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

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

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Rubi [C]  time = 0.43, 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} \begin {gather*} -\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}} \end {gather*}

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 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])

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])

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*}

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Mathematica [A]  time = 5.20, size = 170, normalized size = 0.93 \begin {gather*} \frac {5\ 2^{2/3} b^{5/3} \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 (2 a+7 b x^3\right )}{x^5}}{30 a^2 d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-3*(a + b*x^3)^(2/3)*(2*a + 7*b*x^3))/x^5 + 5*2^(2/3)*b^(5/3)*(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)]))/(30*a^2*d)

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IntegrateAlgebraic [A]  time = 0.49, size = 216, normalized size = 1.19 \begin {gather*} -\frac {2^{2/3} b^{5/3} \log \left (2^{2/3} \sqrt [3]{a+b x^3}-2 \sqrt [3]{b} x\right )}{3 a^2 d}+\frac {2^{2/3} b^{5/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{b} x}\right )}{\sqrt {3} a^2 d}+\frac {b^{5/3} \log \left (2^{2/3} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}+2 b^{2/3} x^2\right )}{3 \sqrt [3]{2} a^2 d}+\frac {\left (-2 a-7 b x^3\right ) \left (a+b x^3\right )^{2/3}}{10 a^2 d x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (b d x^{3} - a d\right )} x^{6}}\,{d x} \end {gather*}

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)

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maple [F]  time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{\left (-b d \,x^{3}+a d \right ) x^{6}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (b d x^{3} - a d\right )} x^{6}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{2/3}}{x^6\,\left (a\,d-b\,d\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{- a x^{6} + b x^{9}}\, dx}{d} \end {gather*}

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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