3.4.10 \(\int x^6 (a+b x)^{9/2} \, dx\)

Optimal. Leaf size=127 \[ \frac {2 a^6 (a+b x)^{11/2}}{11 b^7}-\frac {12 a^5 (a+b x)^{13/2}}{13 b^7}+\frac {2 a^4 (a+b x)^{15/2}}{b^7}-\frac {40 a^3 (a+b x)^{17/2}}{17 b^7}+\frac {30 a^2 (a+b x)^{19/2}}{19 b^7}+\frac {2 (a+b x)^{23/2}}{23 b^7}-\frac {4 a (a+b x)^{21/2}}{7 b^7} \]

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Rubi [A]  time = 0.04, antiderivative size = 127, 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 {30 a^2 (a+b x)^{19/2}}{19 b^7}-\frac {40 a^3 (a+b x)^{17/2}}{17 b^7}+\frac {2 a^4 (a+b x)^{15/2}}{b^7}-\frac {12 a^5 (a+b x)^{13/2}}{13 b^7}+\frac {2 a^6 (a+b x)^{11/2}}{11 b^7}+\frac {2 (a+b x)^{23/2}}{23 b^7}-\frac {4 a (a+b x)^{21/2}}{7 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x)^(9/2),x]

[Out]

(2*a^6*(a + b*x)^(11/2))/(11*b^7) - (12*a^5*(a + b*x)^(13/2))/(13*b^7) + (2*a^4*(a + b*x)^(15/2))/b^7 - (40*a^
3*(a + b*x)^(17/2))/(17*b^7) + (30*a^2*(a + b*x)^(19/2))/(19*b^7) - (4*a*(a + b*x)^(21/2))/(7*b^7) + (2*(a + b
*x)^(23/2))/(23*b^7)

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^6 (a+b x)^{9/2} \, dx &=\int \left (\frac {a^6 (a+b x)^{9/2}}{b^6}-\frac {6 a^5 (a+b x)^{11/2}}{b^6}+\frac {15 a^4 (a+b x)^{13/2}}{b^6}-\frac {20 a^3 (a+b x)^{15/2}}{b^6}+\frac {15 a^2 (a+b x)^{17/2}}{b^6}-\frac {6 a (a+b x)^{19/2}}{b^6}+\frac {(a+b x)^{21/2}}{b^6}\right ) \, dx\\ &=\frac {2 a^6 (a+b x)^{11/2}}{11 b^7}-\frac {12 a^5 (a+b x)^{13/2}}{13 b^7}+\frac {2 a^4 (a+b x)^{15/2}}{b^7}-\frac {40 a^3 (a+b x)^{17/2}}{17 b^7}+\frac {30 a^2 (a+b x)^{19/2}}{19 b^7}-\frac {4 a (a+b x)^{21/2}}{7 b^7}+\frac {2 (a+b x)^{23/2}}{23 b^7}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 79, normalized size = 0.62 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (1024 a^6-5632 a^5 b x+18304 a^4 b^2 x^2-45760 a^3 b^3 x^3+97240 a^2 b^4 x^4-184756 a b^5 x^5+323323 b^6 x^6\right )}{7436429 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(1024*a^6 - 5632*a^5*b*x + 18304*a^4*b^2*x^2 - 45760*a^3*b^3*x^3 + 97240*a^2*b^4*x^4 - 184
756*a*b^5*x^5 + 323323*b^6*x^6))/(7436429*b^7)

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IntegrateAlgebraic [A]  time = 0.03, size = 101, normalized size = 0.80 \begin {gather*} \frac {2 \left (676039 a^6 (a+b x)^{11/2}-3432198 a^5 (a+b x)^{13/2}+7436429 a^4 (a+b x)^{15/2}-8748740 a^3 (a+b x)^{17/2}+5870865 a^2 (a+b x)^{19/2}+323323 (a+b x)^{23/2}-2124694 a (a+b x)^{21/2}\right )}{7436429 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^6*(a + b*x)^(9/2),x]

[Out]

(2*(676039*a^6*(a + b*x)^(11/2) - 3432198*a^5*(a + b*x)^(13/2) + 7436429*a^4*(a + b*x)^(15/2) - 8748740*a^3*(a
+ b*x)^(17/2) + 5870865*a^2*(a + b*x)^(19/2) - 2124694*a*(a + b*x)^(21/2) + 323323*(a + b*x)^(23/2)))/(743642
9*b^7)

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fricas [A]  time = 0.73, size = 130, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (323323 \, b^{11} x^{11} + 1431859 \, a b^{10} x^{10} + 2406690 \, a^{2} b^{9} x^{9} + 1826110 \, a^{3} b^{8} x^{8} + 530959 \, a^{4} b^{7} x^{7} + 231 \, a^{5} b^{6} x^{6} - 252 \, a^{6} b^{5} x^{5} + 280 \, a^{7} b^{4} x^{4} - 320 \, a^{8} b^{3} x^{3} + 384 \, a^{9} b^{2} x^{2} - 512 \, a^{10} b x + 1024 \, a^{11}\right )} \sqrt {b x + a}}{7436429 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^(9/2),x, algorithm="fricas")

[Out]

2/7436429*(323323*b^11*x^11 + 1431859*a*b^10*x^10 + 2406690*a^2*b^9*x^9 + 1826110*a^3*b^8*x^8 + 530959*a^4*b^7
*x^7 + 231*a^5*b^6*x^6 - 252*a^6*b^5*x^5 + 280*a^7*b^4*x^4 - 320*a^8*b^3*x^3 + 384*a^9*b^2*x^2 - 512*a^10*b*x
+ 1024*a^11)*sqrt(b*x + a)/b^7

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giac [B]  time = 1.32, size = 709, normalized size = 5.58

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^(9/2),x, algorithm="giac")

[Out]

2/66927861*(22287*(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^5/b^6 + 52003*(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^4/b^6 +
6118*(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^3/b^6 + 2898*(12155*(b*x + a)^(19/2) - 122265*(b*x + a)^(17/2)*a + 5
54268*(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*sqr
t(b*x + a)*a^9)*a^2/b^6 + 345*(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/b^6 + 33*(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)/b^6)/b

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maple [A]  time = 0.01, size = 76, normalized size = 0.60 \begin {gather*} \frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (323323 x^{6} b^{6}-184756 a \,x^{5} b^{5}+97240 a^{2} x^{4} b^{4}-45760 a^{3} x^{3} b^{3}+18304 a^{4} x^{2} b^{2}-5632 a^{5} x b +1024 a^{6}\right )}{7436429 b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x+a)^(9/2),x)

[Out]

2/7436429*(b*x+a)^(11/2)*(323323*b^6*x^6-184756*a*b^5*x^5+97240*a^2*b^4*x^4-45760*a^3*b^3*x^3+18304*a^4*b^2*x^
2-5632*a^5*b*x+1024*a^6)/b^7

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maxima [A]  time = 1.34, size = 101, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {23}{2}}}{23 \, b^{7}} - \frac {4 \, {\left (b x + a\right )}^{\frac {21}{2}} a}{7 \, b^{7}} + \frac {30 \, {\left (b x + a\right )}^{\frac {19}{2}} a^{2}}{19 \, b^{7}} - \frac {40 \, {\left (b x + a\right )}^{\frac {17}{2}} a^{3}}{17 \, b^{7}} + \frac {2 \, {\left (b x + a\right )}^{\frac {15}{2}} a^{4}}{b^{7}} - \frac {12 \, {\left (b x + a\right )}^{\frac {13}{2}} a^{5}}{13 \, b^{7}} + \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{6}}{11 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^(9/2),x, algorithm="maxima")

[Out]

2/23*(b*x + a)^(23/2)/b^7 - 4/7*(b*x + a)^(21/2)*a/b^7 + 30/19*(b*x + a)^(19/2)*a^2/b^7 - 40/17*(b*x + a)^(17/
2)*a^3/b^7 + 2*(b*x + a)^(15/2)*a^4/b^7 - 12/13*(b*x + a)^(13/2)*a^5/b^7 + 2/11*(b*x + a)^(11/2)*a^6/b^7

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mupad [B]  time = 0.03, size = 101, normalized size = 0.80 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{23/2}}{23\,b^7}+\frac {2\,a^6\,{\left (a+b\,x\right )}^{11/2}}{11\,b^7}-\frac {12\,a^5\,{\left (a+b\,x\right )}^{13/2}}{13\,b^7}+\frac {2\,a^4\,{\left (a+b\,x\right )}^{15/2}}{b^7}-\frac {40\,a^3\,{\left (a+b\,x\right )}^{17/2}}{17\,b^7}+\frac {30\,a^2\,{\left (a+b\,x\right )}^{19/2}}{19\,b^7}-\frac {4\,a\,{\left (a+b\,x\right )}^{21/2}}{7\,b^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(a + b*x)^(9/2),x)

[Out]

(2*(a + b*x)^(23/2))/(23*b^7) + (2*a^6*(a + b*x)^(11/2))/(11*b^7) - (12*a^5*(a + b*x)^(13/2))/(13*b^7) + (2*a^
4*(a + b*x)^(15/2))/b^7 - (40*a^3*(a + b*x)^(17/2))/(17*b^7) + (30*a^2*(a + b*x)^(19/2))/(19*b^7) - (4*a*(a +
b*x)^(21/2))/(7*b^7)

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sympy [A]  time = 36.61, size = 257, normalized size = 2.02 \begin {gather*} \begin {cases} \frac {2048 a^{11} \sqrt {a + b x}}{7436429 b^{7}} - \frac {1024 a^{10} x \sqrt {a + b x}}{7436429 b^{6}} + \frac {768 a^{9} x^{2} \sqrt {a + b x}}{7436429 b^{5}} - \frac {640 a^{8} x^{3} \sqrt {a + b x}}{7436429 b^{4}} + \frac {80 a^{7} x^{4} \sqrt {a + b x}}{1062347 b^{3}} - \frac {72 a^{6} x^{5} \sqrt {a + b x}}{1062347 b^{2}} + \frac {6 a^{5} x^{6} \sqrt {a + b x}}{96577 b} + \frac {7426 a^{4} x^{7} \sqrt {a + b x}}{52003} + \frac {25540 a^{3} b x^{8} \sqrt {a + b x}}{52003} + \frac {1980 a^{2} b^{2} x^{9} \sqrt {a + b x}}{3059} + \frac {62 a b^{3} x^{10} \sqrt {a + b x}}{161} + \frac {2 b^{4} x^{11} \sqrt {a + b x}}{23} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{7}}{7} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x+a)**(9/2),x)

[Out]

Piecewise((2048*a**11*sqrt(a + b*x)/(7436429*b**7) - 1024*a**10*x*sqrt(a + b*x)/(7436429*b**6) + 768*a**9*x**2
*sqrt(a + b*x)/(7436429*b**5) - 640*a**8*x**3*sqrt(a + b*x)/(7436429*b**4) + 80*a**7*x**4*sqrt(a + b*x)/(10623
47*b**3) - 72*a**6*x**5*sqrt(a + b*x)/(1062347*b**2) + 6*a**5*x**6*sqrt(a + b*x)/(96577*b) + 7426*a**4*x**7*sq
rt(a + b*x)/52003 + 25540*a**3*b*x**8*sqrt(a + b*x)/52003 + 1980*a**2*b**2*x**9*sqrt(a + b*x)/3059 + 62*a*b**3
*x**10*sqrt(a + b*x)/161 + 2*b**4*x**11*sqrt(a + b*x)/23, Ne(b, 0)), (a**(9/2)*x**7/7, True))

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