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

Optimal. Leaf size=54 \[ -\frac{a (A b-a B) \log \left (a+b x^2\right )}{2 b^3}+\frac{x^2 (A b-a B)}{2 b^2}+\frac{B x^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.133595, antiderivative size = 54, 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 \[ -\frac{a (A b-a B) \log \left (a+b x^2\right )}{2 b^3}+\frac{x^2 (A b-a B)}{2 b^2}+\frac{B x^4}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B \int ^{x^{2}} x\, dx}{2 b} - \frac{a \left (A b - B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} + \left (\frac{A b}{2} - \frac{B a}{2}\right ) \int ^{x^{2}} \frac{1}{b^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0311052, size = 47, normalized size = 0.87 \[ \frac{b x^2 \left (-2 a B+2 A b+b B x^2\right )+2 a (a B-A b) \log \left (a+b x^2\right )}{4 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 62, normalized size = 1.2 \[{\frac{B{x}^{4}}{4\,b}}+{\frac{A{x}^{2}}{2\,b}}-{\frac{B{x}^{2}a}{2\,{b}^{2}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ) A}{2\,{b}^{2}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) B}{2\,{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x^2+A)/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.35021, size = 68, normalized size = 1.26 \[ \frac{B b x^{4} - 2 \,{\left (B a - A b\right )} x^{2}}{4 \, b^{2}} + \frac{{\left (B a^{2} - A a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22928, size = 69, normalized size = 1.28 \[ \frac{B b^{2} x^{4} - 2 \,{\left (B a b - A b^{2}\right )} x^{2} + 2 \,{\left (B a^{2} - A a b\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.69101, size = 44, normalized size = 0.81 \[ \frac{B x^{4}}{4 b} + \frac{a \left (- A b + B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} - \frac{x^{2} \left (- A b + B a\right )}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229871, size = 70, normalized size = 1.3 \[ \frac{B b x^{4} - 2 \, B a x^{2} + 2 \, A b x^{2}}{4 \, b^{2}} + \frac{{\left (B a^{2} - A a b\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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