\(\int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx\) [2024]

Optimal result

Integrand size = 15, antiderivative size = 294 \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {91 b^2 \sqrt {a+\frac {b}{x^3}} x^2}{320 a^3}-\frac {13 b \sqrt {a+\frac {b}{x^3}} x^5}{80 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^8}{8 a}+\frac {91 \sqrt {2+\sqrt {3}} b^{8/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{320 \sqrt [4]{3} a^3 \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]

[Out]

91/320*b^2*x^2*(a+b/x^3)^(1/2)/a^3-13/80*b*x^5*(a+b/x^3)^(1/2)/a^2+1/8*x^8*(a+b/x^3)^(1/2)/a+91/960*b^(8/3)*(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)*3^(3/
4)/a^3/(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 = 0.12 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {342, 331, 224} \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {91 b^2 x^2 \sqrt {a+\frac {b}{x^3}}}{320 a^3}-\frac {13 b x^5 \sqrt {a+\frac {b}{x^3}}}{80 a^2}+\frac {91 \sqrt {2+\sqrt {3}} b^{8/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt {3}\right )}{320 \sqrt [4]{3} a^3 \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {x^8 \sqrt {a+\frac {b}{x^3}}}{8 a} \]

[In]

Int[x^7/Sqrt[a + b/x^3],x]

[Out]

(91*b^2*Sqrt[a + b/x^3]*x^2)/(320*a^3) - (13*b*Sqrt[a + b/x^3]*x^5)/(80*a^2) + (Sqrt[a + b/x^3]*x^8)/(8*a) + (
91*Sqrt[2 + Sqrt[3]]*b^(8/3)*(a^(1/3) + b^(1/3)/x)*Sqrt[(a^(2/3) + b^(2/3)/x^2 - (a^(1/3)*b^(1/3))/x)/((1 + Sq
rt[3])*a^(1/3) + b^(1/3)/x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b
^(1/3)/x)], -7 - 4*Sqrt[3]])/(320*3^(1/4)*a^3*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 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 {1}{x^9 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {\sqrt {a+\frac {b}{x^3}} x^8}{8 a}+\frac {(13 b) \text {Subst}\left (\int \frac {1}{x^6 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{16 a} \\ & = -\frac {13 b \sqrt {a+\frac {b}{x^3}} x^5}{80 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^8}{8 a}-\frac {\left (91 b^2\right ) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{160 a^2} \\ & = \frac {91 b^2 \sqrt {a+\frac {b}{x^3}} x^2}{320 a^3}-\frac {13 b \sqrt {a+\frac {b}{x^3}} x^5}{80 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^8}{8 a}+\frac {\left (91 b^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{640 a^3} \\ & = \frac {91 b^2 \sqrt {a+\frac {b}{x^3}} x^2}{320 a^3}-\frac {13 b \sqrt {a+\frac {b}{x^3}} x^5}{80 a^2}+\frac {\sqrt {a+\frac {b}{x^3}} x^8}{8 a}+\frac {91 \sqrt {2+\sqrt {3}} b^{8/3} \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 )}{320 \sqrt [4]{3} a^3 \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 = 10.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.31 \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\frac {91 b^3+39 a b^2 x^3-12 a^2 b x^6+40 a^3 x^9-91 b^3 \sqrt {1+\frac {a x^3}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {a x^3}{b}\right )}{320 a^3 \sqrt {a+\frac {b}{x^3}} x} \]

[In]

Integrate[x^7/Sqrt[a + b/x^3],x]

[Out]

(91*b^3 + 39*a*b^2*x^3 - 12*a^2*b*x^6 + 40*a^3*x^9 - 91*b^3*Sqrt[1 + (a*x^3)/b]*Hypergeometric2F1[1/6, 1/2, 7/
6, -((a*x^3)/b)])/(320*a^3*Sqrt[a + b/x^3]*x)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 752 vs. \(2 (227 ) = 454\).

Time = 3.56 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.56

method	result	size
risch	\(\frac {\left (40 a^{2} x^{6}-52 a b \,x^{3}+91 b^{2}\right ) \left (a \,x^{3}+b \right )}{320 a^{3} x \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}-\frac {91 b^{3} \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 ) \sqrt {x \left (a \,x^{3}+b \right )}}{320 a^{2} \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 )}\, x^{2} \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}\)	\(753\)
default	\(\text {Expression too large to display}\)	\(2233\)

[In]

int(x^7/(a+b/x^3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/320*(40*a^2*x^6-52*a*b*x^3+91*b^2)/a^3/x*(a*x^3+b)/((a*x^3+b)/x^3)^(1/2)-91/320*b^3/a^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^2/((a*x^3+b)/x^3)^(1/2)*(x*(a*x^3+b))^(1/2)

Fricas [F]

\[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{7}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]

[In]

integrate(x^7/(a+b/x^3)^(1/2),x, algorithm="fricas")

[Out]

integral(x^10*sqrt((a*x^3 + b)/x^3)/(a*x^3 + b), x)

Sympy [A] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.16 \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=- \frac {x^{8} \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt {a} \Gamma \left (- \frac {5}{3}\right )} \]

[In]

integrate(x**7/(a+b/x**3)**(1/2),x)

[Out]

-x**8*gamma(-8/3)*hyper((-8/3, 1/2), (-5/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqrt(a)*gamma(-5/3))

Maxima [F]

\[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{7}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]

[In]

integrate(x^7/(a+b/x^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^7/sqrt(a + b/x^3), x)

Giac [F]

\[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int { \frac {x^{7}}{\sqrt {a + \frac {b}{x^{3}}}} \,d x } \]

[In]

integrate(x^7/(a+b/x^3)^(1/2),x, algorithm="giac")

[Out]

integrate(x^7/sqrt(a + b/x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \, dx=\int \frac {x^7}{\sqrt {a+\frac {b}{x^3}}} \,d x \]

[In]

int(x^7/(a + b/x^3)^(1/2),x)

[Out]

int(x^7/(a + b/x^3)^(1/2), x)