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

Optimal. Leaf size=75 \[ \frac{a^3}{7 b^4 \left (a+b x^2\right )^{7/2}}-\frac{3 a^2}{5 b^4 \left (a+b x^2\right )^{5/2}}+\frac{a}{b^4 \left (a+b x^2\right )^{3/2}}-\frac{1}{b^4 \sqrt{a+b x^2}} \]

[Out]

a^3/(7*b^4*(a + b*x^2)^(7/2)) - (3*a^2)/(5*b^4*(a + b*x^2)^(5/2)) + a/(b^4*(a + b*x^2)^(3/2)) - 1/(b^4*Sqrt[a
+ b*x^2])

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Rubi [A]  time = 0.0447213, antiderivative size = 75, 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} \[ \frac{a^3}{7 b^4 \left (a+b x^2\right )^{7/2}}-\frac{3 a^2}{5 b^4 \left (a+b x^2\right )^{5/2}}+\frac{a}{b^4 \left (a+b x^2\right )^{3/2}}-\frac{1}{b^4 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^2)^(9/2),x]

[Out]

a^3/(7*b^4*(a + b*x^2)^(7/2)) - (3*a^2)/(5*b^4*(a + b*x^2)^(5/2)) + a/(b^4*(a + b*x^2)^(3/2)) - 1/(b^4*Sqrt[a
+ b*x^2])

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

Mathematica [A]  time = 0.0236671, size = 50, normalized size = 0.67 \[ \frac{-56 a^2 b x^2-16 a^3-70 a b^2 x^4-35 b^3 x^6}{35 b^4 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^2)^(9/2),x]

[Out]

(-16*a^3 - 56*a^2*b*x^2 - 70*a*b^2*x^4 - 35*b^3*x^6)/(35*b^4*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.004, size = 47, normalized size = 0.6 \begin{align*} -{\frac{35\,{b}^{3}{x}^{6}+70\,a{b}^{2}{x}^{4}+56\,{a}^{2}b{x}^{2}+16\,{a}^{3}}{35\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{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]

-1/35*(35*b^3*x^6+70*a*b^2*x^4+56*a^2*b*x^2+16*a^3)/(b*x^2+a)^(7/2)/b^4

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Maxima [A]  time = 1.08614, size = 99, normalized size = 1.32 \begin{align*} -\frac{x^{6}}{{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{2 \, a x^{4}}{{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{8 \, a^{2} x^{2}}{5 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} - \frac{16 \, a^{3}}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} 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="maxima")

[Out]

-x^6/((b*x^2 + a)^(7/2)*b) - 2*a*x^4/((b*x^2 + a)^(7/2)*b^2) - 8/5*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) - 16/35*a^3
/((b*x^2 + a)^(7/2)*b^4)

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Fricas [A]  time = 1.31997, size = 190, normalized size = 2.53 \begin{align*} -\frac{{\left (35 \, b^{3} x^{6} + 70 \, a b^{2} x^{4} + 56 \, a^{2} b x^{2} + 16 \, a^{3}\right )} \sqrt{b x^{2} + a}}{35 \,{\left (b^{8} x^{8} + 4 \, a b^{7} x^{6} + 6 \, a^{2} b^{6} x^{4} + 4 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}} \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]

-1/35*(35*b^3*x^6 + 70*a*b^2*x^4 + 56*a^2*b*x^2 + 16*a^3)*sqrt(b*x^2 + a)/(b^8*x^8 + 4*a*b^7*x^6 + 6*a^2*b^6*x
^4 + 4*a^3*b^5*x^2 + a^4*b^4)

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Sympy [A]  time = 5.45232, size = 364, normalized size = 4.85 \begin{align*} \begin{cases} - \frac{16 a^{3}}{35 a^{3} b^{4} \sqrt{a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt{a + b x^{2}} + 105 a b^{6} x^{4} \sqrt{a + b x^{2}} + 35 b^{7} x^{6} \sqrt{a + b x^{2}}} - \frac{56 a^{2} b x^{2}}{35 a^{3} b^{4} \sqrt{a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt{a + b x^{2}} + 105 a b^{6} x^{4} \sqrt{a + b x^{2}} + 35 b^{7} x^{6} \sqrt{a + b x^{2}}} - \frac{70 a b^{2} x^{4}}{35 a^{3} b^{4} \sqrt{a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt{a + b x^{2}} + 105 a b^{6} x^{4} \sqrt{a + b x^{2}} + 35 b^{7} x^{6} \sqrt{a + b x^{2}}} - \frac{35 b^{3} x^{6}}{35 a^{3} b^{4} \sqrt{a + b x^{2}} + 105 a^{2} b^{5} x^{2} \sqrt{a + b x^{2}} + 105 a b^{6} x^{4} \sqrt{a + b x^{2}} + 35 b^{7} x^{6} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 a^{\frac{9}{2}}} & \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**3/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqr
t(a + b*x**2) + 35*b**7*x**6*sqrt(a + b*x**2)) - 56*a**2*b*x**2/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b**5
*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b**7*x**6*sqrt(a + b*x**2)) - 70*a*b**2*x**4/(3
5*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b**
7*x**6*sqrt(a + b*x**2)) - 35*b**3*x**6/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b**5*x**2*sqrt(a + b*x**2) +
105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b**7*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**8/(8*a**(9/2)), True))

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Giac [A]  time = 2.27929, size = 74, normalized size = 0.99 \begin{align*} -\frac{35 \,{\left (b x^{2} + a\right )}^{3} - 35 \,{\left (b x^{2} + a\right )}^{2} a + 21 \,{\left (b x^{2} + a\right )} a^{2} - 5 \, a^{3}}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} 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="giac")

[Out]

-1/35*(35*(b*x^2 + a)^3 - 35*(b*x^2 + a)^2*a + 21*(b*x^2 + a)*a^2 - 5*a^3)/((b*x^2 + a)^(7/2)*b^4)