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]

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

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Rubi [A]  time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.86, size = 67, normalized size = 0.88 \[ \frac {{\left (4 \, b^{2} x^{7} + 7 \, a b x^{4} + 7 \, a^{2} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{7 \, {\left (a b^{3} x^{9} + 3 \, a^{2} b^{2} x^{6} + 3 \, a^{3} b x^{3} + a^{4}\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {10}{3}}}\,{d x} \]

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)

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maple [A]  time = 0.04, size = 37, normalized size = 0.49 \[ \frac {\left (4 b^{2} x^{6}+7 a b \,x^{3}+7 a^{2}\right ) x}{7 \left (b \,x^{3}+a \right )^{\frac {7}{3}} a} \]

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 = 0.50, size = 105, normalized size = 1.38 \[ \frac {{\left (4 \, b - \frac {7 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} b x^{7}}{14 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a} + \frac {b^{2} x^{7}}{7 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a} + \frac {{\left (2 \, b^{2} - \frac {7 \, {\left (b x^{3} + a\right )} b}{x^{3}} + \frac {14 \, {\left (b x^{3} + a\right )}^{2}}{x^{6}}\right )} x^{7}}{14 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a} \]

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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mupad [B]  time = 1.43, size = 44, normalized size = 0.58 \[ \frac {4\,x\,{\left (b\,x^3+a\right )}^2+4\,a^2\,x-a\,x\,\left (b\,x^3+a\right )}{7\,a\,{\left (b\,x^3+a\right )}^{7/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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