\(\int \frac {(a-b x^3)^2}{(a+b x^3)^{7/3}} \, dx\) [43]

Optimal result

Integrand size = 22, antiderivative size = 110 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\frac {x \left (a-b x^3\right )}{2 \left (a+b x^3\right )^{4/3}}-\frac {x}{2 \sqrt [3]{a+b x^3}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b}} \]

[Out]

1/2*x*(-b*x^3+a)/(b*x^3+a)^(4/3)-1/2*x/(b*x^3+a)^(1/3)-1/2*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(1/3)+1/3*arctan(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 = 110, 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, 393, 245} \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {x}{2 \sqrt [3]{a+b x^3}}+\frac {x \left (a-b x^3\right )}{2 \left (a+b x^3\right )^{4/3}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}} \]

[In]

Int[(a - b*x^3)^2/(a + b*x^3)^(7/3),x]

[Out]

(x*(a - b*x^3))/(2*(a + b*x^3)^(4/3)) - x/(2*(a + b*x^3)^(1/3)) + ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))
/Sqrt[3]]/(Sqrt[3]*b^(1/3)) - Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)]/(2*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 393

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

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\frac {-\frac {6 b^{4/3} x^4}{\left (a+b x^3\right )^{4/3}}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+\log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}} \]

[In]

Integrate[(a - b*x^3)^2/(a + b*x^3)^(7/3),x]

[Out]

((-6*b^(4/3)*x^4)/(a + b*x^3)^(4/3) + 2*Sqrt[3]*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))]
- 2*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)] + Log[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)]
)/(6*b^(1/3))

Maple [A] (verified)

Time = 4.38 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.16

method	result	size
pseudoelliptic	\(\frac {-6 b^{\frac {4}{3}} x^{4}+\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )\right )}{6 b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {4}{3}}}\)	\(128\)

[In]

int((-b*x^3+a)^2/(b*x^3+a)^(7/3),x,method=_RETURNVERBOSE)

[Out]

1/6/b^(1/3)/(b*x^3+a)^(4/3)*(-6*b^(4/3)*x^4+(b*x^3+a)^(4/3)*(-2*3^(1/2)*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3
+a)^(1/3))/b^(1/3)/x)+ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)-2*ln((-b^(1/3)*x+(b*x^3+
a)^(1/3))/x)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (87) = 174\).

Time = 0.35 (sec) , antiderivative size = 521, normalized size of antiderivative = 4.74 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\left [-\frac {6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{2} x^{4} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} b x^{3} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} + 2 \, a\right ) + 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )}}, -\frac {6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{2} x^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{x}\right ) + 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )}}\right ] \]

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(7/3),x, algorithm="fricas")

[Out]

[-1/6*(6*(b*x^3 + a)^(2/3)*b^2*x^4 - 3*sqrt(1/3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*sqrt((-b)^(1/3)/b)*log(3*b*x^
3 - 3*(b*x^3 + a)^(1/3)*(-b)^(2/3)*x^2 - 3*sqrt(1/3)*((-b)^(1/3)*b*x^3 - (b*x^3 + a)^(1/3)*b*x^2 + 2*(b*x^3 +
a)^(2/3)*(-b)^(2/3)*x)*sqrt((-b)^(1/3)/b) + 2*a) + 2*(b^2*x^6 + 2*a*b*x^3 + a^2)*(-b)^(2/3)*log(((-b)^(1/3)*x
+ (b*x^3 + a)^(1/3))/x) - (b^2*x^6 + 2*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))/(b^3*x^6 + 2*a*b^2*x^3 + a^2*b), -1/6*(6*(b*x^3 + a)^(2/3)*b^2*x^4 + 6*sqrt
(1/3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*sqrt(-(-b)^(1/3)/b)*arctan(-sqrt(1/3)*((-b)^(1/3)*x - 2*(b*x^3 + a)^(1/3
))*sqrt(-(-b)^(1/3)/b)/x) + 2*(b^2*x^6 + 2*a*b*x^3 + a^2)*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x)
- (b^2*x^6 + 2*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))/(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)]

Sympy [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\int \frac {\left (- a + b x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac {7}{3}}}\, dx \]

[In]

integrate((-b*x**3+a)**2/(b*x**3+a)**(7/3),x)

[Out]

Integral((-a + b*x**3)**2/(a + b*x**3)**(7/3), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (87) = 174\).

Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=-\frac {{\left (b - \frac {4 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} x^{4}}{4 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}}} - \frac {b x^{4}}{2 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}}} - \frac {1}{12} \, {\left (\frac {3 \, {\left (b + \frac {4 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} x^{4}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2}} + \frac {4 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {7}{3}}} - \frac {2 \, \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {7}{3}}} + \frac {4 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {7}{3}}}\right )} b^{2} \]

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(7/3),x, algorithm="maxima")

[Out]

-1/4*(b - 4*(b*x^3 + a)/x^3)*x^4/(b*x^3 + a)^(4/3) - 1/2*b*x^4/(b*x^3 + a)^(4/3) - 1/12*(3*(b + 4*(b*x^3 + a)/
x^3)*x^4/((b*x^3 + a)^(4/3)*b^2) + 4*sqrt(3)*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3))/b^(
7/3) - 2*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*log(-b^(1/3) + (b*x^3
+ a)^(1/3)/x)/b^(7/3))*b^2

Giac [F]

\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\int { \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {7}{3}}} \,d x } \]

[In]

integrate((-b*x^3+a)^2/(b*x^3+a)^(7/3),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx=\int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{7/3}} \,d x \]

[In]

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

[Out]

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