3.531 \(\int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^5} \, dx\)

Optimal. Leaf size=115 \[ -\frac{\left (a+b x^2\right )^{3/2} (4 a B+A b)}{8 a x^2}+\frac{3 b \sqrt{a+b x^2} (4 a B+A b)}{8 a}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{A \left (a+b x^2\right )^{5/2}}{4 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.23321, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\left (a+b x^2\right )^{3/2} (4 a B+A b)}{8 a x^2}+\frac{3 b \sqrt{a+b x^2} (4 a B+A b)}{8 a}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{A \left (a+b x^2\right )^{5/2}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 19.2761, size = 104, normalized size = 0.9 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{4 a x^{4}} + \frac{3 b \sqrt{a + b x^{2}} \left (A b + 4 B a\right )}{8 a} - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (A b + 4 B a\right )}{8 a x^{2}} - \frac{3 b \left (A b + 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.222913, size = 100, normalized size = 0.87 \[ -\frac{3 b (4 a B+A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{8 \sqrt{a}}+\sqrt{a+b x^2} \left (\frac{-4 a B-5 A b}{8 x^2}-\frac{a A}{4 x^4}+b B\right )+\frac{3 b \log (x) (4 a B+A b)}{8 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^5,x]

[Out]

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

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Maple [A]  time = 0.014, size = 184, normalized size = 1.6 \[ -{\frac{A}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Ab}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}A}{8\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}A}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{3\,{b}^{2}A}{8\,a}\sqrt{b{x}^{2}+a}}-{\frac{B}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bb}{2\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,Bb}{2}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{3\,Bb}{2}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)*(B*x^2+A)/x^5,x)

[Out]

-1/4*A*(b*x^2+a)^(5/2)/a/x^4-1/8*A*b/a^2/x^2*(b*x^2+a)^(5/2)+1/8*A*b^2/a^2*(b*x^
2+a)^(3/2)-3/8*A*b^2/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+3/8*A*b^2/a*(
b*x^2+a)^(1/2)-1/2*B/a/x^2*(b*x^2+a)^(5/2)+1/2*B*b/a*(b*x^2+a)^(3/2)-3/2*B*b*a^(
1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+3/2*B*b*(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)*(b*x^2 + a)^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22874, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, B a b + A b^{2}\right )} x^{4} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (8 \, B b x^{4} -{\left (4 \, B a + 5 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{a}}{16 \, \sqrt{a} x^{4}}, -\frac{3 \,{\left (4 \, B a b + A b^{2}\right )} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (8 \, B b x^{4} -{\left (4 \, B a + 5 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{8 \, \sqrt{-a} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^5,x, algorithm="fricas")

[Out]

[1/16*(3*(4*B*a*b + A*b^2)*x^4*log(-((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*x^2 + a)*a
)/x^2) + 2*(8*B*b*x^4 - (4*B*a + 5*A*b)*x^2 - 2*A*a)*sqrt(b*x^2 + a)*sqrt(a))/(s
qrt(a)*x^4), -1/8*(3*(4*B*a*b + A*b^2)*x^4*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (8
*B*b*x^4 - (4*B*a + 5*A*b)*x^2 - 2*A*a)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*x^4)
]

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Sympy [A]  time = 89.0397, size = 216, normalized size = 1.88 \[ - \frac{A a^{2}}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 A a \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} - \frac{A b^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 A b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 \sqrt{a}} - \frac{3 B \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2} - \frac{B a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} + \frac{B a \sqrt{b}}{x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{\frac{3}{2}} x}{\sqrt{\frac{a}{b x^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**5,x)

[Out]

-A*a**2/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*A*a*sqrt(b)/(8*x**3*sqrt(a/(b*
x**2) + 1)) - A*b**(3/2)*sqrt(a/(b*x**2) + 1)/(2*x) - A*b**(3/2)/(8*x*sqrt(a/(b*
x**2) + 1)) - 3*A*b**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*sqrt(a)) - 3*B*sqrt(a)*b*as
inh(sqrt(a)/(sqrt(b)*x))/2 - B*a*sqrt(b)*sqrt(a/(b*x**2) + 1)/(2*x) + B*a*sqrt(b
)/(x*sqrt(a/(b*x**2) + 1)) + B*b**(3/2)*x/sqrt(a/(b*x**2) + 1)

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GIAC/XCAS [A]  time = 0.230297, size = 177, normalized size = 1.54 \[ \frac{8 \, \sqrt{b x^{2} + a} B b^{2} + \frac{3 \,{\left (4 \, B a b^{2} + A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x^{2} + a} B a^{2} b^{2} + 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{3} - 3 \, \sqrt{b x^{2} + a} A a b^{3}}{b^{2} x^{4}}}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^5,x, algorithm="giac")

[Out]

1/8*(8*sqrt(b*x^2 + a)*B*b^2 + 3*(4*B*a*b^2 + A*b^3)*arctan(sqrt(b*x^2 + a)/sqrt
(-a))/sqrt(-a) - (4*(b*x^2 + a)^(3/2)*B*a*b^2 - 4*sqrt(b*x^2 + a)*B*a^2*b^2 + 5*
(b*x^2 + a)^(3/2)*A*b^3 - 3*sqrt(b*x^2 + a)*A*a*b^3)/(b^2*x^4))/b