3.25.41 \(\int \frac {1}{(a+b (c x^n)^{3/n})^3} \, dx\)

Optimal. Leaf size=235 \[ -\frac {5 x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac {5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2} \]

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Rubi [A]  time = 0.12, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {254, 199, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {5 x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^n)^(3/n))^(-3),x]

[Out]

x/(6*a*(a + b*(c*x^n)^(3/n))^2) + (5*x)/(18*a^2*(a + b*(c*x^n)^(3/n))) - (5*x*ArcTan[(a^(1/3) - 2*b^(1/3)*(c*x
^n)^n^(-1))/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(1/3)*(c*x^n)^n^(-1)) + (5*x*Log[a^(1/3) + b^(1/3)*(c*x^n
)^n^(-1)])/(27*a^(8/3)*b^(1/3)*(c*x^n)^n^(-1)) - (5*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c*x^n)^n^(-1) + b^(2/3)*(
c*x^n)^(2/n)])/(54*a^(8/3)*b^(1/3)*(c*x^n)^n^(-1))

Rule 31

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 254

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> Dist[x/(c*x^q)^(1/q), Subst[Int[(a + b*x^(n*q))
^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

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}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^3\right )^3} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )\\ &=\frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^3\right )^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{6 a}\\ &=\frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{9 a^2}\\ &=\frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3}}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\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,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3}}\\ &=\frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{18 a^{7/3}}-\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\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,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{54 a^{8/3} \sqrt [3]{b}}\\ &=\frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}\right )}{9 a^{8/3} \sqrt [3]{b}}\\ &=\frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}-\frac {5 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{54 a^{8/3} \sqrt [3]{b}}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 215, normalized size = 0.91 \begin {gather*} \frac {x \left (-\frac {5 \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{\sqrt [3]{b}}+\frac {15 a^{2/3}}{a+b \left (c x^n\right )^{3/n}}+\frac {9 a^{5/3}}{\left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {10 \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{\sqrt [3]{b}}-\frac {10 \sqrt {3} \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}\right )}{54 a^{8/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x^n)^(3/n))^(-3),x]

[Out]

(x*((9*a^(5/3))/(a + b*(c*x^n)^(3/n))^2 + (15*a^(2/3))/(a + b*(c*x^n)^(3/n)) - (10*Sqrt[3]*ArcTan[(1 - (2*b^(1
/3)*(c*x^n)^n^(-1))/a^(1/3))/Sqrt[3]])/(b^(1/3)*(c*x^n)^n^(-1)) + (10*Log[a^(1/3) + b^(1/3)*(c*x^n)^n^(-1)])/(
b^(1/3)*(c*x^n)^n^(-1)) - (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c*x^n)^n^(-1) + b^(2/3)*(c*x^n)^(2/n)])/(b^(1/3)*(
c*x^n)^n^(-1))))/(54*a^(8/3))

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IntegrateAlgebraic [F]  time = 0.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*(c*x^n)^(3/n))^(-3),x]

[Out]

Defer[IntegrateAlgebraic][(a + b*(c*x^n)^(3/n))^(-3), x]

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fricas [B]  time = 0.73, size = 885, normalized size = 3.77 \begin {gather*} \left [\frac {15 \, a^{2} b^{2} c^{\frac {6}{n}} x^{4} + 24 \, a^{3} b c^{\frac {3}{n}} x + 15 \, \sqrt {\frac {1}{3}} {\left (a b^{3} c^{\frac {9}{n}} x^{6} + 2 \, a^{2} b^{2} c^{\frac {6}{n}} x^{3} + a^{3} b c^{\frac {3}{n}}\right )} \sqrt {-\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}} \log \left (\frac {2 \, a b c^{\frac {3}{n}} x^{3} - 3 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b c^{\frac {3}{n}} x^{2} + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}}}{b c^{\frac {3}{n}} x^{3} + a}\right ) - 5 \, {\left (b^{2} c^{\frac {6}{n}} x^{6} + 2 \, a b c^{\frac {3}{n}} x^{3} + a^{2}\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x^{2} - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right ) + 10 \, {\left (b^{2} c^{\frac {6}{n}} x^{6} + 2 \, a b c^{\frac {3}{n}} x^{3} + a^{2}\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{4} b^{3} c^{\frac {9}{n}} x^{6} + 2 \, a^{5} b^{2} c^{\frac {6}{n}} x^{3} + a^{6} b c^{\frac {3}{n}}\right )}}, \frac {15 \, a^{2} b^{2} c^{\frac {6}{n}} x^{4} + 24 \, a^{3} b c^{\frac {3}{n}} x + 30 \, \sqrt {\frac {1}{3}} {\left (a b^{3} c^{\frac {9}{n}} x^{6} + 2 \, a^{2} b^{2} c^{\frac {6}{n}} x^{3} + a^{3} b c^{\frac {3}{n}}\right )} \sqrt {\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}}}{a^{2}}\right ) - 5 \, {\left (b^{2} c^{\frac {6}{n}} x^{6} + 2 \, a b c^{\frac {3}{n}} x^{3} + a^{2}\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x^{2} - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right ) + 10 \, {\left (b^{2} c^{\frac {6}{n}} x^{6} + 2 \, a b c^{\frac {3}{n}} x^{3} + a^{2}\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{4} b^{3} c^{\frac {9}{n}} x^{6} + 2 \, a^{5} b^{2} c^{\frac {6}{n}} x^{3} + a^{6} b c^{\frac {3}{n}}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/54*(15*a^2*b^2*c^(6/n)*x^4 + 24*a^3*b*c^(3/n)*x + 15*sqrt(1/3)*(a*b^3*c^(9/n)*x^6 + 2*a^2*b^2*c^(6/n)*x^3 +
a^3*b*c^(3/n))*sqrt(-(a^2*b*c^(3/n))^(1/3)/(b*c^(3/n)))*log((2*a*b*c^(3/n)*x^3 - 3*(a^2*b*c^(3/n))^(1/3)*a*x
- a^2 + 3*sqrt(1/3)*(2*a*b*c^(3/n)*x^2 + (a^2*b*c^(3/n))^(2/3)*x - (a^2*b*c^(3/n))^(1/3)*a)*sqrt(-(a^2*b*c^(3/
n))^(1/3)/(b*c^(3/n))))/(b*c^(3/n)*x^3 + a)) - 5*(b^2*c^(6/n)*x^6 + 2*a*b*c^(3/n)*x^3 + a^2)*(a^2*b*c^(3/n))^(
2/3)*log(a*b*c^(3/n)*x^2 - (a^2*b*c^(3/n))^(2/3)*x + (a^2*b*c^(3/n))^(1/3)*a) + 10*(b^2*c^(6/n)*x^6 + 2*a*b*c^
(3/n)*x^3 + a^2)*(a^2*b*c^(3/n))^(2/3)*log(a*b*c^(3/n)*x + (a^2*b*c^(3/n))^(2/3)))/(a^4*b^3*c^(9/n)*x^6 + 2*a^
5*b^2*c^(6/n)*x^3 + a^6*b*c^(3/n)), 1/54*(15*a^2*b^2*c^(6/n)*x^4 + 24*a^3*b*c^(3/n)*x + 30*sqrt(1/3)*(a*b^3*c^
(9/n)*x^6 + 2*a^2*b^2*c^(6/n)*x^3 + a^3*b*c^(3/n))*sqrt((a^2*b*c^(3/n))^(1/3)/(b*c^(3/n)))*arctan(sqrt(1/3)*(2
*(a^2*b*c^(3/n))^(2/3)*x - (a^2*b*c^(3/n))^(1/3)*a)*sqrt((a^2*b*c^(3/n))^(1/3)/(b*c^(3/n)))/a^2) - 5*(b^2*c^(6
/n)*x^6 + 2*a*b*c^(3/n)*x^3 + a^2)*(a^2*b*c^(3/n))^(2/3)*log(a*b*c^(3/n)*x^2 - (a^2*b*c^(3/n))^(2/3)*x + (a^2*
b*c^(3/n))^(1/3)*a) + 10*(b^2*c^(6/n)*x^6 + 2*a*b*c^(3/n)*x^3 + a^2)*(a^2*b*c^(3/n))^(2/3)*log(a*b*c^(3/n)*x +
(a^2*b*c^(3/n))^(2/3)))/(a^4*b^3*c^(9/n)*x^6 + 2*a^5*b^2*c^(6/n)*x^3 + a^6*b*c^(3/n))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x^n)^(3/n)*b + a)^(-3), x)

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maple [C]  time = 0.40, size = 981, normalized size = 4.17

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*x*(5*b*c^(3/n)*(x^n)^(3/n)*exp(3/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(-csgn(I*x^n)+csgn(I*c*x^n))/n*csgn(I*c
*x^n))+8*a)/a^2/(b*c^(3/n)*(x^n)^(3/n)*exp(3/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(-csgn(I*x^n)+csgn(I*c*x^n))/n*c
sgn(I*c*x^n))+a)^2+5/27/a^2/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(-csgn(I*x^n
)+csgn(I*c*x^n))/n*csgn(I*c*x^n))/(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(-c
sgn(I*x^n)+csgn(I*c*x^n))/n*csgn(I*c*x^n)))^(2/3)*ln(x+(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*(csgn(I*
c)-csgn(I*c*x^n))*(-csgn(I*x^n)+csgn(I*c*x^n))/n*csgn(I*c*x^n)))^(1/3))-5/54/a^2/b/(c^(3/n))/((x^n)^(3/n))*x^3
*exp(-3/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(-csgn(I*x^n)+csgn(I*c*x^n))/n*csgn(I*c*x^n))/(a/b/(c^(3/n))/((x^n)^(
3/n))*x^3*exp(-3/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(-csgn(I*x^n)+csgn(I*c*x^n))/n*csgn(I*c*x^n)))^(2/3)*ln(x^2-
(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(-csgn(I*x^n)+csgn(I*c*x^n))/n*csgn(I
*c*x^n)))^(1/3)*x+(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(-csgn(I*x^n)+csgn(
I*c*x^n))/n*csgn(I*c*x^n)))^(2/3))+5/27/a^2/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*(csgn(I*c)-csgn(I*c*x^
n))*(-csgn(I*x^n)+csgn(I*c*x^n))/n*csgn(I*c*x^n))/(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*(csgn(I*c)-cs
gn(I*c*x^n))*(-csgn(I*x^n)+csgn(I*c*x^n))/n*csgn(I*c*x^n)))^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b/(c^(3/n))
/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*(csgn(I*c)-csgn(I*c*x^n))*(-csgn(I*x^n)+csgn(I*c*x^n))/n*csgn(I*c*x^n)))^(1/3
)*x-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {5 \, b c^{\frac {3}{n}} x {\left (x^{n}\right )}^{\frac {3}{n}} + 8 \, a x}{18 \, {\left (a^{2} b^{2} c^{\frac {6}{n}} {\left (x^{n}\right )}^{\frac {6}{n}} + 2 \, a^{3} b c^{\frac {3}{n}} {\left (x^{n}\right )}^{\frac {3}{n}} + a^{4}\right )}} + 5 \, \int \frac {1}{9 \, {\left (a^{2} b c^{\frac {3}{n}} {\left (x^{n}\right )}^{\frac {3}{n}} + a^{3}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(5*b*c^(3/n)*x*(x^n)^(3/n) + 8*a*x)/(a^2*b^2*c^(6/n)*(x^n)^(6/n) + 2*a^3*b*c^(3/n)*(x^n)^(3/n) + a^4) + 5
*integrate(1/9/(a^2*b*c^(3/n)*(x^n)^(3/n) + a^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*(c*x^n)^(3/n))^3,x)

[Out]

int(1/(a + b*(c*x^n)^(3/n))^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*(c*x**n)**(3/n))**(-3), x)

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