3.413 \(\int x^7 (a+b x^2)^{9/2} \, dx\)

Optimal. Leaf size=80 \[ \frac{3 a^2 \left (a+b x^2\right )^{13/2}}{13 b^4}-\frac{a^3 \left (a+b x^2\right )^{11/2}}{11 b^4}+\frac{\left (a+b x^2\right )^{17/2}}{17 b^4}-\frac{a \left (a+b x^2\right )^{15/2}}{5 b^4} \]

[Out]

-(a^3*(a + b*x^2)^(11/2))/(11*b^4) + (3*a^2*(a + b*x^2)^(13/2))/(13*b^4) - (a*(a + b*x^2)^(15/2))/(5*b^4) + (a
+ b*x^2)^(17/2)/(17*b^4)

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Rubi [A]  time = 0.0458746, antiderivative size = 80, 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} \[ \frac{3 a^2 \left (a+b x^2\right )^{13/2}}{13 b^4}-\frac{a^3 \left (a+b x^2\right )^{11/2}}{11 b^4}+\frac{\left (a+b x^2\right )^{17/2}}{17 b^4}-\frac{a \left (a+b x^2\right )^{15/2}}{5 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^2)^(9/2),x]

[Out]

-(a^3*(a + b*x^2)^(11/2))/(11*b^4) + (3*a^2*(a + b*x^2)^(13/2))/(13*b^4) - (a*(a + b*x^2)^(15/2))/(5*b^4) + (a
+ b*x^2)^(17/2)/(17*b^4)

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^7 \left (a+b x^2\right )^{9/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x)^{9/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{9/2}}{b^3}+\frac{3 a^2 (a+b x)^{11/2}}{b^3}-\frac{3 a (a+b x)^{13/2}}{b^3}+\frac{(a+b x)^{15/2}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3 \left (a+b x^2\right )^{11/2}}{11 b^4}+\frac{3 a^2 \left (a+b x^2\right )^{13/2}}{13 b^4}-\frac{a \left (a+b x^2\right )^{15/2}}{5 b^4}+\frac{\left (a+b x^2\right )^{17/2}}{17 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0296801, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^2\right )^{11/2} \left (88 a^2 b x^2-16 a^3-286 a b^2 x^4+715 b^3 x^6\right )}{12155 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^2)^(9/2),x]

[Out]

((a + b*x^2)^(11/2)*(-16*a^3 + 88*a^2*b*x^2 - 286*a*b^2*x^4 + 715*b^3*x^6))/(12155*b^4)

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Maple [A]  time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-715\,{b}^{3}{x}^{6}+286\,a{b}^{2}{x}^{4}-88\,{a}^{2}b{x}^{2}+16\,{a}^{3}}{12155\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b*x^2+a)^(9/2),x)

[Out]

-1/12155*(b*x^2+a)^(11/2)*(-715*b^3*x^6+286*a*b^2*x^4-88*a^2*b*x^2+16*a^3)/b^4

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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^7*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.7245, size = 238, normalized size = 2.98 \begin{align*} \frac{{\left (715 \, b^{8} x^{16} + 3289 \, a b^{7} x^{14} + 5808 \, a^{2} b^{6} x^{12} + 4714 \, a^{3} b^{5} x^{10} + 1515 \, a^{4} b^{4} x^{8} + 5 \, a^{5} b^{3} x^{6} - 6 \, a^{6} b^{2} x^{4} + 8 \, a^{7} b x^{2} - 16 \, a^{8}\right )} \sqrt{b x^{2} + a}}{12155 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/12155*(715*b^8*x^16 + 3289*a*b^7*x^14 + 5808*a^2*b^6*x^12 + 4714*a^3*b^5*x^10 + 1515*a^4*b^4*x^8 + 5*a^5*b^3
*x^6 - 6*a^6*b^2*x^4 + 8*a^7*b*x^2 - 16*a^8)*sqrt(b*x^2 + a)/b^4

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Sympy [A]  time = 34.4877, size = 204, normalized size = 2.55 \begin{align*} \begin{cases} - \frac{16 a^{8} \sqrt{a + b x^{2}}}{12155 b^{4}} + \frac{8 a^{7} x^{2} \sqrt{a + b x^{2}}}{12155 b^{3}} - \frac{6 a^{6} x^{4} \sqrt{a + b x^{2}}}{12155 b^{2}} + \frac{a^{5} x^{6} \sqrt{a + b x^{2}}}{2431 b} + \frac{303 a^{4} x^{8} \sqrt{a + b x^{2}}}{2431} + \frac{4714 a^{3} b x^{10} \sqrt{a + b x^{2}}}{12155} + \frac{528 a^{2} b^{2} x^{12} \sqrt{a + b x^{2}}}{1105} + \frac{23 a b^{3} x^{14} \sqrt{a + b x^{2}}}{85} + \frac{b^{4} x^{16} \sqrt{a + b x^{2}}}{17} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{8}}{8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(b*x**2+a)**(9/2),x)

[Out]

Piecewise((-16*a**8*sqrt(a + b*x**2)/(12155*b**4) + 8*a**7*x**2*sqrt(a + b*x**2)/(12155*b**3) - 6*a**6*x**4*sq
rt(a + b*x**2)/(12155*b**2) + a**5*x**6*sqrt(a + b*x**2)/(2431*b) + 303*a**4*x**8*sqrt(a + b*x**2)/2431 + 4714
*a**3*b*x**10*sqrt(a + b*x**2)/12155 + 528*a**2*b**2*x**12*sqrt(a + b*x**2)/1105 + 23*a*b**3*x**14*sqrt(a + b*
x**2)/85 + b**4*x**16*sqrt(a + b*x**2)/17, Ne(b, 0)), (a**(9/2)*x**8/8, True))

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Giac [B]  time = 2.64018, size = 595, normalized size = 7.44 \begin{align*} \frac{\frac{2431 \,{\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^{4}}{b^{3}} + \frac{884 \,{\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^{3}}{b^{3}} + \frac{510 \,{\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^{2}}{b^{3}} + \frac{68 \,{\left (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}\right )} a}{b^{3}} + \frac{7 \,{\left (6435 \,{\left (b x^{2} + a\right )}^{\frac{17}{2}} - 51051 \,{\left (b x^{2} + a\right )}^{\frac{15}{2}} a + 176715 \,{\left (b x^{2} + a\right )}^{\frac{13}{2}} a^{2} - 348075 \,{\left (b x^{2} + a\right )}^{\frac{11}{2}} a^{3} + 425425 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} a^{4} - 328185 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a^{5} + 153153 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{6} - 36465 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{7}\right )}}{b^{3}}}{765765 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/765765*(2431*(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^4/b^3 + 884*(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^3/b^3 + 510*(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^2/b^3 + 68*(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)*a/b^3 + 7*(6435*(b*x^2 + a)^(17/2) - 51051*(b*x^2 + a)^(15/2)*a + 176715*(b*x^2 + a)^(13/2)*
a^2 - 348075*(b*x^2 + a)^(11/2)*a^3 + 425425*(b*x^2 + a)^(9/2)*a^4 - 328185*(b*x^2 + a)^(7/2)*a^5 + 153153*(b*
x^2 + a)^(5/2)*a^6 - 36465*(b*x^2 + a)^(3/2)*a^7)/b^3)/b