3.519 \(\int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^{10}} \, dx\)

Optimal. Leaf size=117 \[ \frac{8 b^2 \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{315 a^4 x^3}-\frac{4 b \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{105 a^3 x^5}+\frac{\left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{21 a^2 x^7}-\frac{A \left (a+b x^2\right )^{3/2}}{9 a x^9} \]

[Out]

-(A*(a + b*x^2)^(3/2))/(9*a*x^9) + ((2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(21*a^2*x
^7) - (4*b*(2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(105*a^3*x^5) + (8*b^2*(2*A*b - 3*
a*B)*(a + b*x^2)^(3/2))/(315*a^4*x^3)

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Rubi [A]  time = 0.166954, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{8 b^2 \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{315 a^4 x^3}-\frac{4 b \left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{105 a^3 x^5}+\frac{\left (a+b x^2\right )^{3/2} (2 A b-3 a B)}{21 a^2 x^7}-\frac{A \left (a+b x^2\right )^{3/2}}{9 a x^9} \]

Antiderivative was successfully verified.

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

[Out]

-(A*(a + b*x^2)^(3/2))/(9*a*x^9) + ((2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(21*a^2*x
^7) - (4*b*(2*A*b - 3*a*B)*(a + b*x^2)^(3/2))/(105*a^3*x^5) + (8*b^2*(2*A*b - 3*
a*B)*(a + b*x^2)^(3/2))/(315*a^4*x^3)

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Rubi in Sympy [A]  time = 16.5738, size = 112, normalized size = 0.96 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{9 a x^{9}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 A b - 3 B a\right )}{21 a^{2} x^{7}} - \frac{4 b \left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 A b - 3 B a\right )}{105 a^{3} x^{5}} + \frac{8 b^{2} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 A b - 3 B a\right )}{315 a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**10,x)

[Out]

-A*(a + b*x**2)**(3/2)/(9*a*x**9) + (a + b*x**2)**(3/2)*(2*A*b - 3*B*a)/(21*a**2
*x**7) - 4*b*(a + b*x**2)**(3/2)*(2*A*b - 3*B*a)/(105*a**3*x**5) + 8*b**2*(a + b
*x**2)**(3/2)*(2*A*b - 3*B*a)/(315*a**4*x**3)

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Mathematica [A]  time = 0.0859574, size = 81, normalized size = 0.69 \[ \frac{\left (a+b x^2\right )^{3/2} \left (-5 a^3 \left (7 A+9 B x^2\right )+6 a^2 b x^2 \left (5 A+6 B x^2\right )-24 a b^2 x^4 \left (A+B x^2\right )+16 A b^3 x^6\right )}{315 a^4 x^9} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^2)^(3/2)*(16*A*b^3*x^6 - 24*a*b^2*x^4*(A + B*x^2) + 6*a^2*b*x^2*(5*A +
6*B*x^2) - 5*a^3*(7*A + 9*B*x^2)))/(315*a^4*x^9)

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Maple [A]  time = 0.01, size = 83, normalized size = 0.7 \[ -{\frac{-16\,A{b}^{3}{x}^{6}+24\,Ba{b}^{2}{x}^{6}+24\,Aa{b}^{2}{x}^{4}-36\,B{a}^{2}b{x}^{4}-30\,A{a}^{2}b{x}^{2}+45\,B{a}^{3}{x}^{2}+35\,A{a}^{3}}{315\,{x}^{9}{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(b*x^2+a)^(1/2)/x^10,x)

[Out]

-1/315*(b*x^2+a)^(3/2)*(-16*A*b^3*x^6+24*B*a*b^2*x^6+24*A*a*b^2*x^4-36*B*a^2*b*x
^4-30*A*a^2*b*x^2+45*B*a^3*x^2+35*A*a^3)/x^9/a^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.321932, size = 142, normalized size = 1.21 \[ -\frac{{\left (8 \,{\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{8} - 4 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{6} + 35 \, A a^{4} + 3 \,{\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{4} + 5 \,{\left (9 \, B a^{4} + A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{315 \, a^{4} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^10,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 13.5257, size = 957, normalized size = 8.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**10,x)

[Out]

-35*A*a**7*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**6*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**5*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**4*b**(25/2)*x**6*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) + 5*A*a**3*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) + 30*A*a**2*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) + 40*A*a*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)
+ 16*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) - 15*B*a**5*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**4*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**3*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**2*b**(1
5/2)*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*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*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)

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GIAC/XCAS [A]  time = 0.23492, size = 464, normalized size = 3.97 \[ \frac{16 \,{\left (210 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} B b^{\frac{7}{2}} - 315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} B a b^{\frac{7}{2}} + 630 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} A b^{\frac{9}{2}} + 63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a^{2} b^{\frac{7}{2}} + 378 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A a b^{\frac{9}{2}} - 42 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{3} b^{\frac{7}{2}} + 168 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a^{2} b^{\frac{9}{2}} + 108 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{4} b^{\frac{7}{2}} - 72 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{3} b^{\frac{9}{2}} - 27 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{5} b^{\frac{7}{2}} + 18 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{4} b^{\frac{9}{2}} + 3 \, B a^{6} b^{\frac{7}{2}} - 2 \, A a^{5} b^{\frac{9}{2}}\right )}}{315 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^10,x, algorithm="giac")

[Out]

16/315*(210*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*b^(7/2) - 315*(sqrt(b)*x - sqrt(b
*x^2 + a))^10*B*a*b^(7/2) + 630*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*b^(9/2) + 63*
(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^2*b^(7/2) + 378*(sqrt(b)*x - sqrt(b*x^2 + a)
)^8*A*a*b^(9/2) - 42*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^3*b^(7/2) + 168*(sqrt(b
)*x - sqrt(b*x^2 + a))^6*A*a^2*b^(9/2) + 108*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a
^4*b^(7/2) - 72*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^3*b^(9/2) - 27*(sqrt(b)*x -
sqrt(b*x^2 + a))^2*B*a^5*b^(7/2) + 18*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^4*b^(9
/2) + 3*B*a^6*b^(7/2) - 2*A*a^5*b^(9/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9