3.703 \(\int \frac{x (a+b x^3)^{4/3}}{c+d x^3} \, dx\)

Optimal. Leaf size=277 \[ \frac{(b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 \sqrt [3]{c} d^2}+\frac{\sqrt [3]{b} (3 b c-4 a d) \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{6 d^2}-\frac{(b c-a d)^{4/3} \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{c} d^2}+\frac{\sqrt [3]{b} (3 b c-4 a d) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} d^2}-\frac{(b c-a d)^{4/3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{c} d^2}+\frac{b x^2 \sqrt [3]{a+b x^3}}{3 d} \]

[Out]

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

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Rubi [C]  time = 0.0439897, antiderivative size = 65, normalized size of antiderivative = 0.23, 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} \[ \frac{a x^2 \sqrt [3]{a+b x^3} F_1\left (\frac{2}{3};-\frac{4}{3},1;\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 c \sqrt [3]{\frac{b x^3}{a}+1}} \]

Warning: Unable to verify antiderivative.

[In]

Int[(x*(a + b*x^3)^(4/3))/(c + d*x^3),x]

[Out]

(a*x^2*(a + b*x^3)^(1/3)*AppellF1[2/3, -4/3, 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 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx &=\frac{\left (a \sqrt [3]{a+b x^3}\right ) \int \frac{x \left (1+\frac{b x^3}{a}\right )^{4/3}}{c+d x^3} \, dx}{\sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{a x^2 \sqrt [3]{a+b x^3} F_1\left (\frac{2}{3};-\frac{4}{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 = 0.215738, size = 198, normalized size = 0.71 \[ \frac{2 b x^5 \left (\frac{b x^3}{a}+1\right )^{2/3} \left (\frac{d x^3}{c}+1\right )^{2/3} (4 a d-3 b c) F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+5 x^2 \left (a \left (\frac{b x^3}{a}+1\right )^{2/3} (3 a d-2 b c) \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )+2 b c \left (a+b x^3\right ) \left (\frac{d x^3}{c}+1\right )^{2/3}\right )}{30 c d \left (a+b x^3\right )^{2/3} \left (\frac{d x^3}{c}+1\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x*(a + b*x^3)^(4/3))/(c + d*x^3),x]

[Out]

(2*b*(-3*b*c + 4*a*d)*x^5*(1 + (b*x^3)/a)^(2/3)*(1 + (d*x^3)/c)^(2/3)*AppellF1[5/3, 2/3, 1, 8/3, -((b*x^3)/a),
-((d*x^3)/c)] + 5*x^2*(2*b*c*(a + b*x^3)*(1 + (d*x^3)/c)^(2/3) + a*(-2*b*c + 3*a*d)*(1 + (b*x^3)/a)^(2/3)*Hyp
ergeometric2F1[2/3, 2/3, 5/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))]))/(30*c*d*(a + b*x^3)^(2/3)*(1 + (d*x^3)/c
)^(2/3))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^3+a)^(4/3)/(d*x^3+c),x)

[Out]

int(x*(b*x^3+a)^(4/3)/(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{4}{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)^(4/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(4/3)*x/(d*x^3 + c), x)

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Fricas [A]  time = 3.50299, size = 968, normalized size = 3.49 \begin{align*} \frac{6 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} b d x^{2} - 6 \, \sqrt{3}{\left (b c - a d\right )} \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 (3 \, b c - 4 \, a d\right )} \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 (3 \, b c - 4 \, a d\right )} \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 ) - 6 \,{\left (b c - a d\right )} \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 (3 \, b c - 4 \, a d\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 ) + 3 \,{\left (b c - a d\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 )}{18 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

1/18*(6*(b*x^3 + a)^(1/3)*b*d*x^2 - 6*sqrt(3)*(b*c - a*d)*((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)*(3*b*c - 4*a*d)*(-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*(3*b*c - 4*a*d)*(-b)^(1/3)*
log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - 6*(b*c - a*d)*((b*c - a*d)/c)^(1/3)*log(-(x*((b*c - a*d)/c)^(1/3)
- (b*x^3 + a)^(1/3))/x) + (3*b*c - 4*a*d)*(-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) + 3*(b*c - a*d)*((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^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (a + b x^{3}\right )^{\frac{4}{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)**(4/3)/(d*x**3+c),x)

[Out]

Integral(x*(a + b*x**3)**(4/3)/(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{4}{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)^(4/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(4/3)*x/(d*x^3 + c), x)