*c*x^7) + (b*c + a*d)/(4*a^2*c^2*x^4) - (b^2*c^2 + a*b*c*d + a^2*d^2)/(a^3*c^3*x) + (b^(10/3)*ArcTan[(a
^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(10/3)*(b*c - a*d)) - (d^(10/3)*ArcTan[(c^(1/3) - 2*d^(1/
3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(10/3)*(b*c - a*d)) + (b^(10/3)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(10/3)*(b*
c - a*d)) - (d^(10/3)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(10/3)*(b*c - a*d)) - (b^(10/3)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/(6*a^(10/3)*(b*c - a*d)) + (d^(10/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6
*c^(10/3)*(b*c - a*d))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 480

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*e*(m + 1)), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=-\frac {1}{7 a c x^7}+\frac {\int \frac {-7 (b c+a d)-7 b d x^3}{x^5 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{7 a c}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {\int \frac {-28 \left (b^2 c^2+a b c d+a^2 d^2\right )-28 b d (b c+a d) x^3}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{28 a^2 c^2}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {\int \frac {x \left (-28 (b c+a d) \left (b^2 c^2+a^2 d^2\right )-28 b d \left (b^2 c^2+a b c d+a^2 d^2\right ) x^3\right )}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{28 a^3 c^3}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {\int \left (-\frac {28 b^4 c^3 x}{(b c-a d) \left (a+b x^3\right )}-\frac {28 a^3 d^4 x}{(-b c+a d) \left (c+d x^3\right )}\right ) \, dx}{28 a^3 c^3}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}-\frac {b^4 \int \frac {x}{a+b x^3} \, dx}{a^3 (b c-a d)}+\frac {d^4 \int \frac {x}{c+d x^3} \, dx}{c^3 (b c-a d)}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {b^{11/3} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{10/3} (b c-a d)}-\frac {b^{11/3} \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{10/3} (b c-a d)}-\frac {d^{11/3} \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{10/3} (b c-a d)}+\frac {d^{11/3} \int \frac {\sqrt [3]{c}+\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 c^{10/3} (b c-a d)}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}-\frac {d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac {b^{10/3} \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{10/3} (b c-a d)}-\frac {b^{11/3} \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^3 (b c-a d)}+\frac {d^{10/3} \int \frac {-\sqrt [3]{c} \sqrt [3]{d}+2 d^{2/3} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{6 c^{10/3} (b c-a d)}+\frac {d^{11/3} \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 c^3 (b c-a d)}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}-\frac {d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac {b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac {d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}-\frac {b^{10/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{10/3} (b c-a d)}+\frac {d^{10/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{10/3} (b c-a d)}\\ &=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {b^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{10/3} (b c-a d)}-\frac {d^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{10/3} (b c-a d)}+\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}-\frac {d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac {b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac {d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 304, normalized size = 0.86 \begin {gather*} \frac {-\frac {28 b^{10/3} x^7 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{10/3}}-\frac {28 \sqrt {3} b^{10/3} x^7 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{10/3}}+\frac {14 b^{10/3} x^7 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{10/3}}+\frac {84 b^3 x^6}{a^3}-\frac {21 b^2 x^3}{a^2}+\frac {12 b}{a}+\frac {28 d^{10/3} x^7 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{10/3}}+\frac {28 \sqrt {3} d^{10/3} x^7 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{10/3}}-\frac {14 d^{10/3} x^7 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{10/3}}-\frac {84 d^3 x^6}{c^3}+\frac {21 d^2 x^3}{c^2}-\frac {12 d}{c}}{84 x^7 (a d-b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(a + b*x^3)*(c + d*x^3)),x]

[Out]

((12*b)/a - (12*d)/c - (21*b^2*x^3)/a^2 + (21*d^2*x^3)/c^2 + (84*b^3*x^6)/a^3 - (84*d^3*x^6)/c^3 - (28*Sqrt[3]
*b^(10/3)*x^7*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(10/3) + (28*Sqrt[3]*d^(10/3)*x^7*ArcTan[(1 - (2*
d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(10/3) - (28*b^(10/3)*x^7*Log[a^(1/3) + b^(1/3)*x])/a^(10/3) + (28*d^(10/3)*x^
7*Log[c^(1/3) + d^(1/3)*x])/c^(10/3) + (14*b^(10/3)*x^7*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(10/
3) - (14*d^(10/3)*x^7*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(10/3))/(84*(-(b*c) + a*d)*x^7)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^8*(a + b*x^3)*(c + d*x^3)),x]

[Out]

IntegrateAlgebraic[1/(x^8*(a + b*x^3)*(c + d*x^3)), x]

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fricas [A]  time = 1.20, size = 332, normalized size = 0.94 \begin {gather*} -\frac {28 \, \sqrt {3} b^{3} c^{3} x^{7} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 28 \, \sqrt {3} a^{3} d^{3} x^{7} \left (\frac {d}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {d}{c}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 14 \, b^{3} c^{3} x^{7} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 14 \, a^{3} d^{3} x^{7} \left (\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x^{2} - c x \left (\frac {d}{c}\right )^{\frac {2}{3}} + c \left (\frac {d}{c}\right )^{\frac {1}{3}}\right ) + 28 \, b^{3} c^{3} x^{7} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 28 \, a^{3} d^{3} x^{7} \left (\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x + c \left (\frac {d}{c}\right )^{\frac {2}{3}}\right ) + 84 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{6} + 12 \, a^{2} b c^{3} - 12 \, a^{3} c^{2} d - 21 \, {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{3}}{84 \, {\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)/(d*x^3+c),x, algorithm="fricas")

[Out]

-1/84*(28*sqrt(3)*b^3*c^3*x^7*(-b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3*sqrt(3)) - 28*sqrt(3)*a^3*d
^3*x^7*(d/c)^(1/3)*arctan(2/3*sqrt(3)*x*(d/c)^(1/3) - 1/3*sqrt(3)) - 14*b^3*c^3*x^7*(-b/a)^(1/3)*log(b*x^2 - a
*x*(-b/a)^(2/3) - a*(-b/a)^(1/3)) - 14*a^3*d^3*x^7*(d/c)^(1/3)*log(d*x^2 - c*x*(d/c)^(2/3) + c*(d/c)^(1/3)) +
28*b^3*c^3*x^7*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)) + 28*a^3*d^3*x^7*(d/c)^(1/3)*log(d*x + c*(d/c)^(2/3)) +
84*(b^3*c^3 - a^3*d^3)*x^6 + 12*a^2*b*c^3 - 12*a^3*c^2*d - 21*(a*b^2*c^3 - a^3*c*d^2)*x^3)/((a^3*b*c^4 - a^4*c
^3*d)*x^7)

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giac [A]  time = 0.22, size = 377, normalized size = 1.07 \begin {gather*} \frac {b^{4} \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a^{4} b c - a^{5} d\right )}} - \frac {d^{4} \left (-\frac {c}{d}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{5} - a c^{4} d\right )}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a^{4} b c - \sqrt {3} a^{5} d} - \frac {\left (-c d^{2}\right )^{\frac {2}{3}} d^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c^{5} - \sqrt {3} a c^{4} d} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} b^{2} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{4} b c - a^{5} d\right )}} + \frac {\left (-c d^{2}\right )^{\frac {2}{3}} d^{2} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{5} - a c^{4} d\right )}} - \frac {28 \, b^{2} c^{2} x^{6} + 28 \, a b c d x^{6} + 28 \, a^{2} d^{2} x^{6} - 7 \, a b c^{2} x^{3} - 7 \, a^{2} c d x^{3} + 4 \, a^{2} c^{2}}{28 \, a^{3} c^{3} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)/(d*x^3+c),x, algorithm="giac")

[Out]

1/3*b^4*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a^4*b*c - a^5*d) - 1/3*d^4*(-c/d)^(2/3)*log(abs(x - (-c/d)^(1
/3)))/(b*c^5 - a*c^4*d) + (-a*b^2)^(2/3)*b^2*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*a^
4*b*c - sqrt(3)*a^5*d) - (-c*d^2)^(2/3)*d^2*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(sqrt(3)*b*c
^5 - sqrt(3)*a*c^4*d) - 1/6*(-a*b^2)^(2/3)*b^2*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b*c - a^5*d) + 1/
6*(-c*d^2)^(2/3)*d^2*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b*c^5 - a*c^4*d) - 1/28*(28*b^2*c^2*x^6 + 28*a*
b*c*d*x^6 + 28*a^2*d^2*x^6 - 7*a*b*c^2*x^3 - 7*a^2*c*d*x^3 + 4*a^2*c^2)/(a^3*c^3*x^7)

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maple [A]  time = 0.05, size = 334, normalized size = 0.95 \begin {gather*} \frac {\sqrt {3}\, b^{3} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {b^{3} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {b^{3} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {\sqrt {3}\, d^{3} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} c^{3}}+\frac {d^{3} \ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} c^{3}}-\frac {d^{3} \ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{3}} c^{3}}-\frac {d^{2}}{a \,c^{3} x}-\frac {b d}{a^{2} c^{2} x}-\frac {b^{2}}{a^{3} c x}+\frac {d}{4 a \,c^{2} x^{4}}+\frac {b}{4 a^{2} c \,x^{4}}-\frac {1}{7 a c \,x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(b*x^3+a)/(d*x^3+c),x)

[Out]

-1/3*b^3/a^3/(a*d-b*c)/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6*b^3/a^3/(a*d-b*c)/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a
/b)^(2/3))+1/3*b^3/a^3/(a*d-b*c)*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+1/3*d^3/c^3/(a*d-
b*c)/(c/d)^(1/3)*ln(x+(c/d)^(1/3))-1/6*d^3/c^3/(a*d-b*c)/(c/d)^(1/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))-1/3*d^3
/c^3/(a*d-b*c)*3^(1/2)/(c/d)^(1/3)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))-1/7/a/c/x^7+1/4/a/c^2/x^4*d+1/4/a^2
/c/x^4*b-1/a/c^3/x*d^2-1/a^2/c^2/x*b*d-1/a^3/c/x*b^2

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maxima [A]  time = 1.16, size = 376, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {3} b^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (a^{3} b c - a^{4} d\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} d^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c^{4} - a c^{3} d\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {b^{3} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{3} b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} + \frac {d^{3} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{4} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c^{3} d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} + \frac {b^{3} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (a^{3} b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} - \frac {d^{3} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c^{4} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c^{3} d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} - \frac {28 \, {\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} x^{6} + 4 \, a^{2} c^{2} - 7 \, {\left (a b c^{2} + a^{2} c d\right )} x^{3}}{28 \, a^{3} c^{3} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^3+a)/(d*x^3+c),x, algorithm="maxima")

[Out]

-1/3*sqrt(3)*b^3*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/((a^3*b*c - a^4*d)*(a/b)^(1/3)) + 1/3*sqr
t(3)*d^3*arctan(1/3*sqrt(3)*(2*x - (c/d)^(1/3))/(c/d)^(1/3))/((b*c^4 - a*c^3*d)*(c/d)^(1/3)) - 1/6*b^3*log(x^2
- x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b*c*(a/b)^(1/3) - a^4*d*(a/b)^(1/3)) + 1/6*d^3*log(x^2 - x*(c/d)^(1/3) +
(c/d)^(2/3))/(b*c^4*(c/d)^(1/3) - a*c^3*d*(c/d)^(1/3)) + 1/3*b^3*log(x + (a/b)^(1/3))/(a^3*b*c*(a/b)^(1/3) - a
^4*d*(a/b)^(1/3)) - 1/3*d^3*log(x + (c/d)^(1/3))/(b*c^4*(c/d)^(1/3) - a*c^3*d*(c/d)^(1/3)) - 1/28*(28*(b^2*c^2
+ a*b*c*d + a^2*d^2)*x^6 + 4*a^2*c^2 - 7*(a*b*c^2 + a^2*c*d)*x^3)/(a^3*c^3*x^7)

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mupad [B]  time = 11.91, size = 1814, normalized size = 5.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(a + b*x^3)*(c + d*x^3)),x)

[Out]

log(((-b^10/(a^10*(a*d - b*c)^3))^(2/3)*(((27*a^21*b^3*c^21*d^3*x*(a^8*d^8 + b^8*c^8)*(a*d - b*c)^2 + 27*a^28*
b^3*c^28*d^3*(a*d + b*c)*(a*d - b*c)^4*(-b^10/(a^10*(a*d - b*c)^3))^(2/3))*(-b^10/(a^10*(a*d - b*c)^3))^(1/3))
/3 - 9*a^19*b^14*c^29*d^4 + 9*a^20*b^13*c^28*d^5 + 9*a^28*b^5*c^20*d^13 - 9*a^29*b^4*c^19*d^14))/9 - a^19*b^11
*c^19*d^11*x*(a*d + b*c))*(-b^10/(27*a^13*d^3 - 27*a^10*b^3*c^3 + 81*a^11*b^2*c^2*d - 81*a^12*b*c*d^2))^(1/3)
+ log(((d^10/(c^10*(a*d - b*c)^3))^(2/3)*(((27*a^21*b^3*c^21*d^3*x*(a^8*d^8 + b^8*c^8)*(a*d - b*c)^2 + 27*a^28
*b^3*c^28*d^3*(a*d + b*c)*(a*d - b*c)^4*(d^10/(c^10*(a*d - b*c)^3))^(2/3))*(d^10/(c^10*(a*d - b*c)^3))^(1/3))/
3 - 9*a^19*b^14*c^29*d^4 + 9*a^20*b^13*c^28*d^5 + 9*a^28*b^5*c^20*d^13 - 9*a^29*b^4*c^19*d^14))/9 - a^19*b^11*
c^19*d^11*x*(a*d + b*c))*(-d^10/(27*b^3*c^13 - 27*a^3*c^10*d^3 + 81*a^2*b*c^11*d^2 - 81*a*b^2*c^12*d))^(1/3) -
(1/(7*a*c) - (x^3*(a*d + b*c))/(4*a^2*c^2) + (x^6*(a^2*d^2 + b^2*c^2 + a*b*c*d))/(a^3*c^3))/x^7 - (log(((3^(1
/2)*1i + 1)^2*(-b^10/(a^10*(a*d - b*c)^3))^(2/3)*(((3^(1/2)*1i + 1)*(27*a^21*b^3*c^21*d^3*x*(a^8*d^8 + b^8*c^8
)*(a*d - b*c)^2 + (27*a^28*b^3*c^28*d^3*(3^(1/2)*1i + 1)^2*(a*d + b*c)*(a*d - b*c)^4*(-b^10/(a^10*(a*d - b*c)^
3))^(2/3))/4)*(-b^10/(a^10*(a*d - b*c)^3))^(1/3))/6 + 9*a^19*b^14*c^29*d^4 - 9*a^20*b^13*c^28*d^5 - 9*a^28*b^5
*c^20*d^13 + 9*a^29*b^4*c^19*d^14))/36 + a^19*b^11*c^19*d^11*x*(a*d + b*c))*(-b^10/(27*a^13*d^3 - 27*a^10*b^3*
c^3 + 81*a^11*b^2*c^2*d - 81*a^12*b*c*d^2))^(1/3)*(3^(1/2)*1i + 1))/2 + (log(((3^(1/2)*1i - 1)^2*(-b^10/(a^10*
(a*d - b*c)^3))^(2/3)*(((3^(1/2)*1i - 1)*(27*a^21*b^3*c^21*d^3*x*(a^8*d^8 + b^8*c^8)*(a*d - b*c)^2 + (27*a^28*
b^3*c^28*d^3*(3^(1/2)*1i - 1)^2*(a*d + b*c)*(a*d - b*c)^4*(-b^10/(a^10*(a*d - b*c)^3))^(2/3))/4)*(-b^10/(a^10*
(a*d - b*c)^3))^(1/3))/6 - 9*a^19*b^14*c^29*d^4 + 9*a^20*b^13*c^28*d^5 + 9*a^28*b^5*c^20*d^13 - 9*a^29*b^4*c^1
9*d^14))/36 - a^19*b^11*c^19*d^11*x*(a*d + b*c))*(-b^10/(27*a^13*d^3 - 27*a^10*b^3*c^3 + 81*a^11*b^2*c^2*d - 8
1*a^12*b*c*d^2))^(1/3)*(3^(1/2)*1i - 1))/2 - (log(((3^(1/2)*1i + 1)^2*(d^10/(c^10*(a*d - b*c)^3))^(2/3)*(((3^(
1/2)*1i + 1)*(27*a^21*b^3*c^21*d^3*x*(a^8*d^8 + b^8*c^8)*(a*d - b*c)^2 + (27*a^28*b^3*c^28*d^3*(3^(1/2)*1i + 1
)^2*(a*d + b*c)*(a*d - b*c)^4*(d^10/(c^10*(a*d - b*c)^3))^(2/3))/4)*(d^10/(c^10*(a*d - b*c)^3))^(1/3))/6 + 9*a
^19*b^14*c^29*d^4 - 9*a^20*b^13*c^28*d^5 - 9*a^28*b^5*c^20*d^13 + 9*a^29*b^4*c^19*d^14))/36 + a^19*b^11*c^19*d
^11*x*(a*d + b*c))*(-d^10/(27*b^3*c^13 - 27*a^3*c^10*d^3 + 81*a^2*b*c^11*d^2 - 81*a*b^2*c^12*d))^(1/3)*(3^(1/2
)*1i + 1))/2 + (log(((3^(1/2)*1i - 1)^2*(d^10/(c^10*(a*d - b*c)^3))^(2/3)*(((3^(1/2)*1i - 1)*(27*a^21*b^3*c^21
*d^3*x*(a^8*d^8 + b^8*c^8)*(a*d - b*c)^2 + (27*a^28*b^3*c^28*d^3*(3^(1/2)*1i - 1)^2*(a*d + b*c)*(a*d - b*c)^4*
(d^10/(c^10*(a*d - b*c)^3))^(2/3))/4)*(d^10/(c^10*(a*d - b*c)^3))^(1/3))/6 - 9*a^19*b^14*c^29*d^4 + 9*a^20*b^1
3*c^28*d^5 + 9*a^28*b^5*c^20*d^13 - 9*a^29*b^4*c^19*d^14))/36 - a^19*b^11*c^19*d^11*x*(a*d + b*c))*(-d^10/(27*
b^3*c^13 - 27*a^3*c^10*d^3 + 81*a^2*b*c^11*d^2 - 81*a*b^2*c^12*d))^(1/3)*(3^(1/2)*1i - 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**8/(b*x**3+a)/(d*x**3+c),x)

[Out]

Timed out

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