3.2.20 \(\int \frac {1}{x^5 (a+b x^3) (c+d x^3)} \, dx\)

Optimal. Leaf size=318 \[ \frac {b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3} (b c-a d)}-\frac {b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} (b c-a d)}-\frac {b^{7/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3} (b c-a d)}+\frac {a d+b c}{a^2 c^2 x}-\frac {d^{7/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{7/3} (b c-a d)}+\frac {d^{7/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{7/3} (b c-a d)}+\frac {d^{7/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{7/3} (b c-a d)}-\frac {1}{4 a c x^4} \]

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Rubi [A]  time = 0.38, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 16, 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 {b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3} (b c-a d)}-\frac {b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} (b c-a d)}-\frac {b^{7/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3} (b c-a d)}+\frac {a d+b c}{a^2 c^2 x}-\frac {d^{7/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{7/3} (b c-a d)}+\frac {d^{7/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{7/3} (b c-a d)}+\frac {d^{7/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{7/3} (b c-a d)}-\frac {1}{4 a c x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(a + b*x^3)*(c + d*x^3)),x]

[Out]

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

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Mathematica [A]  time = 0.26, size = 282, normalized size = 0.89 \begin {gather*} \frac {\frac {4 b^{7/3} x^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{7/3}}+\frac {4 \sqrt {3} b^{7/3} x^4 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{7/3}}-\frac {2 b^{7/3} x^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{7/3}}-\frac {12 b^2 x^3}{a^2}+\frac {3 b}{a}-\frac {4 d^{7/3} x^4 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{7/3}}-\frac {4 \sqrt {3} d^{7/3} x^4 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{7/3}}+\frac {2 d^{7/3} x^4 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{7/3}}+\frac {12 d^2 x^3}{c^2}-\frac {3 d}{c}}{12 x^4 (a d-b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(a + b*x^3)*(c + d*x^3)),x]

[Out]

((3*b)/a - (3*d)/c - (12*b^2*x^3)/a^2 + (12*d^2*x^3)/c^2 + (4*Sqrt[3]*b^(7/3)*x^4*ArcTan[(1 - (2*b^(1/3)*x)/a^
(1/3))/Sqrt[3]])/a^(7/3) - (4*Sqrt[3]*d^(7/3)*x^4*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(7/3) + (4*b^
(7/3)*x^4*Log[a^(1/3) + b^(1/3)*x])/a^(7/3) - (4*d^(7/3)*x^4*Log[c^(1/3) + d^(1/3)*x])/c^(7/3) - (2*b^(7/3)*x^
4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(7/3) + (2*d^(7/3)*x^4*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d
^(2/3)*x^2])/c^(7/3))/(12*(-(b*c) + a*d)*x^4)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^5 \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^5*(a + b*x^3)*(c + d*x^3)),x]

[Out]

IntegrateAlgebraic[1/(x^5*(a + b*x^3)*(c + d*x^3)), x]

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fricas [A]  time = 3.41, size = 305, normalized size = 0.96 \begin {gather*} \frac {4 \, \sqrt {3} b^{2} c^{2} x^{4} \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 ) - 4 \, \sqrt {3} a^{2} d^{2} x^{4} \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 ) + 2 \, b^{2} c^{2} x^{4} \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 ) + 2 \, a^{2} d^{2} x^{4} \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 ) - 4 \, b^{2} c^{2} x^{4} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) - 4 \, a^{2} d^{2} x^{4} \left (-\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x + c \left (-\frac {d}{c}\right )^{\frac {2}{3}}\right ) - 3 \, a b c^{2} + 3 \, a^{2} c d + 12 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{3}}{12 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^3+a)/(d*x^3+c),x, algorithm="fricas")

[Out]

1/12*(4*sqrt(3)*b^2*c^2*x^4*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(b/a)^(1/3) - 1/3*sqrt(3)) - 4*sqrt(3)*a^2*d^2*x^
4*(-d/c)^(1/3)*arctan(2/3*sqrt(3)*x*(-d/c)^(1/3) + 1/3*sqrt(3)) + 2*b^2*c^2*x^4*(b/a)^(1/3)*log(b*x^2 - a*x*(b
/a)^(2/3) + a*(b/a)^(1/3)) + 2*a^2*d^2*x^4*(-d/c)^(1/3)*log(d*x^2 - c*x*(-d/c)^(2/3) - c*(-d/c)^(1/3)) - 4*b^2
*c^2*x^4*(b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)) - 4*a^2*d^2*x^4*(-d/c)^(1/3)*log(d*x + c*(-d/c)^(2/3)) - 3*a*b*c
^2 + 3*a^2*c*d + 12*(b^2*c^2 - a^2*d^2)*x^3)/((a^2*b*c^3 - a^3*c^2*d)*x^4)

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giac [A]  time = 0.24, size = 328, normalized size = 1.03 \begin {gather*} -\frac {b^{3} \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^{3} b c - a^{4} d\right )}} + \frac {d^{3} \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^{4} - a c^{3} d\right )}} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} b \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^{3} b c - \sqrt {3} a^{4} d} + \frac {\left (-c d^{2}\right )^{\frac {2}{3}} d \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^{4} - \sqrt {3} a c^{3} d} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} b \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 - a^{4} d\right )}} - \frac {\left (-c d^{2}\right )^{\frac {2}{3}} d \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} - a c^{3} d\right )}} + \frac {4 \, b c x^{3} + 4 \, a d x^{3} - a c}{4 \, a^{2} c^{2} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^3+a)/(d*x^3+c),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 291, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {3}\, b^{2} \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^{2}}+\frac {b^{2} \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^{2}}-\frac {b^{2} \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^{2}}+\frac {\sqrt {3}\, d^{2} \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^{2}}-\frac {d^{2} \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^{2}}+\frac {d^{2} \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^{2}}+\frac {d}{a \,c^{2} x}+\frac {b}{a^{2} c x}-\frac {1}{4 a c \,x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x^3+a)/(d*x^3+c),x)

[Out]

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

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maxima [A]  time = 1.13, size = 341, normalized size = 1.07 \begin {gather*} \frac {\sqrt {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 )}{3 \, {\left (a^{2} b c - a^{3} d\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\sqrt {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 )}{3 \, {\left (b c^{3} - a c^{2} d\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} + \frac {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^{2} b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} - \frac {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^{3} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c^{2} d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} - \frac {b^{2} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (a^{2} b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} + \frac {d^{2} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c^{3} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c^{2} d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} + \frac {4 \, {\left (b c + a d\right )} x^{3} - a c}{4 \, a^{2} c^{2} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^3+a)/(d*x^3+c),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 11.37, size = 1734, normalized size = 5.45

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a + b*x^3)*(c + d*x^3)),x)

[Out]

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

[Out]

Timed out

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