Integrand size = 13, antiderivative size = 136 \[ \int \left (a+\frac {b}{x^{4/3}}\right )^{3/2} \, dx=-\frac {b \sqrt {a+\frac {b}{x^{4/3}}}}{\sqrt [3]{x}}+a \sqrt {a+\frac {b}{x^{4/3}}} x-\frac {2 a^{3/4} b^{3/4} \sqrt {\frac {a+\frac {b}{x^{4/3}}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^{2/3}}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^{2/3}}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{\sqrt {a+\frac {b}{x^{4/3}}}} \] Output:
-b*(a+b/x^(4/3))^(1/2)/x^(1/3)+a*(a+b/x^(4/3))^(1/2)*x-2*a^(3/4)*b^(3/4)*( (a+b/x^(4/3))/(a^(1/2)+b^(1/2)/x^(2/3))^2)^(1/2)*(a^(1/2)+b^(1/2)/x^(2/3)) *InverseJacobiAM(2*arccot(a^(1/4)*x^(1/3)/b^(1/4)),1/2*2^(1/2))/(a+b/x^(4/ 3))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.39 \[ \int \left (a+\frac {b}{x^{4/3}}\right )^{3/2} \, dx=\frac {a \sqrt {a+\frac {b}{x^{4/3}}} x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},\frac {1}{4},-\frac {b}{a x^{4/3}}\right )}{\sqrt {1+\frac {b}{a x^{4/3}}}} \] Input:
Integrate[(a + b/x^(4/3))^(3/2),x]
Output:
(a*Sqrt[a + b/x^(4/3)]*x*Hypergeometric2F1[-3/2, -3/4, 1/4, -(b/(a*x^(4/3) ))])/Sqrt[1 + b/(a*x^(4/3))]
Time = 0.39 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {774, 858, 809, 748, 761}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a+\frac {b}{x^{4/3}}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 3 \int \left (a+\frac {b}{x^{4/3}}\right )^{3/2} x^{2/3}d\sqrt [3]{x}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -3 \int \frac {\left (b x^{4/3}+a\right )^{3/2}}{x^{4/3}}d\frac {1}{\sqrt [3]{x}}\) |
\(\Big \downarrow \) 809 |
\(\displaystyle -3 \left (2 b \int \sqrt {b x^{4/3}+a}d\frac {1}{\sqrt [3]{x}}-\frac {\left (a+b x^{4/3}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 748 |
\(\displaystyle -3 \left (2 b \left (\frac {2}{3} a \int \frac {1}{\sqrt {b x^{4/3}+a}}d\frac {1}{\sqrt [3]{x}}+\frac {\sqrt {a+b x^{4/3}}}{3 \sqrt [3]{x}}\right )-\frac {\left (a+b x^{4/3}\right )^{3/2}}{3 x}\right )\) |
\(\Big \downarrow \) 761 |
\(\displaystyle -3 \left (2 b \left (\frac {a^{3/4} \left (\sqrt {a}+\sqrt {b} x^{2/3}\right ) \sqrt {\frac {a+b x^{4/3}}{\left (\sqrt {a}+\sqrt {b} x^{2/3}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b}}{\sqrt [4]{a} \sqrt [3]{x}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {a+b x^{4/3}}}+\frac {\sqrt {a+b x^{4/3}}}{3 \sqrt [3]{x}}\right )-\frac {\left (a+b x^{4/3}\right )^{3/2}}{3 x}\right )\) |
Input:
Int[(a + b/x^(4/3))^(3/2),x]
Output:
-3*(-1/3*(a + b*x^(4/3))^(3/2)/x + 2*b*(Sqrt[a + b*x^(4/3)]/(3*x^(1/3)) + (a^(3/4)*(Sqrt[a] + Sqrt[b]*x^(2/3))*Sqrt[(a + b*x^(4/3))/(Sqrt[a] + Sqrt[ b]*x^(2/3))^2]*EllipticF[2*ArcTan[b^(1/4)/(a^(1/4)*x^(1/3))], 1/2])/(3*b^( 1/4)*Sqrt[a + b*x^(4/3)])))
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Simp[a*n*(p/(n*p + 1)) Int[(a + b*x^n)^(p - 1), x], x] /; Fre eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat or[p + 1/n], Denominator[p]])
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
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] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(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] && IntB inomialQ[a, b, c, n, m, p, x]
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] && Int egerQ[m]
Result contains complex when optimal does not.
Time = 1.02 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(\frac {\left (\frac {b +a \,x^{\frac {4}{3}}}{x^{\frac {4}{3}}}\right )^{\frac {3}{2}} x \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{2} x^{\frac {8}{3}}+4 a b \sqrt {-\frac {i \sqrt {a}\, x^{\frac {2}{3}}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{\frac {2}{3}}+\sqrt {b}}{\sqrt {b}}}\, \operatorname {EllipticF}\left (x^{\frac {1}{3}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) x -\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b^{2}\right )}{\left (b +a \,x^{\frac {4}{3}}\right )^{2} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}\) | \(135\) |
default | \(\frac {\left (\frac {b +a \,x^{\frac {4}{3}}}{x^{\frac {4}{3}}}\right )^{\frac {3}{2}} x \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{2} x^{\frac {8}{3}}+4 a b \sqrt {-\frac {i \sqrt {a}\, x^{\frac {2}{3}}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{\frac {2}{3}}+\sqrt {b}}{\sqrt {b}}}\, \operatorname {EllipticF}\left (x^{\frac {1}{3}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) x -\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b^{2}\right )}{\left (b +a \,x^{\frac {4}{3}}\right )^{2} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}\) | \(135\) |
Input:
int((a+b/x^(4/3))^(3/2),x,method=_RETURNVERBOSE)
Output:
((b+a*x^(4/3))/x^(4/3))^(3/2)*x*((I*a^(1/2)/b^(1/2))^(1/2)*a^2*x^(8/3)+4*a *b*(-(I*a^(1/2)*x^(2/3)-b^(1/2))/b^(1/2))^(1/2)*((I*a^(1/2)*x^(2/3)+b^(1/2 ))/b^(1/2))^(1/2)*EllipticF(x^(1/3)*(I*a^(1/2)/b^(1/2))^(1/2),I)*x-(I*a^(1 /2)/b^(1/2))^(1/2)*b^2)/(b+a*x^(4/3))^2/(I*a^(1/2)/b^(1/2))^(1/2)
Timed out. \[ \int \left (a+\frac {b}{x^{4/3}}\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate((a+b/x^(4/3))^(3/2),x, algorithm="fricas")
Output:
Timed out
Result contains complex when optimal does not.
Time = 3.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.34 \[ \int \left (a+\frac {b}{x^{4/3}}\right )^{3/2} \, dx=- \frac {3 a^{\frac {3}{2}} x \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{\frac {4}{3}}}} \right )}}{4 \Gamma \left (\frac {1}{4}\right )} \] Input:
integrate((a+b/x**(4/3))**(3/2),x)
Output:
-3*a**(3/2)*x*gamma(-3/4)*hyper((-3/2, -3/4), (1/4,), b*exp_polar(I*pi)/(a *x**(4/3)))/(4*gamma(1/4))
\[ \int \left (a+\frac {b}{x^{4/3}}\right )^{3/2} \, dx=\int { {\left (a + \frac {b}{x^{\frac {4}{3}}}\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b/x^(4/3))^(3/2),x, algorithm="maxima")
Output:
integrate((a + b/x^(4/3))^(3/2), x)
\[ \int \left (a+\frac {b}{x^{4/3}}\right )^{3/2} \, dx=\int { {\left (a + \frac {b}{x^{\frac {4}{3}}}\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b/x^(4/3))^(3/2),x, algorithm="giac")
Output:
integrate((a + b/x^(4/3))^(3/2), x)
Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.28 \[ \int \left (a+\frac {b}{x^{4/3}}\right )^{3/2} \, dx=-\frac {x\,{\left (a+\frac {b}{x^{4/3}}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {3}{4};\ \frac {1}{4};\ -\frac {a\,x^{4/3}}{b}\right )}{{\left (\frac {a\,x^{4/3}}{b}+1\right )}^{3/2}} \] Input:
int((a + b/x^(4/3))^(3/2),x)
Output:
-(x*(a + b/x^(4/3))^(3/2)*hypergeom([-3/2, -3/4], 1/4, -(a*x^(4/3))/b))/(( a*x^(4/3))/b + 1)^(3/2)
\[ \int \left (a+\frac {b}{x^{4/3}}\right )^{3/2} \, dx=\frac {x^{\frac {4}{3}} \sqrt {x^{\frac {4}{3}} a +b}\, a -5 \sqrt {x^{\frac {4}{3}} a +b}\, b -4 \left (\int \frac {\sqrt {x^{\frac {4}{3}} a +b}}{x^{\frac {10}{3}} a +b \,x^{2}}d x \right ) b^{2} x}{x} \] Input:
int((a+b/x^(4/3))^(3/2),x)
Output:
(x**(1/3)*sqrt(x**(1/3)*a*x + b)*a*x - 5*sqrt(x**(1/3)*a*x + b)*b - 4*int( sqrt(x**(1/3)*a*x + b)/(x**(1/3)*a*x**3 + b*x**2),x)*b**2*x)/x