3.536 \(\int \frac{(a+b x^2)^{3/2} (A+B x^2)}{x^{10}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{2 b \left (a+b x^2\right )^{5/2} (4 A b-9 a B)}{315 a^3 x^5}+\frac{\left (a+b x^2\right )^{5/2} (4 A b-9 a B)}{63 a^2 x^7}-\frac{A \left (a+b x^2\right )^{5/2}}{9 a x^9} \]

[Out]

-(A*(a + b*x^2)^(5/2))/(9*a*x^9) + ((4*A*b - 9*a*B)*(a + b*x^2)^(5/2))/(63*a^2*x^7) - (2*b*(4*A*b - 9*a*B)*(a
+ b*x^2)^(5/2))/(315*a^3*x^5)

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Rubi [A]  time = 0.0361016, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {453, 271, 264} \[ -\frac{2 b \left (a+b x^2\right )^{5/2} (4 A b-9 a B)}{315 a^3 x^5}+\frac{\left (a+b x^2\right )^{5/2} (4 A b-9 a B)}{63 a^2 x^7}-\frac{A \left (a+b x^2\right )^{5/2}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^10,x]

[Out]

-(A*(a + b*x^2)^(5/2))/(9*a*x^9) + ((4*A*b - 9*a*B)*(a + b*x^2)^(5/2))/(63*a^2*x^7) - (2*b*(4*A*b - 9*a*B)*(a
+ b*x^2)^(5/2))/(315*a^3*x^5)

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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^{10}} \, dx &=-\frac{A \left (a+b x^2\right )^{5/2}}{9 a x^9}-\frac{(4 A b-9 a B) \int \frac{\left (a+b x^2\right )^{3/2}}{x^8} \, dx}{9 a}\\ &=-\frac{A \left (a+b x^2\right )^{5/2}}{9 a x^9}+\frac{(4 A b-9 a B) \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}+\frac{(2 b (4 A b-9 a B)) \int \frac{\left (a+b x^2\right )^{3/2}}{x^6} \, dx}{63 a^2}\\ &=-\frac{A \left (a+b x^2\right )^{5/2}}{9 a x^9}+\frac{(4 A b-9 a B) \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}-\frac{2 b (4 A b-9 a B) \left (a+b x^2\right )^{5/2}}{315 a^3 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0330151, size = 63, normalized size = 0.75 \[ \frac{\left (a+b x^2\right )^{5/2} \left (-5 a^2 \left (7 A+9 B x^2\right )+2 a b x^2 \left (10 A+9 B x^2\right )-8 A b^2 x^4\right )}{315 a^3 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^10,x]

[Out]

((a + b*x^2)^(5/2)*(-8*A*b^2*x^4 - 5*a^2*(7*A + 9*B*x^2) + 2*a*b*x^2*(10*A + 9*B*x^2)))/(315*a^3*x^9)

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Maple [A]  time = 0.005, size = 59, normalized size = 0.7 \begin{align*} -{\frac{8\,A{b}^{2}{x}^{4}-18\,B{x}^{4}ab-20\,aAb{x}^{2}+45\,B{x}^{2}{a}^{2}+35\,A{a}^{2}}{315\,{x}^{9}{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(3/2)*(B*x^2+A)/x^10,x)

[Out]

-1/315*(b*x^2+a)^(5/2)*(8*A*b^2*x^4-18*B*a*b*x^4-20*A*a*b*x^2+45*B*a^2*x^2+35*A*a^2)/x^9/a^3

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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)^(3/2)*(B*x^2+A)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.80658, size = 230, normalized size = 2.74 \begin{align*} \frac{{\left (2 \,{\left (9 \, B a b^{3} - 4 \, A b^{4}\right )} x^{8} -{\left (9 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} - 35 \, A a^{4} - 3 \,{\left (24 \, B a^{3} b + A a^{2} b^{2}\right )} x^{4} - 5 \,{\left (9 \, B a^{4} + 10 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{315 \, a^{3} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)*(B*x^2+A)/x^10,x, algorithm="fricas")

[Out]

1/315*(2*(9*B*a*b^3 - 4*A*b^4)*x^8 - (9*B*a^2*b^2 - 4*A*a*b^3)*x^6 - 35*A*a^4 - 3*(24*B*a^3*b + A*a^2*b^2)*x^4
- 5*(9*B*a^4 + 10*A*a^3*b)*x^2)*sqrt(b*x^2 + a)/(a^3*x^9)

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Sympy [B]  time = 6.33846, size = 1408, normalized size = 16.76 \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)**(3/2)*(B*x**2+A)/x**10,x)

[Out]

-35*A*a**8*b**(19/2)*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 +
315*a**4*b**12*x**14) - 110*A*a**7*b**(21/2)*x**2*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x*
*10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 114*A*a**6*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(315*a**7*
b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 40*A*a**5*b**(25/2)*x**6*sqr
t(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) -
15*A*a**5*b**(11/2)*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) + 5*A
*a**4*b**(27/2)*x**8*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 +
315*a**4*b**12*x**14) - 33*A*a**4*b**(13/2)*x**2*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8
+ 105*a**3*b**6*x**10) + 30*A*a**3*b**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*
x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 17*A*a**3*b**(15/2)*x**4*sqrt(a/(b*x**2) + 1)/(105*a**5
*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) + 40*A*a**2*b**(31/2)*x**12*sqrt(a/(b*x**2) + 1)/(315*a
**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14) - 3*A*a**2*b**(17/2)*x**6*
sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) + 16*A*a*b**(33/2)*x**14*
sqrt(a/(b*x**2) + 1)/(315*a**7*b**9*x**8 + 945*a**6*b**10*x**10 + 945*a**5*b**11*x**12 + 315*a**4*b**12*x**14)
- 12*A*a*b**(19/2)*x**8*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10)
- 8*A*b**(21/2)*x**10*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 1
5*B*a**6*b**(9/2)*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 33*B*
a**5*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 17*
B*a**4*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 3
*B*a**3*b**(15/2)*x**6*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) -
12*B*a**2*b**(17/2)*x**8*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10)
- 8*B*a*b**(19/2)*x**10*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) -
B*b**(3/2)*sqrt(a/(b*x**2) + 1)/(5*x**4) - B*b**(5/2)*sqrt(a/(b*x**2) + 1)/(15*a*x**2) + 2*B*b**(7/2)*sqrt(a/
(b*x**2) + 1)/(15*a**2)

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Giac [B]  time = 1.18314, size = 540, normalized size = 6.43 \begin{align*} \frac{4 \,{\left (315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} B b^{\frac{7}{2}} - 315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} B a b^{\frac{7}{2}} + 840 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} A b^{\frac{9}{2}} + 315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} B a^{2} b^{\frac{7}{2}} + 1260 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} A a b^{\frac{9}{2}} - 819 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a^{3} b^{\frac{7}{2}} + 1764 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A a^{2} b^{\frac{9}{2}} + 441 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{4} b^{\frac{7}{2}} + 504 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a^{3} b^{\frac{9}{2}} - 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{5} b^{\frac{7}{2}} + 144 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{4} b^{\frac{9}{2}} + 81 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{6} b^{\frac{7}{2}} - 36 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{5} b^{\frac{9}{2}} - 9 \, B a^{7} b^{\frac{7}{2}} + 4 \, A a^{6} b^{\frac{9}{2}}\right )}}{315 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(3/2)*(B*x^2+A)/x^10,x, algorithm="giac")

[Out]

4/315*(315*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*b^(7/2) - 315*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a*b^(7/2) + 840
*(sqrt(b)*x - sqrt(b*x^2 + a))^12*A*b^(9/2) + 315*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^2*b^(7/2) + 1260*(sqrt(
b)*x - sqrt(b*x^2 + a))^10*A*a*b^(9/2) - 819*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^3*b^(7/2) + 1764*(sqrt(b)*x -
sqrt(b*x^2 + a))^8*A*a^2*b^(9/2) + 441*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^4*b^(7/2) + 504*(sqrt(b)*x - sqrt(
b*x^2 + a))^6*A*a^3*b^(9/2) - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^5*b^(7/2) + 144*(sqrt(b)*x - sqrt(b*x^2 +
a))^4*A*a^4*b^(9/2) + 81*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^6*b^(7/2) - 36*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*
a^5*b^(9/2) - 9*B*a^7*b^(7/2) + 4*A*a^6*b^(9/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9