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

Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^2\right )^{11/2}}{11 b^3}+\frac{\left (a+b x^2\right )^{15/2}}{15 b^3}-\frac{2 a \left (a+b x^2\right )^{13/2}}{13 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0350742, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a^2 \left (a+b x^2\right )^{11/2}}{11 b^3}+\frac{\left (a+b x^2\right )^{15/2}}{15 b^3}-\frac{2 a \left (a+b x^2\right )^{13/2}}{13 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^2)^(9/2),x]

[Out]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 \left (a+b x^2\right )^{9/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b x)^{9/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\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,x,x^2\right )\\ &=\frac{a^2 \left (a+b x^2\right )^{11/2}}{11 b^3}-\frac{2 a \left (a+b x^2\right )^{13/2}}{13 b^3}+\frac{\left (a+b x^2\right )^{15/2}}{15 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0215029, size = 39, normalized size = 0.66 \[ \frac{\left (a+b x^2\right )^{11/2} \left (8 a^2-44 a b x^2+143 b^2 x^4\right )}{2145 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^2)^(9/2),x]

[Out]

((a + b*x^2)^(11/2)*(8*a^2 - 44*a*b*x^2 + 143*b^2*x^4))/(2145*b^3)

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Maple [A]  time = 0.005, size = 36, normalized size = 0.6 \begin{align*}{\frac{143\,{b}^{2}{x}^{4}-44\,ab{x}^{2}+8\,{a}^{2}}{2145\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^2+a)^(9/2),x)

[Out]

1/2145*(b*x^2+a)^(11/2)*(143*b^2*x^4-44*a*b*x^2+8*a^2)/b^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.6103, size = 209, normalized size = 3.54 \begin{align*} \frac{{\left (143 \, b^{7} x^{14} + 671 \, a b^{6} x^{12} + 1218 \, a^{2} b^{5} x^{10} + 1030 \, a^{3} b^{4} x^{8} + 355 \, a^{4} b^{3} x^{6} + 3 \, a^{5} b^{2} x^{4} - 4 \, a^{6} b x^{2} + 8 \, a^{7}\right )} \sqrt{b x^{2} + a}}{2145 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2145*(143*b^7*x^14 + 671*a*b^6*x^12 + 1218*a^2*b^5*x^10 + 1030*a^3*b^4*x^8 + 355*a^4*b^3*x^6 + 3*a^5*b^2*x^4
- 4*a^6*b*x^2 + 8*a^7)*sqrt(b*x^2 + a)/b^3

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Sympy [A]  time = 25.2969, size = 180, normalized size = 3.05 \begin{align*} \begin{cases} \frac{8 a^{7} \sqrt{a + b x^{2}}}{2145 b^{3}} - \frac{4 a^{6} x^{2} \sqrt{a + b x^{2}}}{2145 b^{2}} + \frac{a^{5} x^{4} \sqrt{a + b x^{2}}}{715 b} + \frac{71 a^{4} x^{6} \sqrt{a + b x^{2}}}{429} + \frac{206 a^{3} b x^{8} \sqrt{a + b x^{2}}}{429} + \frac{406 a^{2} b^{2} x^{10} \sqrt{a + b x^{2}}}{715} + \frac{61 a b^{3} x^{12} \sqrt{a + b x^{2}}}{195} + \frac{b^{4} x^{14} \sqrt{a + b x^{2}}}{15} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x**2+a)**(9/2),x)

[Out]

Piecewise((8*a**7*sqrt(a + b*x**2)/(2145*b**3) - 4*a**6*x**2*sqrt(a + b*x**2)/(2145*b**2) + a**5*x**4*sqrt(a +
b*x**2)/(715*b) + 71*a**4*x**6*sqrt(a + b*x**2)/429 + 206*a**3*b*x**8*sqrt(a + b*x**2)/429 + 406*a**2*b**2*x*
*10*sqrt(a + b*x**2)/715 + 61*a*b**3*x**12*sqrt(a + b*x**2)/195 + b**4*x**14*sqrt(a + b*x**2)/15, Ne(b, 0)), (
a**(9/2)*x**6/6, True))

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Giac [B]  time = 2.76058, size = 500, normalized size = 8.47 \begin{align*} \frac{\frac{429 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} a^{4}}{b^{2}} + \frac{572 \,{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} a^{3}}{b^{2}} + \frac{78 \,{\left (315 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{4}\right )} a^{2}}{b^{2}} + \frac{20 \,{\left (693 \,{\left (b x^{2} + a\right )}^{\frac{13}{2}} - 4095 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a^{2} - 12870 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{4} - 3003 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{5}\right )} a}{b^{2}} + \frac{3003 \,{\left (b x^{2} + a\right )}^{\frac{15}{2}} - 20790 \,{\left (b x^{2} + a\right )}^{\frac{13}{2}} a + 61425 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} a^{2} - 100100 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a^{3} + 96525 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{4} - 54054 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{5} + 15015 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{6}}{b^{2}}}{45045 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45045*(429*(15*(b*x^2 + a)^(7/2) - 42*(b*x^2 + a)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)*a^4/b^2 + 572*(35*(b*x
^2 + a)^(9/2) - 135*(b*x^2 + a)^(7/2)*a + 189*(b*x^2 + a)^(5/2)*a^2 - 105*(b*x^2 + a)^(3/2)*a^3)*a^3/b^2 + 78*
(315*(b*x^2 + a)^(11/2) - 1540*(b*x^2 + a)^(9/2)*a + 2970*(b*x^2 + a)^(7/2)*a^2 - 2772*(b*x^2 + a)^(5/2)*a^3 +
1155*(b*x^2 + a)^(3/2)*a^4)*a^2/b^2 + 20*(693*(b*x^2 + a)^(13/2) - 4095*(b*x^2 + a)^(11/2)*a + 10010*(b*x^2 +
a)^(9/2)*a^2 - 12870*(b*x^2 + a)^(7/2)*a^3 + 9009*(b*x^2 + a)^(5/2)*a^4 - 3003*(b*x^2 + a)^(3/2)*a^5)*a/b^2 +
(3003*(b*x^2 + a)^(15/2) - 20790*(b*x^2 + a)^(13/2)*a + 61425*(b*x^2 + a)^(11/2)*a^2 - 100100*(b*x^2 + a)^(9/
2)*a^3 + 96525*(b*x^2 + a)^(7/2)*a^4 - 54054*(b*x^2 + a)^(5/2)*a^5 + 15015*(b*x^2 + a)^(3/2)*a^6)/b^2)/b