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]

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

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Rubi [A]  time = 0.04, antiderivative size = 59, 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} \[ \frac {a^2 \left (a+b x^2\right )^{11/2}}{11 b^3}+\frac {\left (a+b x^2\right )^{15/2}}{15 b^3}-\frac {2 a \left (a+b x^2\right )^{13/2}}{13 b^3} \]

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 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^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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.87, size = 90, normalized size = 1.53 \[ \frac {{\left (143 \, b^{7} x^{14} + 671 \, a b^{6} x^{12} + 1218 \, a^{2} b^{5} x^{10} + 1030 \, a^{3} b^{4} x^{8} + 355 \, a^{4} b^{3} x^{6} + 3 \, a^{5} b^{2} x^{4} - 4 \, a^{6} b x^{2} + 8 \, a^{7}\right )} \sqrt {b x^{2} + a}}{2145 \, b^{3}} \]

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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giac [A]  time = 0.96, size = 43, normalized size = 0.73 \[ \frac {143 \, {\left (b x^{2} + a\right )}^{\frac {15}{2}} - 330 \, {\left (b x^{2} + a\right )}^{\frac {13}{2}} a + 195 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{2}}{2145 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2145*(143*(b*x^2 + a)^(15/2) - 330*(b*x^2 + a)^(13/2)*a + 195*(b*x^2 + a)^(11/2)*a^2)/b^3

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maple [A]  time = 0.00, size = 36, normalized size = 0.61 \[ \frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} \left (143 b^{2} x^{4}-44 a b \,x^{2}+8 a^{2}\right )}{2145 b^{3}} \]

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 [A]  time = 1.39, size = 53, normalized size = 0.90 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} x^{4}}{15 \, b} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a x^{2}}{195 \, b^{2}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} a^{2}}{2145 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(b*x^2 + a)^(11/2)*x^4/b - 4/195*(b*x^2 + a)^(11/2)*a*x^2/b^2 + 8/2145*(b*x^2 + a)^(11/2)*a^2/b^3

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mupad [B]  time = 4.60, size = 86, normalized size = 1.46 \[ \sqrt {b\,x^2+a}\,\left (\frac {8\,a^7}{2145\,b^3}+\frac {71\,a^4\,x^6}{429}+\frac {b^4\,x^{14}}{15}+\frac {206\,a^3\,b\,x^8}{429}+\frac {61\,a\,b^3\,x^{12}}{195}+\frac {a^5\,x^4}{715\,b}-\frac {4\,a^6\,x^2}{2145\,b^2}+\frac {406\,a^2\,b^2\,x^{10}}{715}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x^2)^(9/2),x)

[Out]

(a + b*x^2)^(1/2)*((8*a^7)/(2145*b^3) + (71*a^4*x^6)/429 + (b^4*x^14)/15 + (206*a^3*b*x^8)/429 + (61*a*b^3*x^1
2)/195 + (a^5*x^4)/(715*b) - (4*a^6*x^2)/(2145*b^2) + (406*a^2*b^2*x^10)/715)

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sympy [A]  time = 27.02, size = 180, normalized size = 3.05 \[ \begin {cases} \frac {8 a^{7} \sqrt {a + b x^{2}}}{2145 b^{3}} - \frac {4 a^{6} x^{2} \sqrt {a + b x^{2}}}{2145 b^{2}} + \frac {a^{5} x^{4} \sqrt {a + b x^{2}}}{715 b} + \frac {71 a^{4} x^{6} \sqrt {a + b x^{2}}}{429} + \frac {206 a^{3} b x^{8} \sqrt {a + b x^{2}}}{429} + \frac {406 a^{2} b^{2} x^{10} \sqrt {a + b x^{2}}}{715} + \frac {61 a b^{3} x^{12} \sqrt {a + b x^{2}}}{195} + \frac {b^{4} x^{14} \sqrt {a + b x^{2}}}{15} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{6}}{6} & \text {otherwise} \end {cases} \]

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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