3.4.89 \(\int x^5 (a+b x^3)^{3/2} \, dx\) [389]

Optimal. Leaf size=38 \[ -\frac {2 a \left (a+b x^3\right )^{5/2}}{15 b^2}+\frac {2 \left (a+b x^3\right )^{7/2}}{21 b^2} \]

[Out]

-2/15*a*(b*x^3+a)^(5/2)/b^2+2/21*(b*x^3+a)^(7/2)/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 )^{7/2}}{21 b^2}-\frac {2 a \left (a+b x^3\right )^{5/2}}{15 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^3)^(3/2),x]

[Out]

(-2*a*(a + b*x^3)^(5/2))/(15*b^2) + (2*(a + b*x^3)^(7/2))/(21*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 \left (a+b x^3\right )^{3/2} \, dx &=\frac {1}{3} \text {Subst}\left (\int x (a+b x)^{3/2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a (a+b x)^{3/2}}{b}+\frac {(a+b x)^{5/2}}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 a \left (a+b x^3\right )^{5/2}}{15 b^2}+\frac {2 \left (a+b x^3\right )^{7/2}}{21 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 28, normalized size = 0.74 \begin {gather*} \frac {2 \left (a+b x^3\right )^{5/2} \left (-2 a+5 b x^3\right )}{105 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^3)^(3/2),x]

[Out]

(2*(a + b*x^3)^(5/2)*(-2*a + 5*b*x^3))/(105*b^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(68\) vs. \(2(30)=60\).
time = 0.14, size = 69, normalized size = 1.82

method	result	size
gosper	\(-\frac {2 \left (b \,x^{3}+a \right )^{\frac {5}{2}} \left (-5 b \,x^{3}+2 a \right )}{105 b^{2}}\)	\(25\)
trager	\(-\frac {2 \left (-5 b^{3} x^{9}-8 a \,b^{2} x^{6}-a^{2} b \,x^{3}+2 a^{3}\right ) \sqrt {b \,x^{3}+a}}{105 b^{2}}\)	\(47\)
risch	\(-\frac {2 \left (-5 b^{3} x^{9}-8 a \,b^{2} x^{6}-a^{2} b \,x^{3}+2 a^{3}\right ) \sqrt {b \,x^{3}+a}}{105 b^{2}}\)	\(47\)
default	\(\frac {2 b \,x^{9} \sqrt {b \,x^{3}+a}}{21}+\frac {16 a \,x^{6} \sqrt {b \,x^{3}+a}}{105}+\frac {2 a^{2} x^{3} \sqrt {b \,x^{3}+a}}{105 b}-\frac {4 a^{3} \sqrt {b \,x^{3}+a}}{105 b^{2}}\)	\(69\)
elliptic	\(\frac {2 b \,x^{9} \sqrt {b \,x^{3}+a}}{21}+\frac {16 a \,x^{6} \sqrt {b \,x^{3}+a}}{105}+\frac {2 a^{2} x^{3} \sqrt {b \,x^{3}+a}}{105 b}-\frac {4 a^{3} \sqrt {b \,x^{3}+a}}{105 b^{2}}\)	\(69\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*b*x^9*(b*x^3+a)^(1/2)+16/105*a*x^6*(b*x^3+a)^(1/2)+2/105/b*a^2*x^3*(b*x^3+a)^(1/2)-4/105*a^3/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 {7}{2}}}{21 \, b^{2}} - \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a}{15 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(b*x^3 + a)^(7/2)/b^2 - 2/15*(b*x^3 + a)^(5/2)*a/b^2

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Fricas [A]
time = 0.34, size = 45, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (5 \, b^{3} x^{9} + 8 \, a b^{2} x^{6} + a^{2} b x^{3} - 2 \, a^{3}\right )} \sqrt {b x^{3} + a}}{105 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(5*b^3*x^9 + 8*a*b^2*x^6 + a^2*b*x^3 - 2*a^3)*sqrt(b*x^3 + a)/b^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (34) = 68\).
time = 0.24, size = 88, normalized size = 2.32 \begin {gather*} \begin {cases} - \frac {4 a^{3} \sqrt {a + b x^{3}}}{105 b^{2}} + \frac {2 a^{2} x^{3} \sqrt {a + b x^{3}}}{105 b} + \frac {16 a x^{6} \sqrt {a + b x^{3}}}{105} + \frac {2 b x^{9} \sqrt {a + b x^{3}}}{21} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{2}} 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)**(3/2),x)

[Out]

Piecewise((-4*a**3*sqrt(a + b*x**3)/(105*b**2) + 2*a**2*x**3*sqrt(a + b*x**3)/(105*b) + 16*a*x**6*sqrt(a + b*x
**3)/105 + 2*b*x**9*sqrt(a + b*x**3)/21, Ne(b, 0)), (a**(3/2)*x**6/6, True))

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Giac [A]
time = 1.92, size = 29, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (5 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} - 7 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a\right )}}{105 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(5*(b*x^3 + a)^(7/2) - 7*(b*x^3 + a)^(5/2)*a)/b^2

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Mupad [B]
time = 1.10, size = 68, normalized size = 1.79 \begin {gather*} \frac {16\,a\,x^6\,\sqrt {b\,x^3+a}}{105}+\frac {2\,b\,x^9\,\sqrt {b\,x^3+a}}{21}-\frac {4\,a^3\,\sqrt {b\,x^3+a}}{105\,b^2}+\frac {2\,a^2\,x^3\,\sqrt {b\,x^3+a}}{105\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x^3)^(3/2),x)

[Out]

(16*a*x^6*(a + b*x^3)^(1/2))/105 + (2*b*x^9*(a + b*x^3)^(1/2))/21 - (4*a^3*(a + b*x^3)^(1/2))/(105*b^2) + (2*a
^2*x^3*(a + b*x^3)^(1/2))/(105*b)

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