3.583 \(\int \frac{\sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx\)

Optimal. Leaf size=416 \[ -\frac{\log \left (2^{2/3}-\frac{\sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}+\frac{\log \left (\frac{2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}-\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}-\frac{\sqrt [3]{2} \log \left (\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d}+\frac{\log \left (\frac{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{\left (a+b x^3\right )^{2/3}}+\frac{2^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+2 \sqrt [3]{2}\right )}{6\ 2^{2/3} \sqrt [3]{a} \sqrt [3]{b} d}-\frac{\sqrt [3]{2} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a} \sqrt [3]{b} d}-\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} d} \]

[Out]

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

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Rubi [C]  time = 0.0311958, antiderivative size = 61, normalized size of antiderivative = 0.15, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {430, 429} \[ \frac{x \sqrt [3]{a+b x^3} F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{a d \sqrt [3]{\frac{b x^3}{a}+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(a + b*x^3)^(1/3)*AppellF1[1/3, -1/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/(a*d*(1 + (b*x^3)/a)^(1/3))

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx &=\frac{\sqrt [3]{a+b x^3} \int \frac{\sqrt [3]{1+\frac{b x^3}{a}}}{a d-b d x^3} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{x \sqrt [3]{a+b x^3} F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{a d \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}

Mathematica [C]  time = 0.145371, size = 154, normalized size = 0.37 \[ \frac{4 a x \sqrt [3]{a+b x^3} F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{d \left (a-b x^3\right ) \left (b x^3 \left (3 F_1\left (\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )+F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )\right )+4 a F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*a*x*(a + b*x^3)^(1/3)*AppellF1[1/3, -1/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a])/(d*(a - b*x^3)*(4*a*AppellF1[1/
3, -1/3, 1, 4/3, -((b*x^3)/a), (b*x^3)/a] + b*x^3*(3*AppellF1[4/3, -1/3, 2, 7/3, -((b*x^3)/a), (b*x^3)/a] + Ap
pellF1[4/3, 2/3, 1, 7/3, -((b*x^3)/a), (b*x^3)/a])))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-bd{x}^{3}+ad}\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)/(-b*d*x^3+a*d),x)

[Out]

int((b*x^3+a)^(1/3)/(-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{1}{3}}}{b d x^{3} - a d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((b*x^3 + a)^(1/3)/(b*d*x^3 - a*d), 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)/(-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{\sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral((a + b*x**3)**(1/3)/(-a + b*x**3), 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{1}{3}}}{b d x^{3} - a d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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