3.532 \(\int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^6} \, dx\)

Optimal. Leaf size=86 \[ -\frac{A \left (a+b x^2\right )^{5/2}}{5 a x^5}+b^{3/2} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b B \sqrt{a+b x^2}}{x}-\frac{B \left (a+b x^2\right )^{3/2}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.110976, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{A \left (a+b x^2\right )^{5/2}}{5 a x^5}+b^{3/2} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b B \sqrt{a+b x^2}}{x}-\frac{B \left (a+b x^2\right )^{3/2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.8667, size = 75, normalized size = 0.87 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 a x^{5}} + B b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )} - \frac{B b \sqrt{a + b x^{2}}}{x} - \frac{B \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.10902, size = 88, normalized size = 1.02 \[ \sqrt{a+b x^2} \left (\frac{-5 a B-6 A b}{15 x^3}-\frac{b (20 a B+3 A b)}{15 a x}-\frac{a A}{5 x^5}\right )+b^{3/2} B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^6,x]

[Out]

(-(a*A)/(5*x^5) + (-6*A*b - 5*a*B)/(15*x^3) - (b*(3*A*b + 20*a*B))/(15*a*x))*Sqr
t[a + b*x^2] + b^(3/2)*B*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]]

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Maple [A]  time = 0.015, size = 115, normalized size = 1.3 \[ -{\frac{A}{5\,a{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,Bb}{3\,{a}^{2}x} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{b}^{2}Bx}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}Bx}{a}\sqrt{b{x}^{2}+a}}+B{b}^{{\frac{3}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)*(B*x^2+A)/x^6,x)

[Out]

-1/5*A*(b*x^2+a)^(5/2)/a/x^5-1/3*B/a/x^3*(b*x^2+a)^(5/2)-2/3*B*b/a^2/x*(b*x^2+a)
^(5/2)+2/3*B*b^2/a^2*x*(b*x^2+a)^(3/2)+B*b^2/a*x*(b*x^2+a)^(1/2)+B*b^(3/2)*ln(x*
b^(1/2)+(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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231343, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, B a b^{\frac{3}{2}} x^{5} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left ({\left (20 \, B a b + 3 \, A b^{2}\right )} x^{4} + 3 \, A a^{2} +{\left (5 \, B a^{2} + 6 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{30 \, a x^{5}}, \frac{15 \, B a \sqrt{-b} b x^{5} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) -{\left ({\left (20 \, B a b + 3 \, A b^{2}\right )} x^{4} + 3 \, A a^{2} +{\left (5 \, B a^{2} + 6 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \, a x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^6,x, algorithm="fricas")

[Out]

[1/30*(15*B*a*b^(3/2)*x^5*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*((
20*B*a*b + 3*A*b^2)*x^4 + 3*A*a^2 + (5*B*a^2 + 6*A*a*b)*x^2)*sqrt(b*x^2 + a))/(a
*x^5), 1/15*(15*B*a*sqrt(-b)*b*x^5*arctan(b*x/(sqrt(b*x^2 + a)*sqrt(-b))) - ((20
*B*a*b + 3*A*b^2)*x^4 + 3*A*a^2 + (5*B*a^2 + 6*A*a*b)*x^2)*sqrt(b*x^2 + a))/(a*x
^5)]

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Sympy [A]  time = 12.7144, size = 184, normalized size = 2.14 \[ - \frac{A a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{2 A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{2}} - \frac{A b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{5 a} - \frac{B \sqrt{a} b}{x \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{B b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3} + B b^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )} - \frac{B b^{2} x}{\sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**6,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.250725, size = 319, normalized size = 3.71 \[ -\frac{1}{2} \, B b^{\frac{3}{2}}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a b^{\frac{3}{2}} + 15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A b^{\frac{5}{2}} - 90 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{2} b^{\frac{3}{2}} + 110 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{3} b^{\frac{3}{2}} + 30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{2} b^{\frac{5}{2}} - 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{4} b^{\frac{3}{2}} + 20 \, B a^{5} b^{\frac{3}{2}} + 3 \, A a^{4} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^6,x, algorithm="giac")

[Out]

-1/2*B*b^(3/2)*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/15*(30*(sqrt(b)*x - sqrt(
b*x^2 + a))^8*B*a*b^(3/2) + 15*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*b^(5/2) - 90*(s
qrt(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(3/2) + 110*(sqrt(b)*x - sqrt(b*x^2 + a))^
4*B*a^3*b^(3/2) + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^2*b^(5/2) - 70*(sqrt(b)
*x - sqrt(b*x^2 + a))^2*B*a^4*b^(3/2) + 20*B*a^5*b^(3/2) + 3*A*a^4*b^(5/2))/((sq
rt(b)*x - sqrt(b*x^2 + a))^2 - a)^5