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

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

[Out]

-1/11*a^3*(b*x^2+a)^(11/2)/b^4+3/13*a^2*(b*x^2+a)^(13/2)/b^4-1/5*a*(b*x^2+a)^(15/2)/b^4+1/17*(b*x^2+a)^(17/2)/
b^4

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Rubi [A]  time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.03, size = 50, normalized size = 0.62 \[ \frac {\left (a+b x^2\right )^{11/2} \left (-16 a^3+88 a^2 b x^2-286 a b^2 x^4+715 b^3 x^6\right )}{12155 b^4} \]

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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fricas [A]  time = 1.22, size = 101, normalized size = 1.26 \[ \frac {{\left (715 \, b^{8} x^{16} + 3289 \, a b^{7} x^{14} + 5808 \, a^{2} b^{6} x^{12} + 4714 \, a^{3} b^{5} x^{10} + 1515 \, a^{4} b^{4} x^{8} + 5 \, a^{5} b^{3} x^{6} - 6 \, a^{6} b^{2} x^{4} + 8 \, a^{7} b x^{2} - 16 \, a^{8}\right )} \sqrt {b x^{2} + a}}{12155 \, b^{4}} \]

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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giac [A]  time = 1.15, size = 57, normalized size = 0.71 \[ \frac {715 \, {\left (b x^{2} + a\right )}^{\frac {17}{2}} - 2431 \, {\left (b x^{2} + a\right )}^{\frac {15}{2}} a + 2805 \, {\left (b x^{2} + a\right )}^{\frac {13}{2}} a^{2} - 1105 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{3}}{12155 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12155*(715*(b*x^2 + a)^(17/2) - 2431*(b*x^2 + a)^(15/2)*a + 2805*(b*x^2 + a)^(13/2)*a^2 - 1105*(b*x^2 + a)^(
11/2)*a^3)/b^4

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maple [A]  time = 0.01, size = 47, normalized size = 0.59 \[ -\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (-715 b^{3} x^{6}+286 a \,b^{2} x^{4}-88 a^{2} b \,x^{2}+16 a^{3}\right )}{12155 b^{4}} \]

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 [A]  time = 1.42, size = 73, normalized size = 0.91 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} x^{6}}{17 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a x^{4}}{85 \, b^{2}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{2} x^{2}}{1105 \, b^{3}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{3}}{12155 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*(b*x^2 + a)^(11/2)*x^6/b - 2/85*(b*x^2 + a)^(11/2)*a*x^4/b^2 + 8/1105*(b*x^2 + a)^(11/2)*a^2*x^2/b^3 - 16
/12155*(b*x^2 + a)^(11/2)*a^3/b^4

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mupad [B]  time = 4.73, size = 97, normalized size = 1.21 \[ \sqrt {b\,x^2+a}\,\left (\frac {303\,a^4\,x^8}{2431}-\frac {16\,a^8}{12155\,b^4}+\frac {b^4\,x^{16}}{17}+\frac {4714\,a^3\,b\,x^{10}}{12155}+\frac {23\,a\,b^3\,x^{14}}{85}+\frac {a^5\,x^6}{2431\,b}-\frac {6\,a^6\,x^4}{12155\,b^2}+\frac {8\,a^7\,x^2}{12155\,b^3}+\frac {528\,a^2\,b^2\,x^{12}}{1105}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a + b*x^2)^(9/2),x)

[Out]

(a + b*x^2)^(1/2)*((303*a^4*x^8)/2431 - (16*a^8)/(12155*b^4) + (b^4*x^16)/17 + (4714*a^3*b*x^10)/12155 + (23*a
*b^3*x^14)/85 + (a^5*x^6)/(2431*b) - (6*a^6*x^4)/(12155*b^2) + (8*a^7*x^2)/(12155*b^3) + (528*a^2*b^2*x^12)/11
05)

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sympy [A]  time = 37.20, size = 204, normalized size = 2.55 \[ \begin {cases} - \frac {16 a^{8} \sqrt {a + b x^{2}}}{12155 b^{4}} + \frac {8 a^{7} x^{2} \sqrt {a + b x^{2}}}{12155 b^{3}} - \frac {6 a^{6} x^{4} \sqrt {a + b x^{2}}}{12155 b^{2}} + \frac {a^{5} x^{6} \sqrt {a + b x^{2}}}{2431 b} + \frac {303 a^{4} x^{8} \sqrt {a + b x^{2}}}{2431} + \frac {4714 a^{3} b x^{10} \sqrt {a + b x^{2}}}{12155} + \frac {528 a^{2} b^{2} x^{12} \sqrt {a + b x^{2}}}{1105} + \frac {23 a b^{3} x^{14} \sqrt {a + b x^{2}}}{85} + \frac {b^{4} x^{16} \sqrt {a + b x^{2}}}{17} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{8}}{8} & \text {otherwise} \end {cases} \]

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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