3.5.67 \(\int \frac {1}{x^9 \sqrt [3]{a+b x^3} (c+d x^3)} \, dx\)

Optimal. Leaf size=262 \[ \frac {\left (a+b x^3\right )^{2/3} (4 a d+3 b c)}{20 a^2 c^2 x^5}-\frac {\left (a+b x^3\right )^{2/3} \left (20 a^2 d^2+12 a b c d+9 b^2 c^2\right )}{40 a^3 c^3 x^2}-\frac {d^3 \log \left (c+d x^3\right )}{6 c^{11/3} \sqrt [3]{b c-a d}}+\frac {d^3 \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{11/3} \sqrt [3]{b c-a d}}-\frac {d^3 \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{11/3} \sqrt [3]{b c-a d}}-\frac {\left (a+b x^3\right )^{2/3}}{8 a c x^8} \]

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Rubi [A]  time = 0.32, antiderivative size = 317, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {494, 461, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {\left (a+b x^3\right )^{2/3} \left (a^2 d^2+a b c d+b^2 c^2\right )}{2 a^3 c^3 x^2}+\frac {\left (a+b x^3\right )^{5/3} (a d+2 b c)}{5 a^3 c^2 x^5}-\frac {\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}+\frac {d^3 \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3} \sqrt [3]{b c-a d}}-\frac {d^3 \log \left (\frac {x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{6 c^{11/3} \sqrt [3]{b c-a d}}-\frac {d^3 \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{11/3} \sqrt [3]{b c-a d}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^3)^(1/3)*(c + d*x^3)),x]

[Out]

-((b^2*c^2 + a*b*c*d + a^2*d^2)*(a + b*x^3)^(2/3))/(2*a^3*c^3*x^2) + ((2*b*c + a*d)*(a + b*x^3)^(5/3))/(5*a^3*
c^2*x^5) - (a + b*x^3)^(8/3)/(8*a^3*c*x^8) - (d^3*ArcTan[(c^(1/3) + (2*(b*c - a*d)^(1/3)*x)/(a + b*x^3)^(1/3))
/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(11/3)*(b*c - a*d)^(1/3)) + (d^3*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(a + b*x^
3)^(1/3)])/(3*c^(11/3)*(b*c - a*d)^(1/3)) - (d^3*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(a + b*x^3)^(2/3) + (c^
(1/3)*(b*c - a*d)^(1/3)*x)/(a + b*x^3)^(1/3)])/(6*c^(11/3)*(b*c - a*d)^(1/3))

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 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 494

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p
+ q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 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^9 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-b x^3\right )^3}{x^9 \left (c-(b c-a d) x^3\right )} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{c x^9}+\frac {-2 b c-a d}{c^2 x^6}+\frac {b^2 c^2+a b c d+a^2 d^2}{c^3 x^3}+\frac {a^3 d^3}{c^3 \left (-c+(b c-a d) x^3\right )}\right ) \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{a^3}\\ &=-\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 a^3 c^3 x^2}+\frac {(2 b c+a d) \left (a+b x^3\right )^{5/3}}{5 a^3 c^2 x^5}-\frac {\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}+\frac {d^3 \operatorname {Subst}\left (\int \frac {1}{-c+(b c-a d) x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{c^3}\\ &=-\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 a^3 c^3 x^2}+\frac {(2 b c+a d) \left (a+b x^3\right )^{5/3}}{5 a^3 c^2 x^5}-\frac {\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}+\frac {d^3 \operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{c}+\sqrt [3]{b c-a d} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3}}+\frac {d^3 \operatorname {Subst}\left (\int \frac {-2 \sqrt [3]{c}-\sqrt [3]{b c-a d} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3}}\\ &=-\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 a^3 c^3 x^2}+\frac {(2 b c+a d) \left (a+b x^3\right )^{5/3}}{5 a^3 c^2 x^5}-\frac {\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}+\frac {d^3 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3} \sqrt [3]{b c-a d}}-\frac {d^3 \operatorname {Subst}\left (\int \frac {1}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{2 c^{10/3}}-\frac {d^3 \operatorname {Subst}\left (\int \frac {\sqrt [3]{c} \sqrt [3]{b c-a d}+2 (b c-a d)^{2/3} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{6 c^{11/3} \sqrt [3]{b c-a d}}\\ &=-\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 a^3 c^3 x^2}+\frac {(2 b c+a d) \left (a+b x^3\right )^{5/3}}{5 a^3 c^2 x^5}-\frac {\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}+\frac {d^3 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3} \sqrt [3]{b c-a d}}-\frac {d^3 \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{6 c^{11/3} \sqrt [3]{b c-a d}}+\frac {d^3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{c^{11/3} \sqrt [3]{b c-a d}}\\ &=-\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{2 a^3 c^3 x^2}+\frac {(2 b c+a d) \left (a+b x^3\right )^{5/3}}{5 a^3 c^2 x^5}-\frac {\left (a+b x^3\right )^{8/3}}{8 a^3 c x^8}-\frac {d^3 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{11/3} \sqrt [3]{b c-a d}}+\frac {d^3 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{11/3} \sqrt [3]{b c-a d}}-\frac {d^3 \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{6 c^{11/3} \sqrt [3]{b c-a d}}\\ \end {align*}

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Mathematica [C]  time = 2.55, size = 821, normalized size = 3.13 \begin {gather*} -\frac {648 b c d^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}-297 a d^4 \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+297 b c d^3 \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^{12}+648 a c d^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+216 b c^2 d^2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9-243 a c d^3 \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+243 b c^2 d^2 \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^9+216 a c^2 d^2 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-72 b c^3 d \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+45 a c^2 d^2 \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6-45 b c^3 d \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^6+40 b c^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-72 a c^3 d \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+9 b c^4 \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-9 a c^3 d \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3-54 (b c-a d) \left (c-3 d x^3\right ) \left (d x^3+c\right )^2 \, _3F_2\left (\frac {4}{3},2,2;1,\frac {7}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+27 (b c-a d) \left (d x^3+c\right )^3 \, _4F_3\left (\frac {4}{3},2,2,2;1,1,\frac {7}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right ) x^3+40 a c^4 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{320 c^5 x^8 \left (b x^3+a\right )^{4/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^9*(a + b*x^3)^(1/3)*(c + d*x^3)),x]

[Out]

-1/320*(40*a*c^4*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 40*b*c^4*x^3*Hypergeometr
ic2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 72*a*c^3*d*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c -
a*d)*x^3)/(c*(a + b*x^3))] - 72*b*c^3*d*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]
+ 216*a*c^2*d^2*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 216*b*c^2*d^2*x^9*Hype
rgeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 648*a*c*d^3*x^9*Hypergeometric2F1[1/3, 1, 4/3,
((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 648*b*c*d^3*x^12*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a
+ b*x^3))] + 9*b*c^4*x^3*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 9*a*c^3*d*x^3*Hyp
ergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 45*b*c^3*d*x^6*Hypergeometric2F1[4/3, 2, 7/3,
((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 45*a*c^2*d^2*x^6*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a
+ b*x^3))] + 243*b*c^2*d^2*x^9*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 243*a*c*d^3
*x^9*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 297*b*c*d^3*x^12*Hypergeometric2F1[4/
3, 2, 7/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 297*a*d^4*x^12*Hypergeometric2F1[4/3, 2, 7/3, ((b*c - a*d)*x^3
)/(c*(a + b*x^3))] - 54*(b*c - a*d)*x^3*(c - 3*d*x^3)*(c + d*x^3)^2*HypergeometricPFQ[{4/3, 2, 2}, {1, 7/3}, (
(b*c - a*d)*x^3)/(c*(a + b*x^3))] + 27*(b*c - a*d)*x^3*(c + d*x^3)^3*HypergeometricPFQ[{4/3, 2, 2, 2}, {1, 1,
7/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^5*x^8*(a + b*x^3)^(4/3))

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IntegrateAlgebraic [C]  time = 3.11, size = 419, normalized size = 1.60 \begin {gather*} \frac {\left (a+b x^3\right )^{2/3} \left (-5 a^2 c^2+8 a^2 c d x^3-20 a^2 d^2 x^6+6 a b c^2 x^3-12 a b c d x^6-9 b^2 c^2 x^6\right )}{40 a^3 c^3 x^8}-\frac {i \left (\sqrt {3} d^3-i d^3\right ) \log \left (2 x \sqrt [3]{b c-a d}+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{6 c^{11/3} \sqrt [3]{b c-a d}}+\frac {\sqrt {-1+i \sqrt {3}} d^3 \tan ^{-1}\left (\frac {3 x \sqrt [3]{b c-a d}}{\sqrt {3} x \sqrt [3]{b c-a d}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{a+b x^3}-3 i \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{\sqrt {6} c^{11/3} \sqrt [3]{b c-a d}}+\frac {\left (d^3+i \sqrt {3} d^3\right ) \log \left (\left (\sqrt {3}+i\right ) c^{2/3} \left (a+b x^3\right )^{2/3}+\sqrt [3]{c} \left (-\sqrt {3} x+i x\right ) \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}-2 i x^2 (b c-a d)^{2/3}\right )}{12 c^{11/3} \sqrt [3]{b c-a d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^9*(a + b*x^3)^(1/3)*(c + d*x^3)),x]

[Out]

((a + b*x^3)^(2/3)*(-5*a^2*c^2 + 6*a*b*c^2*x^3 + 8*a^2*c*d*x^3 - 9*b^2*c^2*x^6 - 12*a*b*c*d*x^6 - 20*a^2*d^2*x
^6))/(40*a^3*c^3*x^8) + (Sqrt[-1 + I*Sqrt[3]]*d^3*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*(b*c - a*d)^(1/3)*x
- (3*I)*c^(1/3)*(a + b*x^3)^(1/3) - Sqrt[3]*c^(1/3)*(a + b*x^3)^(1/3))])/(Sqrt[6]*c^(11/3)*(b*c - a*d)^(1/3))
- ((I/6)*((-I)*d^3 + Sqrt[3]*d^3)*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3)])/(c^(
11/3)*(b*c - a*d)^(1/3)) + ((d^3 + I*Sqrt[3]*d^3)*Log[(-2*I)*(b*c - a*d)^(2/3)*x^2 + c^(1/3)*(b*c - a*d)^(1/3)
*(I*x - Sqrt[3]*x)*(a + b*x^3)^(1/3) + (I + Sqrt[3])*c^(2/3)*(a + b*x^3)^(2/3)])/(12*c^(11/3)*(b*c - a*d)^(1/3
))

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )} x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(1/3)*(d*x^3 + c)*x^9), x)

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maple [F]  time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (d \,x^{3}+c \right ) x^{9}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^9/(b*x^3+a)^(1/3)/(d*x^3+c),x)

[Out]

int(1/x^9/(b*x^3+a)^(1/3)/(d*x^3+c),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )} x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(b*x^3+a)^(1/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(1/3)*(d*x^3 + c)*x^9), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^9\,{\left (b\,x^3+a\right )}^{1/3}\,\left (d\,x^3+c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^9*(a + b*x^3)^(1/3)*(c + d*x^3)),x)

[Out]

int(1/(x^9*(a + b*x^3)^(1/3)*(c + d*x^3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{9} \sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(b*x**3+a)**(1/3)/(d*x**3+c),x)

[Out]

Integral(1/(x**9*(a + b*x**3)**(1/3)*(c + d*x**3)), x)

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