3.1.36 \(\int \frac {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx\) [36]

Optimal. Leaf size=398 \[ -\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}}+\frac {\log \left (1+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\sqrt [3]{2} \log \left (1+\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}+\frac {\log \left (2 \sqrt [3]{2}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}\right )}{6\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}} \]

[Out]

-1/6*ln(2^(2/3)+(-a^(1/3)-b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(1/3)/b^(1/3)+1/6*ln(1+2^(2/3)*(a^(1/3)+b^(1/3
)*x)^2/(b*x^3+a)^(2/3)-2^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(1/3)/b^(1/3)-1/3*2^(1/3)*ln(1+2
^(1/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))/a^(1/3)/b^(1/3)+1/12*ln(2*2^(1/3)+(a^(1/3)+b^(1/3)*x)^2/(b*x^3+a)^
(2/3)+2^(2/3)*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*2^(1/3)/a^(1/3)/b^(1/3)-1/3*2^(1/3)*arctan(1/3*(1-2*2^(1/3)
*(a^(1/3)+b^(1/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))/a^(1/3)/b^(1/3)*3^(1/2)-1/6*arctan(1/3*(1+2^(1/3)*(a^(1/3)+b^(1
/3)*x)/(b*x^3+a)^(1/3))*3^(1/2))*2^(1/3)/a^(1/3)/b^(1/3)*3^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {420, 493, 298, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\sqrt [3]{2} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\text {ArcTan}\left (\frac {\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (2^{2/3}-\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}}+\frac {\log \left (\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}}-\frac {\sqrt [3]{2} \log \left (\frac {\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{a} \sqrt [3]{b}}+\frac {\log \left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac {2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(1/3)/(a - b*x^3),x]

[Out]

-((2^(1/3)*ArcTan[(1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*a^(1/3)*b^(1/3)
)) - ArcTan[(1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]*a^(1/3)*b^(1/3))
- Log[2^(2/3) - (a^(1/3) + b^(1/3)*x)/(a + b*x^3)^(1/3)]/(3*2^(2/3)*a^(1/3)*b^(1/3)) + Log[1 + (2^(2/3)*(a^(1
/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)]/(3*2^(2/3)*a^(1/3)*
b^(1/3)) - (2^(1/3)*Log[1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)])/(3*a^(1/3)*b^(1/3)) + Log[2*2^
(1/3) + (a^(1/3) + b^(1/3)*x)^2/(a + b*x^3)^(2/3) + (2^(2/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)]/(6*2^(2
/3)*a^(1/3)*b^(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 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 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(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 420

Int[((a_) + (b_.)*(x_)^3)^(1/3)/((c_) + (d_.)*(x_)^3), x_Symbol] :> With[{q = Rt[b/a, 3]}, Dist[9*(a/(c*q)), S
ubst[Int[x/((4 - a*x^3)*(1 + 2*a*x^3)), x], x, (1 + q*x)/(a + b*x^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[b*c - a*d, 0] && EqQ[b*c + a*d, 0]

Rule 493

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), I
nt[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 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]

Rule 642

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 648

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 {\sqrt [3]{a+b x^3}}{a-b x^3} \, dx &=\frac {\sqrt [3]{a+b x^3} \int \frac {\sqrt [3]{1+\frac {b x^3}{a}}}{a-b x^3} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {x \sqrt [3]{a+b x^3} F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {b x^3}{a},-\frac {b x^3}{a}\right )}{a \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}

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Mathematica [A]
time = 2.58, size = 428, normalized size = 1.08 \begin {gather*} \frac {4 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a+b x^3}}{-2 \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a+b x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a+b x^3}}{\sqrt [3]{2} \sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a+b x^3}}\right )-4 \log \left (\sqrt [3]{2} \sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )-2 \log \left (-\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}\right )+\log \left (2^{2/3} a^{2/3}+2^{2/3} b^{2/3} x^2+2 \sqrt [3]{2} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+4 \left (a+b x^3\right )^{2/3}+2 \sqrt [3]{2} \sqrt [3]{a} \left (\sqrt [3]{2} \sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )\right )+2 \log \left (2^{2/3} a^{2/3}+2^{2/3} b^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}+\sqrt [3]{a} \left (2\ 2^{2/3} \sqrt [3]{b} x-\sqrt [3]{2} \sqrt [3]{a+b x^3}\right )\right )}{6\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3]*ArcTan[(Sqrt[3]*(a + b*x^3)^(1/3))/(-2*2^(1/3)*a^(1/3) - 2*2^(1/3)*b^(1/3)*x + (a + b*x^3)^(1/3))]
+ 2*Sqrt[3]*ArcTan[(Sqrt[3]*(a + b*x^3)^(1/3))/(2^(1/3)*a^(1/3) + 2^(1/3)*b^(1/3)*x + (a + b*x^3)^(1/3))] - 4*
Log[2^(1/3)*a^(1/3) + 2^(1/3)*b^(1/3)*x + (a + b*x^3)^(1/3)] - 2*Log[-(2^(1/3)*a^(1/3)) - 2^(1/3)*b^(1/3)*x +
2*(a + b*x^3)^(1/3)] + Log[2^(2/3)*a^(2/3) + 2^(2/3)*b^(2/3)*x^2 + 2*2^(1/3)*b^(1/3)*x*(a + b*x^3)^(1/3) + 4*(
a + b*x^3)^(2/3) + 2*2^(1/3)*a^(1/3)*(2^(1/3)*b^(1/3)*x + (a + b*x^3)^(1/3))] + 2*Log[2^(2/3)*a^(2/3) + 2^(2/3
)*b^(2/3)*x^2 - 2^(1/3)*b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3) + a^(1/3)*(2*2^(2/3)*b^(1/3)*x - 2^(1/
3)*(a + b*x^3)^(1/3))])/(6*2^(2/3)*a^(1/3)*b^(1/3))

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{-b \,x^{3}+a}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (284) = 568\).
time = 25.42, size = 644, normalized size = 1.62 \begin {gather*} -\frac {1}{18} \, \sqrt {3} 2^{\frac {1}{3}} \left (-\frac {1}{a b}\right )^{\frac {1}{3}} \arctan \left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (a b^{6} x^{16} + 33 \, a^{2} b^{5} x^{13} + 110 \, a^{3} b^{4} x^{10} + 110 \, a^{4} b^{3} x^{7} + 33 \, a^{5} b^{2} x^{4} + a^{6} b x\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{a b}\right )^{\frac {2}{3}} + 24 \, \sqrt {3} 2^{\frac {1}{3}} {\left (a b^{5} x^{14} + 2 \, a^{2} b^{4} x^{11} - 6 \, a^{3} b^{3} x^{8} + 2 \, a^{4} b^{2} x^{5} + a^{5} b x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-\frac {1}{a b}\right )^{\frac {1}{3}} - \sqrt {3} {\left (b^{6} x^{18} - 42 \, a b^{5} x^{15} - 417 \, a^{2} b^{4} x^{12} - 812 \, a^{3} b^{3} x^{9} - 417 \, a^{4} b^{2} x^{6} - 42 \, a^{5} b x^{3} + a^{6}\right )}}{3 \, {\left (b^{6} x^{18} + 102 \, a b^{5} x^{15} + 447 \, a^{2} b^{4} x^{12} + 628 \, a^{3} b^{3} x^{9} + 447 \, a^{4} b^{2} x^{6} + 102 \, a^{5} b x^{3} + a^{6}\right )}}\right ) - \frac {1}{36} \cdot 2^{\frac {1}{3}} \left (-\frac {1}{a b}\right )^{\frac {1}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (a b^{3} x^{8} + 4 \, a^{2} b^{2} x^{5} + a^{3} b x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-\frac {1}{a b}\right )^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (b^{4} x^{12} + 32 \, a b^{3} x^{9} + 78 \, a^{2} b^{2} x^{6} + 32 \, a^{3} b x^{3} + a^{4}\right )} \left (-\frac {1}{a b}\right )^{\frac {1}{3}} + 6 \, {\left (b^{3} x^{10} + 11 \, a b^{2} x^{7} + 11 \, a^{2} b x^{4} + a^{3} x\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b^{4} x^{12} - 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} - 4 \, a^{3} b x^{3} + a^{4}}\right ) + \frac {1}{18} \cdot 2^{\frac {1}{3}} \left (-\frac {1}{a b}\right )^{\frac {1}{3}} \log \left (-\frac {12 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{2} + 2^{\frac {2}{3}} {\left (b^{2} x^{6} - 2 \, a b x^{3} + a^{2}\right )} \left (-\frac {1}{a b}\right )^{\frac {2}{3}} + 6 \cdot 2^{\frac {1}{3}} {\left (b x^{4} + a x\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{a b}\right )^{\frac {1}{3}}}{b^{2} x^{6} - 2 \, a b x^{3} + a^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*sqrt(3)*2^(1/3)*(-1/(a*b))^(1/3)*arctan(1/3*(6*sqrt(3)*2^(2/3)*(a*b^6*x^16 + 33*a^2*b^5*x^13 + 110*a^3*b
^4*x^10 + 110*a^4*b^3*x^7 + 33*a^5*b^2*x^4 + a^6*b*x)*(b*x^3 + a)^(1/3)*(-1/(a*b))^(2/3) + 24*sqrt(3)*2^(1/3)*
(a*b^5*x^14 + 2*a^2*b^4*x^11 - 6*a^3*b^3*x^8 + 2*a^4*b^2*x^5 + a^5*b*x^2)*(b*x^3 + a)^(2/3)*(-1/(a*b))^(1/3) -
sqrt(3)*(b^6*x^18 - 42*a*b^5*x^15 - 417*a^2*b^4*x^12 - 812*a^3*b^3*x^9 - 417*a^4*b^2*x^6 - 42*a^5*b*x^3 + a^6
))/(b^6*x^18 + 102*a*b^5*x^15 + 447*a^2*b^4*x^12 + 628*a^3*b^3*x^9 + 447*a^4*b^2*x^6 + 102*a^5*b*x^3 + a^6)) -
1/36*2^(1/3)*(-1/(a*b))^(1/3)*log((12*2^(2/3)*(a*b^3*x^8 + 4*a^2*b^2*x^5 + a^3*b*x^2)*(b*x^3 + a)^(2/3)*(-1/(
a*b))^(2/3) - 2^(1/3)*(b^4*x^12 + 32*a*b^3*x^9 + 78*a^2*b^2*x^6 + 32*a^3*b*x^3 + a^4)*(-1/(a*b))^(1/3) + 6*(b^
3*x^10 + 11*a*b^2*x^7 + 11*a^2*b*x^4 + a^3*x)*(b*x^3 + a)^(1/3))/(b^4*x^12 - 4*a*b^3*x^9 + 6*a^2*b^2*x^6 - 4*a
^3*b*x^3 + a^4)) + 1/18*2^(1/3)*(-1/(a*b))^(1/3)*log(-(12*(b*x^3 + a)^(2/3)*x^2 + 2^(2/3)*(b^2*x^6 - 2*a*b*x^3
+ a^2)*(-1/(a*b))^(2/3) + 6*2^(1/3)*(b*x^4 + a*x)*(b*x^3 + a)^(1/3)*(-1/(a*b))^(1/3))/(b^2*x^6 - 2*a*b*x^3 +
a^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(1/3)/(a - b*x^3),x)

[Out]

int((a + b*x^3)^(1/3)/(a - b*x^3), x)

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