3.5.12 \(\int x^9 (a+b x^2)^{9/2} \, dx\) [412]

Optimal. Leaf size=101 \[ \frac {a^4 \left (a+b x^2\right )^{11/2}}{11 b^5}-\frac {4 a^3 \left (a+b x^2\right )^{13/2}}{13 b^5}+\frac {2 a^2 \left (a+b x^2\right )^{15/2}}{5 b^5}-\frac {4 a \left (a+b x^2\right )^{17/2}}{17 b^5}+\frac {\left (a+b x^2\right )^{19/2}}{19 b^5} \]

[Out]

1/11*a^4*(b*x^2+a)^(11/2)/b^5-4/13*a^3*(b*x^2+a)^(13/2)/b^5+2/5*a^2*(b*x^2+a)^(15/2)/b^5-4/17*a*(b*x^2+a)^(17/
2)/b^5+1/19*(b*x^2+a)^(19/2)/b^5

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Rubi [A]
time = 0.04, antiderivative size = 101, 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 {a^4 \left (a+b x^2\right )^{11/2}}{11 b^5}-\frac {4 a^3 \left (a+b x^2\right )^{13/2}}{13 b^5}+\frac {2 a^2 \left (a+b x^2\right )^{15/2}}{5 b^5}+\frac {\left (a+b x^2\right )^{19/2}}{19 b^5}-\frac {4 a \left (a+b x^2\right )^{17/2}}{17 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^9*(a + b*x^2)^(9/2),x]

[Out]

(a^4*(a + b*x^2)^(11/2))/(11*b^5) - (4*a^3*(a + b*x^2)^(13/2))/(13*b^5) + (2*a^2*(a + b*x^2)^(15/2))/(5*b^5) -
(4*a*(a + b*x^2)^(17/2))/(17*b^5) + (a + b*x^2)^(19/2)/(19*b^5)

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^9 \left (a+b x^2\right )^{9/2} \, dx &=\frac {1}{2} \text {Subst}\left (\int x^4 (a+b x)^{9/2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^4 (a+b x)^{9/2}}{b^4}-\frac {4 a^3 (a+b x)^{11/2}}{b^4}+\frac {6 a^2 (a+b x)^{13/2}}{b^4}-\frac {4 a (a+b x)^{15/2}}{b^4}+\frac {(a+b x)^{17/2}}{b^4}\right ) \, dx,x,x^2\right )\\ &=\frac {a^4 \left (a+b x^2\right )^{11/2}}{11 b^5}-\frac {4 a^3 \left (a+b x^2\right )^{13/2}}{13 b^5}+\frac {2 a^2 \left (a+b x^2\right )^{15/2}}{5 b^5}-\frac {4 a \left (a+b x^2\right )^{17/2}}{17 b^5}+\frac {\left (a+b x^2\right )^{19/2}}{19 b^5}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 61, normalized size = 0.60 \begin {gather*} \frac {\left (a+b x^2\right )^{11/2} \left (128 a^4-704 a^3 b x^2+2288 a^2 b^2 x^4-5720 a b^3 x^6+12155 b^4 x^8\right )}{230945 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^9*(a + b*x^2)^(9/2),x]

[Out]

((a + b*x^2)^(11/2)*(128*a^4 - 704*a^3*b*x^2 + 2288*a^2*b^2*x^4 - 5720*a*b^3*x^6 + 12155*b^4*x^8))/(230945*b^5
)

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Maple [A]
time = 0.08, size = 106, normalized size = 1.05

method	result	size
gosper	\(\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (12155 b^{4} x^{8}-5720 a \,b^{3} x^{6}+2288 a^{2} b^{2} x^{4}-704 a^{3} b \,x^{2}+128 a^{4}\right )}{230945 b^{5}}\)	\(58\)
default	\(\frac {x^{8} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{19 b}-\frac {8 a \left (\frac {x^{6} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{17 b}-\frac {6 a \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{15 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{13 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {11}{2}}}{143 b^{2}}\right )}{15 b}\right )}{17 b}\right )}{19 b}\)	\(106\)
trager	\(\frac {\left (12155 b^{9} x^{18}+55055 a \,b^{8} x^{16}+95238 a^{2} b^{7} x^{14}+75086 a^{3} b^{6} x^{12}+23063 a^{4} b^{5} x^{10}+35 a^{5} b^{4} x^{8}-40 a^{6} b^{3} x^{6}+48 a^{7} b^{2} x^{4}-64 a^{8} b \,x^{2}+128 a^{9}\right ) \sqrt {b \,x^{2}+a}}{230945 b^{5}}\)	\(113\)
risch	\(\frac {\left (12155 b^{9} x^{18}+55055 a \,b^{8} x^{16}+95238 a^{2} b^{7} x^{14}+75086 a^{3} b^{6} x^{12}+23063 a^{4} b^{5} x^{10}+35 a^{5} b^{4} x^{8}-40 a^{6} b^{3} x^{6}+48 a^{7} b^{2} x^{4}-64 a^{8} b \,x^{2}+128 a^{9}\right ) \sqrt {b \,x^{2}+a}}{230945 b^{5}}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/19*x^8*(b*x^2+a)^(11/2)/b-8/19*a/b*(1/17*x^6*(b*x^2+a)^(11/2)/b-6/17*a/b*(1/15*x^4*(b*x^2+a)^(11/2)/b-4/15*a
/b*(1/13*x^2*(b*x^2+a)^(11/2)/b-2/143*a*(b*x^2+a)^(11/2)/b^2)))

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Maxima [A]
time = 0.31, size = 93, normalized size = 0.92 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} x^{8}}{19 \, b} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a x^{6}}{323 \, b^{2}} + \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{2} x^{4}}{1615 \, b^{3}} - \frac {64 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{3} x^{2}}{20995 \, b^{4}} + \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{4}}{230945 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

1/19*(b*x^2 + a)^(11/2)*x^8/b - 8/323*(b*x^2 + a)^(11/2)*a*x^6/b^2 + 16/1615*(b*x^2 + a)^(11/2)*a^2*x^4/b^3 -
64/20995*(b*x^2 + a)^(11/2)*a^3*x^2/b^4 + 128/230945*(b*x^2 + a)^(11/2)*a^4/b^5

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Fricas [A]
time = 0.95, size = 112, normalized size = 1.11 \begin {gather*} \frac {{\left (12155 \, b^{9} x^{18} + 55055 \, a b^{8} x^{16} + 95238 \, a^{2} b^{7} x^{14} + 75086 \, a^{3} b^{6} x^{12} + 23063 \, a^{4} b^{5} x^{10} + 35 \, a^{5} b^{4} x^{8} - 40 \, a^{6} b^{3} x^{6} + 48 \, a^{7} b^{2} x^{4} - 64 \, a^{8} b x^{2} + 128 \, a^{9}\right )} \sqrt {b x^{2} + a}}{230945 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/230945*(12155*b^9*x^18 + 55055*a*b^8*x^16 + 95238*a^2*b^7*x^14 + 75086*a^3*b^6*x^12 + 23063*a^4*b^5*x^10 + 3
5*a^5*b^4*x^8 - 40*a^6*b^3*x^6 + 48*a^7*b^2*x^4 - 64*a^8*b*x^2 + 128*a^9)*sqrt(b*x^2 + a)/b^5

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (92) = 184\).
time = 1.79, size = 230, normalized size = 2.28 \begin {gather*} \begin {cases} \frac {128 a^{9} \sqrt {a + b x^{2}}}{230945 b^{5}} - \frac {64 a^{8} x^{2} \sqrt {a + b x^{2}}}{230945 b^{4}} + \frac {48 a^{7} x^{4} \sqrt {a + b x^{2}}}{230945 b^{3}} - \frac {8 a^{6} x^{6} \sqrt {a + b x^{2}}}{46189 b^{2}} + \frac {7 a^{5} x^{8} \sqrt {a + b x^{2}}}{46189 b} + \frac {23063 a^{4} x^{10} \sqrt {a + b x^{2}}}{230945} + \frac {6826 a^{3} b x^{12} \sqrt {a + b x^{2}}}{20995} + \frac {666 a^{2} b^{2} x^{14} \sqrt {a + b x^{2}}}{1615} + \frac {77 a b^{3} x^{16} \sqrt {a + b x^{2}}}{323} + \frac {b^{4} x^{18} \sqrt {a + b x^{2}}}{19} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{10}}{10} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*(b*x**2+a)**(9/2),x)

[Out]

Piecewise((128*a**9*sqrt(a + b*x**2)/(230945*b**5) - 64*a**8*x**2*sqrt(a + b*x**2)/(230945*b**4) + 48*a**7*x**
4*sqrt(a + b*x**2)/(230945*b**3) - 8*a**6*x**6*sqrt(a + b*x**2)/(46189*b**2) + 7*a**5*x**8*sqrt(a + b*x**2)/(4
6189*b) + 23063*a**4*x**10*sqrt(a + b*x**2)/230945 + 6826*a**3*b*x**12*sqrt(a + b*x**2)/20995 + 666*a**2*b**2*
x**14*sqrt(a + b*x**2)/1615 + 77*a*b**3*x**16*sqrt(a + b*x**2)/323 + b**4*x**18*sqrt(a + b*x**2)/19, Ne(b, 0))
, (a**(9/2)*x**10/10, True))

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Giac [A]
time = 1.02, size = 71, normalized size = 0.70 \begin {gather*} \frac {12155 \, {\left (b x^{2} + a\right )}^{\frac {19}{2}} - 54340 \, {\left (b x^{2} + a\right )}^{\frac {17}{2}} a + 92378 \, {\left (b x^{2} + a\right )}^{\frac {15}{2}} a^{2} - 71060 \, {\left (b x^{2} + a\right )}^{\frac {13}{2}} a^{3} + 20995 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{4}}{230945 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/230945*(12155*(b*x^2 + a)^(19/2) - 54340*(b*x^2 + a)^(17/2)*a + 92378*(b*x^2 + a)^(15/2)*a^2 - 71060*(b*x^2
+ a)^(13/2)*a^3 + 20995*(b*x^2 + a)^(11/2)*a^4)/b^5

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Mupad [B]
time = 4.74, size = 108, normalized size = 1.07 \begin {gather*} \sqrt {b\,x^2+a}\,\left (\frac {128\,a^9}{230945\,b^5}+\frac {23063\,a^4\,x^{10}}{230945}+\frac {b^4\,x^{18}}{19}+\frac {6826\,a^3\,b\,x^{12}}{20995}+\frac {77\,a\,b^3\,x^{16}}{323}+\frac {7\,a^5\,x^8}{46189\,b}-\frac {8\,a^6\,x^6}{46189\,b^2}+\frac {48\,a^7\,x^4}{230945\,b^3}-\frac {64\,a^8\,x^2}{230945\,b^4}+\frac {666\,a^2\,b^2\,x^{14}}{1615}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(a + b*x^2)^(9/2),x)

[Out]

(a + b*x^2)^(1/2)*((128*a^9)/(230945*b^5) + (23063*a^4*x^10)/230945 + (b^4*x^18)/19 + (6826*a^3*b*x^12)/20995
+ (77*a*b^3*x^16)/323 + (7*a^5*x^8)/(46189*b) - (8*a^6*x^6)/(46189*b^2) + (48*a^7*x^4)/(230945*b^3) - (64*a^8*
x^2)/(230945*b^4) + (666*a^2*b^2*x^14)/1615)

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