3.6.67 \(\int \frac {x^7}{(a+b x^3)^{2/3}} \, dx\) [567]

Optimal. Leaf size=123 \[ -\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac {5 a^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {5 a^2 \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{8/3}} \]

[Out]

-5/18*a*x^2*(b*x^3+a)^(1/3)/b^2+1/6*x^5*(b*x^3+a)^(1/3)/b-5/18*a^2*ln(b^(1/3)*x-(b*x^3+a)^(1/3))/b^(8/3)-5/27*
a^2*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(8/3)*3^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {327, 337} \begin {gather*} -\frac {5 a^2 \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {5 a^2 \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{8/3}}-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^3)^(2/3),x]

[Out]

(-5*a*x^2*(a + b*x^3)^(1/3))/(18*b^2) + (x^5*(a + b*x^3)^(1/3))/(6*b) - (5*a^2*ArcTan[(1 + (2*b^(1/3)*x)/(a +
b*x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(8/3)) - (5*a^2*Log[b^(1/3)*x - (a + b*x^3)^(1/3)])/(18*b^(8/3))

Rule 327

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

Rule 337

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

Rubi steps

\begin {align*} \int \frac {x^7}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac {(5 a) \int \frac {x^4}{\left (a+b x^3\right )^{2/3}} \, dx}{6 b}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}+\frac {\left (5 a^2\right ) \int \frac {x}{\left (a+b x^3\right )^{2/3}} \, dx}{9 b^2}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {x}{1-b x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 b^2}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{b} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{7/3}}-\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1-\sqrt [3]{b} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{7/3}}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac {5 a^2 \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{8/3}}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {\sqrt [3]{b}+2 b^{2/3} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{54 b^{8/3}}-\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{18 b^{7/3}}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac {5 a^2 \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{8/3}}+\frac {5 a^2 \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{54 b^{8/3}}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{8/3}}\\ &=-\frac {5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac {5 a^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {5 a^2 \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{8/3}}+\frac {5 a^2 \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{54 b^{8/3}}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 170, normalized size = 1.38 \begin {gather*} \frac {\sqrt [3]{a+b x^3} \left (-5 a x^2+3 b x^5\right )}{18 b^2}-\frac {5 a^2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {5 a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{27 b^{8/3}}+\frac {5 a^2 \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 )}{54 b^{8/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^3)^(2/3),x]

[Out]

((a + b*x^3)^(1/3)*(-5*a*x^2 + 3*b*x^5))/(18*b^2) - (5*a^2*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^
3)^(1/3))])/(9*Sqrt[3]*b^(8/3)) - (5*a^2*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/(27*b^(8/3)) + (5*a^2*Log[b^(2
/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(54*b^(8/3))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{7}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 184, normalized size = 1.50 \begin {gather*} \frac {5 \, \sqrt {3} a^{2} \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 )}{27 \, b^{\frac {8}{3}}} + \frac {5 \, a^{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 )}{54 \, b^{\frac {8}{3}}} - \frac {5 \, a^{2} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{27 \, b^{\frac {8}{3}}} + \frac {\frac {8 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2} b}{x} - \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a^{2}}{x^{4}}}{18 \, {\left (b^{4} - \frac {2 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/27*sqrt(3)*a^2*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3))/b^(8/3) + 5/54*a^2*log(b^(2/3)
+ (b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a)^(2/3)/x^2)/b^(8/3) - 5/27*a^2*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/
b^(8/3) + 1/18*(8*(b*x^3 + a)^(1/3)*a^2*b/x - 5*(b*x^3 + a)^(4/3)*a^2/x^4)/(b^4 - 2*(b*x^3 + a)*b^3/x^3 + (b*x
^3 + a)^2*b^2/x^6)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (96) = 192\).
time = 0.37, size = 212, normalized size = 1.72 \begin {gather*} \frac {10 \, \sqrt {3} a^{2} b \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} b x - 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}}}{3 \, b^{2} x}\right ) - 10 \, \left (-b^{2}\right )^{\frac {2}{3}} a^{2} \log \left (-\frac {\left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) + 5 \, \left (-b^{2}\right )^{\frac {2}{3}} a^{2} \log \left (-\frac {\left (-b^{2}\right )^{\frac {1}{3}} b x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right ) + 3 \, {\left (3 \, b^{3} x^{5} - 5 \, a b^{2} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{54 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(10*sqrt(3)*a^2*b*sqrt(-(-b^2)^(1/3))*arctan(-1/3*(sqrt(3)*(-b^2)^(1/3)*b*x - 2*sqrt(3)*(b*x^3 + a)^(1/3)
*(-b^2)^(2/3))*sqrt(-(-b^2)^(1/3))/(b^2*x)) - 10*(-b^2)^(2/3)*a^2*log(-((-b^2)^(2/3)*x - (b*x^3 + a)^(1/3)*b)/
x) + 5*(-b^2)^(2/3)*a^2*log(-((-b^2)^(1/3)*b*x^2 - (b*x^3 + a)^(1/3)*(-b^2)^(2/3)*x - (b*x^3 + a)^(2/3)*b)/x^2
) + 3*(3*b^3*x^5 - 5*a*b^2*x^2)*(b*x^3 + a)^(1/3))/b^4

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Sympy [C] Result contains complex when optimal does not.
time = 1.47, size = 37, normalized size = 0.30 \begin {gather*} \frac {x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {11}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**8*gamma(8/3)*hyper((2/3, 8/3), (11/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(2/3)*gamma(11/3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^7/(b*x^3 + a)^(2/3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^7}{{\left (b\,x^3+a\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x^3)^(2/3),x)

[Out]

int(x^7/(a + b*x^3)^(2/3), x)

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