3.309 \(\int x^7 (a+b x)^{9/2} \, dx\)

Optimal. Leaf size=146 \[ -\frac{2 a^7 (a+b x)^{11/2}}{11 b^8}+\frac{14 a^6 (a+b x)^{13/2}}{13 b^8}-\frac{14 a^5 (a+b x)^{15/2}}{5 b^8}+\frac{70 a^4 (a+b x)^{17/2}}{17 b^8}-\frac{70 a^3 (a+b x)^{19/2}}{19 b^8}+\frac{2 a^2 (a+b x)^{21/2}}{b^8}+\frac{2 (a+b x)^{25/2}}{25 b^8}-\frac{14 a (a+b x)^{23/2}}{23 b^8} \]

[Out]

(-2*a^7*(a + b*x)^(11/2))/(11*b^8) + (14*a^6*(a + b*x)^(13/2))/(13*b^8) - (14*a^
5*(a + b*x)^(15/2))/(5*b^8) + (70*a^4*(a + b*x)^(17/2))/(17*b^8) - (70*a^3*(a +
b*x)^(19/2))/(19*b^8) + (2*a^2*(a + b*x)^(21/2))/b^8 - (14*a*(a + b*x)^(23/2))/(
23*b^8) + (2*(a + b*x)^(25/2))/(25*b^8)

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Rubi [A]  time = 0.103377, antiderivative size = 146, 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 \[ -\frac{2 a^7 (a+b x)^{11/2}}{11 b^8}+\frac{14 a^6 (a+b x)^{13/2}}{13 b^8}-\frac{14 a^5 (a+b x)^{15/2}}{5 b^8}+\frac{70 a^4 (a+b x)^{17/2}}{17 b^8}-\frac{70 a^3 (a+b x)^{19/2}}{19 b^8}+\frac{2 a^2 (a+b x)^{21/2}}{b^8}+\frac{2 (a+b x)^{25/2}}{25 b^8}-\frac{14 a (a+b x)^{23/2}}{23 b^8} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a + b*x)^(9/2),x]

[Out]

(-2*a^7*(a + b*x)^(11/2))/(11*b^8) + (14*a^6*(a + b*x)^(13/2))/(13*b^8) - (14*a^
5*(a + b*x)^(15/2))/(5*b^8) + (70*a^4*(a + b*x)^(17/2))/(17*b^8) - (70*a^3*(a +
b*x)^(19/2))/(19*b^8) + (2*a^2*(a + b*x)^(21/2))/b^8 - (14*a*(a + b*x)^(23/2))/(
23*b^8) + (2*(a + b*x)^(25/2))/(25*b^8)

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Rubi in Sympy [A]  time = 24.394, size = 141, normalized size = 0.97 \[ - \frac{2 a^{7} \left (a + b x\right )^{\frac{11}{2}}}{11 b^{8}} + \frac{14 a^{6} \left (a + b x\right )^{\frac{13}{2}}}{13 b^{8}} - \frac{14 a^{5} \left (a + b x\right )^{\frac{15}{2}}}{5 b^{8}} + \frac{70 a^{4} \left (a + b x\right )^{\frac{17}{2}}}{17 b^{8}} - \frac{70 a^{3} \left (a + b x\right )^{\frac{19}{2}}}{19 b^{8}} + \frac{2 a^{2} \left (a + b x\right )^{\frac{21}{2}}}{b^{8}} - \frac{14 a \left (a + b x\right )^{\frac{23}{2}}}{23 b^{8}} + \frac{2 \left (a + b x\right )^{\frac{25}{2}}}{25 b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b*x+a)**(9/2),x)

[Out]

-2*a**7*(a + b*x)**(11/2)/(11*b**8) + 14*a**6*(a + b*x)**(13/2)/(13*b**8) - 14*a
**5*(a + b*x)**(15/2)/(5*b**8) + 70*a**4*(a + b*x)**(17/2)/(17*b**8) - 70*a**3*(
a + b*x)**(19/2)/(19*b**8) + 2*a**2*(a + b*x)**(21/2)/b**8 - 14*a*(a + b*x)**(23
/2)/(23*b**8) + 2*(a + b*x)**(25/2)/(25*b**8)

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Mathematica [A]  time = 0.07308, size = 90, normalized size = 0.62 \[ \frac{2 (a+b x)^{11/2} \left (-2048 a^7+11264 a^6 b x-36608 a^5 b^2 x^2+91520 a^4 b^3 x^3-194480 a^3 b^4 x^4+369512 a^2 b^5 x^5-646646 a b^6 x^6+1062347 b^7 x^7\right )}{26558675 b^8} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(-2048*a^7 + 11264*a^6*b*x - 36608*a^5*b^2*x^2 + 91520*a^4*b
^3*x^3 - 194480*a^3*b^4*x^4 + 369512*a^2*b^5*x^5 - 646646*a*b^6*x^6 + 1062347*b^
7*x^7))/(26558675*b^8)

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Maple [A]  time = 0.01, size = 87, normalized size = 0.6 \[ -{\frac{-2124694\,{b}^{7}{x}^{7}+1293292\,a{b}^{6}{x}^{6}-739024\,{a}^{2}{b}^{5}{x}^{5}+388960\,{a}^{3}{b}^{4}{x}^{4}-183040\,{a}^{4}{b}^{3}{x}^{3}+73216\,{a}^{5}{b}^{2}{x}^{2}-22528\,{a}^{6}bx+4096\,{a}^{7}}{26558675\,{b}^{8}} \left ( bx+a \right ) ^{{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x+a)^(9/2),x)

[Out]

-2/26558675*(b*x+a)^(11/2)*(-1062347*b^7*x^7+646646*a*b^6*x^6-369512*a^2*b^5*x^5
+194480*a^3*b^4*x^4-91520*a^4*b^3*x^3+36608*a^5*b^2*x^2-11264*a^6*b*x+2048*a^7)/
b^8

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Maxima [A]  time = 1.34133, size = 157, normalized size = 1.08 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{25}{2}}}{25 \, b^{8}} - \frac{14 \,{\left (b x + a\right )}^{\frac{23}{2}} a}{23 \, b^{8}} + \frac{2 \,{\left (b x + a\right )}^{\frac{21}{2}} a^{2}}{b^{8}} - \frac{70 \,{\left (b x + a\right )}^{\frac{19}{2}} a^{3}}{19 \, b^{8}} + \frac{70 \,{\left (b x + a\right )}^{\frac{17}{2}} a^{4}}{17 \, b^{8}} - \frac{14 \,{\left (b x + a\right )}^{\frac{15}{2}} a^{5}}{5 \, b^{8}} + \frac{14 \,{\left (b x + a\right )}^{\frac{13}{2}} a^{6}}{13 \, b^{8}} - \frac{2 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{7}}{11 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)*x^7,x, algorithm="maxima")

[Out]

2/25*(b*x + a)^(25/2)/b^8 - 14/23*(b*x + a)^(23/2)*a/b^8 + 2*(b*x + a)^(21/2)*a^
2/b^8 - 70/19*(b*x + a)^(19/2)*a^3/b^8 + 70/17*(b*x + a)^(17/2)*a^4/b^8 - 14/5*(
b*x + a)^(15/2)*a^5/b^8 + 14/13*(b*x + a)^(13/2)*a^6/b^8 - 2/11*(b*x + a)^(11/2)
*a^7/b^8

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Fricas [A]  time = 0.20902, size = 190, normalized size = 1.3 \[ \frac{2 \,{\left (1062347 \, b^{12} x^{12} + 4665089 \, a b^{11} x^{11} + 7759752 \, a^{2} b^{10} x^{10} + 5810090 \, a^{3} b^{9} x^{9} + 1659515 \, a^{4} b^{8} x^{8} + 429 \, a^{5} b^{7} x^{7} - 462 \, a^{6} b^{6} x^{6} + 504 \, a^{7} b^{5} x^{5} - 560 \, a^{8} b^{4} x^{4} + 640 \, a^{9} b^{3} x^{3} - 768 \, a^{10} b^{2} x^{2} + 1024 \, a^{11} b x - 2048 \, a^{12}\right )} \sqrt{b x + a}}{26558675 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)*x^7,x, algorithm="fricas")

[Out]

2/26558675*(1062347*b^12*x^12 + 4665089*a*b^11*x^11 + 7759752*a^2*b^10*x^10 + 58
10090*a^3*b^9*x^9 + 1659515*a^4*b^8*x^8 + 429*a^5*b^7*x^7 - 462*a^6*b^6*x^6 + 50
4*a^7*b^5*x^5 - 560*a^8*b^4*x^4 + 640*a^9*b^3*x^3 - 768*a^10*b^2*x^2 + 1024*a^11
*b*x - 2048*a^12)*sqrt(b*x + a)/b^8

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Sympy [A]  time = 115.681, size = 279, normalized size = 1.91 \[ \begin{cases} - \frac{4096 a^{12} \sqrt{a + b x}}{26558675 b^{8}} + \frac{2048 a^{11} x \sqrt{a + b x}}{26558675 b^{7}} - \frac{1536 a^{10} x^{2} \sqrt{a + b x}}{26558675 b^{6}} + \frac{256 a^{9} x^{3} \sqrt{a + b x}}{5311735 b^{5}} - \frac{224 a^{8} x^{4} \sqrt{a + b x}}{5311735 b^{4}} + \frac{1008 a^{7} x^{5} \sqrt{a + b x}}{26558675 b^{3}} - \frac{84 a^{6} x^{6} \sqrt{a + b x}}{2414425 b^{2}} + \frac{6 a^{5} x^{7} \sqrt{a + b x}}{185725 b} + \frac{4642 a^{4} x^{8} \sqrt{a + b x}}{37145} + \frac{956 a^{3} b x^{9} \sqrt{a + b x}}{2185} + \frac{336 a^{2} b^{2} x^{10} \sqrt{a + b x}}{575} + \frac{202 a b^{3} x^{11} \sqrt{a + b x}}{575} + \frac{2 b^{4} x^{12} \sqrt{a + b x}}{25} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{8}}{8} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b*x+a)**(9/2),x)

[Out]

Piecewise((-4096*a**12*sqrt(a + b*x)/(26558675*b**8) + 2048*a**11*x*sqrt(a + b*x
)/(26558675*b**7) - 1536*a**10*x**2*sqrt(a + b*x)/(26558675*b**6) + 256*a**9*x**
3*sqrt(a + b*x)/(5311735*b**5) - 224*a**8*x**4*sqrt(a + b*x)/(5311735*b**4) + 10
08*a**7*x**5*sqrt(a + b*x)/(26558675*b**3) - 84*a**6*x**6*sqrt(a + b*x)/(2414425
*b**2) + 6*a**5*x**7*sqrt(a + b*x)/(185725*b) + 4642*a**4*x**8*sqrt(a + b*x)/371
45 + 956*a**3*b*x**9*sqrt(a + b*x)/2185 + 336*a**2*b**2*x**10*sqrt(a + b*x)/575
+ 202*a*b**3*x**11*sqrt(a + b*x)/575 + 2*b**4*x**12*sqrt(a + b*x)/25, Ne(b, 0)),
(a**(9/2)*x**8/8, True))

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GIAC/XCAS [A]  time = 0.217403, 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^7,x, algorithm="giac")

[Out]

Done