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

Optimal. Leaf size=80 \[ -\frac{a (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4}+\frac{x^2 (b c-a d)^2}{2 b^3}+\frac{d x^4 (2 b c-a d)}{4 b^2}+\frac{d^2 x^6}{6 b} \]

[Out]

((b*c - a*d)^2*x^2)/(2*b^3) + (d*(2*b*c - a*d)*x^4)/(4*b^2) + (d^2*x^6)/(6*b) -
(a*(b*c - a*d)^2*Log[a + b*x^2])/(2*b^4)

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Rubi [A]  time = 0.186262, antiderivative size = 80, 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 \[ -\frac{a (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4}+\frac{x^2 (b c-a d)^2}{2 b^3}+\frac{d x^4 (2 b c-a d)}{4 b^2}+\frac{d^2 x^6}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*x^2)/(2*b^3) + (d*(2*b*c - a*d)*x^4)/(4*b^2) + (d^2*x^6)/(6*b) -
(a*(b*c - a*d)^2*Log[a + b*x^2])/(2*b^4)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a \left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{\left (a d - b c\right )^{2} \int ^{x^{2}} \frac{1}{b^{3}}\, dx}{2} + \frac{d^{2} x^{6}}{6 b} - \frac{d \left (a d - 2 b c\right ) \int ^{x^{2}} x\, dx}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(a*d - b*c)**2*log(a + b*x**2)/(2*b**4) + (a*d - b*c)**2*Integral(b**(-3), (x
, x**2))/2 + d**2*x**6/(6*b) - d*(a*d - 2*b*c)*Integral(x, (x, x**2))/(2*b**2)

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Mathematica [A]  time = 0.0632728, size = 82, normalized size = 1.02 \[ \frac{b x^2 \left (6 a^2 d^2-3 a b d \left (4 c+d x^2\right )+2 b^2 \left (3 c^2+3 c d x^2+d^2 x^4\right )\right )-6 a (b c-a d)^2 \log \left (a+b x^2\right )}{12 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(b*x^2*(6*a^2*d^2 - 3*a*b*d*(4*c + d*x^2) + 2*b^2*(3*c^2 + 3*c*d*x^2 + d^2*x^4))
- 6*a*(b*c - a*d)^2*Log[a + b*x^2])/(12*b^4)

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Maple [A]  time = 0.003, size = 124, normalized size = 1.6 \[{\frac{{d}^{2}{x}^{6}}{6\,b}}-{\frac{{x}^{4}a{d}^{2}}{4\,{b}^{2}}}+{\frac{c{x}^{4}d}{2\,b}}+{\frac{{x}^{2}{a}^{2}{d}^{2}}{2\,{b}^{3}}}-{\frac{a{x}^{2}cd}{{b}^{2}}}+{\frac{{x}^{2}{c}^{2}}{2\,b}}-{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ){d}^{2}}{2\,{b}^{4}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) cd}{{b}^{3}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ){c}^{2}}{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35213, size = 136, normalized size = 1.7 \[ \frac{2 \, b^{2} d^{2} x^{6} + 3 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{4} + 6 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}}{12 \, b^{3}} - \frac{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(2*b^2*d^2*x^6 + 3*(2*b^2*c*d - a*b*d^2)*x^4 + 6*(b^2*c^2 - 2*a*b*c*d + a^2
*d^2)*x^2)/b^3 - 1/2*(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*log(b*x^2 + a)/b^4

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Fricas [A]  time = 0.227416, size = 138, normalized size = 1.72 \[ \frac{2 \, b^{3} d^{2} x^{6} + 3 \,{\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{4} + 6 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2} - 6 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(2*b^3*d^2*x^6 + 3*(2*b^3*c*d - a*b^2*d^2)*x^4 + 6*(b^3*c^2 - 2*a*b^2*c*d +
a^2*b*d^2)*x^2 - 6*(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*log(b*x^2 + a))/b^4

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Sympy [A]  time = 2.16224, size = 83, normalized size = 1.04 \[ - \frac{a \left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{d^{2} x^{6}}{6 b} - \frac{x^{4} \left (a d^{2} - 2 b c d\right )}{4 b^{2}} + \frac{x^{2} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221629, size = 144, normalized size = 1.8 \[ \frac{2 \, b^{2} d^{2} x^{6} + 6 \, b^{2} c d x^{4} - 3 \, a b d^{2} x^{4} + 6 \, b^{2} c^{2} x^{2} - 12 \, a b c d x^{2} + 6 \, a^{2} d^{2} x^{2}}{12 \, b^{3}} - \frac{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(2*b^2*d^2*x^6 + 6*b^2*c*d*x^4 - 3*a*b*d^2*x^4 + 6*b^2*c^2*x^2 - 12*a*b*c*d
*x^2 + 6*a^2*d^2*x^2)/b^3 - 1/2*(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*ln(abs(b*x^2
+ a))/b^4