\(\int \frac {(a-b x^3)^2}{(a+b x^3)^{4/3}} \, dx\) [42]

Optimal result

Integrand size = 22, antiderivative size = 113 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac {7}{3} x \left (a+b x^3\right )^{2/3}-\frac {10 a \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b}}+\frac {5 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{b}} \]

[Out]

2*x*(-b*x^3+a)/(b*x^3+a)^(1/3)+7/3*x*(b*x^3+a)^(2/3)+5/3*a*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(1/3)-10/9*a*arcta
n(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(1/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {424, 396, 245} \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=-\frac {10 a \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b}}+\frac {7}{3} x \left (a+b x^3\right )^{2/3}+\frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac {5 a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{b}} \]

[In]

Int[(a - b*x^3)^2/(a + b*x^3)^(4/3),x]

[Out]

(2*x*(a - b*x^3))/(a + b*x^3)^(1/3) + (7*x*(a + b*x^3)^(2/3))/3 - (10*a*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^
(1/3))/Sqrt[3]])/(3*Sqrt[3]*b^(1/3)) + (5*a*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/(3*b^(1/3))

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 396

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

Rule 424

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac {\int \frac {-a^2 b+7 a b^2 x^3}{\sqrt [3]{a+b x^3}} \, dx}{a b} \\ & = \frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac {7}{3} x \left (a+b x^3\right )^{2/3}-\frac {1}{3} (10 a) \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx \\ & = \frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac {7}{3} x \left (a+b x^3\right )^{2/3}-\frac {10 a \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b}}+\frac {5 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {1}{9} \left (\frac {3 \left (13 a x+b x^4\right )}{\sqrt [3]{a+b x^3}}-\frac {10 \sqrt {3} a \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{b}}+\frac {10 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{b}}-\frac {5 a \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{\sqrt [3]{b}}\right ) \]

[In]

Integrate[(a - b*x^3)^2/(a + b*x^3)^(4/3),x]

[Out]

((3*(13*a*x + b*x^4))/(a + b*x^3)^(1/3) - (10*Sqrt[3]*a*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^
(1/3))])/b^(1/3) + (10*a*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/b^(1/3) - (5*a*Log[b^(2/3)*x^2 + b^(1/3)*x*(a
+ b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/b^(1/3))/9

Maple [A] (verified)

Time = 4.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.38

method	result	size
pseudoelliptic	\(\frac {3 b^{\frac {4}{3}} x^{4}+10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) a \left (b \,x^{3}+a \right )^{\frac {1}{3}}+39 x a \,b^{\frac {1}{3}}+10 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a \left (b \,x^{3}+a \right )^{\frac {1}{3}}-5 \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{9 b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\)	\(156\)

[In]

int((-b*x^3+a)^2/(b*x^3+a)^(4/3),x,method=_RETURNVERBOSE)

[Out]

1/9*(3*b^(4/3)*x^4+10*3^(1/2)*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3+a)^(1/3))/b^(1/3)/x)*a*(b*x^3+a)^(1/3)+39
*x*a*b^(1/3)+10*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)*a*(b*x^3+a)^(1/3)-5*ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)
*x+(b*x^3+a)^(2/3))/x^2)*a*(b*x^3+a)^(1/3))/b^(1/3)/(b*x^3+a)^(1/3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (89) = 178\).

Time = 0.36 (sec) , antiderivative size = 412, normalized size of antiderivative = 3.65 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\left [\frac {15 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{3} + a^{2} b\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{\frac {4}{3}} x^{3} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{\frac {2}{3}} x\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} + 2 \, a\right ) + 10 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 5 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (b^{2} x^{4} + 13 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{9 \, {\left (b^{2} x^{3} + a b\right )}}, \frac {10 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 5 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {30 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{3} + a^{2} b\right )} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (b^{\frac {1}{3}} x + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}} x}\right )}{b^{\frac {1}{3}}} + 3 \, {\left (b^{2} x^{4} + 13 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{9 \, {\left (b^{2} x^{3} + a b\right )}}\right ] \]

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(4/3),x, algorithm="fricas")

[Out]

[1/9*(15*sqrt(1/3)*(a*b^2*x^3 + a^2*b)*sqrt(-1/b^(2/3))*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*b^(2/3)*x^2 - 3*sqrt
(1/3)*(b^(4/3)*x^3 + (b*x^3 + a)^(1/3)*b*x^2 - 2*(b*x^3 + a)^(2/3)*b^(2/3)*x)*sqrt(-1/b^(2/3)) + 2*a) + 10*(a*
b*x^3 + a^2)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) - 5*(a*b*x^3 + a^2)*b^(2/3)*log((b^(2/3)*x^2 + (b
*x^3 + a)^(1/3)*b^(1/3)*x + (b*x^3 + a)^(2/3))/x^2) + 3*(b^2*x^4 + 13*a*b*x)*(b*x^3 + a)^(2/3))/(b^2*x^3 + a*b
), 1/9*(10*(a*b*x^3 + a^2)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) - 5*(a*b*x^3 + a^2)*b^(2/3)*log((b^
(2/3)*x^2 + (b*x^3 + a)^(1/3)*b^(1/3)*x + (b*x^3 + a)^(2/3))/x^2) + 30*sqrt(1/3)*(a*b^2*x^3 + a^2*b)*arctan(sq
rt(1/3)*(b^(1/3)*x + 2*(b*x^3 + a)^(1/3))/(b^(1/3)*x))/b^(1/3) + 3*(b^2*x^4 + 13*a*b*x)*(b*x^3 + a)^(2/3))/(b^
2*x^3 + a*b)]

Sympy [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\int \frac {\left (- a + b x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac {4}{3}}}\, dx \]

[In]

integrate((-b*x**3+a)**2/(b*x**3+a)**(4/3),x)

[Out]

Integral((-a + b*x**3)**2/(a + b*x**3)**(4/3), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (89) = 178\).

Time = 0.29 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.62 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {1}{9} \, b^{2} {\left (\frac {4 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {7}{3}}} + \frac {3 \, {\left (3 \, a b - \frac {4 \, {\left (b x^{3} + a\right )} a}{x^{3}}\right )}}{\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{3}}{x} - \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2}}{x^{4}}} - \frac {2 \, a \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {7}{3}}} + \frac {4 \, a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {7}{3}}}\right )} + \frac {1}{3} \, a b {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}}} + \frac {6 \, x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} b} - \frac {\log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {4}{3}}} + \frac {2 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}}\right )} + \frac {a x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}}} \]

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(4/3),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\int { \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}}} \,d x } \]

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(4/3),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{4/3}} \,d x \]

[In]

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

[Out]

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