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

Optimal. Leaf size=66 \[ -\frac{A b-2 a B}{2 b^3 \left (a+b x^2\right )}+\frac{a (A b-a B)}{4 b^3 \left (a+b x^2\right )^2}+\frac{B \log \left (a+b x^2\right )}{2 b^3} \]

[Out]

(a*(A*b - a*B))/(4*b^3*(a + b*x^2)^2) - (A*b - 2*a*B)/(2*b^3*(a + b*x^2)) + (B*L
og[a + b*x^2])/(2*b^3)

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Rubi [A]  time = 0.167587, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{A b-2 a B}{2 b^3 \left (a+b x^2\right )}+\frac{a (A b-a B)}{4 b^3 \left (a+b x^2\right )^2}+\frac{B \log \left (a+b x^2\right )}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(a*(A*b - a*B))/(4*b^3*(a + b*x^2)^2) - (A*b - 2*a*B)/(2*b^3*(a + b*x^2)) + (B*L
og[a + b*x^2])/(2*b^3)

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Rubi in Sympy [A]  time = 20.8252, size = 56, normalized size = 0.85 \[ \frac{B \log{\left (a + b x^{2} \right )}}{2 b^{3}} + \frac{a \left (A b - B a\right )}{4 b^{3} \left (a + b x^{2}\right )^{2}} - \frac{A b - 2 B a}{2 b^{3} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*log(a + b*x**2)/(2*b**3) + a*(A*b - B*a)/(4*b**3*(a + b*x**2)**2) - (A*b - 2*B
*a)/(2*b**3*(a + b*x**2))

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Mathematica [A]  time = 0.0413053, size = 64, normalized size = 0.97 \[ \frac{3 a^2 B-a b \left (A-4 B x^2\right )+2 B \left (a+b x^2\right )^2 \log \left (a+b x^2\right )-2 A b^2 x^2}{4 b^3 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 80, normalized size = 1.2 \[{\frac{Aa}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{{a}^{2}B}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{B\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{3}}}-{\frac{A}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{Ba}{{b}^{3} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34407, size = 97, normalized size = 1.47 \[ \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x^{2}}{4 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac{B \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218774, size = 120, normalized size = 1.82 \[ \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x^{2} + 2 \,{\left (B b^{2} x^{4} + 2 \, B a b x^{2} + B a^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.21695, size = 70, normalized size = 1.06 \[ \frac{B \log{\left (a + b x^{2} \right )}}{2 b^{3}} + \frac{- A a b + 3 B a^{2} + x^{2} \left (- 2 A b^{2} + 4 B a b\right )}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.233296, size = 82, normalized size = 1.24 \[ \frac{B{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} + \frac{2 \,{\left (2 \, B a - A b\right )} x^{2} + \frac{3 \, B a^{2} - A a b}{b}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*B*ln(abs(b*x^2 + a))/b^3 + 1/4*(2*(2*B*a - A*b)*x^2 + (3*B*a^2 - A*a*b)/b)/(
(b*x^2 + a)^2*b^2)