3.311 \(\int x^5 (a+b x)^{9/2} \, dx\)

Optimal. Leaf size=110 \[ \frac{20 a^2 (a+b x)^{17/2}}{17 b^6}-\frac{4 a^3 (a+b x)^{15/2}}{3 b^6}+\frac{10 a^4 (a+b x)^{13/2}}{13 b^6}-\frac{2 a^5 (a+b x)^{11/2}}{11 b^6}+\frac{2 (a+b x)^{21/2}}{21 b^6}-\frac{10 a (a+b x)^{19/2}}{19 b^6} \]

[Out]

(-2*a^5*(a + b*x)^(11/2))/(11*b^6) + (10*a^4*(a + b*x)^(13/2))/(13*b^6) - (4*a^3*(a + b*x)^(15/2))/(3*b^6) + (
20*a^2*(a + b*x)^(17/2))/(17*b^6) - (10*a*(a + b*x)^(19/2))/(19*b^6) + (2*(a + b*x)^(21/2))/(21*b^6)

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Rubi [A]  time = 0.0321136, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{20 a^2 (a+b x)^{17/2}}{17 b^6}-\frac{4 a^3 (a+b x)^{15/2}}{3 b^6}+\frac{10 a^4 (a+b x)^{13/2}}{13 b^6}-\frac{2 a^5 (a+b x)^{11/2}}{11 b^6}+\frac{2 (a+b x)^{21/2}}{21 b^6}-\frac{10 a (a+b x)^{19/2}}{19 b^6} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x)^(9/2),x]

[Out]

(-2*a^5*(a + b*x)^(11/2))/(11*b^6) + (10*a^4*(a + b*x)^(13/2))/(13*b^6) - (4*a^3*(a + b*x)^(15/2))/(3*b^6) + (
20*a^2*(a + b*x)^(17/2))/(17*b^6) - (10*a*(a + b*x)^(19/2))/(19*b^6) + (2*(a + b*x)^(21/2))/(21*b^6)

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])

Rubi steps

\begin{align*} \int x^5 (a+b x)^{9/2} \, dx &=\int \left (-\frac{a^5 (a+b x)^{9/2}}{b^5}+\frac{5 a^4 (a+b x)^{11/2}}{b^5}-\frac{10 a^3 (a+b x)^{13/2}}{b^5}+\frac{10 a^2 (a+b x)^{15/2}}{b^5}-\frac{5 a (a+b x)^{17/2}}{b^5}+\frac{(a+b x)^{19/2}}{b^5}\right ) \, dx\\ &=-\frac{2 a^5 (a+b x)^{11/2}}{11 b^6}+\frac{10 a^4 (a+b x)^{13/2}}{13 b^6}-\frac{4 a^3 (a+b x)^{15/2}}{3 b^6}+\frac{20 a^2 (a+b x)^{17/2}}{17 b^6}-\frac{10 a (a+b x)^{19/2}}{19 b^6}+\frac{2 (a+b x)^{21/2}}{21 b^6}\\ \end{align*}

Mathematica [A]  time = 0.0646423, size = 68, normalized size = 0.62 \[ \frac{2 (a+b x)^{11/2} \left (-4576 a^3 b^2 x^2+11440 a^2 b^3 x^3+1408 a^4 b x-256 a^5-24310 a b^4 x^4+46189 b^5 x^5\right )}{969969 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(-256*a^5 + 1408*a^4*b*x - 4576*a^3*b^2*x^2 + 11440*a^2*b^3*x^3 - 24310*a*b^4*x^4 + 46189*
b^5*x^5))/(969969*b^6)

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Maple [A]  time = 0.005, size = 65, normalized size = 0.6 \begin{align*} -{\frac{-92378\,{b}^{5}{x}^{5}+48620\,a{b}^{4}{x}^{4}-22880\,{a}^{2}{b}^{3}{x}^{3}+9152\,{a}^{3}{b}^{2}{x}^{2}-2816\,{a}^{4}bx+512\,{a}^{5}}{969969\,{b}^{6}} \left ( bx+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x+a)^(9/2),x)

[Out]

-2/969969*(b*x+a)^(11/2)*(-46189*b^5*x^5+24310*a*b^4*x^4-11440*a^2*b^3*x^3+4576*a^3*b^2*x^2-1408*a^4*b*x+256*a
^5)/b^6

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Maxima [A]  time = 1.05034, size = 116, normalized size = 1.05 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{21}{2}}}{21 \, b^{6}} - \frac{10 \,{\left (b x + a\right )}^{\frac{19}{2}} a}{19 \, b^{6}} + \frac{20 \,{\left (b x + a\right )}^{\frac{17}{2}} a^{2}}{17 \, b^{6}} - \frac{4 \,{\left (b x + a\right )}^{\frac{15}{2}} a^{3}}{3 \, b^{6}} + \frac{10 \,{\left (b x + a\right )}^{\frac{13}{2}} a^{4}}{13 \, b^{6}} - \frac{2 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{5}}{11 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^(9/2),x, algorithm="maxima")

[Out]

2/21*(b*x + a)^(21/2)/b^6 - 10/19*(b*x + a)^(19/2)*a/b^6 + 20/17*(b*x + a)^(17/2)*a^2/b^6 - 4/3*(b*x + a)^(15/
2)*a^3/b^6 + 10/13*(b*x + a)^(13/2)*a^4/b^6 - 2/11*(b*x + a)^(11/2)*a^5/b^6

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Fricas [A]  time = 1.53445, size = 297, normalized size = 2.7 \begin{align*} \frac{2 \,{\left (46189 \, b^{10} x^{10} + 206635 \, a b^{9} x^{9} + 351780 \, a^{2} b^{8} x^{8} + 271414 \, a^{3} b^{7} x^{7} + 80773 \, a^{4} b^{6} x^{6} + 63 \, a^{5} b^{5} x^{5} - 70 \, a^{6} b^{4} x^{4} + 80 \, a^{7} b^{3} x^{3} - 96 \, a^{8} b^{2} x^{2} + 128 \, a^{9} b x - 256 \, a^{10}\right )} \sqrt{b x + a}}{969969 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^(9/2),x, algorithm="fricas")

[Out]

2/969969*(46189*b^10*x^10 + 206635*a*b^9*x^9 + 351780*a^2*b^8*x^8 + 271414*a^3*b^7*x^7 + 80773*a^4*b^6*x^6 + 6
3*a^5*b^5*x^5 - 70*a^6*b^4*x^4 + 80*a^7*b^3*x^3 - 96*a^8*b^2*x^2 + 128*a^9*b*x - 256*a^10)*sqrt(b*x + a)/b^6

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Sympy [A]  time = 36.2401, size = 235, normalized size = 2.14 \begin{align*} \begin{cases} - \frac{512 a^{10} \sqrt{a + b x}}{969969 b^{6}} + \frac{256 a^{9} x \sqrt{a + b x}}{969969 b^{5}} - \frac{64 a^{8} x^{2} \sqrt{a + b x}}{323323 b^{4}} + \frac{160 a^{7} x^{3} \sqrt{a + b x}}{969969 b^{3}} - \frac{20 a^{6} x^{4} \sqrt{a + b x}}{138567 b^{2}} + \frac{6 a^{5} x^{5} \sqrt{a + b x}}{46189 b} + \frac{2098 a^{4} x^{6} \sqrt{a + b x}}{12597} + \frac{3796 a^{3} b x^{7} \sqrt{a + b x}}{6783} + \frac{1640 a^{2} b^{2} x^{8} \sqrt{a + b x}}{2261} + \frac{170 a b^{3} x^{9} \sqrt{a + b x}}{399} + \frac{2 b^{4} x^{10} \sqrt{a + b x}}{21} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x+a)**(9/2),x)

[Out]

Piecewise((-512*a**10*sqrt(a + b*x)/(969969*b**6) + 256*a**9*x*sqrt(a + b*x)/(969969*b**5) - 64*a**8*x**2*sqrt
(a + b*x)/(323323*b**4) + 160*a**7*x**3*sqrt(a + b*x)/(969969*b**3) - 20*a**6*x**4*sqrt(a + b*x)/(138567*b**2)
+ 6*a**5*x**5*sqrt(a + b*x)/(46189*b) + 2098*a**4*x**6*sqrt(a + b*x)/12597 + 3796*a**3*b*x**7*sqrt(a + b*x)/6
783 + 1640*a**2*b**2*x**8*sqrt(a + b*x)/2261 + 170*a*b**3*x**9*sqrt(a + b*x)/399 + 2*b**4*x**10*sqrt(a + b*x)/
21, Ne(b, 0)), (a**(9/2)*x**6/6, True))

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Giac [B]  time = 1.27336, size = 676, normalized size = 6.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^(9/2),x, algorithm="giac")

[Out]

2/14549535*(1615*(693*(b*x + a)^(13/2) - 4095*(b*x + a)^(11/2)*a + 10010*(b*x + a)^(9/2)*a^2 - 12870*(b*x + a)
^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 3003*(b*x + a)^(3/2)*a^5)*a^4/b^5 + 1292*(3003*(b*x + a)^(15/2) - 2079
0*(b*x + a)^(13/2)*a + 61425*(b*x + a)^(11/2)*a^2 - 100100*(b*x + a)^(9/2)*a^3 + 96525*(b*x + a)^(7/2)*a^4 - 5
4054*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6)*a^3/b^5 + 798*(6435*(b*x + a)^(17/2) - 51051*(b*x + a)^(
15/2)*a + 176715*(b*x + a)^(13/2)*a^2 - 348075*(b*x + a)^(11/2)*a^3 + 425425*(b*x + a)^(9/2)*a^4 - 328185*(b*x
+ a)^(7/2)*a^5 + 153153*(b*x + a)^(5/2)*a^6 - 36465*(b*x + a)^(3/2)*a^7)*a^2/b^5 + 28*(109395*(b*x + a)^(19/2
) - 978120*(b*x + a)^(17/2)*a + 3879876*(b*x + a)^(15/2)*a^2 - 8953560*(b*x + a)^(13/2)*a^3 + 13226850*(b*x +
a)^(11/2)*a^4 - 12932920*(b*x + a)^(9/2)*a^5 + 8314020*(b*x + a)^(7/2)*a^6 - 3325608*(b*x + a)^(5/2)*a^7 + 692
835*(b*x + a)^(3/2)*a^8)*a/b^5 + 3*(230945*(b*x + a)^(21/2) - 2297295*(b*x + a)^(19/2)*a + 10270260*(b*x + a)^
(17/2)*a^2 - 27159132*(b*x + a)^(15/2)*a^3 + 47006190*(b*x + a)^(13/2)*a^4 - 55552770*(b*x + a)^(11/2)*a^5 + 4
5265220*(b*x + a)^(9/2)*a^6 - 24942060*(b*x + a)^(7/2)*a^7 + 8729721*(b*x + a)^(5/2)*a^8 - 1616615*(b*x + a)^(
3/2)*a^9)/b^5)/b