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

Optimal. Leaf size=34 \[ \frac{2 (a+b x)^{13/2}}{13 b^2}-\frac{2 a (a+b x)^{11/2}}{11 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0080125, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{2 (a+b x)^{13/2}}{13 b^2}-\frac{2 a (a+b x)^{11/2}}{11 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^(9/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0289238, size = 24, normalized size = 0.71 \[ \frac{2 (a+b x)^{11/2} (11 b x-2 a)}{143 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2)*(-2*a + 11*b*x))/(143*b^2)

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Maple [A]  time = 0.003, size = 21, normalized size = 0.6 \begin{align*} -{\frac{-22\,bx+4\,a}{143\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/143*(b*x+a)^(11/2)*(-11*b*x+2*a)/b^2

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Maxima [A]  time = 1.02162, size = 35, normalized size = 1.03 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{13}{2}}}{13 \, b^{2}} - \frac{2 \,{\left (b x + a\right )}^{\frac{11}{2}} a}{11 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.48734, size = 166, normalized size = 4.88 \begin{align*} \frac{2 \,{\left (11 \, b^{6} x^{6} + 53 \, a b^{5} x^{5} + 100 \, a^{2} b^{4} x^{4} + 90 \, a^{3} b^{3} x^{3} + 35 \, a^{4} b^{2} x^{2} + a^{5} b x - 2 \, a^{6}\right )} \sqrt{b x + a}}{143 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/143*(11*b^6*x^6 + 53*a*b^5*x^5 + 100*a^2*b^4*x^4 + 90*a^3*b^3*x^3 + 35*a^4*b^2*x^2 + a^5*b*x - 2*a^6)*sqrt(b
*x + a)/b^2

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Sympy [A]  time = 17.7795, size = 146, normalized size = 4.29 \begin{align*} \begin{cases} - \frac{4 a^{6} \sqrt{a + b x}}{143 b^{2}} + \frac{2 a^{5} x \sqrt{a + b x}}{143 b} + \frac{70 a^{4} x^{2} \sqrt{a + b x}}{143} + \frac{180 a^{3} b x^{3} \sqrt{a + b x}}{143} + \frac{200 a^{2} b^{2} x^{4} \sqrt{a + b x}}{143} + \frac{106 a b^{3} x^{5} \sqrt{a + b x}}{143} + \frac{2 b^{4} x^{6} \sqrt{a + b x}}{13} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-4*a**6*sqrt(a + b*x)/(143*b**2) + 2*a**5*x*sqrt(a + b*x)/(143*b) + 70*a**4*x**2*sqrt(a + b*x)/143
+ 180*a**3*b*x**3*sqrt(a + b*x)/143 + 200*a**2*b**2*x**4*sqrt(a + b*x)/143 + 106*a*b**3*x**5*sqrt(a + b*x)/143
+ 2*b**4*x**6*sqrt(a + b*x)/13, Ne(b, 0)), (a**(9/2)*x**2/2, True))

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Giac [B]  time = 1.22893, size = 352, normalized size = 10.35 \begin{align*} \frac{2 \,{\left (\frac{3003 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} a^{4}}{b} + \frac{1716 \,{\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^{3}}{b} + \frac{858 \,{\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^{2}}{b} + \frac{52 \,{\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}{b} + \frac{5 \,{\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 )}}{b}\right )}}{45045 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*a^4/b + 1716*(15*(b*x + a)^(7/2) - 42*(b*x + a)^(5/2)*
a + 35*(b*x + a)^(3/2)*a^2)*a^3/b + 858*(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^2/b + 52*(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/b + 5*(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)/b)/b