3.598 \(\int \frac{(a+b x^3)^{2/3}}{x^9 (a d-b d x^3)} \, dx\)

Optimal. Leaf size=209 \[ -\frac{5 b^2 \left (a+b x^3\right )^{2/3}}{8 a^3 d x^2}+\frac{b^{8/3} \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} a^3 d}-\frac{b^{8/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^3 d}+\frac{2^{2/3} b^{8/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} a^3 d}-\frac{b \left (a+b x^3\right )^{2/3}}{4 a^2 d x^5}-\frac{\left (a+b x^3\right )^{2/3}}{8 a d x^8} \]

[Out]

-(a + b*x^3)^(2/3)/(8*a*d*x^8) - (b*(a + b*x^3)^(2/3))/(4*a^2*d*x^5) - (5*b^2*(a + b*x^3)^(2/3))/(8*a^3*d*x^2)
+ (2^(2/3)*b^(8/3)*ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*a^3*d) + (b^(8/3)*
Log[a*d - b*d*x^3])/(3*2^(1/3)*a^3*d) - (b^(8/3)*Log[2^(1/3)*b^(1/3)*x - (a + b*x^3)^(1/3)])/(2^(1/3)*a^3*d)

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Rubi [C]  time = 10.6764, antiderivative size = 244, normalized size of antiderivative = 1.17, 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} \[ -\frac{-18 b x^3 \left (a-b x^3\right )^2 \, _3F_2\left (\frac{1}{3},2,2;1,\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )-4 b x^3 \left (5 a^2+6 a b x^3+9 b^2 x^6\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+42 a^2 b x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+11 a^2 b x^3+5 a^3-54 b^3 x^9 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+12 a b^2 x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{b x^3+a}\right )+15 a b^2 x^6+9 b^3 x^9}{40 a^3 d x^8 \sqrt [3]{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(2/3)/(x^9*(a*d - b*d*x^3)),x]

[Out]

-(5*a^3 + 11*a^2*b*x^3 + 15*a*b^2*x^6 + 9*b^3*x^9 - 4*b*x^3*(5*a^2 + 6*a*b*x^3 + 9*b^2*x^6)*Hypergeometric2F1[
1/3, 1, 4/3, (2*b*x^3)/(a + b*x^3)] + 42*a^2*b*x^3*Hypergeometric2F1[1/3, 2, 4/3, (2*b*x^3)/(a + b*x^3)] + 12*
a*b^2*x^6*Hypergeometric2F1[1/3, 2, 4/3, (2*b*x^3)/(a + b*x^3)] - 54*b^3*x^9*Hypergeometric2F1[1/3, 2, 4/3, (2
*b*x^3)/(a + b*x^3)] - 18*b*x^3*(a - b*x^3)^2*HypergeometricPFQ[{1/3, 2, 2}, {1, 4/3}, (2*b*x^3)/(a + b*x^3)])
/(40*a^3*d*x^8*(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^9 \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^9 \left (a d-b d x^3\right )} \, dx}{\left (1+\frac{b x^3}{a}\right )^{2/3}}\\ &=-\frac{5 a^3+11 a^2 b x^3+15 a b^2 x^6+9 b^3 x^9-4 b x^3 \left (5 a^2+6 a b x^3+9 b^2 x^6\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )+42 a^2 b x^3 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )+12 a b^2 x^6 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )-54 b^3 x^9 \, _2F_1\left (\frac{1}{3},2;\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )-18 b x^3 \left (a-b x^3\right )^2 \, _3F_2\left (\frac{1}{3},2,2;1,\frac{4}{3};\frac{2 b x^3}{a+b x^3}\right )}{40 a^3 d x^8 \sqrt [3]{a+b x^3}}\\ \end{align*}

Mathematica [A]  time = 5.21285, size = 179, normalized size = 0.86 \[ \frac{4\ 2^{2/3} b^{8/3} \left (\log \left (\frac{2^{2/3} b^{2/3} x^2}{\left (a x^3+b\right )^{2/3}}+\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a x^3+b}}+1}{\sqrt{3}}\right )\right )-\frac{3 \left (a+b x^3\right )^{2/3} \left (a^2+2 a b x^3+5 b^2 x^6\right )}{x^8}}{24 a^3 d} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(2/3)/(x^9*(a*d - b*d*x^3)),x]

[Out]

((-3*(a + b*x^3)^(2/3)*(a^2 + 2*a*b*x^3 + 5*b^2*x^6))/x^8 + 4*2^(2/3)*b^(8/3)*(2*Sqrt[3]*ArcTan[(1 + (2*2^(1/3
)*b^(1/3)*x)/(b + a*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)] + Log[1 + (2^(2/3)
*b^(2/3)*x^2)/(b + a*x^3)^(2/3) + (2^(1/3)*b^(1/3)*x)/(b + a*x^3)^(1/3)]))/(24*a^3*d)

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{9} \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^9/(-b*d*x^3+a*d),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^9/(-b*d*x^3+a*d),x, algorithm="maxima")

[Out]

-integrate((b*x^3 + a)^(2/3)/((b*d*x^3 - a*d)*x^9), 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^9/(-b*d*x^3+a*d),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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**9/(-b*d*x**3+a*d),x)

[Out]

Timed out

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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^{9}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^9/(-b*d*x^3+a*d),x, algorithm="giac")

[Out]

integrate(-(b*x^3 + a)^(2/3)/((b*d*x^3 - a*d)*x^9), x)