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

Optimal. Leaf size=201 \[ \frac{\log \left (a d-b d x^3\right )}{3\ 2^{2/3} b^{2/3} d}+\frac{\log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3} d}-\frac{\log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^{2/3} d}+\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} d}-\frac{\sqrt [3]{2} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3} d} \]

[Out]

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

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Rubi [C]  time = 0.042767, antiderivative size = 66, normalized size of antiderivative = 0.33, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {511, 510} \[ \frac{x^2 \sqrt [3]{a+b x^3} F_1\left (\frac{2}{3};-\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{2 a d \sqrt [3]{\frac{b x^3}{a}+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Mathematica [C]  time = 0.0380564, size = 63, normalized size = 0.31 \[ \frac{x^2 \left (\frac{b x^3}{a}+1\right )^{2/3} F_1\left (\frac{2}{3};-\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},\frac{b x^3}{a}\right )}{2 d \left (a+b x^3\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^2*(1 + (b*x^3)/a)^(2/3)*AppellF1[2/3, -1/3, 1, 5/3, -((b*x^3)/a), (b*x^3)/a])/(2*d*(a + b*x^3)^(2/3))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{-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(x*(b*x^3+a)^(1/3)/(-b*d*x^3+a*d),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.3942, size = 869, normalized size = 4.32 \begin{align*} -\frac{2 \, \sqrt{3} 2^{\frac{1}{3}} b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} 2^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} b \left (-\frac{1}{b^{2}}\right )^{\frac{2}{3}} + \sqrt{3} x}{3 \, x}\right ) - 2 \cdot 2^{\frac{1}{3}} b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \log \left (\frac{2^{\frac{1}{3}} b x \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) + 2^{\frac{1}{3}} b^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} \log \left (\frac{2^{\frac{2}{3}} b^{2} x^{2} \left (-\frac{1}{b^{2}}\right )^{\frac{2}{3}} - 2^{\frac{1}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} b x \left (-\frac{1}{b^{2}}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) + 2 \, \sqrt{3}{\left (b^{2}\right )}^{\frac{1}{6}} b \arctan \left (\frac{{\left (\sqrt{3}{\left (b^{2}\right )}^{\frac{1}{3}} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{2}{3}}\right )}{\left (b^{2}\right )}^{\frac{1}{6}}}{3 \, b^{2} x}\right ) - 2 \,{\left (b^{2}\right )}^{\frac{2}{3}} \log \left (-\frac{{\left (b^{2}\right )}^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b}{x}\right ) +{\left (b^{2}\right )}^{\frac{2}{3}} \log \left (\frac{{\left (b^{2}\right )}^{\frac{1}{3}} b x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{2}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}} b}{x^{2}}\right )}{6 \, b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*sqrt(3)*2^(1/3)*b^2*(-1/b^2)^(1/3)*arctan(1/3*(sqrt(3)*2^(2/3)*(b*x^3 + a)^(1/3)*b*(-1/b^2)^(2/3) + sq
rt(3)*x)/x) - 2*2^(1/3)*b^2*(-1/b^2)^(1/3)*log((2^(1/3)*b*x*(-1/b^2)^(1/3) + (b*x^3 + a)^(1/3))/x) + 2^(1/3)*b
^2*(-1/b^2)^(1/3)*log((2^(2/3)*b^2*x^2*(-1/b^2)^(2/3) - 2^(1/3)*(b*x^3 + a)^(1/3)*b*x*(-1/b^2)^(1/3) + (b*x^3
+ a)^(2/3))/x^2) + 2*sqrt(3)*(b^2)^(1/6)*b*arctan(1/3*(sqrt(3)*(b^2)^(1/3)*b*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)*(
b^2)^(2/3))*(b^2)^(1/6)/(b^2*x)) - 2*(b^2)^(2/3)*log(-((b^2)^(2/3)*x - (b*x^3 + a)^(1/3)*b)/x) + (b^2)^(2/3)*l
og(((b^2)^(1/3)*b*x^2 + (b*x^3 + a)^(1/3)*(b^2)^(2/3)*x + (b*x^3 + a)^(2/3)*b)/x^2))/(b^2*d)

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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