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

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

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Rubi [A]  time = 0.03, antiderivative size = 110, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 68, normalized size = 0.62 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.03, size = 75, normalized size = 0.68 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (-88179 a^5+373065 a^4 (a+b x)-646646 a^3 (a+b x)^2+570570 a^2 (a+b x)^3-255255 a (a+b x)^4+46189 (a+b x)^5\right )}{969969 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^5*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(-88179*a^5 + 373065*a^4*(a + b*x) - 646646*a^3*(a + b*x)^2 + 570570*a^2*(a + b*x)^3 - 255
255*a*(a + b*x)^4 + 46189*(a + b*x)^5))/(969969*b^6)

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fricas [A]  time = 0.91, size = 119, normalized size = 1.08 \begin {gather*} \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 {gather*}

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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giac [B]  time = 1.05, size = 637, normalized size = 5.79 \begin {gather*} \frac {2 \, {\left (\frac {4199 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} a^{5}}{b^{5}} + \frac {4845 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} a^{4}}{b^{5}} + \frac {4522 \, {\left (429 \, {\left (b x + a\right )}^{\frac {15}{2}} - 3465 \, {\left (b x + a\right )}^{\frac {13}{2}} a + 12285 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2} - 25025 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{3} + 32175 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4} - 27027 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} + 15015 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} - 6435 \, \sqrt {b x + a} a^{7}\right )} a^{3}}{b^{5}} + \frac {266 \, {\left (6435 \, {\left (b x + a\right )}^{\frac {17}{2}} - 58344 \, {\left (b x + a\right )}^{\frac {15}{2}} a + 235620 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{2} - 556920 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{3} + 850850 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{4} - 875160 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{5} + 612612 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{6} - 291720 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{7} + 109395 \, \sqrt {b x + a} a^{8}\right )} a^{2}}{b^{5}} + \frac {63 \, {\left (12155 \, {\left (b x + a\right )}^{\frac {19}{2}} - 122265 \, {\left (b x + a\right )}^{\frac {17}{2}} a + 554268 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{2} - 1492260 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{3} + 2645370 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{4} - 3233230 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{5} + 2771340 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{6} - 1662804 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{7} + 692835 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{8} - 230945 \, \sqrt {b x + a} a^{9}\right )} a}{b^{5}} + \frac {3 \, {\left (46189 \, {\left (b x + a\right )}^{\frac {21}{2}} - 510510 \, {\left (b x + a\right )}^{\frac {19}{2}} a + 2567565 \, {\left (b x + a\right )}^{\frac {17}{2}} a^{2} - 7759752 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{3} + 15668730 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{4} - 22221108 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{5} + 22632610 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{6} - 16628040 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{7} + 8729721 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{8} - 3233230 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{9} + 969969 \, \sqrt {b x + a} a^{10}\right )}}{b^{5}}\right )}}{2909907 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2909907*(4199*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/2)*
a^3 + 1155*(b*x + a)^(3/2)*a^4 - 693*sqrt(b*x + a)*a^5)*a^5/b^5 + 4845*(231*(b*x + a)^(13/2) - 1638*(b*x + a)^
(11/2)*a + 5005*(b*x + a)^(9/2)*a^2 - 8580*(b*x + a)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 6006*(b*x + a)^(3/
2)*a^5 + 3003*sqrt(b*x + a)*a^6)*a^4/b^5 + 4522*(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*sqrt(b*x + a)*a^7)*a^3/b^5 + 266*(6435*(b*x + a)^(17/2) - 58344*(b*x + a)^(15/2)*a + 2
35620*(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^2/b^5 + 63*(12155
*(b*x + a)^(19/2) - 122265*(b*x + a)^(17/2)*a + 554268*(b*x + a)^(15/2)*a^2 - 1492260*(b*x + a)^(13/2)*a^3 + 2
645370*(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/b^5 + 3*(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)/b^5)/b

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maple [A]  time = 0.01, size = 65, normalized size = 0.59 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (-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}\right )}{969969 b^{6}} \end {gather*}

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 = 1.35, size = 86, normalized size = 0.78 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.03, size = 86, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{21/2}}{21\,b^6}-\frac {2\,a^5\,{\left (a+b\,x\right )}^{11/2}}{11\,b^6}+\frac {10\,a^4\,{\left (a+b\,x\right )}^{13/2}}{13\,b^6}-\frac {4\,a^3\,{\left (a+b\,x\right )}^{15/2}}{3\,b^6}+\frac {20\,a^2\,{\left (a+b\,x\right )}^{17/2}}{17\,b^6}-\frac {10\,a\,{\left (a+b\,x\right )}^{19/2}}{19\,b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x)^(9/2),x)

[Out]

(2*(a + b*x)^(21/2))/(21*b^6) - (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)

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sympy [A]  time = 28.76, size = 235, normalized size = 2.14 \begin {gather*} \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 {gather*}

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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