3.4.62 \(\int \frac {x^2 (a+b x^3)^{2/3}}{a d-b d x^3} \, dx\)

Optimal. Leaf size=153 \[ \frac {a^{2/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b d}-\frac {a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b d}-\frac {2^{2/3} a^{2/3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b d}-\frac {\left (a+b x^3\right )^{2/3}}{2 b d} \]

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Rubi [A]  time = 0.13, antiderivative size = 153, 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 = {444, 50, 55, 617, 204, 31} \begin {gather*} \frac {a^{2/3} \log \left (a-b x^3\right )}{3 \sqrt [3]{2} b d}-\frac {a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b d}-\frac {2^{2/3} a^{2/3} \tan ^{-1}\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 {\left (a+b x^3\right )^{2/3}}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + b*x^3)^(2/3)/(2*b*d) - (2^(2/3)*a^(2/3)*ArcTan[(a^(1/3) + 2^(2/3)*(a + b*x^3)^(1/3))/(Sqrt[3]*a^(1/3))])
/(Sqrt[3]*b*d) + (a^(2/3)*Log[a - b*x^3])/(3*2^(1/3)*b*d) - (a^(2/3)*Log[2^(1/3)*a^(1/3) - (a + b*x^3)^(1/3)])
/(2^(1/3)*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 50

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 55

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), 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 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 444

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 130, normalized size = 0.85 \begin {gather*} \frac {2^{2/3} a^{2/3} \log \left (a-b x^3\right )-3 \left (2^{2/3} a^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+\left (a+b x^3\right )^{2/3}\right )-2\ 2^{2/3} \sqrt {3} a^{2/3} \tan ^{-1}\left (\frac {\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{6 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.19, size = 193, normalized size = 1.26 \begin {gather*} -\frac {2^{2/3} a^{2/3} \log \left (2^{2/3} \sqrt [3]{a+b x^3}-2 \sqrt [3]{a}\right )}{3 b d}+\frac {a^{2/3} \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 )}{3 \sqrt [3]{2} b d}-\frac {2^{2/3} a^{2/3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} b d}-\frac {\left (a+b x^3\right )^{2/3}}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.17, size = 167, normalized size = 1.09 \begin {gather*} -\frac {2 \cdot 4^{\frac {1}{3}} \sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{3}} \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + 4^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} \log \left (4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a - 2 \cdot 4^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} a\right ) - 2 \cdot 4^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} \log \left (-4^{\frac {2}{3}} \left (-a^{2}\right )^{\frac {2}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a\right ) + 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{6 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*4^(1/3)*sqrt(3)*(-a^2)^(1/3)*arctan(1/3*(4^(1/3)*sqrt(3)*(b*x^3 + a)^(1/3)*(-a^2)^(1/3) - sqrt(3)*a)/a
) + 4^(1/3)*(-a^2)^(1/3)*log(4^(2/3)*(b*x^3 + a)^(1/3)*(-a^2)^(2/3) + 2*(b*x^3 + a)^(2/3)*a - 2*4^(1/3)*(-a^2)
^(1/3)*a) - 2*4^(1/3)*(-a^2)^(1/3)*log(-4^(2/3)*(-a^2)^(2/3) + 2*(b*x^3 + a)^(1/3)*a) + 3*(b*x^3 + a)^(2/3))/(
b*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)^(2/3)/(-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 rootofAlgebraic extensions not allowed in a rootofAlgebraic 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 rootofAlgebraic extensions not allowed in a rootofAlgebraic 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 rootofAlgebraic extensions not allowed in a rootofAlgebraic 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 rootofAlgebraic extensions not allowed in a rootofAlgebraic 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 rootofAlgebraic extensions not allowed in a rootofAlgebraic 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 rootofAlgebraic extensions not allowed in a rootofAlgebraic 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 rootofAlgebraic extensions not allowed in a rootofAlgebraic 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 rootofAlgebraic extensions not allowed in a rootofAlgebraic 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 rootofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootof((2*a)^(1/3))^2*1/
6/b/d*ln(((a+b*x^3)^(1/3))^2+(2*a)^(1/3)*(a+b*x^3)^(1/3)+(2*a)^(1/3)*(2*a)^(1/3))-((2*a)^(1/3))^2/sqrt(3)/b/d*
atan(((a+b*x^3)^(1/3)+1/2*(2*a)^(1/3))/sqrt(3)*2/(2*a)^(1/3))-2*(2*a)^(1/3)*a*b^2*d^2*(2*a)^(1/3)*1/6/a/b^3/d^
3*ln(abs((a+b*x^3)^(1/3)-(2*a)^(1/3)))-1/2*((a+b*x^3)^(1/3))^2*b*d/b^2/d^2

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maple [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{2}}{-b d \,x^{3}+a d}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.31, size = 140, normalized size = 0.92 \begin {gather*} -\frac {\frac {2 \, \sqrt {3} 2^{\frac {2}{3}} a^{\frac {2}{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 {2}{3}} a^{\frac {2}{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 {2}{3}} a^{\frac {2}{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 (b x^{3} + a\right )}^{\frac {2}{3}}}{d}}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.83, size = 186, normalized size = 1.22 \begin {gather*} -\frac {{\left (b\,x^3+a\right )}^{2/3}}{2\,b\,d}-\frac {4^{1/3}\,a^{2/3}\,\ln \left ({\left (b\,x^3+a\right )}^{1/3}-2^{1/3}\,a^{1/3}\right )}{3\,b\,d}-\frac {4^{1/3}\,a^{2/3}\,\ln \left (\frac {4\,a^2\,{\left (b\,x^3+a\right )}^{1/3}}{b^2\,d^2}-\frac {2\,4^{2/3}\,a^{7/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^2\,d^2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b\,d}+\frac {4^{1/3}\,a^{2/3}\,\ln \left (\frac {4\,a^2\,{\left (b\,x^3+a\right )}^{1/3}}{b^2\,d^2}-\frac {18\,4^{2/3}\,a^{7/3}\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2}{b^2\,d^2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

-Integral(x**2*(a + b*x**3)**(2/3)/(-a + b*x**3), x)/d

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