3.583 \(\int \frac{A+B x^2}{x^8 (a+b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{16 b^3 x (8 A b-7 a B)}{35 a^5 \sqrt{a+b x^2}}+\frac{8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt{a+b x^2}}-\frac{2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt{a+b x^2}}+\frac{8 A b-7 a B}{35 a^2 x^5 \sqrt{a+b x^2}}-\frac{A}{7 a x^7 \sqrt{a+b x^2}} \]

[Out]

-A/(7*a*x^7*Sqrt[a + b*x^2]) + (8*A*b - 7*a*B)/(35*a^2*x^5*Sqrt[a + b*x^2]) - (2*b*(8*A*b - 7*a*B))/(35*a^3*x^
3*Sqrt[a + b*x^2]) + (8*b^2*(8*A*b - 7*a*B))/(35*a^4*x*Sqrt[a + b*x^2]) + (16*b^3*(8*A*b - 7*a*B)*x)/(35*a^5*S
qrt[a + b*x^2])

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Rubi [A]  time = 0.0612387, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {453, 271, 191} \[ \frac{16 b^3 x (8 A b-7 a B)}{35 a^5 \sqrt{a+b x^2}}+\frac{8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt{a+b x^2}}-\frac{2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt{a+b x^2}}+\frac{8 A b-7 a B}{35 a^2 x^5 \sqrt{a+b x^2}}-\frac{A}{7 a x^7 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^8*(a + b*x^2)^(3/2)),x]

[Out]

-A/(7*a*x^7*Sqrt[a + b*x^2]) + (8*A*b - 7*a*B)/(35*a^2*x^5*Sqrt[a + b*x^2]) - (2*b*(8*A*b - 7*a*B))/(35*a^3*x^
3*Sqrt[a + b*x^2]) + (8*b^2*(8*A*b - 7*a*B))/(35*a^4*x*Sqrt[a + b*x^2]) + (16*b^3*(8*A*b - 7*a*B)*x)/(35*a^5*S
qrt[a + b*x^2])

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -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]

Rule 191

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

Rubi steps

\begin{align*} \int \frac{A+B x^2}{x^8 \left (a+b x^2\right )^{3/2}} \, dx &=-\frac{A}{7 a x^7 \sqrt{a+b x^2}}-\frac{(8 A b-7 a B) \int \frac{1}{x^6 \left (a+b x^2\right )^{3/2}} \, dx}{7 a}\\ &=-\frac{A}{7 a x^7 \sqrt{a+b x^2}}+\frac{8 A b-7 a B}{35 a^2 x^5 \sqrt{a+b x^2}}+\frac{(6 b (8 A b-7 a B)) \int \frac{1}{x^4 \left (a+b x^2\right )^{3/2}} \, dx}{35 a^2}\\ &=-\frac{A}{7 a x^7 \sqrt{a+b x^2}}+\frac{8 A b-7 a B}{35 a^2 x^5 \sqrt{a+b x^2}}-\frac{2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt{a+b x^2}}-\frac{\left (8 b^2 (8 A b-7 a B)\right ) \int \frac{1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{35 a^3}\\ &=-\frac{A}{7 a x^7 \sqrt{a+b x^2}}+\frac{8 A b-7 a B}{35 a^2 x^5 \sqrt{a+b x^2}}-\frac{2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt{a+b x^2}}+\frac{8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt{a+b x^2}}+\frac{\left (16 b^3 (8 A b-7 a B)\right ) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{35 a^4}\\ &=-\frac{A}{7 a x^7 \sqrt{a+b x^2}}+\frac{8 A b-7 a B}{35 a^2 x^5 \sqrt{a+b x^2}}-\frac{2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt{a+b x^2}}+\frac{8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt{a+b x^2}}+\frac{16 b^3 (8 A b-7 a B) x}{35 a^5 \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0273841, size = 71, normalized size = 0.48 \[ \frac{x^2 \left (-2 a^2 b x^2+a^3+8 a b^2 x^4+16 b^3 x^6\right ) (8 A b-7 a B)-5 a^4 A}{35 a^5 x^7 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^8*(a + b*x^2)^(3/2)),x]

[Out]

(-5*a^4*A + (8*A*b - 7*a*B)*x^2*(a^3 - 2*a^2*b*x^2 + 8*a*b^2*x^4 + 16*b^3*x^6))/(35*a^5*x^7*Sqrt[a + b*x^2])

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Maple [A]  time = 0.005, size = 107, normalized size = 0.7 \begin{align*} -{\frac{-128\,A{b}^{4}{x}^{8}+112\,Ba{b}^{3}{x}^{8}-64\,Aa{b}^{3}{x}^{6}+56\,B{a}^{2}{b}^{2}{x}^{6}+16\,A{a}^{2}{b}^{2}{x}^{4}-14\,B{a}^{3}b{x}^{4}-8\,A{a}^{3}b{x}^{2}+7\,B{a}^{4}{x}^{2}+5\,A{a}^{4}}{35\,{x}^{7}{a}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^8/(b*x^2+a)^(3/2),x)

[Out]

-1/35*(-128*A*b^4*x^8+112*B*a*b^3*x^8-64*A*a*b^3*x^6+56*B*a^2*b^2*x^6+16*A*a^2*b^2*x^4-14*B*a^3*b*x^4-8*A*a^3*
b*x^2+7*B*a^4*x^2+5*A*a^4)/(b*x^2+a)^(1/2)/x^7/a^5

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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((B*x^2+A)/x^8/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77955, size = 246, normalized size = 1.66 \begin{align*} -\frac{{\left (16 \,{\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{8} + 8 \,{\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 5 \, A a^{4} - 2 \,{\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} +{\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{35 \,{\left (a^{5} b x^{9} + a^{6} x^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^8/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

-1/35*(16*(7*B*a*b^3 - 8*A*b^4)*x^8 + 8*(7*B*a^2*b^2 - 8*A*a*b^3)*x^6 + 5*A*a^4 - 2*(7*B*a^3*b - 8*A*a^2*b^2)*
x^4 + (7*B*a^4 - 8*A*a^3*b)*x^2)*sqrt(b*x^2 + a)/(a^5*b*x^9 + a^6*x^7)

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Sympy [B]  time = 25.4645, size = 1030, normalized size = 6.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**8/(b*x**2+a)**(3/2),x)

[Out]

A*(-5*a**7*b**(33/2)*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 1
40*a**6*b**19*x**12 + 35*a**5*b**20*x**14) - 7*a**6*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 +
140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) - 7*a**5*b**(37/2)*x*
*4*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**1
2 + 35*a**5*b**20*x**14) + 35*a**4*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x*
*8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) + 280*a**3*b**(41/2)*x**8*sqrt(a/(b*x*
*2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**
20*x**14) + 560*a**2*b**(43/2)*x**10*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7
*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) + 448*a*b**(45/2)*x**12*sqrt(a/(b*x**2) + 1)/(35*a*
*9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) + 128
*b**(47/2)*x**14*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a
**6*b**19*x**12 + 35*a**5*b**20*x**14)) + B*(-a**5*b**(19/2)*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*
b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 5*a**3*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(5*a**7*b**
9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 30*a**2*b**(25/2)*x**6*sqrt(a/(b*x**2
) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 40*a*b**(27/2)*x**8
*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 16*b
**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12
*x**10))

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Giac [B]  time = 1.1738, size = 549, normalized size = 3.71 \begin{align*} -\frac{{\left (B a b^{3} - A b^{4}\right )} x}{\sqrt{b x^{2} + a} a^{5}} + \frac{2 \,{\left (35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} B a b^{\frac{5}{2}} - 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} A b^{\frac{7}{2}} - 280 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} B a^{2} b^{\frac{5}{2}} + 280 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} A a b^{\frac{7}{2}} + 1015 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a^{3} b^{\frac{5}{2}} - 1015 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A a^{2} b^{\frac{7}{2}} - 1680 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{4} b^{\frac{5}{2}} + 2240 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a^{3} b^{\frac{7}{2}} + 1337 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{5} b^{\frac{5}{2}} - 1673 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{4} b^{\frac{7}{2}} - 504 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{6} b^{\frac{5}{2}} + 616 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{5} b^{\frac{7}{2}} + 77 \, B a^{7} b^{\frac{5}{2}} - 93 \, A a^{6} b^{\frac{7}{2}}\right )}}{35 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^8/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

-(B*a*b^3 - A*b^4)*x/(sqrt(b*x^2 + a)*a^5) + 2/35*(35*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a*b^(5/2) - 35*(sqrt(
b)*x - sqrt(b*x^2 + a))^12*A*b^(7/2) - 280*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^2*b^(5/2) + 280*(sqrt(b)*x - s
qrt(b*x^2 + a))^10*A*a*b^(7/2) + 1015*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^3*b^(5/2) - 1015*(sqrt(b)*x - sqrt(b
*x^2 + a))^8*A*a^2*b^(7/2) - 1680*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^4*b^(5/2) + 2240*(sqrt(b)*x - sqrt(b*x^2
+ a))^6*A*a^3*b^(7/2) + 1337*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^5*b^(5/2) - 1673*(sqrt(b)*x - sqrt(b*x^2 + a
))^4*A*a^4*b^(7/2) - 504*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^6*b^(5/2) + 616*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A
*a^5*b^(7/2) + 77*B*a^7*b^(5/2) - 93*A*a^6*b^(7/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7*a^4)