\(\int \sqrt {a+\frac {b}{x^3}} x^4 \, dx\) [1999]

Optimal result

Integrand size = 15, antiderivative size = 267 \[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\frac {3 b \sqrt {a+\frac {b}{x^3}} x^2}{20 a}+\frac {1}{5} \sqrt {a+\frac {b}{x^3}} x^5+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{20 a \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]

[Out]

3/20*b*x^2*(a+b/x^3)^(1/2)/a+1/5*x^5*(a+b/x^3)^(1/2)+1/20*3^(3/4)*b^(5/3)*(a^(1/3)+b^(1/3)/x)*EllipticF((b^(1/
3)/x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)/x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)+1/2*2^(1/2))*((a^(2/3)+b
^(2/3)/x^2-a^(1/3)*b^(1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)/a/(a+b/x^3)^(1/2)/(a^(1/3)*(a^(1/3)+b^(
1/3)/x)/(b^(1/3)/x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {342, 283, 331, 224} \[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{20 a \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {1}{5} x^5 \sqrt {a+\frac {b}{x^3}}+\frac {3 b x^2 \sqrt {a+\frac {b}{x^3}}}{20 a} \]

[In]

Int[Sqrt[a + b/x^3]*x^4,x]

[Out]

(3*b*Sqrt[a + b/x^3]*x^2)/(20*a) + (Sqrt[a + b/x^3]*x^5)/5 + (3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(5/3)*(a^(1/3) + b^(
1/3)/x)*Sqrt[(a^(2/3) + b^(2/3)/x^2 - (a^(1/3)*b^(1/3))/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2]*EllipticF[Ar
cSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(20*a*Sqrt[a +
b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2])

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)
], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 283

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

Rule 331

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

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt {a+b x^3}}{x^6} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{5} \sqrt {a+\frac {b}{x^3}} x^5-\frac {1}{10} (3 b) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {3 b \sqrt {a+\frac {b}{x^3}} x^2}{20 a}+\frac {1}{5} \sqrt {a+\frac {b}{x^3}} x^5+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{40 a} \\ & = \frac {3 b \sqrt {a+\frac {b}{x^3}} x^2}{20 a}+\frac {1}{5} \sqrt {a+\frac {b}{x^3}} x^5+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{5/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{20 a \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.30 \[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\frac {\sqrt {a+\frac {b}{x^3}} x^2 \left (\left (b+a x^3\right ) \sqrt {1+\frac {a x^3}{b}}-b \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{6},\frac {7}{6},-\frac {a x^3}{b}\right )\right )}{5 a \sqrt {1+\frac {a x^3}{b}}} \]

[In]

Integrate[Sqrt[a + b/x^3]*x^4,x]

[Out]

(Sqrt[a + b/x^3]*x^2*((b + a*x^3)*Sqrt[1 + (a*x^3)/b] - b*Hypergeometric2F1[-1/2, 1/6, 7/6, -((a*x^3)/b)]))/(5
*a*Sqrt[1 + (a*x^3)/b])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (204 ) = 408\).

Time = 3.47 (sec) , antiderivative size = 739, normalized size of antiderivative = 2.77

method	result	size
risch	\(\frac {\left (4 a \,x^{3}+3 b \right ) x^{2} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{20 a}-\frac {3 b^{2} \left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {\frac {\left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) x}{\left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, {\left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}^{2} \sqrt {\frac {\left (-a^{2} b \right )^{\frac {1}{3}} \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{a \left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, \sqrt {\frac {\left (-a^{2} b \right )^{\frac {1}{3}} \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{a \left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}\, F\left (\sqrt {\frac {\left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) x}{\left (-\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}}, \sqrt {\frac {\left (\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}{\left (\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right ) x \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \sqrt {x \left (a \,x^{3}+b \right )}}{20 \left (-\frac {3 \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {a x \left (x -\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right ) \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \left (x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}\, \left (a \,x^{3}+b \right )}\)	\(739\)
default	\(\text {Expression too large to display}\)	\(2010\)

[In]

int(x^4*(a+b/x^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/20*(4*a*x^3+3*b)/a*x^2*((a*x^3+b)/x^3)^(1/2)-3/20*b^2*(1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*
((-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*x/(-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3
))/(x-1/a*(-a^2*b)^(1/3)))^(1/2)*(x-1/a*(-a^2*b)^(1/3))^2*(1/a*(-a^2*b)^(1/3)*(x+1/2/a*(-a^2*b)^(1/3)+1/2*I*3^
(1/2)/a*(-a^2*b)^(1/3))/(-1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(x-1/a*(-a^2*b)^(1/3)))^(1/2)*(
1/a*(-a^2*b)^(1/3)*(x+1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2
)/a*(-a^2*b)^(1/3))/(x-1/a*(-a^2*b)^(1/3)))^(1/2)/(-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(-a^2
*b)^(1/3)/(a*x*(x-1/a*(-a^2*b)^(1/3))*(x+1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*(x+1/2/a*(-a^2*b
)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3)))^(1/2)*EllipticF(((-3/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3
))*x/(-1/2/a*(-a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(x-1/a*(-a^2*b)^(1/3)))^(1/2),((3/2/a*(-a^2*b)^(1/
3)+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))*(1/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(1/2/a*(-a^2*b)^(1/3)
+1/2*I*3^(1/2)/a*(-a^2*b)^(1/3))/(3/2/a*(-a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(-a^2*b)^(1/3)))^(1/2))*x*((a*x^3+b)/x^
3)^(1/2)*(x*(a*x^3+b))^(1/2)/(a*x^3+b)

Fricas [F]

\[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\int { \sqrt {a + \frac {b}{x^{3}}} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b/x^3)^(1/2),x, algorithm="fricas")

[Out]

integral(x^4*sqrt((a*x^3 + b)/x^3), x)

Sympy [A] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.18 \[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=- \frac {\sqrt {a} x^{5} \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \Gamma \left (- \frac {2}{3}\right )} \]

[In]

integrate(x**4*(a+b/x**3)**(1/2),x)

[Out]

-sqrt(a)*x**5*gamma(-5/3)*hyper((-5/3, -1/2), (-2/3,), b*exp_polar(I*pi)/(a*x**3))/(3*gamma(-2/3))

Maxima [F]

\[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\int { \sqrt {a + \frac {b}{x^{3}}} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b/x^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a + b/x^3)*x^4, x)

Giac [F]

\[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\int { \sqrt {a + \frac {b}{x^{3}}} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b/x^3)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a + b/x^3)*x^4, x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+\frac {b}{x^3}} x^4 \, dx=\int x^4\,\sqrt {a+\frac {b}{x^3}} \,d x \]

[In]

int(x^4*(a + b/x^3)^(1/2),x)

[Out]

int(x^4*(a + b/x^3)^(1/2), x)