3.2277 \(\int \frac {x^5}{(a+b x^{3/2})^{2/3}} \, dx\)

Optimal. Leaf size=86 \[ -\frac {2 a^3 \sqrt [3]{a+b x^{3/2}}}{b^4}+\frac {3 a^2 \left (a+b x^{3/2}\right )^{4/3}}{2 b^4}+\frac {\left (a+b x^{3/2}\right )^{10/3}}{5 b^4}-\frac {6 a \left (a+b x^{3/2}\right )^{7/3}}{7 b^4} \]

[Out]

-2*a^3*(a+b*x^(3/2))^(1/3)/b^4+3/2*a^2*(a+b*x^(3/2))^(4/3)/b^4-6/7*a*(a+b*x^(3/2))^(7/3)/b^4+1/5*(a+b*x^(3/2))
^(10/3)/b^4

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac {3 a^2 \left (a+b x^{3/2}\right )^{4/3}}{2 b^4}-\frac {2 a^3 \sqrt [3]{a+b x^{3/2}}}{b^4}+\frac {\left (a+b x^{3/2}\right )^{10/3}}{5 b^4}-\frac {6 a \left (a+b x^{3/2}\right )^{7/3}}{7 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x^(3/2))^(2/3),x]

[Out]

(-2*a^3*(a + b*x^(3/2))^(1/3))/b^4 + (3*a^2*(a + b*x^(3/2))^(4/3))/(2*b^4) - (6*a*(a + b*x^(3/2))^(7/3))/(7*b^
4) + (a + b*x^(3/2))^(10/3)/(5*b^4)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^5}{\left (a+b x^{3/2}\right )^{2/3}} \, dx &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {x^3}{(a+b x)^{2/3}} \, dx,x,x^{3/2}\right )\\ &=\frac {2}{3} \operatorname {Subst}\left (\int \left (-\frac {a^3}{b^3 (a+b x)^{2/3}}+\frac {3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac {3 a (a+b x)^{4/3}}{b^3}+\frac {(a+b x)^{7/3}}{b^3}\right ) \, dx,x,x^{3/2}\right )\\ &=-\frac {2 a^3 \sqrt [3]{a+b x^{3/2}}}{b^4}+\frac {3 a^2 \left (a+b x^{3/2}\right )^{4/3}}{2 b^4}-\frac {6 a \left (a+b x^{3/2}\right )^{7/3}}{7 b^4}+\frac {\left (a+b x^{3/2}\right )^{10/3}}{5 b^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 56, normalized size = 0.65 \[ \frac {\sqrt [3]{a+b x^{3/2}} \left (-81 a^3+27 a^2 b x^{3/2}-18 a b^2 x^3+14 b^3 x^{9/2}\right )}{70 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x^(3/2))^(2/3),x]

[Out]

((a + b*x^(3/2))^(1/3)*(-81*a^3 + 27*a^2*b*x^(3/2) - 18*a*b^2*x^3 + 14*b^3*x^(9/2)))/(70*b^4)

________________________________________________________________________________________

fricas [A]  time = 1.29, size = 50, normalized size = 0.58 \[ -\frac {{\left (18 \, a b^{2} x^{3} + 81 \, a^{3} - {\left (14 \, b^{3} x^{4} + 27 \, a^{2} b x\right )} \sqrt {x}\right )} {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{70 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*x^(3/2))^(2/3),x, algorithm="fricas")

[Out]

-1/70*(18*a*b^2*x^3 + 81*a^3 - (14*b^3*x^4 + 27*a^2*b*x)*sqrt(x))*(b*x^(3/2) + a)^(1/3)/b^4

________________________________________________________________________________________

giac [A]  time = 0.21, size = 61, normalized size = 0.71 \[ -\frac {2 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a^{3}}{b^{4}} + \frac {14 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {10}{3}} - 60 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {7}{3}} a + 105 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} a^{2}}{70 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*x^(3/2))^(2/3),x, algorithm="giac")

[Out]

-2*(b*x^(3/2) + a)^(1/3)*a^3/b^4 + 1/70*(14*(b*x^(3/2) + a)^(10/3) - 60*(b*x^(3/2) + a)^(7/3)*a + 105*(b*x^(3/
2) + a)^(4/3)*a^2)/b^4

________________________________________________________________________________________

maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\left (b \,x^{\frac {3}{2}}+a \right )^{\frac {2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^(3/2)+a)^(2/3),x)

[Out]

int(x^5/(b*x^(3/2)+a)^(2/3),x)

________________________________________________________________________________________

maxima [A]  time = 0.90, size = 64, normalized size = 0.74 \[ \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {10}{3}}}{5 \, b^{4}} - \frac {6 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {7}{3}} a}{7 \, b^{4}} + \frac {3 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} a^{2}}{2 \, b^{4}} - \frac {2 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a^{3}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b*x^(3/2))^(2/3),x, algorithm="maxima")

[Out]

1/5*(b*x^(3/2) + a)^(10/3)/b^4 - 6/7*(b*x^(3/2) + a)^(7/3)*a/b^4 + 3/2*(b*x^(3/2) + a)^(4/3)*a^2/b^4 - 2*(b*x^
(3/2) + a)^(1/3)*a^3/b^4

________________________________________________________________________________________

mupad [B]  time = 1.42, size = 64, normalized size = 0.74 \[ \frac {{\left (a+b\,x^{3/2}\right )}^{10/3}}{5\,b^4}-\frac {6\,a\,{\left (a+b\,x^{3/2}\right )}^{7/3}}{7\,b^4}-\frac {2\,a^3\,{\left (a+b\,x^{3/2}\right )}^{1/3}}{b^4}+\frac {3\,a^2\,{\left (a+b\,x^{3/2}\right )}^{4/3}}{2\,b^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b*x^(3/2))^(2/3),x)

[Out]

(a + b*x^(3/2))^(10/3)/(5*b^4) - (6*a*(a + b*x^(3/2))^(7/3))/(7*b^4) - (2*a^3*(a + b*x^(3/2))^(1/3))/b^4 + (3*
a^2*(a + b*x^(3/2))^(4/3))/(2*b^4)

________________________________________________________________________________________

sympy [A]  time = 39.17, size = 102, normalized size = 1.19 \[ \begin {cases} - \frac {81 a^{3} \sqrt [3]{a + b x^{\frac {3}{2}}}}{70 b^{4}} + \frac {27 a^{2} x^{\frac {3}{2}} \sqrt [3]{a + b x^{\frac {3}{2}}}}{70 b^{3}} - \frac {9 a x^{3} \sqrt [3]{a + b x^{\frac {3}{2}}}}{35 b^{2}} + \frac {x^{\frac {9}{2}} \sqrt [3]{a + b x^{\frac {3}{2}}}}{5 b} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {2}{3}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a+b*x**(3/2))**(2/3),x)

[Out]

Piecewise((-81*a**3*(a + b*x**(3/2))**(1/3)/(70*b**4) + 27*a**2*x**(3/2)*(a + b*x**(3/2))**(1/3)/(70*b**3) - 9
*a*x**3*(a + b*x**(3/2))**(1/3)/(35*b**2) + x**(9/2)*(a + b*x**(3/2))**(1/3)/(5*b), Ne(b, 0)), (x**6/(6*a**(2/
3)), True))

________________________________________________________________________________________