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

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*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.184789, 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 \[ \frac{16 b^3 x (8 A b-7 a B)}{35 a^5 \sqrt{a+b x^2}}+\frac{8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt{a+b x^2}}-\frac{2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt{a+b x^2}}+\frac{8 A b-7 a B}{35 a^2 x^5 \sqrt{a+b x^2}}-\frac{A}{7 a x^7 \sqrt{a+b x^2}} \]

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*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 20.1098, size = 143, normalized size = 0.97 \[ - \frac{A}{7 a x^{7} \sqrt{a + b x^{2}}} + \frac{8 A b - 7 B a}{35 a^{2} x^{5} \sqrt{a + b x^{2}}} - \frac{2 b \left (8 A b - 7 B a\right )}{35 a^{3} x^{3} \sqrt{a + b x^{2}}} + \frac{8 b^{2} \left (8 A b - 7 B a\right )}{35 a^{4} x \sqrt{a + b x^{2}}} + \frac{16 b^{3} x \left (8 A b - 7 B a\right )}{35 a^{5} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.108209, size = 105, normalized size = 0.71 \[ \frac{-a^4 \left (5 A+7 B x^2\right )+2 a^3 b x^2 \left (4 A+7 B x^2\right )-8 a^2 b^2 x^4 \left (2 A+7 B x^2\right )+16 a b^3 x^6 \left (4 A-7 B x^2\right )+128 A b^4 x^8}{35 a^5 x^7 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 107, normalized size = 0.7 \[ -{\frac{-128\,A{b}^{4}{x}^{8}+112\,Ba{b}^{3}{x}^{8}-64\,Aa{b}^{3}{x}^{6}+56\,B{a}^{2}{b}^{2}{x}^{6}+16\,A{a}^{2}{b}^{2}{x}^{4}-14\,B{a}^{3}b{x}^{4}-8\,A{a}^{3}b{x}^{2}+7\,B{a}^{4}{x}^{2}+5\,A{a}^{4}}{35\,{x}^{7}{a}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

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]  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^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289299, size = 158, normalized size = 1.07 \[ -\frac{{\left (16 \,{\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{8} + 8 \,{\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 5 \, A a^{4} - 2 \,{\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} +{\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{35 \,{\left (a^{5} b x^{9} + a^{6} x^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^8),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 [A]  time = 83.8316, size = 1030, normalized size = 6.96 \[ \text{result too large to display} \]

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 + 140*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 + 21
0*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) + 35*a**4*b**(3
9/2)*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*sqr
t(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 + 1
5*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**1
1*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/XCAS [A]  time = 0.250755, size = 549, normalized size = 3.71 \[ -\frac{{\left (B a b^{3} - A b^{4}\right )} x}{\sqrt{b x^{2} + a} a^{5}} + \frac{2 \,{\left (35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} B a b^{\frac{5}{2}} - 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} A b^{\frac{7}{2}} - 280 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} B a^{2} b^{\frac{5}{2}} + 280 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} A a b^{\frac{7}{2}} + 1015 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a^{3} b^{\frac{5}{2}} - 1015 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A a^{2} b^{\frac{7}{2}} - 1680 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{4} b^{\frac{5}{2}} + 2240 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a^{3} b^{\frac{7}{2}} + 1337 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{5} b^{\frac{5}{2}} - 1673 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{4} b^{\frac{7}{2}} - 504 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{6} b^{\frac{5}{2}} + 616 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{5} b^{\frac{7}{2}} + 77 \, B a^{7} b^{\frac{5}{2}} - 93 \, A a^{6} b^{\frac{7}{2}}\right )}}{35 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^8),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 - sqrt(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)