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

Optimal. Leaf size=53 \[ \frac{2 a^2 (a+b x)^{11/2}}{11 b^3}+\frac{2 (a+b x)^{15/2}}{15 b^3}-\frac{4 a (a+b x)^{13/2}}{13 b^3} \]

[Out]

(2*a^2*(a + b*x)^(11/2))/(11*b^3) - (4*a*(a + b*x)^(13/2))/(13*b^3) + (2*(a + b*x)^(15/2))/(15*b^3)

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Rubi [A]  time = 0.0134202, antiderivative size = 53, 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} \[ \frac{2 a^2 (a+b x)^{11/2}}{11 b^3}+\frac{2 (a+b x)^{15/2}}{15 b^3}-\frac{4 a (a+b x)^{13/2}}{13 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^(9/2),x]

[Out]

(2*a^2*(a + b*x)^(11/2))/(11*b^3) - (4*a*(a + b*x)^(13/2))/(13*b^3) + (2*(a + b*x)^(15/2))/(15*b^3)

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^2 (a+b x)^{9/2} \, dx &=\int \left (\frac{a^2 (a+b x)^{9/2}}{b^2}-\frac{2 a (a+b x)^{11/2}}{b^2}+\frac{(a+b x)^{13/2}}{b^2}\right ) \, dx\\ &=\frac{2 a^2 (a+b x)^{11/2}}{11 b^3}-\frac{4 a (a+b x)^{13/2}}{13 b^3}+\frac{2 (a+b x)^{15/2}}{15 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0493958, size = 35, normalized size = 0.66 \[ \frac{2 (a+b x)^{11/2} \left (8 a^2-44 a b x+143 b^2 x^2\right )}{2145 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(8*a^2 - 44*a*b*x + 143*b^2*x^2))/(2145*b^3)

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Maple [A]  time = 0.004, size = 32, normalized size = 0.6 \begin{align*}{\frac{286\,{b}^{2}{x}^{2}-88\,abx+16\,{a}^{2}}{2145\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2145*(b*x+a)^(11/2)*(143*b^2*x^2-44*a*b*x+8*a^2)/b^3

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Maxima [A]  time = 1.06994, size = 55, normalized size = 1.04 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{15}{2}}}{15 \, b^{3}} - \frac{4 \,{\left (b x + a\right )}^{\frac{13}{2}} a}{13 \, b^{3}} + \frac{2 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{2}}{11 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(b*x + a)^(15/2)/b^3 - 4/13*(b*x + a)^(13/2)*a/b^3 + 2/11*(b*x + a)^(11/2)*a^2/b^3

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Fricas [B]  time = 1.49029, size = 200, normalized size = 3.77 \begin{align*} \frac{2 \,{\left (143 \, b^{7} x^{7} + 671 \, a b^{6} x^{6} + 1218 \, a^{2} b^{5} x^{5} + 1030 \, a^{3} b^{4} x^{4} + 355 \, a^{4} b^{3} x^{3} + 3 \, a^{5} b^{2} x^{2} - 4 \, a^{6} b x + 8 \, a^{7}\right )} \sqrt{b x + a}}{2145 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2145*(143*b^7*x^7 + 671*a*b^6*x^6 + 1218*a^2*b^5*x^5 + 1030*a^3*b^4*x^4 + 355*a^4*b^3*x^3 + 3*a^5*b^2*x^2 -
4*a^6*b*x + 8*a^7)*sqrt(b*x + a)/b^3

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Sympy [A]  time = 23.4346, size = 168, normalized size = 3.17 \begin{align*} \begin{cases} \frac{16 a^{7} \sqrt{a + b x}}{2145 b^{3}} - \frac{8 a^{6} x \sqrt{a + b x}}{2145 b^{2}} + \frac{2 a^{5} x^{2} \sqrt{a + b x}}{715 b} + \frac{142 a^{4} x^{3} \sqrt{a + b x}}{429} + \frac{412 a^{3} b x^{4} \sqrt{a + b x}}{429} + \frac{812 a^{2} b^{2} x^{5} \sqrt{a + b x}}{715} + \frac{122 a b^{3} x^{6} \sqrt{a + b x}}{195} + \frac{2 b^{4} x^{7} \sqrt{a + b x}}{15} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*a**7*sqrt(a + b*x)/(2145*b**3) - 8*a**6*x*sqrt(a + b*x)/(2145*b**2) + 2*a**5*x**2*sqrt(a + b*x)/
(715*b) + 142*a**4*x**3*sqrt(a + b*x)/429 + 412*a**3*b*x**4*sqrt(a + b*x)/429 + 812*a**2*b**2*x**5*sqrt(a + b*
x)/715 + 122*a*b**3*x**6*sqrt(a + b*x)/195 + 2*b**4*x**7*sqrt(a + b*x)/15, Ne(b, 0)), (a**(9/2)*x**3/3, True))

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Giac [B]  time = 1.19632, size = 432, normalized size = 8.15 \begin{align*} \frac{2 \,{\left (\frac{429 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} a^{4}}{b^{2}} + \frac{572 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} a^{3}}{b^{2}} + \frac{78 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}\right )} a^{2}}{b^{2}} + \frac{20 \,{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} - 4095 \,{\left (b x + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} - 12870 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} - 3003 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5}\right )} a}{b^{2}} + \frac{3003 \,{\left (b x + a\right )}^{\frac{15}{2}} - 20790 \,{\left (b x + a\right )}^{\frac{13}{2}} a + 61425 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{2} - 100100 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{3} + 96525 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{4} - 54054 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{5} + 15015 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{6}}{b^{2}}\right )}}{45045 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(429*(15*(b*x + a)^(7/2) - 42*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2)*a^4/b^2 + 572*(35*(b*x + a)^
(9/2) - 135*(b*x + a)^(7/2)*a + 189*(b*x + a)^(5/2)*a^2 - 105*(b*x + a)^(3/2)*a^3)*a^3/b^2 + 78*(315*(b*x + a)
^(11/2) - 1540*(b*x + a)^(9/2)*a + 2970*(b*x + a)^(7/2)*a^2 - 2772*(b*x + a)^(5/2)*a^3 + 1155*(b*x + a)^(3/2)*
a^4)*a^2/b^2 + 20*(693*(b*x + a)^(13/2) - 4095*(b*x + a)^(11/2)*a + 10010*(b*x + a)^(9/2)*a^2 - 12870*(b*x + a
)^(7/2)*a^3 + 9009*(b*x + a)^(5/2)*a^4 - 3003*(b*x + a)^(3/2)*a^5)*a/b^2 + (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)/b^2)/b