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

Optimal. Leaf size=157 \[ \frac{b^{2/3} \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} a d}-\frac{b^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a d}+\frac{2^{2/3} b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} a d}-\frac{\left (a+b x^3\right )^{2/3}}{2 a d x^2} \]

[Out]

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

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Rubi [C]  time = 0.0707691, antiderivative size = 79, normalized size of antiderivative = 0.5, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {511, 510} \[ -\frac{\left (a+b x^3\right )^{2/3} \left (1-\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (-\frac{2}{3},-\frac{2}{3};\frac{1}{3};-\frac{2 b x^3}{a-b x^3}\right )}{2 a d x^2 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a + b*x^3)^(2/3)*(1 - (b*x^3)/a)^(2/3)*Hypergeometric2F1[-2/3, -2/3, 1/3, (-2*b*x^3)/(a - b*x^3)])/(2*a*d*x
^2*(1 + (b*x^3)/a)^(2/3))

Rule 511

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

Rule 510

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^{2/3}}{x^3 \left (a d-b d x^3\right )} \, dx &=\frac{\left (a+b x^3\right )^{2/3} \int \frac{\left (1+\frac{b x^3}{a}\right )^{2/3}}{x^3 \left (a d-b d x^3\right )} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=-\frac{\left (a+b x^3\right )^{2/3} \left (1-\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (-\frac{2}{3},-\frac{2}{3};\frac{1}{3};-\frac{2 b x^3}{a-b x^3}\right )}{2 a d x^2 \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0161884, size = 47, normalized size = 0.3 \[ -\frac{\left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac{2}{3},1;\frac{1}{3};\frac{2 b x^3}{b x^3+a}\right )}{2 a d x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x^3)^(2/3)*Hypergeometric2F1[-2/3, 1, 1/3, (2*b*x^3)/(a + b*x^3)])/(2*a*d*x^2)

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( -bd{x}^{3}+ad \right ) } \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{- a x^{3} + b x^{6}}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral((a + b*x**3)**(2/3)/(-a*x**3 + b*x**6), x)/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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