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

Optimal. Leaf size=117 \[ \frac{a (b c-a d)^3}{2 b^5 \left (a+b x^2\right )}+\frac{(b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5}+\frac{3 d x^2 (b c-a d)^2}{2 b^4}+\frac{d^2 x^4 (3 b c-2 a d)}{4 b^3}+\frac{d^3 x^6}{6 b^2} \]

[Out]

(3*d*(b*c - a*d)^2*x^2)/(2*b^4) + (d^2*(3*b*c - 2*a*d)*x^4)/(4*b^3) + (d^3*x^6)/
(6*b^2) + (a*(b*c - a*d)^3)/(2*b^5*(a + b*x^2)) + ((b*c - 4*a*d)*(b*c - a*d)^2*L
og[a + b*x^2])/(2*b^5)

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Rubi [A]  time = 0.328727, antiderivative size = 117, 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 \[ \frac{a (b c-a d)^3}{2 b^5 \left (a+b x^2\right )}+\frac{(b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5}+\frac{3 d x^2 (b c-a d)^2}{2 b^4}+\frac{d^2 x^4 (3 b c-2 a d)}{4 b^3}+\frac{d^3 x^6}{6 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*d*(b*c - a*d)^2*x^2)/(2*b^4) + (d^2*(3*b*c - 2*a*d)*x^4)/(4*b^3) + (d^3*x^6)/
(6*b^2) + (a*(b*c - a*d)^3)/(2*b^5*(a + b*x^2)) + ((b*c - 4*a*d)*(b*c - a*d)^2*L
og[a + b*x^2])/(2*b^5)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a \left (a d - b c\right )^{3}}{2 b^{5} \left (a + b x^{2}\right )} + \frac{d^{3} x^{6}}{6 b^{2}} - \frac{d^{2} \left (2 a d - 3 b c\right ) \int ^{x^{2}} x\, dx}{2 b^{3}} + \frac{3 d x^{2} \left (a d - b c\right )^{2}}{2 b^{4}} - \frac{\left (a d - b c\right )^{2} \left (4 a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.152384, size = 106, normalized size = 0.91 \[ \frac{3 b^2 d^2 x^4 (3 b c-2 a d)+18 b d x^2 (b c-a d)^2-\frac{6 a (a d-b c)^3}{a+b x^2}+6 (b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )+2 b^3 d^3 x^6}{12 b^5} \]

Antiderivative was successfully verified.

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

[Out]

(18*b*d*(b*c - a*d)^2*x^2 + 3*b^2*d^2*(3*b*c - 2*a*d)*x^4 + 2*b^3*d^3*x^6 - (6*a
*(-(b*c) + a*d)^3)/(a + b*x^2) + 6*(b*c - 4*a*d)*(b*c - a*d)^2*Log[a + b*x^2])/(
12*b^5)

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Maple [B]  time = 0.017, size = 229, normalized size = 2. \[{\frac{{d}^{3}{x}^{6}}{6\,{b}^{2}}}-{\frac{{d}^{3}{x}^{4}a}{2\,{b}^{3}}}+{\frac{3\,{d}^{2}{x}^{4}c}{4\,{b}^{2}}}+{\frac{3\,{d}^{3}{x}^{2}{a}^{2}}{2\,{b}^{4}}}-3\,{\frac{{d}^{2}{x}^{2}ac}{{b}^{3}}}+{\frac{3\,d{x}^{2}{c}^{2}}{2\,{b}^{2}}}-2\,{\frac{\ln \left ( b{x}^{2}+a \right ){a}^{3}{d}^{3}}{{b}^{5}}}+{\frac{9\,\ln \left ( b{x}^{2}+a \right ){a}^{2}{d}^{2}c}{2\,{b}^{4}}}-3\,{\frac{\ln \left ( b{x}^{2}+a \right ) ad{c}^{2}}{{b}^{3}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,{b}^{2}}}-{\frac{{a}^{4}{d}^{3}}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,{a}^{3}{d}^{2}c}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,{a}^{2}{c}^{2}d}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{a{c}^{3}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*d^3*x^6/b^2-1/2*d^3/b^3*x^4*a+3/4*d^2/b^2*x^4*c+3/2*d^3/b^4*x^2*a^2-3*d^2/b^
3*x^2*a*c+3/2*d/b^2*x^2*c^2-2/b^5*ln(b*x^2+a)*a^3*d^3+9/2/b^4*ln(b*x^2+a)*a^2*d^
2*c-3/b^3*ln(b*x^2+a)*a*d*c^2+1/2/b^2*ln(b*x^2+a)*c^3-1/2/b^5*a^4/(b*x^2+a)*d^3+
3/2/b^4*a^3/(b*x^2+a)*d^2*c-3/2/b^3*a^2/(b*x^2+a)*d*c^2+1/2/b^2*a/(b*x^2+a)*c^3

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Maxima [A]  time = 1.34705, size = 235, normalized size = 2.01 \[ \frac{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}}{2 \,{\left (b^{6} x^{2} + a b^{5}\right )}} + \frac{2 \, b^{2} d^{3} x^{6} + 3 \,{\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{4} + 18 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{12 \, b^{4}} + \frac{{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)/(b^6*x^2 + a*b^5) +
1/12*(2*b^2*d^3*x^6 + 3*(3*b^2*c*d^2 - 2*a*b*d^3)*x^4 + 18*(b^2*c^2*d - 2*a*b*c*
d^2 + a^2*d^3)*x^2)/b^4 + 1/2*(b^3*c^3 - 6*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 4*a^3*d
^3)*log(b*x^2 + a)/b^5

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Fricas [A]  time = 0.225204, size = 343, normalized size = 2.93 \[ \frac{2 \, b^{4} d^{3} x^{8} + 6 \, a b^{3} c^{3} - 18 \, a^{2} b^{2} c^{2} d + 18 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} +{\left (9 \, b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{6} + 3 \,{\left (6 \, b^{4} c^{2} d - 9 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x^{4} + 18 \,{\left (a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{2} + 6 \,{\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{6} x^{2} + a b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(2*b^4*d^3*x^8 + 6*a*b^3*c^3 - 18*a^2*b^2*c^2*d + 18*a^3*b*c*d^2 - 6*a^4*d^
3 + (9*b^4*c*d^2 - 4*a*b^3*d^3)*x^6 + 3*(6*b^4*c^2*d - 9*a*b^3*c*d^2 + 4*a^2*b^2
*d^3)*x^4 + 18*(a*b^3*c^2*d - 2*a^2*b^2*c*d^2 + a^3*b*d^3)*x^2 + 6*(a*b^3*c^3 -
6*a^2*b^2*c^2*d + 9*a^3*b*c*d^2 - 4*a^4*d^3 + (b^4*c^3 - 6*a*b^3*c^2*d + 9*a^2*b
^2*c*d^2 - 4*a^3*b*d^3)*x^2)*log(b*x^2 + a))/(b^6*x^2 + a*b^5)

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Sympy [A]  time = 6.1864, size = 158, normalized size = 1.35 \[ - \frac{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}}{2 a b^{5} + 2 b^{6} x^{2}} + \frac{d^{3} x^{6}}{6 b^{2}} - \frac{x^{4} \left (2 a d^{3} - 3 b c d^{2}\right )}{4 b^{3}} + \frac{x^{2} \left (3 a^{2} d^{3} - 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{2 b^{4}} - \frac{\left (a d - b c\right )^{2} \left (4 a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.247111, size = 336, normalized size = 2.87 \[ \frac{\frac{{\left (2 \, d^{3} + \frac{3 \,{\left (3 \, b^{2} c d^{2} - 4 \, a b d^{3}\right )}}{{\left (b x^{2} + a\right )} b} + \frac{18 \,{\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x^{2} + a\right )}^{2} b^{2}}\right )}{\left (b x^{2} + a\right )}^{3}}{b^{4}} - \frac{6 \,{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )}{\rm ln}\left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} + \frac{6 \,{\left (\frac{a b^{6} c^{3}}{b x^{2} + a} - \frac{3 \, a^{2} b^{5} c^{2} d}{b x^{2} + a} + \frac{3 \, a^{3} b^{4} c d^{2}}{b x^{2} + a} - \frac{a^{4} b^{3} d^{3}}{b x^{2} + a}\right )}}{b^{7}}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*((2*d^3 + 3*(3*b^2*c*d^2 - 4*a*b*d^3)/((b*x^2 + a)*b) + 18*(b^4*c^2*d - 3*a
*b^3*c*d^2 + 2*a^2*b^2*d^3)/((b*x^2 + a)^2*b^2))*(b*x^2 + a)^3/b^4 - 6*(b^3*c^3
- 6*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 4*a^3*d^3)*ln(abs(b*x^2 + a)/((b*x^2 + a)^2*ab
s(b)))/b^4 + 6*(a*b^6*c^3/(b*x^2 + a) - 3*a^2*b^5*c^2*d/(b*x^2 + a) + 3*a^3*b^4*
c*d^2/(b*x^2 + a) - a^4*b^3*d^3/(b*x^2 + a))/b^7)/b