3.1998 \(\int \sqrt{a+\frac{b}{x^3}} x^7 \, dx\)
Optimal. Leaf size=291 \[ -\frac{7\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{8/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right ),-7-4 \sqrt{3}\right )}{320 a^2 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{21 b^2 x^2 \sqrt{a+\frac{b}{x^3}}}{320 a^2}+\frac{1}{8} x^8 \sqrt{a+\frac{b}{x^3}}+\frac{3 b x^5 \sqrt{a+\frac{b}{x^3}}}{80 a} \]
[Out]
(-21*b^2*Sqrt[a + b/x^3]*x^2)/(320*a^2) + (3*b*Sqrt[a + b/x^3]*x^5)/(80*a) + (Sqrt[a + b/x^3]*x^8)/8 - (7*3^(3
/4)*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 + S
qrt[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*a^2*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])
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Rubi [A] time = 0.190527, antiderivative size = 291, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used =
{335, 277, 325, 218} \[ -\frac{21 b^2 x^2 \sqrt{a+\frac{b}{x^3}}}{320 a^2}-\frac{7\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{8/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{320 a^2 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{1}{8} x^8 \sqrt{a+\frac{b}{x^3}}+\frac{3 b x^5 \sqrt{a+\frac{b}{x^3}}}{80 a} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + b/x^3]*x^7,x]
[Out]
(-21*b^2*Sqrt[a + b/x^3]*x^2)/(320*a^2) + (3*b*Sqrt[a + b/x^3]*x^5)/(80*a) + (Sqrt[a + b/x^3]*x^8)/8 - (7*3^(3
/4)*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 + S
qrt[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*a^2*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 335
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]
Rule 277
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 325
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 218
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((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]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]
Rubi steps
\begin{align*} \int \sqrt{a+\frac{b}{x^3}} x^7 \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^3}}{x^9} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{8} \sqrt{a+\frac{b}{x^3}} x^8-\frac{1}{16} (3 b) \operatorname{Subst}\left (\int \frac{1}{x^6 \sqrt{a+b x^3}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{3 b \sqrt{a+\frac{b}{x^3}} x^5}{80 a}+\frac{1}{8} \sqrt{a+\frac{b}{x^3}} x^8+\frac{\left (21 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x^3}} \, dx,x,\frac{1}{x}\right )}{160 a}\\ &=-\frac{21 b^2 \sqrt{a+\frac{b}{x^3}} x^2}{320 a^2}+\frac{3 b \sqrt{a+\frac{b}{x^3}} x^5}{80 a}+\frac{1}{8} \sqrt{a+\frac{b}{x^3}} x^8-\frac{\left (21 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^3}} \, dx,x,\frac{1}{x}\right )}{640 a^2}\\ &=-\frac{21 b^2 \sqrt{a+\frac{b}{x^3}} x^2}{320 a^2}+\frac{3 b \sqrt{a+\frac{b}{x^3}} x^5}{80 a}+\frac{1}{8} \sqrt{a+\frac{b}{x^3}} x^8-\frac{7\ 3^{3/4} \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 a^2 \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] time = 0.070027, size = 96, normalized size = 0.33 \[ \frac{x^2 \sqrt{a+\frac{b}{x^3}} \left (\sqrt{\frac{a x^3}{b}+1} \left (10 a^2 x^6+3 a b x^3-7 b^2\right )+7 b^2 \, _2F_1\left (-\frac{1}{2},\frac{1}{6};\frac{7}{6};-\frac{a x^3}{b}\right )\right )}{80 a^2 \sqrt{\frac{a x^3}{b}+1}} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + b/x^3]*x^7,x]
[Out]
(Sqrt[a + b/x^3]*x^2*(Sqrt[1 + (a*x^3)/b]*(-7*b^2 + 3*a*b*x^3 + 10*a^2*x^6) + 7*b^2*Hypergeometric2F1[-1/2, 1/
6, 7/6, -((a*x^3)/b)]))/(80*a^2*Sqrt[1 + (a*x^3)/b])
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Maple [B] time = 0.023, size = 2226, normalized size = 7.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^7*(a+b/x^3)^(1/2),x)
[Out]
1/320*((a*x^3+b)/x^3)^(1/2)*x^2/a^3/(-b*a^2)^(1/3)*(40*I*(a*x^4+b*x)^(1/2)*(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3
^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)*(-b*a^2)^(1
/3)*3^(1/2)*x^6*a^3-42*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^
(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^
2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a
^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*3^(1/2)*x^2*a^2*b^3-120*x^
6*(a*x^4+b*x)^(1/2)*a^3*(-b*a^2)^(1/3)*(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)
^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)+84*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(
-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1
/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(
I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^
(1/2)-3))^(1/2))*(-b*a^2)^(1/3)*3^(1/2)*x*a*b^3+12*I*(a*x^4+b*x)^(1/2)*(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/
2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)*(-b*a^2)^(1/3)*
3^(1/2)*x^3*a^2*b-42*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1
/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)
^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2
)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(2/3)*3^(1/2)*b^3+4
2*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1
/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2)
)/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3
^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*x^2*a^2*b^3-84*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))
/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1
/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*Ellip
ticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2
))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(1/3)*x*a*b^3-36*b*x^3*(a*x^4+b*x)^(1/2)*a^2*(-b*a^2)^(1/3)*(1/a^2*x*(-a*x+(
-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))
)^(1/2)+42*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-
b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+
I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1
/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(2/3)*b^3-21*I*(a*x^4+b*x)^(1/2
)*(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a
*x-(-b*a^2)^(1/3)))^(1/2)*(-b*a^2)^(1/3)*3^(1/2)*a*b^2+63*b^2*(a*x^4+b*x)^(1/2)*a*(-b*a^2)^(1/3)*(1/a^2*x*(-a*
x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/
3)))^(1/2))/(x*(a*x^3+b))^(1/2)/(I*3^(1/2)-3)/(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(
-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + \frac{b}{x^{3}}} x^{7}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7*(a+b/x^3)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(a + b/x^3)*x^7, x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{7} \sqrt{\frac{a x^{3} + b}{x^{3}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7*(a+b/x^3)^(1/2),x, algorithm="fricas")
[Out]
integral(x^7*sqrt((a*x^3 + b)/x^3), x)
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Sympy [A] time = 1.72813, size = 48, normalized size = 0.16 \begin{align*} - \frac{\sqrt{a} 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 \Gamma \left (- \frac{5}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**7*(a+b/x**3)**(1/2),x)
[Out]
-sqrt(a)*x**8*gamma(-8/3)*hyper((-8/3, -1/2), (-5/3,), b*exp_polar(I*pi)/(a*x**3))/(3*gamma(-5/3))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + \frac{b}{x^{3}}} x^{7}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^7*(a+b/x^3)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(a + b/x^3)*x^7, x)