3.4.70 \(\int x^5 \sqrt {a+b x^3} \, dx\) [370]

Optimal. Leaf size=38 \[ -\frac {2 a \left (a+b x^3\right )^{3/2}}{9 b^2}+\frac {2 \left (a+b x^3\right )^{5/2}}{15 b^2} \]

[Out]

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

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Rubi [A]
time = 0.02, 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 = {272, 45} \begin {gather*} \frac {2 \left (a+b x^3\right )^{5/2}}{15 b^2}-\frac {2 a \left (a+b x^3\right )^{3/2}}{9 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[a + b*x^3],x]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[a + b*x^3],x]

[Out]

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

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Maple [A]
time = 0.13, size = 51, normalized size = 1.34

method	result	size
gosper	\(-\frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (-3 b \,x^{3}+2 a \right )}{45 b^{2}}\)	\(25\)
trager	\(-\frac {2 \left (-3 b^{2} x^{6}-a b \,x^{3}+2 a^{2}\right ) \sqrt {b \,x^{3}+a}}{45 b^{2}}\)	\(36\)
risch	\(-\frac {2 \left (-3 b^{2} x^{6}-a b \,x^{3}+2 a^{2}\right ) \sqrt {b \,x^{3}+a}}{45 b^{2}}\)	\(36\)
default	\(\frac {2 x^{6} \sqrt {b \,x^{3}+a}}{15}+\frac {2 a \,x^{3} \sqrt {b \,x^{3}+a}}{45 b}-\frac {4 a^{2} \sqrt {b \,x^{3}+a}}{45 b^{2}}\)	\(51\)
elliptic	\(\frac {2 x^{6} \sqrt {b \,x^{3}+a}}{15}+\frac {2 a \,x^{3} \sqrt {b \,x^{3}+a}}{45 b}-\frac {4 a^{2} \sqrt {b \,x^{3}+a}}{45 b^{2}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*x^6*(b*x^3+a)^(1/2)+2/45*a/b*x^3*(b*x^3+a)^(1/2)-4/45*a^2/b^2*(b*x^3+a)^(1/2)

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Maxima [A]
time = 0.29, size = 30, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}}}{15 \, b^{2}} - \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a}{9 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.34, size = 34, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (3 \, b^{2} x^{6} + a b x^{3} - 2 \, a^{2}\right )} \sqrt {b x^{3} + a}}{45 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.13, size = 66, normalized size = 1.74 \begin {gather*} \begin {cases} - \frac {4 a^{2} \sqrt {a + b x^{3}}}{45 b^{2}} + \frac {2 a x^{3} \sqrt {a + b x^{3}}}{45 b} + \frac {2 x^{6} \sqrt {a + b x^{3}}}{15} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-4*a**2*sqrt(a + b*x**3)/(45*b**2) + 2*a*x**3*sqrt(a + b*x**3)/(45*b) + 2*x**6*sqrt(a + b*x**3)/15,
Ne(b, 0)), (sqrt(a)*x**6/6, True))

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Giac [A]
time = 1.15, size = 29, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} - 5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a\right )}}{45 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.09, size = 29, normalized size = 0.76 \begin {gather*} -\frac {10\,a\,{\left (b\,x^3+a\right )}^{3/2}-6\,{\left (b\,x^3+a\right )}^{5/2}}{45\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x^3)^(1/2),x)

[Out]

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

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