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

Optimal. Leaf size=88 \[ -\frac{a^2 (A b-a B)}{4 b^4 \left (a+b x^2\right )^2}+\frac{a (2 A b-3 a B)}{2 b^4 \left (a+b x^2\right )}+\frac{(A b-3 a B) \log \left (a+b x^2\right )}{2 b^4}+\frac{B x^2}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.224423, antiderivative size = 88, 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 \[ -\frac{a^2 (A b-a B)}{4 b^4 \left (a+b x^2\right )^2}+\frac{a (2 A b-3 a B)}{2 b^4 \left (a+b x^2\right )}+\frac{(A b-3 a B) \log \left (a+b x^2\right )}{2 b^4}+\frac{B x^2}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \left (A b - B a\right )}{4 b^{4} \left (a + b x^{2}\right )^{2}} + \frac{a \left (2 A b - 3 B a\right )}{2 b^{4} \left (a + b x^{2}\right )} + \frac{\int ^{x^{2}} B\, dx}{2 b^{3}} + \frac{\left (A b - 3 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0621631, size = 92, normalized size = 1.05 \[ \frac{2 a A b-3 a^2 B}{2 b^4 \left (a+b x^2\right )}+\frac{a^3 B-a^2 A b}{4 b^4 \left (a+b x^2\right )^2}+\frac{(A b-3 a B) \log \left (a+b x^2\right )}{2 b^4}+\frac{B x^2}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 109, normalized size = 1.2 \[{\frac{B{x}^{2}}{2\,{b}^{3}}}-{\frac{{a}^{2}A}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{B{a}^{3}}{4\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ) A}{2\,{b}^{3}}}-{\frac{3\,\ln \left ( b{x}^{2}+a \right ) Ba}{2\,{b}^{4}}}+{\frac{aA}{{b}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,{a}^{2}B}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.33848, size = 127, normalized size = 1.44 \[ -\frac{5 \, B a^{3} - 3 \, A a^{2} b + 2 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{2}}{4 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} + \frac{B x^{2}}{2 \, b^{3}} - \frac{{\left (3 \, B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(5*B*a^3 - 3*A*a^2*b + 2*(3*B*a^2*b - 2*A*a*b^2)*x^2)/(b^6*x^4 + 2*a*b^5*x^
2 + a^2*b^4) + 1/2*B*x^2/b^3 - 1/2*(3*B*a - A*b)*log(b*x^2 + a)/b^4

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Fricas [A]  time = 0.223146, size = 192, normalized size = 2.18 \[ \frac{2 \, B b^{3} x^{6} + 4 \, B a b^{2} x^{4} - 5 \, B a^{3} + 3 \, A a^{2} b - 4 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} - 2 \,{\left ({\left (3 \, B a b^{2} - A b^{3}\right )} x^{4} + 3 \, B a^{3} - A a^{2} b + 2 \,{\left (3 \, B a^{2} b - A a b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(2*B*b^3*x^6 + 4*B*a*b^2*x^4 - 5*B*a^3 + 3*A*a^2*b - 4*(B*a^2*b - A*a*b^2)*x
^2 - 2*((3*B*a*b^2 - A*b^3)*x^4 + 3*B*a^3 - A*a^2*b + 2*(3*B*a^2*b - A*a*b^2)*x^
2)*log(b*x^2 + a))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)

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Sympy [A]  time = 5.53614, size = 94, normalized size = 1.07 \[ \frac{B x^{2}}{2 b^{3}} - \frac{- 3 A a^{2} b + 5 B a^{3} + x^{2} \left (- 4 A a b^{2} + 6 B a^{2} b\right )}{4 a^{2} b^{4} + 8 a b^{5} x^{2} + 4 b^{6} x^{4}} - \frac{\left (- A b + 3 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**2/(2*b**3) - (-3*A*a**2*b + 5*B*a**3 + x**2*(-4*A*a*b**2 + 6*B*a**2*b))/(4*
a**2*b**4 + 8*a*b**5*x**2 + 4*b**6*x**4) - (-A*b + 3*B*a)*log(a + b*x**2)/(2*b**
4)

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GIAC/XCAS [A]  time = 0.226637, size = 126, normalized size = 1.43 \[ \frac{B x^{2}}{2 \, b^{3}} - \frac{{\left (3 \, B a - A b\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac{9 \, B a b^{2} x^{4} - 3 \, A b^{3} x^{4} + 12 \, B a^{2} b x^{2} - 2 \, A a b^{2} x^{2} + 4 \, B a^{3}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*B*x^2/b^3 - 1/2*(3*B*a - A*b)*ln(abs(b*x^2 + a))/b^4 + 1/4*(9*B*a*b^2*x^4 -
3*A*b^3*x^4 + 12*B*a^2*b*x^2 - 2*A*a*b^2*x^2 + 4*B*a^3)/((b*x^2 + a)^2*b^4)