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

Optimal. Leaf size=68 \[ \frac {8 b^2 x^7}{105 a^3 \left (a+b x^2\right )^{7/2}}+\frac {4 b x^5}{15 a^2 \left (a+b x^2\right )^{7/2}}+\frac {x^3}{3 a \left (a+b x^2\right )^{7/2}} \]

[Out]

1/3*x^3/a/(b*x^2+a)^(7/2)+4/15*b*x^5/a^2/(b*x^2+a)^(7/2)+8/105*b^2*x^7/a^3/(b*x^2+a)^(7/2)

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Rubi [A]  time = 0.02, antiderivative size = 68, 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 = {271, 264} \[ \frac {8 b^2 x^7}{105 a^3 \left (a+b x^2\right )^{7/2}}+\frac {4 b x^5}{15 a^2 \left (a+b x^2\right )^{7/2}}+\frac {x^3}{3 a \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^2)^(9/2),x]

[Out]

x^3/(3*a*(a + b*x^2)^(7/2)) + (4*b*x^5)/(15*a^2*(a + b*x^2)^(7/2)) + (8*b^2*x^7)/(105*a^3*(a + b*x^2)^(7/2))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac {x^3}{3 a \left (a+b x^2\right )^{7/2}}+\frac {(4 b) \int \frac {x^4}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a}\\ &=\frac {x^3}{3 a \left (a+b x^2\right )^{7/2}}+\frac {4 b x^5}{15 a^2 \left (a+b x^2\right )^{7/2}}+\frac {\left (8 b^2\right ) \int \frac {x^6}{\left (a+b x^2\right )^{9/2}} \, dx}{15 a^2}\\ &=\frac {x^3}{3 a \left (a+b x^2\right )^{7/2}}+\frac {4 b x^5}{15 a^2 \left (a+b x^2\right )^{7/2}}+\frac {8 b^2 x^7}{105 a^3 \left (a+b x^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 42, normalized size = 0.62 \[ \frac {x^3 \left (35 a^2+28 a b x^2+8 b^2 x^4\right )}{105 a^3 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^2)^(9/2),x]

[Out]

(x^3*(35*a^2 + 28*a*b*x^2 + 8*b^2*x^4))/(105*a^3*(a + b*x^2)^(7/2))

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fricas [A]  time = 0.81, size = 82, normalized size = 1.21 \[ \frac {{\left (8 \, b^{2} x^{7} + 28 \, a b x^{5} + 35 \, a^{2} x^{3}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{3} b^{4} x^{8} + 4 \, a^{4} b^{3} x^{6} + 6 \, a^{5} b^{2} x^{4} + 4 \, a^{6} b x^{2} + a^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(8*b^2*x^7 + 28*a*b*x^5 + 35*a^2*x^3)*sqrt(b*x^2 + a)/(a^3*b^4*x^8 + 4*a^4*b^3*x^6 + 6*a^5*b^2*x^4 + 4*a
^6*b*x^2 + a^7)

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giac [A]  time = 1.08, size = 43, normalized size = 0.63 \[ \frac {{\left (4 \, x^{2} {\left (\frac {2 \, b^{2} x^{2}}{a^{3}} + \frac {7 \, b}{a^{2}}\right )} + \frac {35}{a}\right )} x^{3}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(4*x^2*(2*b^2*x^2/a^3 + 7*b/a^2) + 35/a)*x^3/(b*x^2 + a)^(7/2)

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maple [A]  time = 0.01, size = 39, normalized size = 0.57 \[ \frac {\left (8 b^{2} x^{4}+28 a b \,x^{2}+35 a^{2}\right ) x^{3}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*x^3*(8*b^2*x^4+28*a*b*x^2+35*a^2)/(b*x^2+a)^(7/2)/a^3

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maxima [A]  time = 1.39, size = 70, normalized size = 1.03 \[ -\frac {x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*x/((b*x^2 + a)^(7/2)*b) + 8/105*x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*x/((b*x^2 + a)^(3/2)*a^2*b) + 1/35*x/((
b*x^2 + a)^(5/2)*a*b)

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mupad [B]  time = 4.76, size = 70, normalized size = 1.03 \[ \frac {8\,x}{105\,a^3\,b\,\sqrt {b\,x^2+a}}-\frac {x}{7\,b\,{\left (b\,x^2+a\right )}^{7/2}}+\frac {4\,x}{105\,a^2\,b\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {x}{35\,a\,b\,{\left (b\,x^2+a\right )}^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x^2)^(9/2),x)

[Out]

(8*x)/(105*a^3*b*(a + b*x^2)^(1/2)) - x/(7*b*(a + b*x^2)^(7/2)) + (4*x)/(105*a^2*b*(a + b*x^2)^(3/2)) + x/(35*
a*b*(a + b*x^2)^(5/2))

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sympy [B]  time = 1.87, size = 517, normalized size = 7.60 \[ \frac {35 a^{5} x^{3}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {63 a^{4} b x^{5}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {36 a^{3} b^{2} x^{7}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {8 a^{2} b^{3} x^{9}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35*a**5*x**3/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b**2*
x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x**2
/a)) + 63*a**4*b*x**5/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15
/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1
+ b*x**2/a)) + 36*a**3*b**2*x**7/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a)
+ 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4
*x**8*sqrt(1 + b*x**2/a)) + 8*a**2*b**3*x**9/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 +
b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**
(11/2)*b**4*x**8*sqrt(1 + b*x**2/a))

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