3.47 \(\int \frac{(a-b x^3)^2}{(a+b x^3)^{19/3}} \, dx\)

Optimal. Leaf size=117 \[ \frac{81 x}{182 a^4 \sqrt [3]{a+b x^3}}+\frac{27 x}{182 a^3 \left (a+b x^3\right )^{4/3}}+\frac{9 x}{91 a^2 \left (a+b x^3\right )^{7/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}} \]

[Out]

(x*(a - b*x^3))/(8*(a + b*x^3)^(16/3)) + (11*x)/(104*(a + b*x^3)^(13/3)) + x/(13*a*(a + b*x^3)^(10/3)) + (9*x)
/(91*a^2*(a + b*x^3)^(7/3)) + (27*x)/(182*a^3*(a + b*x^3)^(4/3)) + (81*x)/(182*a^4*(a + b*x^3)^(1/3))

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Rubi [A]  time = 0.0442149, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {413, 385, 192, 191} \[ \frac{81 x}{182 a^4 \sqrt [3]{a+b x^3}}+\frac{27 x}{182 a^3 \left (a+b x^3\right )^{4/3}}+\frac{9 x}{91 a^2 \left (a+b x^3\right )^{7/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^3)^2/(a + b*x^3)^(19/3),x]

[Out]

(x*(a - b*x^3))/(8*(a + b*x^3)^(16/3)) + (11*x)/(104*(a + b*x^3)^(13/3)) + x/(13*a*(a + b*x^3)^(10/3)) + (9*x)
/(91*a^2*(a + b*x^3)^(7/3)) + (27*x)/(182*a^3*(a + b*x^3)^(4/3)) + (81*x)/(182*a^4*(a + b*x^3)^(1/3))

Rule 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

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{\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{\int \frac{14 a^2 b-8 a b^2 x^3}{\left (a+b x^3\right )^{16/3}} \, dx}{16 a b}\\ &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{10}{13} \int \frac{1}{\left (a+b x^3\right )^{13/3}} \, dx\\ &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{9 \int \frac{1}{\left (a+b x^3\right )^{10/3}} \, dx}{13 a}\\ &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{9 x}{91 a^2 \left (a+b x^3\right )^{7/3}}+\frac{54 \int \frac{1}{\left (a+b x^3\right )^{7/3}} \, dx}{91 a^2}\\ &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{9 x}{91 a^2 \left (a+b x^3\right )^{7/3}}+\frac{27 x}{182 a^3 \left (a+b x^3\right )^{4/3}}+\frac{81 \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx}{182 a^3}\\ &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{9 x}{91 a^2 \left (a+b x^3\right )^{7/3}}+\frac{27 x}{182 a^3 \left (a+b x^3\right )^{4/3}}+\frac{81 x}{182 a^4 \sqrt [3]{a+b x^3}}\\ \end{align*}

Mathematica [A]  time = 0.0373948, size = 73, normalized size = 0.62 \[ \frac{x \left (1872 a^2 b^3 x^9+2080 a^3 b^2 x^6+1183 a^4 b x^3+364 a^5+864 a b^4 x^{12}+162 b^5 x^{15}\right )}{364 a^4 \left (a+b x^3\right )^{16/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^3)^2/(a + b*x^3)^(19/3),x]

[Out]

(x*(364*a^5 + 1183*a^4*b*x^3 + 2080*a^3*b^2*x^6 + 1872*a^2*b^3*x^9 + 864*a*b^4*x^12 + 162*b^5*x^15))/(364*a^4*
(a + b*x^3)^(16/3))

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Maple [A]  time = 0.007, size = 70, normalized size = 0.6 \begin{align*}{\frac{x \left ( 162\,{b}^{5}{x}^{15}+864\,{b}^{4}{x}^{12}a+1872\,{b}^{3}{x}^{9}{a}^{2}+2080\,{a}^{3}{b}^{2}{x}^{6}+1183\,b{x}^{3}{a}^{4}+364\,{a}^{5} \right ) }{364\,{a}^{4}} \left ( b{x}^{3}+a \right ) ^{-{\frac{16}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^3+a)^2/(b*x^3+a)^(19/3),x)

[Out]

1/364*x*(162*b^5*x^15+864*a*b^4*x^12+1872*a^2*b^3*x^9+2080*a^3*b^2*x^6+1183*a^4*b*x^3+364*a^5)/(b*x^3+a)^(16/3
)/a^4

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Maxima [B]  time = 0.980965, size = 347, normalized size = 2.97 \begin{align*} -\frac{{\left (455 \, b^{3} - \frac{1680 \,{\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac{2184 \,{\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac{1040 \,{\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} b^{2} x^{16}}{7280 \,{\left (b x^{3} + a\right )}^{\frac{16}{3}} a^{4}} - \frac{{\left (455 \, b^{4} - \frac{2240 \,{\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac{4368 \,{\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac{4160 \,{\left (b x^{3} + a\right )}^{3} b}{x^{9}} + \frac{1820 \,{\left (b x^{3} + a\right )}^{4}}{x^{12}}\right )} b x^{16}}{3640 \,{\left (b x^{3} + a\right )}^{\frac{16}{3}} a^{4}} - \frac{{\left (91 \, b^{5} - \frac{560 \,{\left (b x^{3} + a\right )} b^{4}}{x^{3}} + \frac{1456 \,{\left (b x^{3} + a\right )}^{2} b^{3}}{x^{6}} - \frac{2080 \,{\left (b x^{3} + a\right )}^{3} b^{2}}{x^{9}} + \frac{1820 \,{\left (b x^{3} + a\right )}^{4} b}{x^{12}} - \frac{1456 \,{\left (b x^{3} + a\right )}^{5}}{x^{15}}\right )} x^{16}}{1456 \,{\left (b x^{3} + a\right )}^{\frac{16}{3}} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(19/3),x, algorithm="maxima")

[Out]

-1/7280*(455*b^3 - 1680*(b*x^3 + a)*b^2/x^3 + 2184*(b*x^3 + a)^2*b/x^6 - 1040*(b*x^3 + a)^3/x^9)*b^2*x^16/((b*
x^3 + a)^(16/3)*a^4) - 1/3640*(455*b^4 - 2240*(b*x^3 + a)*b^3/x^3 + 4368*(b*x^3 + a)^2*b^2/x^6 - 4160*(b*x^3 +
a)^3*b/x^9 + 1820*(b*x^3 + a)^4/x^12)*b*x^16/((b*x^3 + a)^(16/3)*a^4) - 1/1456*(91*b^5 - 560*(b*x^3 + a)*b^4/
x^3 + 1456*(b*x^3 + a)^2*b^3/x^6 - 2080*(b*x^3 + a)^3*b^2/x^9 + 1820*(b*x^3 + a)^4*b/x^12 - 1456*(b*x^3 + a)^5
/x^15)*x^16/((b*x^3 + a)^(16/3)*a^4)

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Fricas [A]  time = 2.18981, size = 311, normalized size = 2.66 \begin{align*} \frac{{\left (162 \, b^{5} x^{16} + 864 \, a b^{4} x^{13} + 1872 \, a^{2} b^{3} x^{10} + 2080 \, a^{3} b^{2} x^{7} + 1183 \, a^{4} b x^{4} + 364 \, a^{5} x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{364 \,{\left (a^{4} b^{6} x^{18} + 6 \, a^{5} b^{5} x^{15} + 15 \, a^{6} b^{4} x^{12} + 20 \, a^{7} b^{3} x^{9} + 15 \, a^{8} b^{2} x^{6} + 6 \, a^{9} b x^{3} + a^{10}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(19/3),x, algorithm="fricas")

[Out]

1/364*(162*b^5*x^16 + 864*a*b^4*x^13 + 1872*a^2*b^3*x^10 + 2080*a^3*b^2*x^7 + 1183*a^4*b*x^4 + 364*a^5*x)*(b*x
^3 + a)^(2/3)/(a^4*b^6*x^18 + 6*a^5*b^5*x^15 + 15*a^6*b^4*x^12 + 20*a^7*b^3*x^9 + 15*a^8*b^2*x^6 + 6*a^9*b*x^3
+ a^10)

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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)**(19/3),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{19}{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)^(19/3),x, algorithm="giac")

[Out]

integrate((b*x^3 - a)^2/(b*x^3 + a)^(19/3), x)