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

Optimal. Leaf size=61 \[ \frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^2 (b c-a d)}{2 b^2}+\frac{\left (c+d x^2\right )^2}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.114315, antiderivative size = 61, 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 \[ \frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^2 (b c-a d)}{2 b^2}+\frac{\left (c+d x^2\right )^2}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0385013, size = 49, normalized size = 0.8 \[ \frac{b d x^2 \left (-2 a d+4 b c+b d x^2\right )+2 (b c-a d)^2 \log \left (a+b x^2\right )}{4 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 85, normalized size = 1.4 \[{\frac{{d}^{2}{x}^{4}}{4\,b}}-{\frac{a{d}^{2}{x}^{2}}{2\,{b}^{2}}}+{\frac{d{x}^{2}c}{b}}+{\frac{\ln \left ( b{x}^{2}+a \right ){a}^{2}{d}^{2}}{2\,{b}^{3}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) cad}{{b}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{2}}{2\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(d*x^2+c)^2/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.34031, size = 89, normalized size = 1.46 \[ \frac{b d^{2} x^{4} + 2 \,{\left (2 \, b c d - a d^{2}\right )} x^{2}}{4 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.234495, size = 90, normalized size = 1.48 \[ \frac{b^{2} d^{2} x^{4} + 2 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.97639, size = 51, normalized size = 0.84 \[ \frac{d^{2} x^{4}}{4 b} - \frac{x^{2} \left (a d^{2} - 2 b c d\right )}{2 b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.225176, size = 90, normalized size = 1.48 \[ \frac{b d^{2} x^{4} + 4 \, b c d x^{2} - 2 \, a d^{2} x^{2}}{4 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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