3.44 \(\int \frac{(a-b x^3)^2}{(a+b x^3)^{10/3}} \, dx\)

Optimal. Leaf size=76 \[ \frac{x \left (a-b x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}}+\frac{3 x \left (a-b x^3\right )}{14 a \left (a+b x^3\right )^{4/3}}+\frac{9 x}{14 a \sqrt [3]{a+b x^3}} \]

[Out]

(x*(a - b*x^3)^2)/(7*a*(a + b*x^3)^(7/3)) + (3*x*(a - b*x^3))/(14*a*(a + b*x^3)^(4/3)) + (9*x)/(14*a*(a + b*x^
3)^(1/3))

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Rubi [A]  time = 0.0212121, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {378, 191} \[ \frac{x \left (a-b x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}}+\frac{3 x \left (a-b x^3\right )}{14 a \left (a+b x^3\right )^{4/3}}+\frac{9 x}{14 a \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^3)^2/(a + b*x^3)^(10/3),x]

[Out]

(x*(a - b*x^3)^2)/(7*a*(a + b*x^3)^(7/3)) + (3*x*(a - b*x^3))/(14*a*(a + b*x^3)^(4/3)) + (9*x)/(14*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{\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{10/3}} \, dx &=\frac{x \left (a-b x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}}+\frac{6}{7} \int \frac{a-b x^3}{\left (a+b x^3\right )^{7/3}} \, dx\\ &=\frac{x \left (a-b x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}}+\frac{3 x \left (a-b x^3\right )}{14 a \left (a+b x^3\right )^{4/3}}+\frac{9}{14} \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx\\ &=\frac{x \left (a-b x^3\right )^2}{7 a \left (a+b x^3\right )^{7/3}}+\frac{3 x \left (a-b x^3\right )}{14 a \left (a+b x^3\right )^{4/3}}+\frac{9 x}{14 a \sqrt [3]{a+b x^3}}\\ \end{align*}

Mathematica [A]  time = 0.0146196, size = 40, normalized size = 0.53 \[ \frac{x \left (7 a^2+7 a b x^3+4 b^2 x^6\right )}{7 a \left (a+b x^3\right )^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^3)^2/(a + b*x^3)^(10/3),x]

[Out]

(x*(7*a^2 + 7*a*b*x^3 + 4*b^2*x^6))/(7*a*(a + b*x^3)^(7/3))

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Maple [A]  time = 0.005, size = 37, normalized size = 0.5 \begin{align*}{\frac{x \left ( 4\,{b}^{2}{x}^{6}+7\,a{x}^{3}b+7\,{a}^{2} \right ) }{7\,a} \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)^2/(b*x^3+a)^(10/3),x)

[Out]

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

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Maxima [A]  time = 1.00197, size = 142, normalized size = 1.87 \begin{align*} \frac{{\left (4 \, b - \frac{7 \,{\left (b x^{3} + a\right )}}{x^{3}}\right )} b x^{7}}{14 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a} + \frac{b^{2} x^{7}}{7 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} a} + \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(10/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.98809, size = 142, normalized size = 1.87 \begin{align*} \frac{{\left (4 \, b^{2} x^{7} + 7 \, a b x^{4} + 7 \, a^{2} x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{7 \,{\left (a b^{3} x^{9} + 3 \, a^{2} b^{2} x^{6} + 3 \, a^{3} b x^{3} + a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(10/3),x, algorithm="fricas")

[Out]

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

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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)**2/(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{{\left (b x^{3} - a\right )}^{2}}{{\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)^2/(b*x^3+a)^(10/3),x, algorithm="giac")

[Out]

integrate((b*x^3 - a)^2/(b*x^3 + a)^(10/3), x)