3.597 \(\int \frac{(a+b x^3)^{2/3}}{x^6 (a d-b d x^3)} \, dx\)

Optimal. Leaf size=182 \[ \frac{b^{5/3} \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} a^2 d}-\frac{b^{5/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^2 d}+\frac{2^{2/3} b^{5/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} a^2 d}-\frac{7 b \left (a+b x^3\right )^{2/3}}{10 a^2 d x^2}-\frac{\left (a+b x^3\right )^{2/3}}{5 a d x^5} \]

[Out]

-(a + b*x^3)^(2/3)/(5*a*d*x^5) - (7*b*(a + b*x^3)^(2/3))/(10*a^2*d*x^2) + (2^(2/3)*b^(5/3)*ArcTan[(1 + (2*2^(1
/3)*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*a^2*d) + (b^(5/3)*Log[a*d - b*d*x^3])/(3*2^(1/3)*a^2*d) -
(b^(5/3)*Log[2^(1/3)*b^(1/3)*x - (a + b*x^3)^(1/3)])/(2^(1/3)*a^2*d)

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Rubi [C]  time = 0.429248, antiderivative size = 121, normalized size of antiderivative = 0.66, 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} \[ -\frac{2 a^2-4 b x^3 \left (2 a+3 b x^3\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+12 b x^3 \left (a-b x^3\right ) \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+5 a b x^3+3 b^2 x^6}{10 a^2 d x^5 \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(2/3)/(x^6*(a*d - b*d*x^3)),x]

[Out]

-(2*a^2 + 5*a*b*x^3 + 3*b^2*x^6 - 4*b*x^3*(2*a + 3*b*x^3)*Hypergeometric2F1[1/3, 1, 4/3, (2*b*x^3)/(a + b*x^3)
] + 12*b*x^3*(a - b*x^3)*Hypergeometric2F1[1/3, 2, 4/3, (2*b*x^3)/(a + b*x^3)])/(10*a^2*d*x^5*(a + b*x^3)^(1/3
))

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^6 \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^6 \left (a d-b d x^3\right )} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=-\frac{2 a^2+5 a b x^3+3 b^2 x^6-4 b x^3 \left (2 a+3 b x^3\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )+12 b x^3 \left (a-b x^3\right ) \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )}{10 a^2 d x^5 \sqrt [3]{a+b x^3}}\\ \end{align*}

Mathematica [C]  time = 4.66952, size = 124, normalized size = 0.68 \[ \frac{\text{Gamma}\left (-\frac{2}{3}\right ) \left (a+b x^3\right )^{5/3} \left (\left (2 a^2-5 a b x^3+3 b^2 x^6\right ) \, _2F_1\left (1,1;\frac{1}{3};-\frac{2 b x^3}{a-b x^3}\right )+18 b x^3 \left (a+b x^3\right ) \, _2F_1\left (2,2;\frac{4}{3};-\frac{2 b x^3}{a-b x^3}\right )\right )}{15 a^2 d x^5 \text{Gamma}\left (\frac{1}{3}\right ) \left (a-b x^3\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(2/3)/(x^6*(a*d - b*d*x^3)),x]

[Out]

((a + b*x^3)^(5/3)*Gamma[-2/3]*((2*a^2 - 5*a*b*x^3 + 3*b^2*x^6)*Hypergeometric2F1[1, 1, 1/3, (-2*b*x^3)/(a - b
*x^3)] + 18*b*x^3*(a + b*x^3)*Hypergeometric2F1[2, 2, 4/3, (-2*b*x^3)/(a - b*x^3)]))/(15*a^2*d*x^5*(a - b*x^3)
^2*Gamma[1/3])

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6} \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^6/(-b*d*x^3+a*d),x)

[Out]

int((b*x^3+a)^(2/3)/x^6/(-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^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^6/(-b*d*x^3+a*d),x, algorithm="maxima")

[Out]

-integrate((b*x^3 + a)^(2/3)/((b*d*x^3 - a*d)*x^6), 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^6/(-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^{6} + b x^{9}}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral((a + b*x**3)**(2/3)/(-a*x**6 + b*x**9), 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^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^6/(-b*d*x^3+a*d),x, algorithm="giac")

[Out]

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