3.200 \(\int \frac{x^3 \left (c+d x^2\right )}{a+b x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.131333, 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 \[ -\frac{a (b c-a d) \log \left (a+b x^2\right )}{2 b^3}+\frac{x^2 (b c-a d)}{2 b^2}+\frac{d x^4}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(d*x**2+c)/(b*x**2+a),x)

[Out]

a*(a*d - b*c)*log(a + b*x**2)/(2*b**3) - (a*d/2 - b*c/2)*Integral(b**(-2), (x, x
**2)) + d*Integral(x, (x, x**2))/(2*b)

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Mathematica [A]  time = 0.0319273, size = 47, normalized size = 0.87 \[ \frac{b x^2 \left (-2 a d+2 b c+b d x^2\right )+2 a (a d-b c) \log \left (a+b x^2\right )}{4 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 62, normalized size = 1.2 \[{\frac{d{x}^{4}}{4\,b}}-{\frac{ad{x}^{2}}{2\,{b}^{2}}}+{\frac{c{x}^{2}}{2\,b}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) d}{2\,{b}^{3}}}-{\frac{ac\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(d*x^2+c)/(b*x^2+a),x)

[Out]

1/4*d*x^4/b-1/2/b^2*a*d*x^2+1/2*c*x^2/b+1/2*a^2/b^3*ln(b*x^2+a)*d-1/2*a*c*ln(b*x
^2+a)/b^2

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Maxima [A]  time = 1.34607, size = 68, normalized size = 1.26 \[ \frac{b d x^{4} + 2 \,{\left (b c - a d\right )} x^{2}}{4 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b*d*x^4 + 2*(b*c - a*d)*x^2)/b^2 - 1/2*(a*b*c - a^2*d)*log(b*x^2 + a)/b^3

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Fricas [A]  time = 0.232231, size = 69, normalized size = 1.28 \[ \frac{b^{2} d x^{4} + 2 \,{\left (b^{2} c - a b d\right )} x^{2} - 2 \,{\left (a b c - a^{2} d\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(b^2*d*x^4 + 2*(b^2*c - a*b*d)*x^2 - 2*(a*b*c - a^2*d)*log(b*x^2 + a))/b^3

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Sympy [A]  time = 1.65353, size = 44, normalized size = 0.81 \[ \frac{a \left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} + \frac{d x^{4}}{4 b} - \frac{x^{2} \left (a d - b c\right )}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(d*x**2+c)/(b*x**2+a),x)

[Out]

a*(a*d - b*c)*log(a + b*x**2)/(2*b**3) + d*x**4/(4*b) - x**2*(a*d - b*c)/(2*b**2
)

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GIAC/XCAS [A]  time = 0.233297, size = 70, normalized size = 1.3 \[ \frac{b d x^{4} + 2 \, b c x^{2} - 2 \, a d x^{2}}{4 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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