3.604 \(\int \frac{(a+b x^3)^{2/3}}{x^5 (a d-b d x^3)} \, dx\)

Optimal. Leaf size=512 \[ \frac{3 b^2 x^2 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{4 a^2 d \sqrt [3]{a+b x^3}}+\frac{b^{4/3} \log \left (\frac{2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{2} a^{5/3} d}-\frac{2^{2/3} b^{4/3} \log \left (\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 a^{5/3} d}-\frac{b^{4/3} \log \left (\frac{\sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac{2^{2/3} \sqrt [3]{b} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{2 \sqrt [3]{2} a^{5/3} d}+\frac{2^{2/3} b^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} a^{5/3} d}+\frac{b^{4/3} \tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} a^{5/3} d}+\frac{b^{4/3} \log \left (\frac{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a}\right )}{6 \sqrt [3]{2} a^{5/3} d}-\frac{3 b \left (a+b x^3\right )^{2/3}}{2 a^2 d x}-\frac{\left (a+b x^3\right )^{2/3}}{4 a d x^4} \]

[Out]

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

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Rubi [C]  time = 0.0689181, antiderivative size = 66, normalized size of antiderivative = 0.13, 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} \[ -\frac{\left (a+b x^3\right )^{2/3} F_1\left (-\frac{4}{3};-\frac{2}{3},1;-\frac{1}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{4 a d x^4 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(2/3)/(x^5*(a*d - b*d*x^3)),x]

[Out]

-((a + b*x^3)^(2/3)*AppellF1[-4/3, -2/3, 1, -1/3, -((b*x^3)/a), (b*x^3)/a])/(4*a*d*x^4*(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^5 \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^5 \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} F_1\left (-\frac{4}{3};-\frac{2}{3},1;-\frac{1}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{4 a d x^4 \left (1+\frac{b x^3}{a}\right )^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0810564, size = 148, normalized size = 0.29 \[ \frac{-5 a \left (a^2+7 a b x^3+6 b^2 x^6\right )-6 b^3 x^9 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{5}{3};\frac{1}{3},1;\frac{8}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )+35 a b^2 x^6 \sqrt [3]{\frac{b x^3}{a}+1} F_1\left (\frac{2}{3};\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{20 a^3 d x^4 \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(2/3)/(x^5*(a*d - b*d*x^3)),x]

[Out]

(-5*a*(a^2 + 7*a*b*x^3 + 6*b^2*x^6) + 35*a*b^2*x^6*(1 + (b*x^3)/a)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, -((b*x^3)/
a), (b*x^3)/a] - 6*b^3*x^9*(1 + (b*x^3)/a)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -((b*x^3)/a), (b*x^3)/a])/(20*a^3*
d*x^4*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5} \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^5/(-b*d*x^3+a*d),x)

[Out]

int((b*x^3+a)^(2/3)/x^5/(-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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^5/(-b*d*x^3+a*d),x, algorithm="maxima")

[Out]

-integrate((b*x^3 + a)^(2/3)/((b*d*x^3 - a*d)*x^5), 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^5/(-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^{5} + b x^{8}}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(2/3)/x**5/(-b*d*x**3+a*d),x)

[Out]

-Integral((a + b*x**3)**(2/3)/(-a*x**5 + b*x**8), 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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^5/(-b*d*x^3+a*d),x, algorithm="giac")

[Out]

integrate(-(b*x^3 + a)^(2/3)/((b*d*x^3 - a*d)*x^5), x)