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

Optimal. Leaf size=88 \[ -\frac{(b c-a d)^2 (2 a d+b c) \log \left (a+b x^2\right )}{2 a^2 b^3}+\frac{c^3 \log (x)}{a^2}+\frac{(b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{d^3 x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.203636, antiderivative size = 88, 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 \[ -\frac{(b c-a d)^2 (2 a d+b c) \log \left (a+b x^2\right )}{2 a^2 b^3}+\frac{c^3 \log (x)}{a^2}+\frac{(b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{d^3 x^2}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.176408, size = 111, normalized size = 1.26 \[ \frac{\frac{\frac{a \left (-a^3 d^3+a^2 b d^2 \left (3 c+d x^2\right )+a b^2 \left (d^3 x^4-3 c^2 d\right )+b^3 c^3\right )}{a+b x^2}-(b c-a d)^2 (2 a d+b c) \log \left (a+b x^2\right )}{b^3}+2 c^3 \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.023, size = 146, normalized size = 1.7 \[{\frac{{d}^{3}{x}^{2}}{2\,{b}^{2}}}+{\frac{{c}^{3}\ln \left ( x \right ) }{{a}^{2}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ){d}^{3}}{{b}^{3}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ){d}^{2}c}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,{a}^{2}}}-{\frac{{a}^{2}{d}^{3}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,a{d}^{2}c}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,d{c}^{2}}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{{c}^{3}}{2\,a \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35421, size = 165, normalized size = 1.88 \[ \frac{d^{3} x^{2}}{2 \, b^{2}} + \frac{c^{3} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{2 \,{\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} - \frac{{\left (b^{3} c^{3} - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239041, size = 240, normalized size = 2.73 \[ \frac{a^{2} b^{2} d^{3} x^{4} + a^{3} b d^{3} x^{2} + a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} -{\left (a b^{3} c^{3} - 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left (b^{4} c^{3} x^{2} + a b^{3} c^{3}\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.2533, size = 110, normalized size = 1.25 \[ - \frac{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{2 a^{2} b^{3} + 2 a b^{4} x^{2}} + \frac{d^{3} x^{2}}{2 b^{2}} + \frac{c^{3} \log{\left (x \right )}}{a^{2}} - \frac{\left (a d - b c\right )^{2} \left (2 a d + b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232261, size = 203, normalized size = 2.31 \[ \frac{d^{3} x^{2}}{2 \, b^{2}} + \frac{c^{3}{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} - \frac{{\left (b^{3} c^{3} - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b^{3}} + \frac{b^{4} c^{3} x^{2} - 3 \, a^{2} b^{2} c d^{2} x^{2} + 2 \, a^{3} b d^{3} x^{2} + 2 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + a^{4} d^{3}}{2 \,{\left (b x^{2} + a\right )} a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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