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]

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

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Rubi [A]  time = 0.04, antiderivative size = 146, normalized size of antiderivative = 1.00, 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} \[ \frac {2 a^2 (a+b x)^{21/2}}{b^8}-\frac {70 a^3 (a+b x)^{19/2}}{19 b^8}+\frac {70 a^4 (a+b x)^{17/2}}{17 b^8}-\frac {14 a^5 (a+b x)^{15/2}}{5 b^8}+\frac {14 a^6 (a+b x)^{13/2}}{13 b^8}-\frac {2 a^7 (a+b x)^{11/2}}{11 b^8}+\frac {2 (a+b x)^{25/2}}{25 b^8}-\frac {14 a (a+b x)^{23/2}}{23 b^8} \]

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)

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^7 (a+b x)^{9/2} \, dx &=\int \left (-\frac {a^7 (a+b x)^{9/2}}{b^7}+\frac {7 a^6 (a+b x)^{11/2}}{b^7}-\frac {21 a^5 (a+b x)^{13/2}}{b^7}+\frac {35 a^4 (a+b x)^{15/2}}{b^7}-\frac {35 a^3 (a+b x)^{17/2}}{b^7}+\frac {21 a^2 (a+b x)^{19/2}}{b^7}-\frac {7 a (a+b x)^{21/2}}{b^7}+\frac {(a+b x)^{23/2}}{b^7}\right ) \, dx\\ &=-\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 {14 a (a+b x)^{23/2}}{23 b^8}+\frac {2 (a+b x)^{25/2}}{25 b^8}\\ \end {align*}

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Mathematica [A]  time = 0.05, 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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fricas [A]  time = 0.46, size = 141, normalized size = 0.97 \[ \frac {2 \, {\left (1062347 \, b^{12} x^{12} + 4665089 \, a b^{11} x^{11} + 7759752 \, a^{2} b^{10} x^{10} + 5810090 \, a^{3} b^{9} x^{9} + 1659515 \, a^{4} b^{8} x^{8} + 429 \, a^{5} b^{7} x^{7} - 462 \, a^{6} b^{6} x^{6} + 504 \, a^{7} b^{5} x^{5} - 560 \, a^{8} b^{4} x^{4} + 640 \, a^{9} b^{3} x^{3} - 768 \, a^{10} b^{2} x^{2} + 1024 \, a^{11} b x - 2048 \, a^{12}\right )} \sqrt {b x + a}}{26558675 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/26558675*(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)*sqrt(b*x + a)/b^8

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giac [B]  time = 1.14, size = 781, normalized size = 5.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x+a)^(9/2),x, algorithm="giac")

[Out]

2/1673196525*(260015*(429*(b*x + a)^(15/2) - 3465*(b*x + a)^(13/2)*a + 12285*(b*x + a)^(11/2)*a^2 - 25025*(b*x
+ a)^(9/2)*a^3 + 32175*(b*x + a)^(7/2)*a^4 - 27027*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6 - 6435*sqr
t(b*x + a)*a^7)*a^5/b^7 + 76475*(6435*(b*x + a)^(17/2) - 58344*(b*x + a)^(15/2)*a + 235620*(b*x + a)^(13/2)*a^
2 - 556920*(b*x + a)^(11/2)*a^3 + 850850*(b*x + a)^(9/2)*a^4 - 875160*(b*x + a)^(7/2)*a^5 + 612612*(b*x + a)^(
5/2)*a^6 - 291720*(b*x + a)^(3/2)*a^7 + 109395*sqrt(b*x + a)*a^8)*a^4/b^7 + 72450*(12155*(b*x + a)^(19/2) - 12
2265*(b*x + a)^(17/2)*a + 554268*(b*x + a)^(15/2)*a^2 - 1492260*(b*x + a)^(13/2)*a^3 + 2645370*(b*x + a)^(11/2
)*a^4 - 3233230*(b*x + a)^(9/2)*a^5 + 2771340*(b*x + a)^(7/2)*a^6 - 1662804*(b*x + a)^(5/2)*a^7 + 692835*(b*x
+ a)^(3/2)*a^8 - 230945*sqrt(b*x + a)*a^9)*a^3/b^7 + 17250*(46189*(b*x + a)^(21/2) - 510510*(b*x + a)^(19/2)*a
+ 2567565*(b*x + a)^(17/2)*a^2 - 7759752*(b*x + a)^(15/2)*a^3 + 15668730*(b*x + a)^(13/2)*a^4 - 22221108*(b*x
+ a)^(11/2)*a^5 + 22632610*(b*x + a)^(9/2)*a^6 - 16628040*(b*x + a)^(7/2)*a^7 + 8729721*(b*x + a)^(5/2)*a^8 -
3233230*(b*x + a)^(3/2)*a^9 + 969969*sqrt(b*x + a)*a^10)*a^2/b^7 + 4125*(88179*(b*x + a)^(23/2) - 1062347*(b*
x + a)^(21/2)*a + 5870865*(b*x + a)^(19/2)*a^2 - 19684665*(b*x + a)^(17/2)*a^3 + 44618574*(b*x + a)^(15/2)*a^4
- 72076158*(b*x + a)^(13/2)*a^5 + 85180914*(b*x + a)^(11/2)*a^6 - 74364290*(b*x + a)^(9/2)*a^7 + 47805615*(b*
x + a)^(7/2)*a^8 - 22309287*(b*x + a)^(5/2)*a^9 + 7436429*(b*x + a)^(3/2)*a^10 - 2028117*sqrt(b*x + a)*a^11)*a
/b^7 + 99*(676039*(b*x + a)^(25/2) - 8817900*(b*x + a)^(23/2)*a + 53117350*(b*x + a)^(21/2)*a^2 - 195695500*(b
*x + a)^(19/2)*a^3 + 492116625*(b*x + a)^(17/2)*a^4 - 892371480*(b*x + a)^(15/2)*a^5 + 1201269300*(b*x + a)^(1
3/2)*a^6 - 1216870200*(b*x + a)^(11/2)*a^7 + 929553625*(b*x + a)^(9/2)*a^8 - 531173500*(b*x + a)^(7/2)*a^9 + 2
23092870*(b*x + a)^(5/2)*a^10 - 67603900*(b*x + a)^(3/2)*a^11 + 16900975*sqrt(b*x + a)*a^12)/b^7)/b

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maple [A]  time = 0.01, size = 87, normalized size = 0.60 \[ -\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (-1062347 b^{7} x^{7}+646646 a \,b^{6} x^{6}-369512 a^{2} b^{5} x^{5}+194480 a^{3} b^{4} x^{4}-91520 a^{4} b^{3} x^{3}+36608 a^{5} b^{2} x^{2}-11264 a^{6} b x +2048 a^{7}\right )}{26558675 b^{8}} \]

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 = 1.33, size = 116, normalized size = 0.79 \[ \frac {2 \, {\left (b x + a\right )}^{\frac {25}{2}}}{25 \, b^{8}} - \frac {14 \, {\left (b x + a\right )}^{\frac {23}{2}} a}{23 \, b^{8}} + \frac {2 \, {\left (b x + a\right )}^{\frac {21}{2}} a^{2}}{b^{8}} - \frac {70 \, {\left (b x + a\right )}^{\frac {19}{2}} a^{3}}{19 \, b^{8}} + \frac {70 \, {\left (b x + a\right )}^{\frac {17}{2}} a^{4}}{17 \, b^{8}} - \frac {14 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{5}}{5 \, b^{8}} + \frac {14 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{6}}{13 \, b^{8}} - \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{7}}{11 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x+a)^(9/2),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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mupad [B]  time = 0.04, size = 116, normalized size = 0.79 \[ \frac {2\,{\left (a+b\,x\right )}^{25/2}}{25\,b^8}-\frac {2\,a^7\,{\left (a+b\,x\right )}^{11/2}}{11\,b^8}+\frac {14\,a^6\,{\left (a+b\,x\right )}^{13/2}}{13\,b^8}-\frac {14\,a^5\,{\left (a+b\,x\right )}^{15/2}}{5\,b^8}+\frac {70\,a^4\,{\left (a+b\,x\right )}^{17/2}}{17\,b^8}-\frac {70\,a^3\,{\left (a+b\,x\right )}^{19/2}}{19\,b^8}+\frac {2\,a^2\,{\left (a+b\,x\right )}^{21/2}}{b^8}-\frac {14\,a\,{\left (a+b\,x\right )}^{23/2}}{23\,b^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a + b*x)^(9/2),x)

[Out]

(2*(a + b*x)^(25/2))/(25*b^8) - (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)

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sympy [A]  time = 40.30, 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) + 1008*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)/37145 + 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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