3.4.88 \(\int x^8 (a+b x^3)^{3/2} \, dx\) [388]

Optimal. Leaf size=59 \[ \frac {2 a^2 \left (a+b x^3\right )^{5/2}}{15 b^3}-\frac {4 a \left (a+b x^3\right )^{7/2}}{21 b^3}+\frac {2 \left (a+b x^3\right )^{9/2}}{27 b^3} \]

[Out]

2/15*a^2*(b*x^3+a)^(5/2)/b^3-4/21*a*(b*x^3+a)^(7/2)/b^3+2/27*(b*x^3+a)^(9/2)/b^3

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Rubi [A]
time = 0.02, antiderivative size = 59, 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 a^2 \left (a+b x^3\right )^{5/2}}{15 b^3}+\frac {2 \left (a+b x^3\right )^{9/2}}{27 b^3}-\frac {4 a \left (a+b x^3\right )^{7/2}}{21 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^3)^(3/2),x]

[Out]

(2*a^2*(a + b*x^3)^(5/2))/(15*b^3) - (4*a*(a + b*x^3)^(7/2))/(21*b^3) + (2*(a + b*x^3)^(9/2))/(27*b^3)

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^8 \left (a+b x^3\right )^{3/2} \, dx &=\frac {1}{3} \text {Subst}\left (\int x^2 (a+b x)^{3/2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^2 (a+b x)^{3/2}}{b^2}-\frac {2 a (a+b x)^{5/2}}{b^2}+\frac {(a+b x)^{7/2}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac {2 a^2 \left (a+b x^3\right )^{5/2}}{15 b^3}-\frac {4 a \left (a+b x^3\right )^{7/2}}{21 b^3}+\frac {2 \left (a+b x^3\right )^{9/2}}{27 b^3}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 39, normalized size = 0.66 \begin {gather*} \frac {2 \left (a+b x^3\right )^{5/2} \left (8 a^2-20 a b x^3+35 b^2 x^6\right )}{945 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^3)^(3/2),x]

[Out]

(2*(a + b*x^3)^(5/2)*(8*a^2 - 20*a*b*x^3 + 35*b^2*x^6))/(945*b^3)

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Maple [A]
time = 0.14, size = 89, normalized size = 1.51

method	result	size
gosper	\(\frac {2 \left (b \,x^{3}+a \right )^{\frac {5}{2}} \left (35 b^{2} x^{6}-20 a b \,x^{3}+8 a^{2}\right )}{945 b^{3}}\)	\(36\)
trager	\(\frac {2 \left (35 b^{4} x^{12}+50 a \,b^{3} x^{9}+3 a^{2} b^{2} x^{6}-4 a^{3} b \,x^{3}+8 a^{4}\right ) \sqrt {b \,x^{3}+a}}{945 b^{3}}\)	\(58\)
risch	\(\frac {2 \left (35 b^{4} x^{12}+50 a \,b^{3} x^{9}+3 a^{2} b^{2} x^{6}-4 a^{3} b \,x^{3}+8 a^{4}\right ) \sqrt {b \,x^{3}+a}}{945 b^{3}}\)	\(58\)
default	\(\frac {2 b \,x^{12} \sqrt {b \,x^{3}+a}}{27}+\frac {20 a \,x^{9} \sqrt {b \,x^{3}+a}}{189}+\frac {2 a^{2} x^{6} \sqrt {b \,x^{3}+a}}{315 b}-\frac {8 a^{3} x^{3} \sqrt {b \,x^{3}+a}}{945 b^{2}}+\frac {16 a^{4} \sqrt {b \,x^{3}+a}}{945 b^{3}}\)	\(89\)
elliptic	\(\frac {2 b \,x^{12} \sqrt {b \,x^{3}+a}}{27}+\frac {20 a \,x^{9} \sqrt {b \,x^{3}+a}}{189}+\frac {2 a^{2} x^{6} \sqrt {b \,x^{3}+a}}{315 b}-\frac {8 a^{3} x^{3} \sqrt {b \,x^{3}+a}}{945 b^{2}}+\frac {16 a^{4} \sqrt {b \,x^{3}+a}}{945 b^{3}}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/27*b*x^12*(b*x^3+a)^(1/2)+20/189*a*x^9*(b*x^3+a)^(1/2)+2/315/b*a^2*x^6*(b*x^3+a)^(1/2)-8/945*a^3/b^2*x^3*(b*
x^3+a)^(1/2)+16/945*a^4/b^3*(b*x^3+a)^(1/2)

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Maxima [A]
time = 0.30, size = 47, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}}}{27 \, b^{3}} - \frac {4 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a}{21 \, b^{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{2}}{15 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^(3/2),x, algorithm="maxima")

[Out]

2/27*(b*x^3 + a)^(9/2)/b^3 - 4/21*(b*x^3 + a)^(7/2)*a/b^3 + 2/15*(b*x^3 + a)^(5/2)*a^2/b^3

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Fricas [A]
time = 0.35, size = 57, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (35 \, b^{4} x^{12} + 50 \, a b^{3} x^{9} + 3 \, a^{2} b^{2} x^{6} - 4 \, a^{3} b x^{3} + 8 \, a^{4}\right )} \sqrt {b x^{3} + a}}{945 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^(3/2),x, algorithm="fricas")

[Out]

2/945*(35*b^4*x^12 + 50*a*b^3*x^9 + 3*a^2*b^2*x^6 - 4*a^3*b*x^3 + 8*a^4)*sqrt(b*x^3 + a)/b^3

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (54) = 108\).
time = 0.39, size = 112, normalized size = 1.90 \begin {gather*} \begin {cases} \frac {16 a^{4} \sqrt {a + b x^{3}}}{945 b^{3}} - \frac {8 a^{3} x^{3} \sqrt {a + b x^{3}}}{945 b^{2}} + \frac {2 a^{2} x^{6} \sqrt {a + b x^{3}}}{315 b} + \frac {20 a x^{9} \sqrt {a + b x^{3}}}{189} + \frac {2 b x^{12} \sqrt {a + b x^{3}}}{27} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{2}} x^{9}}{9} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(b*x**3+a)**(3/2),x)

[Out]

Piecewise((16*a**4*sqrt(a + b*x**3)/(945*b**3) - 8*a**3*x**3*sqrt(a + b*x**3)/(945*b**2) + 2*a**2*x**6*sqrt(a
+ b*x**3)/(315*b) + 20*a*x**9*sqrt(a + b*x**3)/189 + 2*b*x**12*sqrt(a + b*x**3)/27, Ne(b, 0)), (a**(3/2)*x**9/
9, True))

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Giac [A]
time = 1.69, size = 43, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (35 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}} - 90 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a + 63 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{2}\right )}}{945 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^(3/2),x, algorithm="giac")

[Out]

2/945*(35*(b*x^3 + a)^(9/2) - 90*(b*x^3 + a)^(7/2)*a + 63*(b*x^3 + a)^(5/2)*a^2)/b^3

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Mupad [B]
time = 1.07, size = 88, normalized size = 1.49 \begin {gather*} \frac {20\,a\,x^9\,\sqrt {b\,x^3+a}}{189}+\frac {2\,b\,x^{12}\,\sqrt {b\,x^3+a}}{27}+\frac {16\,a^4\,\sqrt {b\,x^3+a}}{945\,b^3}-\frac {8\,a^3\,x^3\,\sqrt {b\,x^3+a}}{945\,b^2}+\frac {2\,a^2\,x^6\,\sqrt {b\,x^3+a}}{315\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a + b*x^3)^(3/2),x)

[Out]

(20*a*x^9*(a + b*x^3)^(1/2))/189 + (2*b*x^12*(a + b*x^3)^(1/2))/27 + (16*a^4*(a + b*x^3)^(1/2))/(945*b^3) - (8
*a^3*x^3*(a + b*x^3)^(1/2))/(945*b^2) + (2*a^2*x^6*(a + b*x^3)^(1/2))/(315*b)

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