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

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

[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 = 0.0786845, 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 \[ -\frac{2 a^5 (a+b x)^{11/2}}{11 b^6}+\frac{10 a^4 (a+b x)^{13/2}}{13 b^6}-\frac{4 a^3 (a+b x)^{15/2}}{3 b^6}+\frac{20 a^2 (a+b x)^{17/2}}{17 b^6}+\frac{2 (a+b x)^{21/2}}{21 b^6}-\frac{10 a (a+b x)^{19/2}}{19 b^6} \]

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)

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Rubi in Sympy [A]  time = 17.7991, size = 105, normalized size = 0.95 \[ - \frac{2 a^{5} \left (a + b x\right )^{\frac{11}{2}}}{11 b^{6}} + \frac{10 a^{4} \left (a + b x\right )^{\frac{13}{2}}}{13 b^{6}} - \frac{4 a^{3} \left (a + b x\right )^{\frac{15}{2}}}{3 b^{6}} + \frac{20 a^{2} \left (a + b x\right )^{\frac{17}{2}}}{17 b^{6}} - \frac{10 a \left (a + b x\right )^{\frac{19}{2}}}{19 b^{6}} + \frac{2 \left (a + b x\right )^{\frac{21}{2}}}{21 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x+a)**(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)

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Mathematica [A]  time = 0.0572699, size = 68, normalized size = 0.62 \[ \frac{2 (a+b x)^{11/2} \left (-256 a^5+1408 a^4 b x-4576 a^3 b^2 x^2+11440 a^2 b^3 x^3-24310 a b^4 x^4+46189 b^5 x^5\right )}{969969 b^6} \]

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 = 0.009, size = 65, normalized size = 0.6 \[ -{\frac{-92378\,{b}^{5}{x}^{5}+48620\,a{b}^{4}{x}^{4}-22880\,{a}^{2}{b}^{3}{x}^{3}+9152\,{a}^{3}{b}^{2}{x}^{2}-2816\,{a}^{4}bx+512\,{a}^{5}}{969969\,{b}^{6}} \left ( bx+a \right ) ^{{\frac{11}{2}}}} \]

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 = 1.34494, size = 116, normalized size = 1.05 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{21}{2}}}{21 \, b^{6}} - \frac{10 \,{\left (b x + a\right )}^{\frac{19}{2}} a}{19 \, b^{6}} + \frac{20 \,{\left (b x + a\right )}^{\frac{17}{2}} a^{2}}{17 \, b^{6}} - \frac{4 \,{\left (b x + a\right )}^{\frac{15}{2}} a^{3}}{3 \, b^{6}} + \frac{10 \,{\left (b x + a\right )}^{\frac{13}{2}} a^{4}}{13 \, b^{6}} - \frac{2 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{5}}{11 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)*x^5,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 = 0.208456, size = 161, normalized size = 1.46 \[ \frac{2 \,{\left (46189 \, b^{10} x^{10} + 206635 \, a b^{9} x^{9} + 351780 \, a^{2} b^{8} x^{8} + 271414 \, a^{3} b^{7} x^{7} + 80773 \, a^{4} b^{6} x^{6} + 63 \, a^{5} b^{5} x^{5} - 70 \, a^{6} b^{4} x^{4} + 80 \, a^{7} b^{3} x^{3} - 96 \, a^{8} b^{2} x^{2} + 128 \, a^{9} b x - 256 \, a^{10}\right )} \sqrt{b x + a}}{969969 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)*x^5,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 + 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)*sqrt(b*x + a)/b^6

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Sympy [A]  time = 82.6622, size = 235, normalized size = 2.14 \[ \begin{cases} - \frac{512 a^{10} \sqrt{a + b x}}{969969 b^{6}} + \frac{256 a^{9} x \sqrt{a + b x}}{969969 b^{5}} - \frac{64 a^{8} x^{2} \sqrt{a + b x}}{323323 b^{4}} + \frac{160 a^{7} x^{3} \sqrt{a + b x}}{969969 b^{3}} - \frac{20 a^{6} x^{4} \sqrt{a + b x}}{138567 b^{2}} + \frac{6 a^{5} x^{5} \sqrt{a + b x}}{46189 b} + \frac{2098 a^{4} x^{6} \sqrt{a + b x}}{12597} + \frac{3796 a^{3} b x^{7} \sqrt{a + b x}}{6783} + \frac{1640 a^{2} b^{2} x^{8} \sqrt{a + b x}}{2261} + \frac{170 a b^{3} x^{9} \sqrt{a + b x}}{399} + \frac{2 b^{4} x^{10} \sqrt{a + b x}}{21} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{6}}{6} & \text{otherwise} \end{cases} \]

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)/(96
9969*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)/6783 + 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/XCAS [A]  time = 0.215117, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done