3.30 \(\int \frac{a-b x^3}{(a+b x^3)^{7/3}} \, dx\)

Optimal. Leaf size=47 \[ \frac{3 x}{4 a \sqrt [3]{a+b x^3}}+\frac{x \left (a-b x^3\right )}{4 a \left (a+b x^3\right )^{4/3}} \]

[Out]

(x*(a - b*x^3))/(4*a*(a + b*x^3)^(4/3)) + (3*x)/(4*a*(a + b*x^3)^(1/3))

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Rubi [A]  time = 0.0093022, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {378, 191} \[ \frac{3 x}{4 a \sqrt [3]{a+b x^3}}+\frac{x \left (a-b x^3\right )}{4 a \left (a+b x^3\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^3)/(a + b*x^3)^(7/3),x]

[Out]

(x*(a - b*x^3))/(4*a*(a + b*x^3)^(4/3)) + (3*x)/(4*a*(a + b*x^3)^(1/3))

Rule 378

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{a-b x^3}{\left (a+b x^3\right )^{7/3}} \, dx &=\frac{x \left (a-b x^3\right )}{4 a \left (a+b x^3\right )^{4/3}}+\frac{3}{4} \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx\\ &=\frac{x \left (a-b x^3\right )}{4 a \left (a+b x^3\right )^{4/3}}+\frac{3 x}{4 a \sqrt [3]{a+b x^3}}\\ \end{align*}

Mathematica [A]  time = 0.0168929, size = 28, normalized size = 0.6 \[ \frac{x \left (2 a+b x^3\right )}{2 a \left (a+b x^3\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^3)/(a + b*x^3)^(7/3),x]

[Out]

(x*(2*a + b*x^3))/(2*a*(a + b*x^3)^(4/3))

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Maple [A]  time = 0.004, size = 25, normalized size = 0.5 \begin{align*}{\frac{x \left ( b{x}^{3}+2\,a \right ) }{2\,a} \left ( b{x}^{3}+a \right ) ^{-{\frac{4}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^3+a)/(b*x^3+a)^(7/3),x)

[Out]

1/2*x*(b*x^3+2*a)/(b*x^3+a)^(4/3)/a

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Maxima [A]  time = 1.09459, size = 68, normalized size = 1.45 \begin{align*} -\frac{{\left (b - \frac{4 \,{\left (b x^{3} + a\right )}}{x^{3}}\right )} x^{4}}{4 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a} - \frac{b x^{4}}{4 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^3+a)/(b*x^3+a)^(7/3),x, algorithm="maxima")

[Out]

-1/4*(b - 4*(b*x^3 + a)/x^3)*x^4/((b*x^3 + a)^(4/3)*a) - 1/4*b*x^4/((b*x^3 + a)^(4/3)*a)

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Fricas [A]  time = 1.61742, size = 96, normalized size = 2.04 \begin{align*} \frac{{\left (b x^{4} + 2 \, a x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{2 \,{\left (a b^{2} x^{6} + 2 \, a^{2} b x^{3} + a^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^3+a)/(b*x^3+a)^(7/3),x, algorithm="fricas")

[Out]

1/2*(b*x^4 + 2*a*x)*(b*x^3 + a)^(2/3)/(a*b^2*x^6 + 2*a^2*b*x^3 + a^3)

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Sympy [B]  time = 100.391, size = 190, normalized size = 4.04 \begin{align*} a \left (\frac{4 a x \Gamma \left (\frac{1}{3}\right )}{9 a^{\frac{10}{3}} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right ) + 9 a^{\frac{7}{3}} b x^{3} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right )} + \frac{3 b x^{4} \Gamma \left (\frac{1}{3}\right )}{9 a^{\frac{10}{3}} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right ) + 9 a^{\frac{7}{3}} b x^{3} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right )}\right ) - \frac{b x^{4} \Gamma \left (\frac{4}{3}\right )}{3 a^{\frac{7}{3}} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right ) + 3 a^{\frac{4}{3}} b x^{3} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{7}{3}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x**3+a)/(b*x**3+a)**(7/3),x)

[Out]

a*(4*a*x*gamma(1/3)/(9*a**(10/3)*(1 + b*x**3/a)**(1/3)*gamma(7/3) + 9*a**(7/3)*b*x**3*(1 + b*x**3/a)**(1/3)*ga
mma(7/3)) + 3*b*x**4*gamma(1/3)/(9*a**(10/3)*(1 + b*x**3/a)**(1/3)*gamma(7/3) + 9*a**(7/3)*b*x**3*(1 + b*x**3/
a)**(1/3)*gamma(7/3))) - b*x**4*gamma(4/3)/(3*a**(7/3)*(1 + b*x**3/a)**(1/3)*gamma(7/3) + 3*a**(4/3)*b*x**3*(1
+ b*x**3/a)**(1/3)*gamma(7/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b x^{3} - a}{{\left (b x^{3} + a\right )}^{\frac{7}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^3+a)/(b*x^3+a)^(7/3),x, algorithm="giac")

[Out]

integrate(-(b*x^3 - a)/(b*x^3 + a)^(7/3), x)