3.21.46 \(\int \frac {x}{(a+\frac {b}{x^3})^{3/2}} \, dx\) [2046]

Optimal. Leaf size=269 \[ -\frac {2 x^2}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {7 \sqrt {a+\frac {b}{x^3}} x^2}{6 a^2}+\frac {7 \sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right ) \sqrt {\frac {a^{2/3}+\frac {b^{2/3}}{x^2}-\frac {\sqrt [3]{a} \sqrt [3]{b}}{x}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt {3}\right )}{6 \sqrt [4]{3} a^2 \sqrt {a+\frac {b}{x^3}} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b}}{x}\right )^2}}} \]

[Out]

-2/3*x^2/a/(a+b/x^3)^(1/2)+7/6*x^2*(a+b/x^3)^(1/2)/a^2+7/18*b^(2/3)*(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^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 = 0.09, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {342, 296, 331, 224} \begin {gather*} \frac {7 x^2 \sqrt {a+\frac {b}{x^3}}}{6 a^2}+\frac {7 \sqrt {2+\sqrt {3}} b^{2/3} \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 (\text {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 )}{6 \sqrt [4]{3} 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 {2 x^2}{3 a \sqrt {a+\frac {b}{x^3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x^3)^(3/2),x]

[Out]

(-2*x^2)/(3*a*Sqrt[a + b/x^3]) + (7*Sqrt[a + b/x^3]*x^2)/(6*a^2) + (7*Sqrt[2 + Sqrt[3]]*b^(2/3)*(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]])/(6*3^(1/4)*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 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 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && 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*} \int \frac {x}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx &=-\text {Subst}\left (\int \frac {1}{x^3 \left (a+b x^3\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 x^2}{3 a \sqrt {a+\frac {b}{x^3}}}-\frac {7 \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{3 a}\\ &=-\frac {2 x^2}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {7 \sqrt {a+\frac {b}{x^3}} x^2}{6 a^2}+\frac {(7 b) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^3}} \, dx,x,\frac {1}{x}\right )}{12 a^2}\\ &=-\frac {2 x^2}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {7 \sqrt {a+\frac {b}{x^3}} x^2}{6 a^2}+\frac {7 \sqrt {2+\sqrt {3}} b^{2/3} \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 )}{6 \sqrt [4]{3} 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] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 67, normalized size = 0.25 \begin {gather*} \frac {7 b+3 a x^3-7 b \sqrt {1+\frac {a x^3}{b}} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};-\frac {a x^3}{b}\right )}{6 a^2 \sqrt {a+\frac {b}{x^3}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x^3)^(3/2),x]

[Out]

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

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2051 vs. \(2 (206 ) = 412\).
time = 0.07, size = 2052, normalized size = 7.63

method result size
risch \(\text {Expression too large to display}\) \(1431\)
default \(\text {Expression too large to display}\) \(2052\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6/((a*x^3+b)/x^3)^(3/2)/x^5*(a*x^3+b)/a^3/(-a^2*b)^(1/3)*(14*I*(x*(a*x^3+b))^(1/2)*(-(I*3^(1/2)-3)*x*a/(-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))/(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))/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1
/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(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)*a^2*b*x^2-28*I*(x*(a*x^3+b))^(1/2)*(-(I*3^(1/2)-3)*x*a/(-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))/(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))/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*Elli
pticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(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))*(-a^2*b)^(1/3)*3^(1/2)*a*b*x+14*I*(x*(a*x^3+b))^(1/2)*(-(I*3^(1/2)-3)*x*a/(-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))/(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))/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*E
llipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(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))*(-a^2*b)^(2/3)*3^(1/2)*b-14*(-(I*3^(1/2)-3)*x*a/(-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))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(x*(a*x
^3+b))^(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)*Elli
pticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(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))*a^2*b*x^2+28*(-(I*3^(1/2)-3)*x*a/(-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))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(x*(a*x^3+b))^(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)*EllipticF((-(I*3^(1/2)
-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(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))*(-a^2*b)^(1/3)*a*b*x-14*(-(I*3^(1/2)-3)*x*a/(-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))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(x*(a*x^3+b))^(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)*EllipticF((-(I*3^(1/2)-3)*x*a
/(-1+I*3^(1/2))/(-a*x+(-a^2*b)^(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))
*(-a^2*b)^(2/3)*b+3*I*(x*(a*x^3+b))^(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^2*b)^(1/3)*3^(1/2)*a+4
*I*(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^2*b)^(1/3)*3^(1/2)*a*b*x-9*(x*(a*x^3+b))^(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^2*b)^(1/3)*a-12*(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^2*b)^(1/3)*a*b*x)/(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(a + b/x^3)^(3/2), x)

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Fricas [F]
time = 0.08, size = 39, normalized size = 0.14 \begin {gather*} {\rm integral}\left (\frac {x^{7} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{a^{2} x^{6} + 2 \, a b x^{3} + b^{2}}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^7*sqrt((a*x^3 + b)/x^3)/(a^2*x^6 + 2*a*b*x^3 + b^2), x)

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Sympy [A]
time = 0.51, size = 42, normalized size = 0.16 \begin {gather*} - \frac {x^{2} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {1}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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/(a+b/x^3)^(3/2),x, algorithm="giac")

[Out]

integrate(x/(a + b/x^3)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b/x^3)^(3/2),x)

[Out]

int(x/(a + b/x^3)^(3/2), x)

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