\(\int \frac {x^5}{\sqrt {b+a x^3}} \, dx\) [301]

Optimal result

Integrand size = 15, antiderivative size = 27 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{9 a^2} \]

[Out]

2/9*(a*x^3-2*b)*(a*x^3+b)^(1/2)/a^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \left (a x^3+b\right )^{3/2}}{9 a^2}-\frac {2 b \sqrt {a x^3+b}}{3 a^2} \]

[In]

Int[x^5/Sqrt[b + a*x^3],x]

[Out]

(-2*b*Sqrt[b + a*x^3])/(3*a^2) + (2*(b + a*x^3)^(3/2))/(9*a^2)

Rule 45

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 272

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*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {x}{\sqrt {b+a x}} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (-\frac {b}{a \sqrt {b+a x}}+\frac {\sqrt {b+a x}}{a}\right ) \, dx,x,x^3\right ) \\ & = -\frac {2 b \sqrt {b+a x^3}}{3 a^2}+\frac {2 \left (b+a x^3\right )^{3/2}}{9 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{9 a^2} \]

[In]

Integrate[x^5/Sqrt[b + a*x^3],x]

[Out]

(2*(-2*b + a*x^3)*Sqrt[b + a*x^3])/(9*a^2)

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89

method	result	size
gosper	\(\frac {2 \left (a \,x^{3}-2 b \right ) \sqrt {a \,x^{3}+b}}{9 a^{2}}\)	\(24\)
trager	\(\frac {2 \left (a \,x^{3}-2 b \right ) \sqrt {a \,x^{3}+b}}{9 a^{2}}\)	\(24\)
risch	\(\frac {2 \left (a \,x^{3}-2 b \right ) \sqrt {a \,x^{3}+b}}{9 a^{2}}\)	\(24\)
pseudoelliptic	\(\frac {2 \left (a \,x^{3}-2 b \right ) \sqrt {a \,x^{3}+b}}{9 a^{2}}\)	\(24\)
default	\(\frac {2 x^{3} \sqrt {a \,x^{3}+b}}{9 a}-\frac {4 b \sqrt {a \,x^{3}+b}}{9 a^{2}}\)	\(34\)
elliptic	\(\frac {2 x^{3} \sqrt {a \,x^{3}+b}}{9 a}-\frac {4 b \sqrt {a \,x^{3}+b}}{9 a^{2}}\)	\(34\)

[In]

int(x^5/(a*x^3+b)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/9*(a*x^3-2*b)*(a*x^3+b)^(1/2)/a^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \, \sqrt {a x^{3} + b} {\left (a x^{3} - 2 \, b\right )}}{9 \, a^{2}} \]

[In]

integrate(x^5/(a*x^3+b)^(1/2),x, algorithm="fricas")

[Out]

2/9*sqrt(a*x^3 + b)*(a*x^3 - 2*b)/a^2

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\begin {cases} \frac {2 x^{3} \sqrt {a x^{3} + b}}{9 a} - \frac {4 b \sqrt {a x^{3} + b}}{9 a^{2}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6 \sqrt {b}} & \text {otherwise} \end {cases} \]

[In]

integrate(x**5/(a*x**3+b)**(1/2),x)

[Out]

Piecewise((2*x**3*sqrt(a*x**3 + b)/(9*a) - 4*b*sqrt(a*x**3 + b)/(9*a**2), Ne(a, 0)), (x**6/(6*sqrt(b)), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}}}{9 \, a^{2}} - \frac {2 \, \sqrt {a x^{3} + b} b}{3 \, a^{2}} \]

[In]

integrate(x^5/(a*x^3+b)^(1/2),x, algorithm="maxima")

[Out]

2/9*(a*x^3 + b)^(3/2)/a^2 - 2/3*sqrt(a*x^3 + b)*b/a^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}}}{9 \, a^{2}} - \frac {2 \, \sqrt {a x^{3} + b} b}{3 \, a^{2}} \]

[In]

integrate(x^5/(a*x^3+b)^(1/2),x, algorithm="giac")

[Out]

2/9*(a*x^3 + b)^(3/2)/a^2 - 2/3*sqrt(a*x^3 + b)*b/a^2

Mupad [B] (verification not implemented)

Time = 5.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=-\frac {2\,\sqrt {a\,x^3+b}\,\left (2\,b-a\,x^3\right )}{9\,a^2} \]

[In]

int(x^5/(b + a*x^3)^(1/2),x)

[Out]

-(2*(b + a*x^3)^(1/2)*(2*b - a*x^3))/(9*a^2)