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

Optimal. Leaf size=27 \[ \frac {2 \left (a x^3-2 b\right ) \sqrt {a x^3+b}}{9 a^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \begin {gather*} \frac {2 \left (a x^3+b\right )^{3/2}}{9 a^2}-\frac {2 b \sqrt {a x^3+b}}{3 a^2} \end {gather*}

Antiderivative was successfully verified.

[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 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}{\sqrt {b+a x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt {b+a x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {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*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} \frac {2 \left (a x^3-2 b\right ) \sqrt {a x^3+b}}{9 a^2} \end {gather*}

Antiderivative was successfully verified.

[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)

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IntegrateAlgebraic [A]  time = 0.03, size = 27, normalized size = 1.00 \begin {gather*} \frac {2 \left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{9 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^5/Sqrt[b + a*x^3],x]

[Out]

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

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fricas [A]  time = 0.46, size = 23, normalized size = 0.85 \begin {gather*} \frac {2 \, \sqrt {a x^{3} + b} {\left (a x^{3} - 2 \, b\right )}}{9 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.59, size = 30, normalized size = 1.11 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.06, size = 24, normalized size = 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	\(\frac {2 \left (a \,x^{3}-2 b \right ) \sqrt {a \,x^{3}+b}}{9 a^{2}}\)	\(24\)
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	\(\frac {2 x^{3} \sqrt {a \,x^{3}+b}}{9 a}-\frac {4 b \sqrt {a \,x^{3}+b}}{9 a^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [A]  time = 0.44, size = 30, normalized size = 1.11 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.31, size = 24, normalized size = 0.89 \begin {gather*} -\frac {2\,\sqrt {a\,x^3+b}\,\left (2\,b-a\,x^3\right )}{9\,a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.60, size = 46, normalized size = 1.70 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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