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

Optimal. Leaf size=77 \[ \frac{x^3 (A b-a B)}{3 a b \left (a+b x^2\right )^{3/2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{5/2}}-\frac{B x}{b^2 \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.107159, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x^3 (A b-a B)}{3 a b \left (a+b x^2\right )^{3/2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{5/2}}-\frac{B x}{b^2 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.7865, size = 66, normalized size = 0.86 \[ - \frac{B x}{b^{2} \sqrt{a + b x^{2}}} + \frac{B \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{5}{2}}} + \frac{x^{3} \left (A b - B a\right )}{3 a b \left (a + b x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.147081, size = 75, normalized size = 0.97 \[ \frac{-3 a^2 B x-4 a b B x^3+A b^2 x^3}{3 a b^2 \left (a+b x^2\right )^{3/2}}+\frac{B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 92, normalized size = 1.2 \[ -{\frac{Ax}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{Ax}{3\,ab}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{{x}^{3}B}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Bx}{{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{B\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238117, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \, B a^{2} x +{\left (4 \, B a b - A b^{2}\right )} x^{3}\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (B a b^{2} x^{4} + 2 \, B a^{2} b x^{2} + B a^{3}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{6 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \sqrt{b}}, -\frac{{\left (3 \, B a^{2} x +{\left (4 \, B a b - A b^{2}\right )} x^{3}\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 3 \,{\left (B a b^{2} x^{4} + 2 \, B a^{2} b x^{2} + B a^{3}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{3 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(2*(3*B*a^2*x + (4*B*a*b - A*b^2)*x^3)*sqrt(b*x^2 + a)*sqrt(b) - 3*(B*a*b^
2*x^4 + 2*B*a^2*b*x^2 + B*a^3)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b
)))/((a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*sqrt(b)), -1/3*((3*B*a^2*x + (4*B*a*b
- A*b^2)*x^3)*sqrt(b*x^2 + a)*sqrt(-b) - 3*(B*a*b^2*x^4 + 2*B*a^2*b*x^2 + B*a^3
)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/((a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*sqr
t(-b))]

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Sympy [A]  time = 44.1447, size = 352, normalized size = 4.57 \[ \frac{A x^{3}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{3}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + B \left (\frac{3 a^{\frac{39}{2}} b^{11} \sqrt{1 + \frac{b x^{2}}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{\frac{37}{2}} b^{12} x^{2} \sqrt{1 + \frac{b x^{2}}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{19} b^{\frac{23}{2}} x}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{4 a^{18} b^{\frac{25}{2}} x^{3}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*x**3/(3*a**(5/2)*sqrt(1 + b*x**2/a) + 3*a**(3/2)*b*x**2*sqrt(1 + b*x**2/a)) +
B*(3*a**(39/2)*b**11*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(3*a**(39/2)*b*
*(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)) + 3*
a**(37/2)*b**12*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(3*a**(39/2)*b*
*(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)) - 3*
a**19*b**(23/2)*x/(3*a**(39/2)*b**(27/2)*sqrt(1 + b*x**2/a) + 3*a**(37/2)*b**(29
/2)*x**2*sqrt(1 + b*x**2/a)) - 4*a**18*b**(25/2)*x**3/(3*a**(39/2)*b**(27/2)*sqr
t(1 + b*x**2/a) + 3*a**(37/2)*b**(29/2)*x**2*sqrt(1 + b*x**2/a)))

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GIAC/XCAS [A]  time = 0.236512, size = 93, normalized size = 1.21 \[ -\frac{x{\left (\frac{3 \, B a}{b^{2}} + \frac{{\left (4 \, B a b^{2} - A b^{3}\right )} x^{2}}{a b^{3}}\right )}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} - \frac{B{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*x*(3*B*a/b^2 + (4*B*a*b^2 - A*b^3)*x^2/(a*b^3))/(b*x^2 + a)^(3/2) - B*ln(ab
s(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)