3.6.70 \(\int \frac {x^2 \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx\) [570]

Optimal. Leaf size=150 \[ -\frac {\sqrt [3]{a+b x^3}}{b d}+\frac {\sqrt [3]{2} \sqrt [3]{a} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b d}+\frac {\sqrt [3]{a} \log \left (a-b x^3\right )}{3\ 2^{2/3} b d}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b d} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {455, 52, 59, 631, 210, 31} \begin {gather*} \frac {\sqrt [3]{2} \sqrt [3]{a} \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 d}-\frac {\sqrt [3]{a+b x^3}}{b d}+\frac {\sqrt [3]{a} \log \left (a-b x^3\right )}{3\ 2^{2/3} b d}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]

[Out]

-((a + b*x^3)^(1/3)/(b*d)) + (2^(1/3)*a^(1/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + b*x^3)^(1/3))/(Sqrt[3]*a^(1/3))])
/(Sqrt[3]*b*d) + (a^(1/3)*Log[a - b*x^3])/(3*2^(2/3)*b*d) - (a^(1/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)])
/(2^(2/3)*b*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 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 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

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

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Mathematica [A]
time = 0.13, size = 169, normalized size = 1.13 \begin {gather*} \frac {-6 \sqrt [3]{a+b x^3}+2 \sqrt [3]{2} \sqrt {3} \sqrt [3]{a} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \sqrt [3]{2} \sqrt [3]{a} \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )+\sqrt [3]{2} \sqrt [3]{a} \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 )}{6 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x]

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{-b d \,x^{3}+a d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^3+a)^(1/3)/(-b*d*x^3+a*d),x)

[Out]

int(x^2*(b*x^3+a)^(1/3)/(-b*d*x^3+a*d),x)

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Maxima [A]
time = 0.53, size = 139, normalized size = 0.93 \begin {gather*} \frac {\frac {2 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {1}{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 {2^{\frac {1}{3}} a^{\frac {1}{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 {2 \cdot 2^{\frac {1}{3}} a^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}{d} - \frac {6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{d}}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^(1/3)/(-b*d*x^3+a*d),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.48, size = 144, normalized size = 0.96 \begin {gather*} -\frac {2 \, \sqrt {3} 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{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 ) + 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{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 ) - 2 \cdot 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} \log \left (2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{6 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^(1/3)/(-b*d*x^3+a*d),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {x^{2} \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**2*(b*x**3+a)**(1/3)/(-b*d*x**3+a*d),x)

[Out]

-Integral(x**2*(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^2*(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 = 4.64, size = 194, normalized size = 1.29 \begin {gather*} \frac {2^{1/3}\,{\left (-a\right )}^{1/3}\,\ln \left (6\,a\,{\left (b\,x^3+a\right )}^{1/3}-6\,2^{1/3}\,{\left (-a\right )}^{4/3}\right )}{3\,b\,d}-\frac {{\left (b\,x^3+a\right )}^{1/3}}{b\,d}+\frac {2^{1/3}\,{\left (-a\right )}^{1/3}\,\ln \left (\frac {6\,a\,{\left (b\,x^3+a\right )}^{1/3}}{b\,d}-\frac {6\,2^{1/3}\,{\left (-a\right )}^{4/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b\,d}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b\,d}-\frac {2^{1/3}\,{\left (-a\right )}^{1/3}\,\ln \left (\frac {6\,a\,{\left (b\,x^3+a\right )}^{1/3}}{b\,d}+\frac {6\,2^{1/3}\,{\left (-a\right )}^{4/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b\,d}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(a + b*x^3)^(1/3))/(a*d - b*d*x^3),x)

[Out]

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

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