3.31 \(\int \frac{a-b x^3}{(a+b x^3)^{10/3}} \, dx\)

Optimal. Leaf size=55 \[ \frac{15 x}{28 a^2 \sqrt [3]{a+b x^3}}+\frac{5 x}{28 a \left (a+b x^3\right )^{4/3}}+\frac{2 x}{7 \left (a+b x^3\right )^{7/3}} \]

[Out]

(2*x)/(7*(a + b*x^3)^(7/3)) + (5*x)/(28*a*(a + b*x^3)^(4/3)) + (15*x)/(28*a^2*(a + b*x^3)^(1/3))

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Rubi [A]  time = 0.0131894, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {385, 192, 191} \[ \frac{15 x}{28 a^2 \sqrt [3]{a+b x^3}}+\frac{5 x}{28 a \left (a+b x^3\right )^{4/3}}+\frac{2 x}{7 \left (a+b x^3\right )^{7/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^3)/(a + b*x^3)^(10/3),x]

[Out]

(2*x)/(7*(a + b*x^3)^(7/3)) + (5*x)/(28*a*(a + b*x^3)^(4/3)) + (15*x)/(28*a^2*(a + b*x^3)^(1/3))

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 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 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 )^{10/3}} \, dx &=\frac{2 x}{7 \left (a+b x^3\right )^{7/3}}+\frac{5}{7} \int \frac{1}{\left (a+b x^3\right )^{7/3}} \, dx\\ &=\frac{2 x}{7 \left (a+b x^3\right )^{7/3}}+\frac{5 x}{28 a \left (a+b x^3\right )^{4/3}}+\frac{15 \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx}{28 a}\\ &=\frac{2 x}{7 \left (a+b x^3\right )^{7/3}}+\frac{5 x}{28 a \left (a+b x^3\right )^{4/3}}+\frac{15 x}{28 a^2 \sqrt [3]{a+b x^3}}\\ \end{align*}

Mathematica [A]  time = 0.0190547, size = 40, normalized size = 0.73 \[ \frac{x \left (28 a^2+35 a b x^3+15 b^2 x^6\right )}{28 a^2 \left (a+b x^3\right )^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^3)/(a + b*x^3)^(10/3),x]

[Out]

(x*(28*a^2 + 35*a*b*x^3 + 15*b^2*x^6))/(28*a^2*(a + b*x^3)^(7/3))

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Maple [A]  time = 0.003, size = 37, normalized size = 0.7 \begin{align*}{\frac{x \left ( 15\,{b}^{2}{x}^{6}+35\,b{x}^{3}a+28\,{a}^{2} \right ) }{28\,{a}^{2}} \left ( b{x}^{3}+a \right ) ^{-{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^3+a)/(b*x^3+a)^(10/3),x)

[Out]

1/28*x*(15*b^2*x^6+35*a*b*x^3+28*a^2)/(b*x^3+a)^(7/3)/a^2

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Maxima [A]  time = 0.972712, size = 115, normalized size = 2.09 \begin{align*} \frac{{\left (4 \, b - \frac{7 \,{\left (b x^{3} + a\right )}}{x^{3}}\right )} b x^{7}}{28 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a^{2}} + \frac{{\left (2 \, b^{2} - \frac{7 \,{\left (b x^{3} + a\right )} b}{x^{3}} + \frac{14 \,{\left (b x^{3} + a\right )}^{2}}{x^{6}}\right )} x^{7}}{14 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^3+a)/(b*x^3+a)^(10/3),x, algorithm="maxima")

[Out]

1/28*(4*b - 7*(b*x^3 + a)/x^3)*b*x^7/((b*x^3 + a)^(7/3)*a^2) + 1/14*(2*b^2 - 7*(b*x^3 + a)*b/x^3 + 14*(b*x^3 +
a)^2/x^6)*x^7/((b*x^3 + a)^(7/3)*a^2)

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Fricas [A]  time = 1.74418, size = 150, normalized size = 2.73 \begin{align*} \frac{{\left (15 \, b^{2} x^{7} + 35 \, a b x^{4} + 28 \, a^{2} x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{28 \,{\left (a^{2} b^{3} x^{9} + 3 \, a^{3} b^{2} x^{6} + 3 \, a^{4} b x^{3} + a^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^3+a)/(b*x^3+a)^(10/3),x, algorithm="fricas")

[Out]

1/28*(15*b^2*x^7 + 35*a*b*x^4 + 28*a^2*x)*(b*x^3 + a)^(2/3)/(a^2*b^3*x^9 + 3*a^3*b^2*x^6 + 3*a^4*b*x^3 + a^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x**3+a)/(b*x**3+a)**(10/3),x)

[Out]

Timed out

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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{10}{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)^(10/3),x, algorithm="giac")

[Out]

integrate(-(b*x^3 - a)/(b*x^3 + a)^(10/3), x)