3.37 \(\int \frac{1}{(a-b x^3) (a+b x^3)^{2/3}} \, dx\)

Optimal. Leaf size=452 \[ -\frac{\log \left (2^{2/3}-\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{6\ 2^{2/3} a^{4/3} \sqrt [3]{b}}+\frac{\log \left (\frac{2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{6\ 2^{2/3} a^{4/3} \sqrt [3]{b}}-\frac{\log \left (\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} a^{4/3} \sqrt [3]{b}}+\frac{\log \left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac{2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{12\ 2^{2/3} a^{4/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} a^{4/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{2\ 2^{2/3} \sqrt{3} a^{4/3} \sqrt [3]{b}}+\frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{2 a \left (a+b x^3\right )^{2/3}} \]

[Out]

-(ArcTan[(1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[3]*a^(4/3)*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^(2/3)*Sqrt[3]*a^(4/3)*b^(1/3))
+ (x*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a)])/(2*a*(a + b*x^3)^(2/3)) - Log[2^(2
/3) - (a^(1/3) + b^(1/3)*x)/(a + b*x^3)^(1/3)]/(6*2^(2/3)*a^(4/3)*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)]/(6*2^(2/3)*a^(4/3)*b^(1/3)) -
Log[1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)]/(3*2^(2/3)*a^(4/3)*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)]/(12*2^(2/3)*a^(4/3)*
b^(1/3))

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Rubi [C]  time = 0.0276162, antiderivative size = 58, normalized size of antiderivative = 0.13, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {430, 429} \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{1}{3};1,\frac{2}{3};\frac{4}{3};\frac{b x^3}{a},-\frac{b x^3}{a}\right )}{a \left (a+b x^3\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(1 + (b*x^3)/a)^(2/3)*AppellF1[1/3, 1, 2/3, 4/3, (b*x^3)/a, -((b*x^3)/a)])/(a*(a + b*x^3)^(2/3))

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a-b x^3\right ) \left (a+b x^3\right )^{2/3}} \, dx &=\frac{\left (1+\frac{b x^3}{a}\right )^{2/3} \int \frac{1}{\left (a-b x^3\right ) \left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=\frac{x \left (1+\frac{b x^3}{a}\right )^{2/3} F_1\left (\frac{1}{3};1,\frac{2}{3};\frac{4}{3};\frac{b x^3}{a},-\frac{b x^3}{a}\right )}{a \left (a+b x^3\right )^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0514956, size = 153, normalized size = 0.34 \[ \frac{4 a x F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{\left (a-b x^3\right ) \left (a+b x^3\right )^{2/3} \left (b x^3 \left (3 F_1\left (\frac{4}{3};\frac{2}{3},2;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )-2 F_1\left (\frac{4}{3};\frac{5}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )\right )+4 a F_1\left (\frac{1}{3};\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*a*x*AppellF1[1/3, 2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/((a - b*x^3)*(a + b*x^3)^(2/3)*(4*a*AppellF1[1/3,
2/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a] + b*x^3*(3*AppellF1[4/3, 2/3, 2, 7/3, -((b*x^3)/a), (b*x^3)/a] - 2*Appel
lF1[4/3, 5/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])))

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Maple [F]  time = 0.428, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-b{x}^{3}+a} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (b x^{3} - a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- a \left (a + b x^{3}\right )^{\frac{2}{3}} + b x^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x**3+a)/(b*x**3+a)**(2/3),x)

[Out]

-Integral(1/(-a*(a + b*x**3)**(2/3) + b*x**3*(a + b*x**3)**(2/3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (b x^{3} - a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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