3.568 \(\int \frac{A+B x^2}{x^8 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{8 b^2 \sqrt{a+b x^2} (6 A b-7 a B)}{105 a^4 x}-\frac{4 b \sqrt{a+b x^2} (6 A b-7 a B)}{105 a^3 x^3}+\frac{\sqrt{a+b x^2} (6 A b-7 a B)}{35 a^2 x^5}-\frac{A \sqrt{a+b x^2}}{7 a x^7} \]

[Out]

-(A*Sqrt[a + b*x^2])/(7*a*x^7) + ((6*A*b - 7*a*B)*Sqrt[a + b*x^2])/(35*a^2*x^5)
- (4*b*(6*A*b - 7*a*B)*Sqrt[a + b*x^2])/(105*a^3*x^3) + (8*b^2*(6*A*b - 7*a*B)*S
qrt[a + b*x^2])/(105*a^4*x)

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Rubi [A]  time = 0.158491, 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 \[ \frac{8 b^2 \sqrt{a+b x^2} (6 A b-7 a B)}{105 a^4 x}-\frac{4 b \sqrt{a+b x^2} (6 A b-7 a B)}{105 a^3 x^3}+\frac{\sqrt{a+b x^2} (6 A b-7 a B)}{35 a^2 x^5}-\frac{A \sqrt{a+b x^2}}{7 a x^7} \]

Antiderivative was successfully verified.

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

[Out]

-(A*Sqrt[a + b*x^2])/(7*a*x^7) + ((6*A*b - 7*a*B)*Sqrt[a + b*x^2])/(35*a^2*x^5)
- (4*b*(6*A*b - 7*a*B)*Sqrt[a + b*x^2])/(105*a^3*x^3) + (8*b^2*(6*A*b - 7*a*B)*S
qrt[a + b*x^2])/(105*a^4*x)

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Rubi in Sympy [A]  time = 16.9291, size = 110, normalized size = 0.94 \[ - \frac{A \sqrt{a + b x^{2}}}{7 a x^{7}} + \frac{\sqrt{a + b x^{2}} \left (6 A b - 7 B a\right )}{35 a^{2} x^{5}} - \frac{4 b \sqrt{a + b x^{2}} \left (6 A b - 7 B a\right )}{105 a^{3} x^{3}} + \frac{8 b^{2} \sqrt{a + b x^{2}} \left (6 A b - 7 B a\right )}{105 a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**8/(b*x**2+a)**(1/2),x)

[Out]

-A*sqrt(a + b*x**2)/(7*a*x**7) + sqrt(a + b*x**2)*(6*A*b - 7*B*a)/(35*a**2*x**5)
- 4*b*sqrt(a + b*x**2)*(6*A*b - 7*B*a)/(105*a**3*x**3) + 8*b**2*sqrt(a + b*x**2
)*(6*A*b - 7*B*a)/(105*a**4*x)

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Mathematica [A]  time = 0.0768327, size = 85, normalized size = 0.73 \[ \sqrt{a+b x^2} \left (-\frac{8 b^2 (7 a B-6 A b)}{105 a^4 x}+\frac{4 b (7 a B-6 A b)}{105 a^3 x^3}+\frac{6 A b-7 a B}{35 a^2 x^5}-\frac{A}{7 a x^7}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-A/(7*a*x^7) + (6*A*b - 7*a*B)/(35*a^2*x^5) + (4*b*(-6*A*b + 7*a*B))/(105*a^3*x
^3) - (8*b^2*(-6*A*b + 7*a*B))/(105*a^4*x))*Sqrt[a + b*x^2]

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Maple [A]  time = 0.008, size = 83, normalized size = 0.7 \[ -{\frac{-48\,A{b}^{3}{x}^{6}+56\,Ba{b}^{2}{x}^{6}+24\,Aa{b}^{2}{x}^{4}-28\,B{a}^{2}b{x}^{4}-18\,A{a}^{2}b{x}^{2}+21\,B{a}^{3}{x}^{2}+15\,A{a}^{3}}{105\,{x}^{7}{a}^{4}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^8/(b*x^2+a)^(1/2),x)

[Out]

-1/105*(b*x^2+a)^(1/2)*(-48*A*b^3*x^6+56*B*a*b^2*x^6+24*A*a*b^2*x^4-28*B*a^2*b*x
^4-18*A*a^2*b*x^2+21*B*a^3*x^2+15*A*a^3)/x^7/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^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268216, size = 111, normalized size = 0.95 \[ -\frac{{\left (8 \,{\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} x^{6} - 4 \,{\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 3 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, a^{4} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*x^8),x, algorithm="fricas")

[Out]

-1/105*(8*(7*B*a*b^2 - 6*A*b^3)*x^6 - 4*(7*B*a^2*b - 6*A*a*b^2)*x^4 + 15*A*a^3 +
3*(7*B*a^3 - 6*A*a^2*b)*x^2)*sqrt(b*x^2 + a)/(a^4*x^7)

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Sympy [A]  time = 10.6679, size = 819, normalized size = 7. \[ - \frac{5 A a^{6} b^{\frac{19}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac{9 A a^{5} b^{\frac{21}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac{5 A a^{4} b^{\frac{23}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac{5 A a^{3} b^{\frac{25}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac{30 A a^{2} b^{\frac{27}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac{40 A a b^{\frac{29}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} + \frac{16 A b^{\frac{31}{2}} x^{12} \sqrt{\frac{a}{b x^{2}} + 1}}{35 a^{7} b^{9} x^{6} + 105 a^{6} b^{10} x^{8} + 105 a^{5} b^{11} x^{10} + 35 a^{4} b^{12} x^{12}} - \frac{3 B a^{4} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{2 B a^{3} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{3 B a^{2} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{12 B a b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac{8 B b^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**8/(b*x**2+a)**(1/2),x)

[Out]

-5*A*a**6*b**(19/2)*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**
8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 9*A*a**5*b**(21/2)*x**2*sqrt(a
/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 +
35*a**4*b**12*x**12) - 5*A*a**4*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(35*a**7*b*
*9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) + 5*
A*a**3*b**(25/2)*x**6*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x
**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) + 30*A*a**2*b**(27/2)*x**8*sqr
t(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**1
0 + 35*a**4*b**12*x**12) + 40*A*a*b**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(35*a**7*
b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) +
16*A*b**(31/2)*x**12*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x*
*8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 3*B*a**4*b**(9/2)*sqrt(a/(b*x
**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 2*B*a**3
*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15
*a**3*b**6*x**8) - 3*B*a**2*b**(13/2)*x**4*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x*
*4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 12*B*a*b**(15/2)*x**6*sqrt(a/(b*x*
*2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 8*B*b**(1
7/2)*x**8*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*
b**6*x**8)

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GIAC/XCAS [A]  time = 0.247798, size = 313, normalized size = 2.68 \[ \frac{16 \,{\left (70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B b^{\frac{5}{2}} - 175 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a b^{\frac{5}{2}} + 210 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A b^{\frac{7}{2}} + 147 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{2} b^{\frac{5}{2}} - 126 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a b^{\frac{7}{2}} - 49 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{3} b^{\frac{5}{2}} + 42 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{2} b^{\frac{7}{2}} + 7 \, B a^{4} b^{\frac{5}{2}} - 6 \, A a^{3} b^{\frac{7}{2}}\right )}}{105 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*x^8),x, algorithm="giac")

[Out]

16/105*(70*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*b^(5/2) - 175*(sqrt(b)*x - sqrt(b*x
^2 + a))^6*B*a*b^(5/2) + 210*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*b^(7/2) + 147*(sq
rt(b)*x - sqrt(b*x^2 + a))^4*B*a^2*b^(5/2) - 126*(sqrt(b)*x - sqrt(b*x^2 + a))^4
*A*a*b^(7/2) - 49*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^3*b^(5/2) + 42*(sqrt(b)*x
- sqrt(b*x^2 + a))^2*A*a^2*b^(7/2) + 7*B*a^4*b^(5/2) - 6*A*a^3*b^(7/2))/((sqrt(b
)*x - sqrt(b*x^2 + a))^2 - a)^7