3.518 \(\int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^9} \, dx\)

Optimal. Leaf size=156 \[ \frac{b^3 (5 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{7/2}}-\frac{b^2 \sqrt{a+b x^2} (5 A b-8 a B)}{128 a^3 x^2}+\frac{b \sqrt{a+b x^2} (5 A b-8 a B)}{192 a^2 x^4}+\frac{\sqrt{a+b x^2} (5 A b-8 a B)}{48 a x^6}-\frac{A \left (a+b x^2\right )^{3/2}}{8 a x^8} \]

[Out]

((5*A*b - 8*a*B)*Sqrt[a + b*x^2])/(48*a*x^6) + (b*(5*A*b - 8*a*B)*Sqrt[a + b*x^2
])/(192*a^2*x^4) - (b^2*(5*A*b - 8*a*B)*Sqrt[a + b*x^2])/(128*a^3*x^2) - (A*(a +
b*x^2)^(3/2))/(8*a*x^8) + (b^3*(5*A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]
)/(128*a^(7/2))

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Rubi [A]  time = 0.314677, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{b^3 (5 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{128 a^{7/2}}-\frac{b^2 \sqrt{a+b x^2} (5 A b-8 a B)}{128 a^3 x^2}+\frac{b \sqrt{a+b x^2} (5 A b-8 a B)}{192 a^2 x^4}+\frac{\sqrt{a+b x^2} (5 A b-8 a B)}{48 a x^6}-\frac{A \left (a+b x^2\right )^{3/2}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^9,x]

[Out]

((5*A*b - 8*a*B)*Sqrt[a + b*x^2])/(48*a*x^6) + (b*(5*A*b - 8*a*B)*Sqrt[a + b*x^2
])/(192*a^2*x^4) - (b^2*(5*A*b - 8*a*B)*Sqrt[a + b*x^2])/(128*a^3*x^2) - (A*(a +
b*x^2)^(3/2))/(8*a*x^8) + (b^3*(5*A*b - 8*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]
)/(128*a^(7/2))

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Rubi in Sympy [A]  time = 25.992, size = 144, normalized size = 0.92 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{8 a x^{8}} + \frac{\sqrt{a + b x^{2}} \left (5 A b - 8 B a\right )}{48 a x^{6}} + \frac{b \sqrt{a + b x^{2}} \left (5 A b - 8 B a\right )}{192 a^{2} x^{4}} - \frac{b^{2} \sqrt{a + b x^{2}} \left (5 A b - 8 B a\right )}{128 a^{3} x^{2}} + \frac{b^{3} \left (5 A b - 8 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{128 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**9,x)

[Out]

-A*(a + b*x**2)**(3/2)/(8*a*x**8) + sqrt(a + b*x**2)*(5*A*b - 8*B*a)/(48*a*x**6)
+ b*sqrt(a + b*x**2)*(5*A*b - 8*B*a)/(192*a**2*x**4) - b**2*sqrt(a + b*x**2)*(5
*A*b - 8*B*a)/(128*a**3*x**2) + b**3*(5*A*b - 8*B*a)*atanh(sqrt(a + b*x**2)/sqrt
(a))/(128*a**(7/2))

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Mathematica [A]  time = 0.192892, size = 147, normalized size = 0.94 \[ \frac{b^3 (5 A b-8 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{128 a^{7/2}}-\frac{b^3 \log (x) (5 A b-8 a B)}{128 a^{7/2}}+\sqrt{a+b x^2} \left (\frac{b^2 (8 a B-5 A b)}{128 a^3 x^2}-\frac{b (8 a B-5 A b)}{192 a^2 x^4}+\frac{-8 a B-A b}{48 a x^6}-\frac{A}{8 x^8}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^9,x]

[Out]

(-A/(8*x^8) + (-(A*b) - 8*a*B)/(48*a*x^6) - (b*(-5*A*b + 8*a*B))/(192*a^2*x^4) +
(b^2*(-5*A*b + 8*a*B))/(128*a^3*x^2))*Sqrt[a + b*x^2] - (b^3*(5*A*b - 8*a*B)*Lo
g[x])/(128*a^(7/2)) + (b^3*(5*A*b - 8*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/(12
8*a^(7/2))

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Maple [A]  time = 0.024, size = 239, normalized size = 1.5 \[ -{\frac{A}{8\,a{x}^{8}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Ab}{48\,{a}^{2}{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{2}A}{64\,{a}^{3}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{b}^{3}}{128\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{5\,A{b}^{4}}{128\,{a}^{4}}\sqrt{b{x}^{2}+a}}-{\frac{B}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bb}{8\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{B{b}^{2}}{16\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{B{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{B{b}^{3}}{16\,{a}^{3}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(b*x^2+a)^(1/2)/x^9,x)

[Out]

-1/8*A*(b*x^2+a)^(3/2)/a/x^8+5/48*A*b/a^2/x^6*(b*x^2+a)^(3/2)-5/64*A*b^2/a^3/x^4
*(b*x^2+a)^(3/2)+5/128*A*b^3/a^4/x^2*(b*x^2+a)^(3/2)+5/128*A*b^4/a^(7/2)*ln((2*a
+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-5/128*A*b^4/a^4*(b*x^2+a)^(1/2)-1/6*B/a/x^6*(b*x^
2+a)^(3/2)+1/8*B*b/a^2/x^4*(b*x^2+a)^(3/2)-1/16*B*b^2/a^3/x^2*(b*x^2+a)^(3/2)-1/
16*B*b^3/a^(5/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+1/16*B*b^3/a^3*(b*x^2+a)^
(1/2)

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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^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289363, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} x^{8} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (3 \,{\left (8 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} - 2 \,{\left (8 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} - 48 \, A a^{3} - 8 \,{\left (8 \, B a^{3} + A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{768 \, a^{\frac{7}{2}} x^{8}}, -\frac{3 \,{\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} x^{8} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \,{\left (8 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} - 2 \,{\left (8 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} - 48 \, A a^{3} - 8 \,{\left (8 \, B a^{3} + A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{384 \, \sqrt{-a} a^{3} x^{8}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^9,x, algorithm="fricas")

[Out]

[-1/768*(3*(8*B*a*b^3 - 5*A*b^4)*x^8*log(-((b*x^2 + 2*a)*sqrt(a) + 2*sqrt(b*x^2
+ a)*a)/x^2) - 2*(3*(8*B*a*b^2 - 5*A*b^3)*x^6 - 2*(8*B*a^2*b - 5*A*a*b^2)*x^4 -
48*A*a^3 - 8*(8*B*a^3 + A*a^2*b)*x^2)*sqrt(b*x^2 + a)*sqrt(a))/(a^(7/2)*x^8), -1
/384*(3*(8*B*a*b^3 - 5*A*b^4)*x^8*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (3*(8*B*a*b
^2 - 5*A*b^3)*x^6 - 2*(8*B*a^2*b - 5*A*a*b^2)*x^4 - 48*A*a^3 - 8*(8*B*a^3 + A*a^
2*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^3*x^8)]

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Sympy [A]  time = 178.175, size = 286, normalized size = 1.83 \[ - \frac{A a}{8 \sqrt{b} x^{9} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{7 A \sqrt{b}}{48 x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{3}{2}}}{192 a x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 A b^{\frac{5}{2}}}{384 a^{2} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 A b^{\frac{7}{2}}}{128 a^{3} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{5 A b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{128 a^{\frac{7}{2}}} - \frac{B a}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 B \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{\frac{3}{2}}}{48 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{\frac{5}{2}}}{16 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{B b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**9,x)

[Out]

-A*a/(8*sqrt(b)*x**9*sqrt(a/(b*x**2) + 1)) - 7*A*sqrt(b)/(48*x**7*sqrt(a/(b*x**2
) + 1)) + A*b**(3/2)/(192*a*x**5*sqrt(a/(b*x**2) + 1)) - 5*A*b**(5/2)/(384*a**2*
x**3*sqrt(a/(b*x**2) + 1)) - 5*A*b**(7/2)/(128*a**3*x*sqrt(a/(b*x**2) + 1)) + 5*
A*b**4*asinh(sqrt(a)/(sqrt(b)*x))/(128*a**(7/2)) - B*a/(6*sqrt(b)*x**7*sqrt(a/(b
*x**2) + 1)) - 5*B*sqrt(b)/(24*x**5*sqrt(a/(b*x**2) + 1)) + B*b**(3/2)/(48*a*x**
3*sqrt(a/(b*x**2) + 1)) + B*b**(5/2)/(16*a**2*x*sqrt(a/(b*x**2) + 1)) - B*b**3*a
sinh(sqrt(a)/(sqrt(b)*x))/(16*a**(5/2))

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GIAC/XCAS [A]  time = 0.235266, size = 262, normalized size = 1.68 \[ \frac{\frac{3 \,{\left (8 \, B a b^{4} - 5 \, A b^{5}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{24 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B a b^{4} - 88 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a^{2} b^{4} + 40 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{3} b^{4} + 24 \, \sqrt{b x^{2} + a} B a^{4} b^{4} - 15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} A b^{5} + 55 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A a b^{5} - 73 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{2} b^{5} - 15 \, \sqrt{b x^{2} + a} A a^{3} b^{5}}{a^{3} b^{4} x^{8}}}{384 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^9,x, algorithm="giac")

[Out]

1/384*(3*(8*B*a*b^4 - 5*A*b^5)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^3) +
(24*(b*x^2 + a)^(7/2)*B*a*b^4 - 88*(b*x^2 + a)^(5/2)*B*a^2*b^4 + 40*(b*x^2 + a)
^(3/2)*B*a^3*b^4 + 24*sqrt(b*x^2 + a)*B*a^4*b^4 - 15*(b*x^2 + a)^(7/2)*A*b^5 + 5
5*(b*x^2 + a)^(5/2)*A*a*b^5 - 73*(b*x^2 + a)^(3/2)*A*a^2*b^5 - 15*sqrt(b*x^2 + a
)*A*a^3*b^5)/(a^3*b^4*x^8))/b