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

Optimal. Leaf size=99 \[ -\frac{d \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 b^{4/3}}+\frac{d \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3}}+\frac{x (b c-a d)}{a b \sqrt [3]{a+b x^3}} \]

[Out]

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

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Rubi [A]  time = 0.0235665, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {385, 239} \[ -\frac{d \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 b^{4/3}}+\frac{d \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3}}+\frac{x (b c-a d)}{a b \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)/(a + b*x^3)^(4/3),x]

[Out]

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

Rule 385

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

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [C]  time = 0.0354563, size = 61, normalized size = 0.62 \[ \frac{d x^4 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )+4 c x}{4 a \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*c*x + d*x^4*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[4/3, 4/3, 7/3, -((b*x^3)/a)])/(4*a*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.221, size = 0, normalized size = 0. \begin{align*} \int{(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((d*x^3+c)/(b*x^3+a)^(4/3),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.98966, size = 1187, normalized size = 11.99 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}}{\left (a b^{2} d x^{3} + a^{2} b d\right )} \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}} x^{2} - 3 \, \sqrt{\frac{1}{3}}{\left (\left (-b\right )^{\frac{1}{3}} b x^{3} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b x^{2} + 2 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} x\right )} \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} + 2 \, a\right ) + 6 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (b^{2} c - a b d\right )} x - 2 \,{\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) +{\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac{2}{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 )}{6 \,{\left (a b^{3} x^{3} + a^{2} b^{2}\right )}}, -\frac{6 \, \sqrt{\frac{1}{3}}{\left (a b^{2} d x^{3} + a^{2} b d\right )} \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (\left (-b\right )^{\frac{1}{3}} x - 2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}}}{x}\right ) - 6 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (b^{2} c - a b d\right )} x + 2 \,{\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) -{\left (a b d x^{3} + a^{2} d\right )} \left (-b\right )^{\frac{2}{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 )}{6 \,{\left (a b^{3} x^{3} + a^{2} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*sqrt(1/3)*(a*b^2*d*x^3 + a^2*b*d)*sqrt((-b)^(1/3)/b)*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*(-b)^(2/3)*x^2
- 3*sqrt(1/3)*((-b)^(1/3)*b*x^3 - (b*x^3 + a)^(1/3)*b*x^2 + 2*(b*x^3 + a)^(2/3)*(-b)^(2/3)*x)*sqrt((-b)^(1/3)/
b) + 2*a) + 6*(b*x^3 + a)^(2/3)*(b^2*c - a*b*d)*x - 2*(a*b*d*x^3 + a^2*d)*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^
3 + a)^(1/3))/x) + (a*b*d*x^3 + a^2*d)*(-b)^(2/3)*log(((-b)^(2/3)*x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/3)*x + (b*x^
3 + a)^(2/3))/x^2))/(a*b^3*x^3 + a^2*b^2), -1/6*(6*sqrt(1/3)*(a*b^2*d*x^3 + a^2*b*d)*sqrt(-(-b)^(1/3)/b)*arcta
n(-sqrt(1/3)*((-b)^(1/3)*x - 2*(b*x^3 + a)^(1/3))*sqrt(-(-b)^(1/3)/b)/x) - 6*(b*x^3 + a)^(2/3)*(b^2*c - a*b*d)
*x + 2*(a*b*d*x^3 + a^2*d)*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - (a*b*d*x^3 + a^2*d)*(-b)^(2/
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))/(a*b^3*x^3 + a^2*b^2)]

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Sympy [C]  time = 10.4044, size = 71, normalized size = 0.72 \begin{align*} \frac{c x \Gamma \left (\frac{1}{3}\right )}{3 a^{\frac{4}{3}} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{4}{3}\right )} + \frac{d x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{4}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{4}{3}} \Gamma \left (\frac{7}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x*gamma(1/3)/(3*a**(4/3)*(1 + b*x**3/a)**(1/3)*gamma(4/3)) + d*x**4*gamma(4/3)*hyper((4/3, 4/3), (7/3,), b*x
**3*exp_polar(I*pi)/a)/(3*a**(4/3)*gamma(7/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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