3.1998 \(\int \sqrt {a+\frac {b}{x^3}} x^7 \, dx\)
Optimal. Leaf size=291 \[ -\frac {21 b^2 x^2 \sqrt {a+\frac {b}{x^3}}}{320 a^2}-\frac {7\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{8/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac {b^{2/3}}{x^2}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{320 a^2 \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}}+\frac {1}{8} x^8 \sqrt {a+\frac {b}{x^3}}+\frac {3 b x^5 \sqrt {a+\frac {b}{x^3}}}{80 a} \]
[Out]
-21/320*b^2*x^2*(a+b/x^3)^(1/2)/a^2+3/80*b*x^5*(a+b/x^3)^(1/2)/a+1/8*x^8*(a+b/x^3)^(1/2)-7/320*3^(3/4)*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)/a^
2/(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)
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Rubi [A] time = 0.19, antiderivative size = 291, normalized size of antiderivative = 1.00,
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 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]
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 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]
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*}
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Mathematica [C] time = 0.08, 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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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{7} \sqrt {\frac {a x^{3} + b}{x^{3}}}, x\right ) \]
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + \frac {b}{x^{3}}} x^{7}\,{d x} \]
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)
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maple [B] time = 0.03, size = 2226, normalized size = 7.65 \[ \text {result too large to display} \]
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/(-a^2*b)^(1/3)*(-40*I*(-a^2*b)^(1/3)*3^(1/2)*(a*x^4+b*x)^(1/2)*(1/a^2*x*(
-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^
(1/3)))^(1/2)*x^6*a^3+42*I*3^(1/2)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*
(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a
*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*
x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*x^2*a^2*b^3-84*I*(-a
^2*b)^(1/3)*3^(1/2)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+
2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/
3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3
)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*x*a*b^3+120*x^6*(a*x^4+b*x)^(1/2)*a
^3*(-a^2*b)^(1/3)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-
a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)+42*I*(-a^2*b)^(2/3)*3^(1/2)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+
(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(
1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(
I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1
/2)-3))^(1/2))*b^3-12*I*(-a^2*b)^(1/3)*3^(1/2)*(a*x^4+b*x)^(1/2)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a
^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*x^3*a^2*b-42*(-(I*3^(
1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3
^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^
2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*
3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*x^2*a^2*b^3+84*(-a^2*b)^(1/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)
/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1
/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*Ellipt
icF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/
(I*3^(1/2)-3))^(1/2))*x*a*b^3-42*(-a^2*b)^(2/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)
*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2
*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^
(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*b^3+36*
b*x^3*(a*x^4+b*x)^(1/2)*a^2*(-a^2*b)^(1/3)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^
2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)+21*I*(-a^2*b)^(1/3)*3^(1/2)*(a*x^4+b*x)^(1/
2)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*
a*x-(-a^2*b)^(1/3)))^(1/2)*a*b^2-63*b^2*(a*x^4+b*x)^(1/2)*a*(-a^2*b)^(1/3)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3
^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2))/(x*(a*x^3+
b))^(1/2)/(I*3^(1/2)-3)/(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1
/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + \frac {b}{x^{3}}} x^{7}\,{d x} \]
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^7\,\sqrt {a+\frac {b}{x^3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^7*(a + b/x^3)^(1/2),x)
[Out]
int(x^7*(a + b/x^3)^(1/2), x)
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sympy [A] time = 1.59, size = 48, normalized size = 0.16 \[ - \frac {\sqrt {a} x^{8} \Gamma \left (- \frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{3}, - \frac {1}{2} \\ - \frac {5}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \Gamma \left (- \frac {5}{3}\right )} \]
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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