3.6.67 \(\int \frac {x^{11} \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx\) [567]

Optimal. Leaf size=220 \[ -\frac {a^3 \sqrt [3]{a+b x^3}}{b^4 d}-\frac {a^2 \left (a+b x^3\right )^{4/3}}{4 b^4 d}+\frac {a \left (a+b x^3\right )^{7/3}}{7 b^4 d}-\frac {\left (a+b x^3\right )^{10/3}}{10 b^4 d}+\frac {\sqrt [3]{2} a^{10/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^4 d}+\frac {a^{10/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^4 d}-\frac {a^{10/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^4 d} \]

[Out]

-a^3*(b*x^3+a)^(1/3)/b^4/d-1/4*a^2*(b*x^3+a)^(4/3)/b^4/d+1/7*a*(b*x^3+a)^(7/3)/b^4/d-1/10*(b*x^3+a)^(10/3)/b^4
/d+1/6*a^(10/3)*ln(-b*x^3+a)*2^(1/3)/b^4/d-1/2*a^(10/3)*ln(2^(1/3)*a^(1/3)-(b*x^3+a)^(1/3))*2^(1/3)/b^4/d+1/3*
2^(1/3)*a^(10/3)*arctan(1/3*(a^(1/3)+2^(2/3)*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))/b^4/d*3^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 90, 52, 59, 631, 210, 31} \begin {gather*} \frac {\sqrt [3]{2} a^{10/3} \text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^4 d}+\frac {a^{10/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^4 d}-\frac {a^{10/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^4 d}-\frac {a^3 \sqrt [3]{a+b x^3}}{b^4 d}-\frac {a^2 \left (a+b x^3\right )^{4/3}}{4 b^4 d}+\frac {a \left (a+b x^3\right )^{7/3}}{7 b^4 d}-\frac {\left (a+b x^3\right )^{10/3}}{10 b^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^11*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]

[Out]

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

Rule 31

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 59

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 210

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

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 631

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]

Rubi steps

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

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Mathematica [A]
time = 0.23, size = 202, normalized size = 0.92 \begin {gather*} \frac {-3 \sqrt [3]{a+b x^3} \left (169 a^3+37 a^2 b x^3+22 a b^2 x^6+14 b^3 x^9\right )+140 \sqrt [3]{2} \sqrt {3} a^{10/3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-140 \sqrt [3]{2} a^{10/3} \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )+70 \sqrt [3]{2} a^{10/3} \log \left (2 a^{2/3}+2^{2/3} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{420 b^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^11*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]

[Out]

(-3*(a + b*x^3)^(1/3)*(169*a^3 + 37*a^2*b*x^3 + 22*a*b^2*x^6 + 14*b^3*x^9) + 140*2^(1/3)*Sqrt[3]*a^(10/3)*ArcT
an[(1 + (2^(2/3)*(a + b*x^3)^(1/3))/a^(1/3))/Sqrt[3]] - 140*2^(1/3)*a^(10/3)*Log[-2*a^(1/3) + 2^(2/3)*(a + b*x
^3)^(1/3)] + 70*2^(1/3)*a^(10/3)*Log[2*a^(2/3) + 2^(2/3)*a^(1/3)*(a + b*x^3)^(1/3) + 2^(1/3)*(a + b*x^3)^(2/3)
])/(420*b^4*d)

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{11} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{-b d \,x^{3}+a d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(b*x^3+a)^(1/3)/(-b*d*x^3+a*d),x)

[Out]

int(x^11*(b*x^3+a)^(1/3)/(-b*d*x^3+a*d),x)

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Maxima [A]
time = 0.51, size = 183, normalized size = 0.83 \begin {gather*} \frac {\frac {140 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {10}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{d} + \frac {70 \cdot 2^{\frac {1}{3}} a^{\frac {10}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right )}{d} - \frac {140 \cdot 2^{\frac {1}{3}} a^{\frac {10}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}{d} - \frac {3 \, {\left (14 \, {\left (b x^{3} + a\right )}^{\frac {10}{3}} - 20 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a + 35 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a^{2} + 140 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{3}\right )}}{d}}{420 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^(1/3)/(-b*d*x^3+a*d),x, algorithm="maxima")

[Out]

1/420*(140*sqrt(3)*2^(1/3)*a^(10/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a^(1/3) + 2*(b*x^3 + a)^(1/3))/a^(1/3)
)/d + 70*2^(1/3)*a^(10/3)*log(2^(2/3)*a^(2/3) + 2^(1/3)*(b*x^3 + a)^(1/3)*a^(1/3) + (b*x^3 + a)^(2/3))/d - 140
*2^(1/3)*a^(10/3)*log(-2^(1/3)*a^(1/3) + (b*x^3 + a)^(1/3))/d - 3*(14*(b*x^3 + a)^(10/3) - 20*(b*x^3 + a)^(7/3
)*a + 35*(b*x^3 + a)^(4/3)*a^2 + 140*(b*x^3 + a)^(1/3)*a^3)/d)/b^4

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Fricas [A]
time = 3.16, size = 186, normalized size = 0.85 \begin {gather*} -\frac {140 \, \sqrt {3} 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a^{3} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + 70 \cdot 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a^{3} \log \left (2^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right ) - 140 \cdot 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a^{3} \log \left (2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (14 \, b^{3} x^{9} + 22 \, a b^{2} x^{6} + 37 \, a^{2} b x^{3} + 169 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{420 \, b^{4} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^(1/3)/(-b*d*x^3+a*d),x, algorithm="fricas")

[Out]

-1/420*(140*sqrt(3)*2^(1/3)*(-a)^(1/3)*a^3*arctan(1/3*(sqrt(3)*2^(2/3)*(b*x^3 + a)^(1/3)*(-a)^(2/3) + sqrt(3)*
a)/a) + 70*2^(1/3)*(-a)^(1/3)*a^3*log(2^(2/3)*(-a)^(2/3) - 2^(1/3)*(b*x^3 + a)^(1/3)*(-a)^(1/3) + (b*x^3 + a)^
(2/3)) - 140*2^(1/3)*(-a)^(1/3)*a^3*log(2^(1/3)*(-a)^(1/3) + (b*x^3 + a)^(1/3)) + 3*(14*b^3*x^9 + 22*a*b^2*x^6
+ 37*a^2*b*x^3 + 169*a^3)*(b*x^3 + a)^(1/3))/(b^4*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {x^{11} \sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(b*x**3+a)**(1/3)/(-b*d*x**3+a*d),x)

[Out]

-Integral(x**11*(a + b*x**3)**(1/3)/(-a + b*x**3), x)/d

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^(1/3)/(-b*d*x^3+a*d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Algebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a r

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Mupad [B]
time = 4.81, size = 240, normalized size = 1.09 \begin {gather*} \frac {a\,{\left (b\,x^3+a\right )}^{7/3}}{7\,b^4\,d}-\frac {a^3\,{\left (b\,x^3+a\right )}^{1/3}}{b^4\,d}-\frac {a^2\,{\left (b\,x^3+a\right )}^{4/3}}{4\,b^4\,d}-\frac {{\left (b\,x^3+a\right )}^{10/3}}{10\,b^4\,d}-\frac {2^{1/3}\,a^{10/3}\,\ln \left ({\left (b\,x^3+a\right )}^{1/3}-2^{1/3}\,a^{1/3}\right )}{3\,b^4\,d}-\frac {2^{1/3}\,a^{10/3}\,\ln \left (\frac {6\,a^4\,{\left (b\,x^3+a\right )}^{1/3}}{b^4\,d}-\frac {6\,2^{1/3}\,a^{13/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^4\,d}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^4\,d}+\frac {2^{1/3}\,a^{10/3}\,\ln \left (\frac {6\,a^4\,{\left (b\,x^3+a\right )}^{1/3}}{b^4\,d}+\frac {18\,2^{1/3}\,a^{13/3}\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^4\,d}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^4\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^11*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x)

[Out]

(a*(a + b*x^3)^(7/3))/(7*b^4*d) - (a^3*(a + b*x^3)^(1/3))/(b^4*d) - (a^2*(a + b*x^3)^(4/3))/(4*b^4*d) - (a + b
*x^3)^(10/3)/(10*b^4*d) - (2^(1/3)*a^(10/3)*log((a + b*x^3)^(1/3) - 2^(1/3)*a^(1/3)))/(3*b^4*d) - (2^(1/3)*a^(
10/3)*log((6*a^4*(a + b*x^3)^(1/3))/(b^4*d) - (6*2^(1/3)*a^(13/3)*((3^(1/2)*1i)/2 - 1/2))/(b^4*d))*((3^(1/2)*1
i)/2 - 1/2))/(3*b^4*d) + (2^(1/3)*a^(10/3)*log((6*a^4*(a + b*x^3)^(1/3))/(b^4*d) + (18*2^(1/3)*a^(13/3)*((3^(1
/2)*1i)/6 + 1/6))/(b^4*d))*((3^(1/2)*1i)/6 + 1/6))/(b^4*d)

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