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

Optimal. Leaf size=38 \[ \frac {2 \sqrt {a x^3+b} \left (3 a^2 x^6+a b x^3-2 b^2\right )}{45 a^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 )^{5/2}}{15 a^2}-\frac {2 b \left (a x^3+b\right )^{3/2}}{9 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[b + a*x^3],x]

[Out]

(-2*b*(b + a*x^3)^(3/2))/(9*a^2) + (2*(b + a*x^3)^(5/2))/(15*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 x^5 \sqrt {b+a x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x \sqrt {b+a x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {b \sqrt {b+a x}}{a}+\frac {(b+a x)^{3/2}}{a}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 b \left (b+a x^3\right )^{3/2}}{9 a^2}+\frac {2 \left (b+a x^3\right )^{5/2}}{15 a^2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 0.74 \begin {gather*} \frac {2 \left (a x^3+b\right )^{3/2} \left (3 a x^3-2 b\right )}{45 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[b + a*x^3],x]

[Out]

(2*(b + a*x^3)^(3/2)*(-2*b + 3*a*x^3))/(45*a^2)

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IntegrateAlgebraic [A]  time = 0.03, size = 38, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {b+a x^3} \left (-2 b^2+a b x^3+3 a^2 x^6\right )}{45 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b + a*x^3]*(-2*b^2 + a*b*x^3 + 3*a^2*x^6))/(45*a^2)

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fricas [A]  time = 0.45, size = 34, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (3 \, a^{2} x^{6} + a b x^{3} - 2 \, b^{2}\right )} \sqrt {a x^{3} + b}}{45 \, 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]

2/45*(3*a^2*x^6 + a*b*x^3 - 2*b^2)*sqrt(a*x^3 + b)/a^2

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giac [A]  time = 0.36, size = 29, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (3 \, {\left (a x^{3} + b\right )}^{\frac {5}{2}} - 5 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}} b\right )}}{45 \, 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]

2/45*(3*(a*x^3 + b)^(5/2) - 5*(a*x^3 + b)^(3/2)*b)/a^2

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maple [A]  time = 0.08, size = 25, normalized size = 0.66

method	result	size
gosper	\(\frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}} \left (3 a \,x^{3}-2 b \right )}{45 a^{2}}\)	\(25\)
trager	\(\frac {2 \sqrt {a \,x^{3}+b}\, \left (3 a^{2} x^{6}+a b \,x^{3}-2 b^{2}\right )}{45 a^{2}}\)	\(35\)
risch	\(\frac {2 \sqrt {a \,x^{3}+b}\, \left (3 a^{2} x^{6}+a b \,x^{3}-2 b^{2}\right )}{45 a^{2}}\)	\(35\)
default	\(\frac {2 x^{6} \sqrt {a \,x^{3}+b}}{15}+\frac {2 b \,x^{3} \sqrt {a \,x^{3}+b}}{45 a}-\frac {4 b^{2} \sqrt {a \,x^{3}+b}}{45 a^{2}}\)	\(51\)
elliptic	\(\frac {2 x^{6} \sqrt {a \,x^{3}+b}}{15}+\frac {2 b \,x^{3} \sqrt {a \,x^{3}+b}}{45 a}-\frac {4 b^{2} \sqrt {a \,x^{3}+b}}{45 a^{2}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*(a*x^3+b)^(3/2)*(3*a*x^3-2*b)/a^2

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maxima [A]  time = 0.32, size = 30, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (a x^{3} + b\right )}^{\frac {5}{2}}}{15 \, a^{2}} - \frac {2 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}} b}{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="maxima")

[Out]

2/15*(a*x^3 + b)^(5/2)/a^2 - 2/9*(a*x^3 + b)^(3/2)*b/a^2

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mupad [B]  time = 0.33, size = 29, normalized size = 0.76 \begin {gather*} -\frac {10\,b\,{\left (a\,x^3+b\right )}^{3/2}-6\,{\left (a\,x^3+b\right )}^{5/2}}{45\,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]

-(10*b*(b + a*x^3)^(3/2) - 6*(b + a*x^3)^(5/2))/(45*a^2)

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sympy [A]  time = 0.47, size = 66, normalized size = 1.74 \begin {gather*} \begin {cases} \frac {2 x^{6} \sqrt {a x^{3} + b}}{15} + \frac {2 b x^{3} \sqrt {a x^{3} + b}}{45 a} - \frac {4 b^{2} \sqrt {a x^{3} + b}}{45 a^{2}} & \text {for}\: a \neq 0 \\\frac {\sqrt {b} x^{6}}{6} & \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**6*sqrt(a*x**3 + b)/15 + 2*b*x**3*sqrt(a*x**3 + b)/(45*a) - 4*b**2*sqrt(a*x**3 + b)/(45*a**2),
Ne(a, 0)), (sqrt(b)*x**6/6, True))

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