3.669 \(\int \frac{\sqrt [3]{a+b x^3}}{x^5 (c+d x^3)} \, dx\)

Optimal. Leaf size=204 \[ -\frac{\sqrt [3]{a+b x^3} (b c-4 a d)}{4 a c^2 x}-\frac{d \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 c^{7/3}}+\frac{d \sqrt [3]{b c-a d} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{7/3}}+\frac{d \sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{7/3}}-\frac{\sqrt [3]{a+b x^3}}{4 c x^4} \]

[Out]

-(a + b*x^3)^(1/3)/(4*c*x^4) - ((b*c - 4*a*d)*(a + b*x^3)^(1/3))/(4*a*c^2*x) + (d*(b*c - a*d)^(1/3)*ArcTan[(1
+ (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*c^(7/3)) - (d*(b*c - a*d)^(1/3)*Log[
c + d*x^3])/(6*c^(7/3)) + (d*(b*c - a*d)^(1/3)*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(2*c^(7
/3))

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Rubi [C]  time = 0.132877, antiderivative size = 145, normalized size of antiderivative = 0.71, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac{x^3 \left (c-3 d x^3\right ) (-(b c-a d)) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+3 x^3 \left (c+d x^3\right ) (b c-a d) \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+2 c \left (a+b x^3\right ) \left (c-3 d x^3\right )}{8 c^3 x^4 \left (a+b x^3\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(a + b*x^3)^(1/3)/(x^5*(c + d*x^3)),x]

[Out]

-(2*c*(a + b*x^3)*(c - 3*d*x^3) - (b*c - a*d)*x^3*(c - 3*d*x^3)*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x^
3)/(c*(a + b*x^3))] + 3*(b*c - a*d)*x^3*(c + d*x^3)*Hypergeometric2F1[2/3, 2, 5/3, ((b*c - a*d)*x^3)/(c*(a + b
*x^3))])/(8*c^3*x^4*(a + b*x^3)^(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{\sqrt [3]{a+b x^3}}{x^5 \left (c+d x^3\right )} \, dx &=\frac{\sqrt [3]{a+b x^3} \int \frac{\sqrt [3]{1+\frac{b x^3}{a}}}{x^5 \left (c+d x^3\right )} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=-\frac{2 c \left (a+b x^3\right ) \left (c-3 d x^3\right )-(b c-a d) x^3 \left (c-3 d x^3\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )+3 (b c-a d) x^3 \left (c+d x^3\right ) \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{8 c^3 x^4 \left (a+b x^3\right )^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.398873, size = 146, normalized size = 0.72 \[ -\frac{x^3 \left (3 d x^3-c\right ) (b c-a d) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+3 x^3 \left (c+d x^3\right ) (b c-a d) \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+2 c \left (a+b x^3\right ) \left (c-3 d x^3\right )}{8 c^3 x^4 \left (a+b x^3\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x^3)^(1/3)/(x^5*(c + d*x^3)),x]

[Out]

-(2*c*(a + b*x^3)*(c - 3*d*x^3) + (b*c - a*d)*x^3*(-c + 3*d*x^3)*Hypergeometric2F1[2/3, 1, 5/3, ((b*c - a*d)*x
^3)/(c*(a + b*x^3))] + 3*(b*c - a*d)*x^3*(c + d*x^3)*Hypergeometric2F1[2/3, 2, 5/3, ((b*c - a*d)*x^3)/(c*(a +
b*x^3))])/(8*c^3*x^4*(a + b*x^3)^(2/3))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5} \left ( d{x}^{3}+c \right ) }\sqrt [3]{b{x}^{3}+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(1/3)/x^5/(d*x^3+c),x)

[Out]

int((b*x^3+a)^(1/3)/x^5/(d*x^3+c),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{1}{3}}}{{\left (d x^{3} + c\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^5/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(1/3)/((d*x^3 + c)*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)^(1/3)/x^5/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{a + b x^{3}}}{x^{5} \left (c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(1/3)/x**5/(d*x**3+c),x)

[Out]

Integral((a + b*x**3)**(1/3)/(x**5*(c + d*x**3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{{\left (d x^{3} + c\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^5/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(1/3)/((d*x^3 + c)*x^5), x)