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

Optimal. Leaf size=234 \[ -\frac{\sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 \sqrt [3]{c} d}+\frac{\sqrt [3]{b c-a d} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{c} d}+\frac{\sqrt [3]{b c-a d} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{c} d}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 d}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.0377606, antiderivative size = 64, normalized size of antiderivative = 0.27, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {511, 510} \[ \frac{x^2 \sqrt [3]{a+b x^3} F_1\left (\frac{2}{3};-\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 c \sqrt [3]{\frac{b x^3}{a}+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

Mathematica [C]  time = 0.0305237, size = 65, normalized size = 0.28 \[ \frac{x^2 \sqrt [3]{a+b x^3} F_1\left (\frac{2}{3};-\frac{1}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 c \sqrt [3]{\frac{a+b x^3}{a}}} \]

Warning: Unable to verify antiderivative.

[In]

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

________________________________________________________________________________________

Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{d{x}^{3}+c}\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)/(d*x^3+c),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.89708, size = 807, normalized size = 3.45 \begin{align*} \frac{2 \, \sqrt{3} \left (\frac{b c - a d}{c}\right )^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3}{\left (b c - a d\right )} x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} c \left (\frac{b c - a d}{c}\right )^{\frac{2}{3}}}{3 \,{\left (b c - a d\right )} x}\right ) - 2 \, \sqrt{3} \left (-b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}}}{3 \, b x}\right ) + 2 \, \left (-b\right )^{\frac{1}{3}} \log \left (\frac{\left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) + 2 \, \left (\frac{b c - a d}{c}\right )^{\frac{1}{3}} \log \left (-\frac{x \left (\frac{b c - a d}{c}\right )^{\frac{1}{3}} -{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) - \left (-b\right )^{\frac{1}{3}} \log \left (\frac{\left (-b\right )^{\frac{2}{3}} x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) - \left (\frac{b c - a d}{c}\right )^{\frac{1}{3}} \log \left (\frac{x^{2} \left (\frac{b c - a d}{c}\right )^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} x \left (\frac{b c - a d}{c}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*sqrt(3)*((b*c - a*d)/c)^(1/3)*arctan(-1/3*(sqrt(3)*(b*c - a*d)*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)*c*((b*c
- a*d)/c)^(2/3))/((b*c - a*d)*x)) - 2*sqrt(3)*(-b)^(1/3)*arctan(1/3*(sqrt(3)*b*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)
*(-b)^(2/3))/(b*x)) + 2*(-b)^(1/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + 2*((b*c - a*d)/c)^(1/3)*log(-(x
*((b*c - a*d)/c)^(1/3) - (b*x^3 + a)^(1/3))/x) - (-b)^(1/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/3)
*x + (b*x^3 + a)^(2/3))/x^2) - ((b*c - a*d)/c)^(1/3)*log((x^2*((b*c - a*d)/c)^(2/3) + (b*x^3 + a)^(1/3)*x*((b*
c - a*d)/c)^(1/3) + (b*x^3 + a)^(2/3))/x^2))/d

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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