3.2.21 \(\int \frac {1}{x^6 (a+b x^3) (c+d x^3)} \, dx\)

Optimal. Leaf size=321 \[ -\frac {b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac {b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac {b^{8/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{8/3} (b c-a d)}+\frac {a d+b c}{2 a^2 c^2 x^2}+\frac {d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}-\frac {d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}+\frac {d^{8/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{8/3} (b c-a d)}-\frac {1}{5 a c x^5} \]

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Rubi [A]  time = 0.46, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {480, 583, 522, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac {b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac {b^{8/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{8/3} (b c-a d)}+\frac {a d+b c}{2 a^2 c^2 x^2}+\frac {d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}-\frac {d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}+\frac {d^{8/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{8/3} (b c-a d)}-\frac {1}{5 a c x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a + b*x^3)*(c + d*x^3)),x]

[Out]

-1/(5*a*c*x^5) + (b*c + a*d)/(2*a^2*c^2*x^2) - (b^(8/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sq
rt[3]*a^(8/3)*(b*c - a*d)) + (d^(8/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(8/3)*(b*c
- a*d)) + (b^(8/3)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(8/3)*(b*c - a*d)) - (d^(8/3)*Log[c^(1/3) + d^(1/3)*x])/(3*
c^(8/3)*(b*c - a*d)) - (b^(8/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(8/3)*(b*c - a*d)) + (d^(
8/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(8/3)*(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 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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 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 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 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, 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 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^6 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx &=-\frac {1}{5 a c x^5}+\frac {\int \frac {-5 (b c+a d)-5 b d x^3}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{5 a c}\\ &=-\frac {1}{5 a c x^5}+\frac {b c+a d}{2 a^2 c^2 x^2}-\frac {\int \frac {-10 \left (b^2 c^2+a b c d+a^2 d^2\right )-10 b d (b c+a d) x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx}{10 a^2 c^2}\\ &=-\frac {1}{5 a c x^5}+\frac {b c+a d}{2 a^2 c^2 x^2}+\frac {b^3 \int \frac {1}{a+b x^3} \, dx}{a^2 (b c-a d)}-\frac {d^3 \int \frac {1}{c+d x^3} \, dx}{c^2 (b c-a d)}\\ &=-\frac {1}{5 a c x^5}+\frac {b c+a d}{2 a^2 c^2 x^2}+\frac {b^3 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{8/3} (b c-a d)}+\frac {b^3 \int \frac {2 \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^{8/3} (b c-a d)}-\frac {d^3 \int \frac {1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{8/3} (b c-a d)}-\frac {d^3 \int \frac {2 \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^{8/3} (b c-a d)}\\ &=-\frac {1}{5 a c x^5}+\frac {b c+a d}{2 a^2 c^2 x^2}+\frac {b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac {d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}-\frac {b^{8/3} \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^{8/3} (b c-a d)}+\frac {b^3 \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{7/3} (b c-a d)}+\frac {d^{8/3} \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^{8/3} (b c-a d)}-\frac {d^3 \int \frac {1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 c^{7/3} (b c-a d)}\\ &=-\frac {1}{5 a c x^5}+\frac {b c+a d}{2 a^2 c^2 x^2}+\frac {b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac {d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}-\frac {b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac {d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}+\frac {b^{8/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{8/3} (b c-a d)}-\frac {d^{8/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{8/3} (b c-a d)}\\ &=-\frac {1}{5 a c x^5}+\frac {b c+a d}{2 a^2 c^2 x^2}-\frac {b^{8/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{8/3} (b c-a d)}+\frac {d^{8/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{8/3} (b c-a d)}+\frac {b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{8/3} (b c-a d)}-\frac {d^{8/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{8/3} (b c-a d)}-\frac {b^{8/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{8/3} (b c-a d)}+\frac {d^{8/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{8/3} (b c-a d)}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 282, normalized size = 0.88 \begin {gather*} \frac {-\frac {10 b^{8/3} x^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{8/3}}+\frac {10 \sqrt {3} b^{8/3} x^5 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{8/3}}+\frac {5 b^{8/3} x^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{8/3}}-\frac {15 b^2 x^3}{a^2}+\frac {6 b}{a}+\frac {10 d^{8/3} x^5 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{8/3}}-\frac {10 \sqrt {3} d^{8/3} x^5 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{8/3}}-\frac {5 d^{8/3} x^5 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{8/3}}+\frac {15 d^2 x^3}{c^2}-\frac {6 d}{c}}{30 x^5 (a d-b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a + b*x^3)*(c + d*x^3)),x]

[Out]

((6*b)/a - (6*d)/c - (15*b^2*x^3)/a^2 + (15*d^2*x^3)/c^2 + (10*Sqrt[3]*b^(8/3)*x^5*ArcTan[(1 - (2*b^(1/3)*x)/a
^(1/3))/Sqrt[3]])/a^(8/3) - (10*Sqrt[3]*d^(8/3)*x^5*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(8/3) - (10
*b^(8/3)*x^5*Log[a^(1/3) + b^(1/3)*x])/a^(8/3) + (10*d^(8/3)*x^5*Log[c^(1/3) + d^(1/3)*x])/c^(8/3) + (5*b^(8/3
)*x^5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(8/3) - (5*d^(8/3)*x^5*Log[c^(2/3) - c^(1/3)*d^(1/3)*x
+ d^(2/3)*x^2])/c^(8/3))/(30*(-(b*c) + a*d)*x^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^6 \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^6*(a + b*x^3)*(c + d*x^3)),x]

[Out]

IntegrateAlgebraic[1/(x^6*(a + b*x^3)*(c + d*x^3)), x]

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fricas [A]  time = 1.04, size = 356, normalized size = 1.11 \begin {gather*} -\frac {10 \, \sqrt {3} b^{2} c^{2} x^{5} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 10 \, \sqrt {3} a^{2} d^{2} x^{5} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} c x \left (\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}} - \sqrt {3} d}{3 \, d}\right ) - 5 \, b^{2} c^{2} x^{5} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - 5 \, a^{2} d^{2} x^{5} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d^{2} x^{2} - c d x \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} + c^{2} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}}\right ) + 10 \, b^{2} c^{2} x^{5} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 10 \, a^{2} d^{2} x^{5} \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d x + c \left (\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}}\right ) + 6 \, a b c^{2} - 6 \, a^{2} c d - 15 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{3}}{30 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^3+a)/(d*x^3+c),x, algorithm="fricas")

[Out]

-1/30*(10*sqrt(3)*b^2*c^2*x^5*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b^2/a^2)^(2/3) - sqrt(3)*b)/b) + 10
*sqrt(3)*a^2*d^2*x^5*(d^2/c^2)^(1/3)*arctan(1/3*(2*sqrt(3)*c*x*(d^2/c^2)^(2/3) - sqrt(3)*d)/d) - 5*b^2*c^2*x^5
*(-b^2/a^2)^(1/3)*log(b^2*x^2 + a*b*x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 5*a^2*d^2*x^5*(d^2/c^2)^(1/3)
*log(d^2*x^2 - c*d*x*(d^2/c^2)^(1/3) + c^2*(d^2/c^2)^(2/3)) + 10*b^2*c^2*x^5*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^
2/a^2)^(1/3)) + 10*a^2*d^2*x^5*(d^2/c^2)^(1/3)*log(d*x + c*(d^2/c^2)^(1/3)) + 6*a*b*c^2 - 6*a^2*c*d - 15*(b^2*
c^2 - a^2*d^2)*x^3)/((a^2*b*c^3 - a^3*c^2*d)*x^5)

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giac [A]  time = 0.21, size = 336, normalized size = 1.05 \begin {gather*} -\frac {b^{3} \left (-\frac {a}{b}\right )^{\frac {1}{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 {1}{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 {1}{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^{3} b c - \sqrt {3} a^{4} d} - \frac {\left (-c d^{2}\right )^{\frac {1}{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^{4} - \sqrt {3} a c^{3} d} + \frac {\left (-a b^{2}\right )^{\frac {1}{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^{3} b c - a^{4} d\right )}} - \frac {\left (-c d^{2}\right )^{\frac {1}{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^{4} - a c^{3} d\right )}} + \frac {5 \, b c x^{3} + 5 \, a d x^{3} - 2 \, a c}{10 \, a^{2} c^{2} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^3+a)/(d*x^3+c),x, algorithm="giac")

[Out]

-1/3*b^3*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*b*c - a^4*d) + 1/3*d^3*(-c/d)^(1/3)*log(abs(x - (-c/d)^(
1/3)))/(b*c^4 - a*c^3*d) + (-a*b^2)^(1/3)*b^2*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)^(1/3)*d^2*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)^(1/3)*b^2*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b*c - a^4*d) - 1
/6*(-c*d^2)^(1/3)*d^2*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b*c^4 - a*c^3*d) + 1/10*(5*b*c*x^3 + 5*a*d*x^3
- 2*a*c)/(a^2*c^2*x^5)

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maple [A]  time = 0.05, size = 293, normalized size = 0.91 \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 {2}{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 {2}{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 {2}{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 {2}{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 {2}{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 {2}{3}} c^{2}}+\frac {d}{2 a \,c^{2} x^{2}}+\frac {b}{2 a^{2} c \,x^{2}}-\frac {1}{5 a c \,x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(b*x^3+a)/(d*x^3+c),x)

[Out]

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

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maxima [A]  time = 1.23, size = 369, normalized size = 1.15 \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 \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{3} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\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} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c^{2} d \left (\frac {c}{d}\right )^{\frac {1}{3}}\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 {2}{3}} - a^{3} d \left (\frac {a}{b}\right )^{\frac {2}{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 {2}{3}} - a c^{2} d \left (\frac {c}{d}\right )^{\frac {2}{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 {2}{3}} - a^{3} d \left (\frac {a}{b}\right )^{\frac {2}{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 {2}{3}} - a c^{2} d \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} + \frac {5 \, {\left (b c + a d\right )} x^{3} - 2 \, a c}{10 \, a^{2} c^{2} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(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/b)^(1/3) - a^3*d*(a/b)^(1/3))
*(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*(c/d)^(1/3) - a*c^
2*d*(c/d)^(1/3))*(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)^(2/3) - a^3*d*(a
/b)^(2/3)) + 1/6*d^2*log(x^2 - x*(c/d)^(1/3) + (c/d)^(2/3))/(b*c^3*(c/d)^(2/3) - a*c^2*d*(c/d)^(2/3)) + 1/3*b^
2*log(x + (a/b)^(1/3))/(a^2*b*c*(a/b)^(2/3) - a^3*d*(a/b)^(2/3)) - 1/3*d^2*log(x + (c/d)^(1/3))/(b*c^3*(c/d)^(
2/3) - a*c^2*d*(c/d)^(2/3)) + 1/10*(5*(b*c + a*d)*x^3 - 2*a*c)/(a^2*c^2*x^5)

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mupad [B]  time = 11.57, size = 1860, normalized size = 5.79

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(a + b*x^3)*(c + d*x^3)),x)

[Out]

log(((-b^8/(a^8*(a*d - b*c)^3))^(1/3)*(9*a^13*b^11*c^19*d^5 - 9*a^12*b^12*c^20*d^4 + 9*a^19*b^5*c^13*d^11 - 9*
a^20*b^4*c^12*d^12 + 9*a^16*b^3*c^16*d^3*(a*d + b*c)*(a*d - b*c)^4*(a^3*c^3*(-b^8/(a^8*(a*d - b*c)^3))^(1/3) +
a^2*d^2*x + b^2*c^2*x)*(-b^8/(a^8*(a*d - b*c)^3))^(2/3)))/3 + 3*a^12*b^7*c^12*d^7*x*(a^4*d^4 + b^4*c^4))*(-b^
8/(27*a^11*d^3 - 27*a^8*b^3*c^3 + 81*a^9*b^2*c^2*d - 81*a^10*b*c*d^2))^(1/3) - (1/(5*a*c) - (x^3*(a*d + b*c))/
(2*a^2*c^2))/x^5 + log(((d^8/(c^8*(a*d - b*c)^3))^(1/3)*(9*a^13*b^11*c^19*d^5 - 9*a^12*b^12*c^20*d^4 + 9*a^19*
b^5*c^13*d^11 - 9*a^20*b^4*c^12*d^12 + 9*a^16*b^3*c^16*d^3*(a*d + b*c)*(a*d - b*c)^4*(a^3*c^3*(d^8/(c^8*(a*d -
b*c)^3))^(1/3) + a^2*d^2*x + b^2*c^2*x)*(d^8/(c^8*(a*d - b*c)^3))^(2/3)))/3 + 3*a^12*b^7*c^12*d^7*x*(a^4*d^4
+ b^4*c^4))*(-d^8/(27*b^3*c^11 - 27*a^3*c^8*d^3 + 81*a^2*b*c^9*d^2 - 81*a*b^2*c^10*d))^(1/3) + (log(((3^(1/2)*
1i - 1)*(-b^8/(a^8*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i - 1)^2*(81*a^16*b^3*c^16*d^3*x*(a*d - b*c)^4*(a^3*d^3 +
b^3*c^3 + a*b^2*c^2*d + a^2*b*c*d^2) + (81*a^19*b^3*c^19*d^3*(3^(1/2)*1i - 1)*(a*d + b*c)*(a*d - b*c)^4*(-b^8
/(a^8*(a*d - b*c)^3))^(1/3))/2)*(-b^8/(a^8*(a*d - b*c)^3))^(2/3))/36 - 9*a^12*b^12*c^20*d^4 + 9*a^13*b^11*c^19
*d^5 + 9*a^19*b^5*c^13*d^11 - 9*a^20*b^4*c^12*d^12))/6 + 3*a^12*b^7*c^12*d^7*x*(a^4*d^4 + b^4*c^4))*(-b^8/(27*
a^11*d^3 - 27*a^8*b^3*c^3 + 81*a^9*b^2*c^2*d - 81*a^10*b*c*d^2))^(1/3)*(3^(1/2)*1i - 1))/2 - (log(((3^(1/2)*1i
+ 1)*(-b^8/(a^8*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i + 1)^2*(81*a^16*b^3*c^16*d^3*x*(a*d - b*c)^4*(a^3*d^3 + b
^3*c^3 + a*b^2*c^2*d + a^2*b*c*d^2) - (81*a^19*b^3*c^19*d^3*(3^(1/2)*1i + 1)*(a*d + b*c)*(a*d - b*c)^4*(-b^8/(
a^8*(a*d - b*c)^3))^(1/3))/2)*(-b^8/(a^8*(a*d - b*c)^3))^(2/3))/36 - 9*a^12*b^12*c^20*d^4 + 9*a^13*b^11*c^19*d
^5 + 9*a^19*b^5*c^13*d^11 - 9*a^20*b^4*c^12*d^12))/6 - 3*a^12*b^7*c^12*d^7*x*(a^4*d^4 + b^4*c^4))*(-b^8/(27*a^
11*d^3 - 27*a^8*b^3*c^3 + 81*a^9*b^2*c^2*d - 81*a^10*b*c*d^2))^(1/3)*(3^(1/2)*1i + 1))/2 + (log(((3^(1/2)*1i -
1)*(d^8/(c^8*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i - 1)^2*(81*a^16*b^3*c^16*d^3*x*(a*d - b*c)^4*(a^3*d^3 + b^3*
c^3 + a*b^2*c^2*d + a^2*b*c*d^2) + (81*a^19*b^3*c^19*d^3*(3^(1/2)*1i - 1)*(a*d + b*c)*(a*d - b*c)^4*(d^8/(c^8*
(a*d - b*c)^3))^(1/3))/2)*(d^8/(c^8*(a*d - b*c)^3))^(2/3))/36 - 9*a^12*b^12*c^20*d^4 + 9*a^13*b^11*c^19*d^5 +
9*a^19*b^5*c^13*d^11 - 9*a^20*b^4*c^12*d^12))/6 + 3*a^12*b^7*c^12*d^7*x*(a^4*d^4 + b^4*c^4))*(-d^8/(27*b^3*c^1
1 - 27*a^3*c^8*d^3 + 81*a^2*b*c^9*d^2 - 81*a*b^2*c^10*d))^(1/3)*(3^(1/2)*1i - 1))/2 - (log(((3^(1/2)*1i + 1)*(
d^8/(c^8*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i + 1)^2*(81*a^16*b^3*c^16*d^3*x*(a*d - b*c)^4*(a^3*d^3 + b^3*c^3 +
a*b^2*c^2*d + a^2*b*c*d^2) - (81*a^19*b^3*c^19*d^3*(3^(1/2)*1i + 1)*(a*d + b*c)*(a*d - b*c)^4*(d^8/(c^8*(a*d
- b*c)^3))^(1/3))/2)*(d^8/(c^8*(a*d - b*c)^3))^(2/3))/36 - 9*a^12*b^12*c^20*d^4 + 9*a^13*b^11*c^19*d^5 + 9*a^1
9*b^5*c^13*d^11 - 9*a^20*b^4*c^12*d^12))/6 - 3*a^12*b^7*c^12*d^7*x*(a^4*d^4 + b^4*c^4))*(-d^8/(27*b^3*c^11 - 2
7*a^3*c^8*d^3 + 81*a^2*b*c^9*d^2 - 81*a*b^2*c^10*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**6/(b*x**3+a)/(d*x**3+c),x)

[Out]

Timed out

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