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

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

[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 = 0.0424934, 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, 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*}

Mathematica [A]  time = 0.12302, size = 90, normalized size = 0.62 \[ \frac{2 (a+b x)^{11/2} \left (-36608 a^5 b^2 x^2+91520 a^4 b^3 x^3-194480 a^3 b^4 x^4+369512 a^2 b^5 x^5+11264 a^6 b x-2048 a^7-646646 a b^6 x^6+1062347 b^7 x^7\right )}{26558675 b^8} \]

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 = 0.005, size = 87, normalized size = 0.6 \begin{align*} -{\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}}}} \end{align*}

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 = 1.09388, size = 157, normalized size = 1.08 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 1.51393, size = 374, normalized size = 2.56 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 53.0196, size = 279, normalized size = 1.91 \begin{align*} \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} \end{align*}

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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Giac [B]  time = 1.24997, size = 838, normalized size = 5.74 \begin{align*} \text{result too large to display} \end{align*}

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*(15295*(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 - 3
6465*(b*x + a)^(3/2)*a^7)*a^4/b^7 + 3220*(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 + 692835*(b*x + a)^(3/2)*a^8)*a^3/b^7 + 2070*(23094
5*(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 + 45265220*(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)*a^2/b^7 + 300*(969969*(b*x + a)
^(23/2) - 10623470*(b*x + a)^(21/2)*a + 52837785*(b*x + a)^(19/2)*a^2 - 157477320*(b*x + a)^(17/2)*a^3 + 31233
0018*(b*x + a)^(15/2)*a^4 - 432456948*(b*x + a)^(13/2)*a^5 + 425904570*(b*x + a)^(11/2)*a^6 - 297457160*(b*x +
a)^(9/2)*a^7 + 143416845*(b*x + a)^(7/2)*a^8 - 44618574*(b*x + a)^(5/2)*a^9 + 7436429*(b*x + a)^(3/2)*a^10)*a
/b^7 + 33*(2028117*(b*x + a)^(25/2) - 24249225*(b*x + a)^(23/2)*a + 132793375*(b*x + a)^(21/2)*a^2 - 440314875
*(b*x + a)^(19/2)*a^3 + 984233250*(b*x + a)^(17/2)*a^4 - 1561650090*(b*x + a)^(15/2)*a^5 + 1801903950*(b*x + a
)^(13/2)*a^6 - 1521087750*(b*x + a)^(11/2)*a^7 + 929553625*(b*x + a)^(9/2)*a^8 - 398380125*(b*x + a)^(7/2)*a^9
+ 111546435*(b*x + a)^(5/2)*a^10 - 16900975*(b*x + a)^(3/2)*a^11)/b^7)/b