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

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

[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)

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Rubi [A]  time = 0.0350039, antiderivative size = 127, 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} \[ \frac{30 a^2 (a+b x)^{19/2}}{19 b^7}-\frac{40 a^3 (a+b x)^{17/2}}{17 b^7}+\frac{2 a^4 (a+b x)^{15/2}}{b^7}-\frac{12 a^5 (a+b x)^{13/2}}{13 b^7}+\frac{2 a^6 (a+b x)^{11/2}}{11 b^7}+\frac{2 (a+b x)^{23/2}}{23 b^7}-\frac{4 a (a+b x)^{21/2}}{7 b^7} \]

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*}

Mathematica [A]  time = 0.0558602, size = 79, normalized size = 0.62 \[ \frac{2 (a+b x)^{11/2} \left (18304 a^4 b^2 x^2-45760 a^3 b^3 x^3+97240 a^2 b^4 x^4-5632 a^5 b x+1024 a^6-184756 a b^5 x^5+323323 b^6 x^6\right )}{7436429 b^7} \]

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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Maple [A]  time = 0.005, size = 76, normalized size = 0.6 \begin{align*}{\frac{646646\,{x}^{6}{b}^{6}-369512\,a{x}^{5}{b}^{5}+194480\,{a}^{2}{x}^{4}{b}^{4}-91520\,{a}^{3}{x}^{3}{b}^{3}+36608\,{a}^{4}{x}^{2}{b}^{2}-11264\,{a}^{5}xb+2048\,{a}^{6}}{7436429\,{b}^{7}} \left ( bx+a \right ) ^{{\frac{11}{2}}}} \end{align*}

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 = 1.0907, size = 136, normalized size = 1.07 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.52381, size = 340, normalized size = 2.68 \begin{align*} \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{align*}

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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Sympy [A]  time = 45.7565, size = 257, normalized size = 2.02 \begin{align*} \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{align*}

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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Giac [B]  time = 1.26437, size = 757, normalized size = 5.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/334639305*(7429*(3003*(b*x + a)^(15/2) - 20790*(b*x + a)^(13/2)*a + 61425*(b*x + a)^(11/2)*a^2 - 100100*(b*x
+ a)^(9/2)*a^3 + 96525*(b*x + a)^(7/2)*a^4 - 54054*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6)*a^4/b^6 +
12236*(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 - 36465*(b*x + a
)^(3/2)*a^7)*a^3/b^6 + 966*(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^2/b^6 + 276*(230945*(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/b^6 + 15*(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 + 312330018*(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 + 14
3416845*(b*x + a)^(7/2)*a^8 - 44618574*(b*x + a)^(5/2)*a^9 + 7436429*(b*x + a)^(3/2)*a^10)/b^6)/b