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

Optimal. Leaf size=90 \[ \frac{c (b c-a d)^2}{2 d^4 \left (c+d x^2\right )}+\frac{(b c-a d) (3 b c-a d) \log \left (c+d x^2\right )}{2 d^4}-\frac{b x^2 (b c-a d)}{d^3}+\frac{b^2 x^4}{4 d^2} \]

[Out]

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

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Rubi [A]  time = 0.257606, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{c (b c-a d)^2}{2 d^4 \left (c+d x^2\right )}+\frac{(b c-a d) (3 b c-a d) \log \left (c+d x^2\right )}{2 d^4}-\frac{b x^2 (b c-a d)}{d^3}+\frac{b^2 x^4}{4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.10544, size = 87, normalized size = 0.97 \[ \frac{2 \left (a^2 d^2-4 a b c d+3 b^2 c^2\right ) \log \left (c+d x^2\right )+4 b d x^2 (a d-b c)+\frac{2 c (b c-a d)^2}{c+d x^2}+b^2 d^2 x^4}{4 d^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 142, normalized size = 1.6 \[{\frac{{b}^{2}{x}^{4}}{4\,{d}^{2}}}+{\frac{ab{x}^{2}}{{d}^{2}}}-{\frac{{b}^{2}c{x}^{2}}{{d}^{3}}}+{\frac{{a}^{2}c}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{ab{c}^{2}}{{d}^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{b}^{2}{c}^{3}}{2\,{d}^{4} \left ( d{x}^{2}+c \right ) }}+{\frac{\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{d}^{2}}}-2\,{\frac{\ln \left ( d{x}^{2}+c \right ) abc}{{d}^{3}}}+{\frac{3\,\ln \left ( d{x}^{2}+c \right ){b}^{2}{c}^{2}}{2\,{d}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.3477, size = 144, normalized size = 1.6 \[ \frac{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}}{2 \,{\left (d^{5} x^{2} + c d^{4}\right )}} + \frac{b^{2} d x^{4} - 4 \,{\left (b^{2} c - a b d\right )} x^{2}}{4 \, d^{3}} + \frac{{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229741, size = 217, normalized size = 2.41 \[ \frac{b^{2} d^{3} x^{6} + 2 \, b^{2} c^{3} - 4 \, a b c^{2} d + 2 \, a^{2} c d^{2} -{\left (3 \, b^{2} c d^{2} - 4 \, a b d^{3}\right )} x^{4} - 4 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} x^{2} + 2 \,{\left (3 \, b^{2} c^{3} - 4 \, a b c^{2} d + a^{2} c d^{2} +{\left (3 \, b^{2} c^{2} d - 4 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (d^{5} x^{2} + c d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.36309, size = 97, normalized size = 1.08 \[ \frac{b^{2} x^{4}}{4 d^{2}} + \frac{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}}{2 c d^{4} + 2 d^{5} x^{2}} + \frac{x^{2} \left (a b d - b^{2} c\right )}{d^{3}} + \frac{\left (a d - 3 b c\right ) \left (a d - b c\right ) \log{\left (c + d x^{2} \right )}}{2 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*x**4/(4*d**2) + (a**2*c*d**2 - 2*a*b*c**2*d + b**2*c**3)/(2*c*d**4 + 2*d**5
*x**2) + x**2*(a*b*d - b**2*c)/d**3 + (a*d - 3*b*c)*(a*d - b*c)*log(c + d*x**2)/
(2*d**4)

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GIAC/XCAS [A]  time = 0.224205, size = 220, normalized size = 2.44 \[ \frac{\frac{{\left (d x^{2} + c\right )}^{2}{\left (b^{2} - \frac{2 \,{\left (3 \, b^{2} c d - 2 \, a b d^{2}\right )}}{{\left (d x^{2} + c\right )} d}\right )}}{d^{3}} - \frac{2 \,{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left (\frac{{\left | d x^{2} + c \right |}}{{\left (d x^{2} + c\right )}^{2}{\left | d \right |}}\right )}{d^{3}} + \frac{2 \,{\left (\frac{b^{2} c^{3} d^{2}}{d x^{2} + c} - \frac{2 \, a b c^{2} d^{3}}{d x^{2} + c} + \frac{a^{2} c d^{4}}{d x^{2} + c}\right )}}{d^{5}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*((d*x^2 + c)^2*(b^2 - 2*(3*b^2*c*d - 2*a*b*d^2)/((d*x^2 + c)*d))/d^3 - 2*(3*
b^2*c^2 - 4*a*b*c*d + a^2*d^2)*ln(abs(d*x^2 + c)/((d*x^2 + c)^2*abs(d)))/d^3 + 2
*(b^2*c^3*d^2/(d*x^2 + c) - 2*a*b*c^2*d^3/(d*x^2 + c) + a^2*c*d^4/(d*x^2 + c))/d
^5)/d