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

Optimal. Leaf size=84 \[ -\frac{2 b \left (a+b x^2\right )^{3/2} (4 A b-7 a B)}{105 a^3 x^3}+\frac{\left (a+b x^2\right )^{3/2} (4 A b-7 a B)}{35 a^2 x^5}-\frac{A \left (a+b x^2\right )^{3/2}}{7 a x^7} \]

[Out]

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

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Rubi [A]  time = 0.118273, 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 \[ -\frac{2 b \left (a+b x^2\right )^{3/2} (4 A b-7 a B)}{105 a^3 x^3}+\frac{\left (a+b x^2\right )^{3/2} (4 A b-7 a B)}{35 a^2 x^5}-\frac{A \left (a+b x^2\right )^{3/2}}{7 a x^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.4272, size = 78, normalized size = 0.93 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{7 a x^{7}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (4 A b - 7 B a\right )}{35 a^{2} x^{5}} - \frac{2 b \left (a + b x^{2}\right )^{\frac{3}{2}} \left (4 A b - 7 B a\right )}{105 a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0691115, size = 63, normalized size = 0.75 \[ \frac{\left (a+b x^2\right )^{3/2} \left (-3 a^2 \left (5 A+7 B x^2\right )+2 a b x^2 \left (6 A+7 B x^2\right )-8 A b^2 x^4\right )}{105 a^3 x^7} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 59, normalized size = 0.7 \[ -{\frac{8\,A{b}^{2}{x}^{4}-14\,Bab{x}^{4}-12\,aAb{x}^{2}+21\,B{a}^{2}{x}^{2}+15\,A{a}^{2}}{105\,{x}^{7}{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(b*x^2+a)^(3/2)*(8*A*b^2*x^4-14*B*a*b*x^4-12*A*a*b*x^2+21*B*a^2*x^2+15*A*
a^2)/x^7/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)*sqrt(b*x^2 + a)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249552, size = 109, normalized size = 1.3 \[ \frac{{\left (2 \,{\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} x^{6} -{\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{4} - 15 \, A a^{3} - 3 \,{\left (7 \, B a^{3} + A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, a^{3} x^{7}} \]

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

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Sympy [A]  time = 8.58804, size = 442, normalized size = 5.26 \[ - \frac{15 A a^{5} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{33 A a^{4} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{17 A a^{3} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{3 A a^{2} b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{12 A a b^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{8 A b^{\frac{19}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{B b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a x^{2}} + \frac{2 B b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*A*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*A*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*A*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*A*a**2*b**(15/2)*x**6*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 21
0*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 12*A*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*A*b**(1
9/2)*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*sqrt(b)*sqrt(a/(b*x**2) + 1)/(5*x**4) - B*b**(3/2)*sqrt(a/(b
*x**2) + 1)/(15*a*x**2) + 2*B*b**(5/2)*sqrt(a/(b*x**2) + 1)/(15*a**2)

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GIAC/XCAS [A]  time = 0.241779, size = 389, normalized size = 4.63 \[ \frac{4 \,{\left (105 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} B b^{\frac{5}{2}} - 175 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a b^{\frac{5}{2}} + 280 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A b^{\frac{7}{2}} + 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{2} b^{\frac{5}{2}} + 140 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a b^{\frac{7}{2}} - 42 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{3} b^{\frac{5}{2}} + 84 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{2} b^{\frac{7}{2}} + 49 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{4} b^{\frac{5}{2}} - 28 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{3} b^{\frac{7}{2}} - 7 \, B a^{5} b^{\frac{5}{2}} + 4 \, A a^{4} b^{\frac{7}{2}}\right )}}{105 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7}} \]

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]

4/105*(105*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*b^(5/2) - 175*(sqrt(b)*x - sqrt(b*
x^2 + a))^8*B*a*b^(5/2) + 280*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*b^(7/2) + 70*(sq
rt(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(5/2) + 140*(sqrt(b)*x - sqrt(b*x^2 + a))^6
*A*a*b^(7/2) - 42*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*b^(5/2) + 84*(sqrt(b)*x
- sqrt(b*x^2 + a))^4*A*a^2*b^(7/2) + 49*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^
(5/2) - 28*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^3*b^(7/2) - 7*B*a^5*b^(5/2) + 4*A
*a^4*b^(7/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7