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

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 = 0.118607, 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 \[ -\frac{2 b \left (a+b x^2\right )^{5/2} (4 A b-9 a B)}{315 a^3 x^5}+\frac{\left (a+b x^2\right )^{5/2} (4 A b-9 a B)}{63 a^2 x^7}-\frac{A \left (a+b x^2\right )^{5/2}}{9 a x^9} \]

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)

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Rubi in Sympy [A]  time = 12.4541, size = 78, normalized size = 0.93 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{9 a x^{9}} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (4 A b - 9 B a\right )}{63 a^{2} x^{7}} - \frac{2 b \left (a + b x^{2}\right )^{\frac{5}{2}} \left (4 A b - 9 B a\right )}{315 a^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0888618, 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 = 0.008, size = 59, normalized size = 0.7 \[ -{\frac{8\,A{b}^{2}{x}^{4}-18\,Bab{x}^{4}-20\,aAb{x}^{2}+45\,B{a}^{2}{x}^{2}+35\,A{a}^{2}}{315\,{x}^{9}{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]

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]  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)*(b*x^2 + a)^(3/2)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.332746, size = 142, normalized size = 1.69 \[ \frac{{\left (2 \,{\left (9 \, B a b^{3} - 4 \, A b^{4}\right )} x^{8} -{\left (9 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} - 35 \, A a^{4} - 3 \,{\left (24 \, B a^{3} b + A a^{2} b^{2}\right )} x^{4} - 5 \,{\left (9 \, B a^{4} + 10 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{315 \, a^{3} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/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 [A]  time = 21.2061, size = 1408, normalized size = 16.76 \[ \text{result too large to display} \]

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*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*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**1
4) - 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 + 3
15*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 + 10
5*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/XCAS [A]  time = 0.256708, size = 540, normalized size = 6.43 \[ \frac{4 \,{\left (315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} B b^{\frac{7}{2}} - 315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} B a b^{\frac{7}{2}} + 840 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} A b^{\frac{9}{2}} + 315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} B a^{2} b^{\frac{7}{2}} + 1260 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} A a b^{\frac{9}{2}} - 819 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a^{3} b^{\frac{7}{2}} + 1764 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A a^{2} b^{\frac{9}{2}} + 441 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{4} b^{\frac{7}{2}} + 504 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a^{3} b^{\frac{9}{2}} - 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{5} b^{\frac{7}{2}} + 144 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{4} b^{\frac{9}{2}} + 81 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{6} b^{\frac{7}{2}} - 36 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{5} b^{\frac{9}{2}} - 9 \, B a^{7} b^{\frac{7}{2}} + 4 \, A a^{6} b^{\frac{9}{2}}\right )}}{315 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/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*(s
qrt(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 - s
qrt(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