3.597 \(\int \frac{A+B x^2}{x^6 (a+b x^2)^{5/2}} \, dx\)

Optimal. Leaf size=146 \[ -\frac{16 b^2 x (8 A b-5 a B)}{15 a^5 \sqrt{a+b x^2}}-\frac{8 b^2 x (8 A b-5 a B)}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac{2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}+\frac{8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac{A}{5 a x^5 \left (a+b x^2\right )^{3/2}} \]

[Out]

-A/(5*a*x^5*(a + b*x^2)^(3/2)) + (8*A*b - 5*a*B)/(15*a^2*x^3*(a + b*x^2)^(3/2)) - (2*b*(8*A*b - 5*a*B))/(5*a^3
*x*(a + b*x^2)^(3/2)) - (8*b^2*(8*A*b - 5*a*B)*x)/(15*a^4*(a + b*x^2)^(3/2)) - (16*b^2*(8*A*b - 5*a*B)*x)/(15*
a^5*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.0568177, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {453, 271, 192, 191} \[ -\frac{16 b^2 x (8 A b-5 a B)}{15 a^5 \sqrt{a+b x^2}}-\frac{8 b^2 x (8 A b-5 a B)}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac{2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}+\frac{8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac{A}{5 a x^5 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^6*(a + b*x^2)^(5/2)),x]

[Out]

-A/(5*a*x^5*(a + b*x^2)^(3/2)) + (8*A*b - 5*a*B)/(15*a^2*x^3*(a + b*x^2)^(3/2)) - (2*b*(8*A*b - 5*a*B))/(5*a^3
*x*(a + b*x^2)^(3/2)) - (8*b^2*(8*A*b - 5*a*B)*x)/(15*a^4*(a + b*x^2)^(3/2)) - (16*b^2*(8*A*b - 5*a*B)*x)/(15*
a^5*Sqrt[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 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -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^6 \left (a+b x^2\right )^{5/2}} \, dx &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{3/2}}-\frac{(8 A b-5 a B) \int \frac{1}{x^4 \left (a+b x^2\right )^{5/2}} \, dx}{5 a}\\ &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac{8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}+\frac{(2 b (8 A b-5 a B)) \int \frac{1}{x^2 \left (a+b x^2\right )^{5/2}} \, dx}{5 a^2}\\ &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac{8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac{2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}-\frac{\left (8 b^2 (8 A b-5 a B)\right ) \int \frac{1}{\left (a+b x^2\right )^{5/2}} \, dx}{5 a^3}\\ &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac{8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac{2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}-\frac{8 b^2 (8 A b-5 a B) x}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac{\left (16 b^2 (8 A b-5 a B)\right ) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{15 a^4}\\ &=-\frac{A}{5 a x^5 \left (a+b x^2\right )^{3/2}}+\frac{8 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{3/2}}-\frac{2 b (8 A b-5 a B)}{5 a^3 x \left (a+b x^2\right )^{3/2}}-\frac{8 b^2 (8 A b-5 a B) x}{15 a^4 \left (a+b x^2\right )^{3/2}}-\frac{16 b^2 (8 A b-5 a B) x}{15 a^5 \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0376436, size = 72, normalized size = 0.49 \[ \frac{a x^2 \left (-6 a^2 b x^2+a^3-24 a b^2 x^4-16 b^3 x^6\right ) (8 A b-5 a B)-3 a^5 A}{15 a^6 x^5 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^6*(a + b*x^2)^(5/2)),x]

[Out]

(-3*a^5*A + a*(8*A*b - 5*a*B)*x^2*(a^3 - 6*a^2*b*x^2 - 24*a*b^2*x^4 - 16*b^3*x^6))/(15*a^6*x^5*(a + b*x^2)^(3/
2))

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Maple [A]  time = 0.004, size = 107, normalized size = 0.7 \begin{align*} -{\frac{128\,A{b}^{4}{x}^{8}-80\,Ba{b}^{3}{x}^{8}+192\,Aa{b}^{3}{x}^{6}-120\,B{a}^{2}{b}^{2}{x}^{6}+48\,A{a}^{2}{b}^{2}{x}^{4}-30\,B{a}^{3}b{x}^{4}-8\,A{a}^{3}b{x}^{2}+5\,B{a}^{4}{x}^{2}+3\,A{a}^{4}}{15\,{x}^{5}{a}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^6/(b*x^2+a)^(5/2),x)

[Out]

-1/15*(128*A*b^4*x^8-80*B*a*b^3*x^8+192*A*a*b^3*x^6-120*B*a^2*b^2*x^6+48*A*a^2*b^2*x^4-30*B*a^3*b*x^4-8*A*a^3*
b*x^2+5*B*a^4*x^2+3*A*a^4)/(b*x^2+a)^(3/2)/x^5/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^6/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.73053, size = 267, normalized size = 1.83 \begin{align*} \frac{{\left (16 \,{\left (5 \, B a b^{3} - 8 \, A b^{4}\right )} x^{8} + 24 \,{\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} - 3 \, A a^{4} + 6 \,{\left (5 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} -{\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/15*(16*(5*B*a*b^3 - 8*A*b^4)*x^8 + 24*(5*B*a^2*b^2 - 8*A*a*b^3)*x^6 - 3*A*a^4 + 6*(5*B*a^3*b - 8*A*a^2*b^2)*
x^4 - (5*B*a^4 - 8*A*a^3*b)*x^2)*sqrt(b*x^2 + a)/(a^5*b^2*x^9 + 2*a^6*b*x^7 + a^7*x^5)

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Sympy [B]  time = 108.123, size = 944, normalized size = 6.47 \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**6/(b*x**2+a)**(5/2),x)

[Out]

A*(-3*a**6*b**(33/2)*sqrt(a/(b*x**2) + 1)/(15*a**9*b**16*x**4 + 60*a**8*b**17*x**6 + 90*a**7*b**18*x**8 + 60*a
**6*b**19*x**10 + 15*a**5*b**20*x**12) + 2*a**5*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(15*a**9*b**16*x**4 + 60*a
**8*b**17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b**19*x**10 + 15*a**5*b**20*x**12) - 35*a**4*b**(37/2)*x**4*sqrt
(a/(b*x**2) + 1)/(15*a**9*b**16*x**4 + 60*a**8*b**17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b**19*x**10 + 15*a**5
*b**20*x**12) - 280*a**3*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(15*a**9*b**16*x**4 + 60*a**8*b**17*x**6 + 90*a**
7*b**18*x**8 + 60*a**6*b**19*x**10 + 15*a**5*b**20*x**12) - 560*a**2*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1)/(15*a
**9*b**16*x**4 + 60*a**8*b**17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b**19*x**10 + 15*a**5*b**20*x**12) - 448*a*
b**(43/2)*x**10*sqrt(a/(b*x**2) + 1)/(15*a**9*b**16*x**4 + 60*a**8*b**17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b
**19*x**10 + 15*a**5*b**20*x**12) - 128*b**(45/2)*x**12*sqrt(a/(b*x**2) + 1)/(15*a**9*b**16*x**4 + 60*a**8*b**
17*x**6 + 90*a**7*b**18*x**8 + 60*a**6*b**19*x**10 + 15*a**5*b**20*x**12)) + B*(-a**4*b**(19/2)*sqrt(a/(b*x**2
) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8) + 5*a**3*b**(21/2)*x**2*
sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x**8) + 30*a**2*
b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3*a**4*b**12*x
**8) + 40*a*b**(25/2)*x**6*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11*x**6 + 3*
a**4*b**12*x**8) + 16*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(3*a**7*b**9*x**2 + 9*a**6*b**10*x**4 + 9*a**5*b**11
*x**6 + 3*a**4*b**12*x**8))

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Giac [B]  time = 1.16016, size = 454, normalized size = 3.11 \begin{align*} \frac{x{\left (\frac{{\left (8 \, B a^{5} b^{4} - 11 \, A a^{4} b^{5}\right )} x^{2}}{a^{9} b} + \frac{3 \,{\left (3 \, B a^{6} b^{3} - 4 \, A a^{5} b^{4}\right )}}{a^{9} b}\right )}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} - \frac{2 \,{\left (30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a b^{\frac{3}{2}} - 45 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A b^{\frac{5}{2}} - 150 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{2} b^{\frac{3}{2}} + 240 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a b^{\frac{5}{2}} + 250 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{3} b^{\frac{3}{2}} - 490 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{2} b^{\frac{5}{2}} - 170 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{4} b^{\frac{3}{2}} + 320 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{3} b^{\frac{5}{2}} + 40 \, B a^{5} b^{\frac{3}{2}} - 73 \, A a^{4} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^6/(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

1/3*x*((8*B*a^5*b^4 - 11*A*a^4*b^5)*x^2/(a^9*b) + 3*(3*B*a^6*b^3 - 4*A*a^5*b^4)/(a^9*b))/(b*x^2 + a)^(3/2) - 2
/15*(30*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a*b^(3/2) - 45*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*b^(5/2) - 150*(sqrt
(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(3/2) + 240*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a*b^(5/2) + 250*(sqrt(b)*x -
sqrt(b*x^2 + a))^4*B*a^3*b^(3/2) - 490*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^2*b^(5/2) - 170*(sqrt(b)*x - sqrt(b
*x^2 + a))^2*B*a^4*b^(3/2) + 320*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^3*b^(5/2) + 40*B*a^5*b^(3/2) - 73*A*a^4*b
^(5/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5*a^4)