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

Optimal. Leaf size=88 \[ -\frac{(b c-a d)^3}{2 b^4 \left (a+b x^2\right )}+\frac{3 d (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4}+\frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac{d^3 x^4}{4 b^2} \]

[Out]

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

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Rubi [A]  time = 0.218004, antiderivative size = 88, 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 \[ -\frac{(b c-a d)^3}{2 b^4 \left (a+b x^2\right )}+\frac{3 d (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4}+\frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac{d^3 x^4}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(d^2*(3*b*c - 2*a*d)*x^2)/(2*b^3) + (d^3*x^4)/(4*b^2) - (b*c - a*d)^3/(2*b^4*(a
+ b*x^2)) + (3*d*(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. \[ - \frac{d^{2} \left (2 a d - 3 b c\right ) \int ^{x^{2}} \frac{1}{b^{3}}\, dx}{2} + \frac{d^{3} \int ^{x^{2}} x\, dx}{2 b^{2}} + \frac{3 d \left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{\left (a d - b c\right )^{3}}{2 b^{4} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0741487, size = 127, normalized size = 1.44 \[ \frac{3 \left (a^2 d^3-2 a b c d^2+b^2 c^2 d\right ) \log \left (a+b x^2\right )}{2 b^4}+\frac{a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3}{2 b^4 \left (a+b x^2\right )}+\frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac{d^3 x^4}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.014, size = 168, normalized size = 1.9 \[{\frac{{d}^{3}{x}^{4}}{4\,{b}^{2}}}-{\frac{a{d}^{3}{x}^{2}}{{b}^{3}}}+{\frac{3\,{d}^{2}{x}^{2}c}{2\,{b}^{2}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ){d}^{3}{a}^{2}}{2\,{b}^{4}}}-3\,{\frac{\ln \left ( b{x}^{2}+a \right ){d}^{2}ca}{{b}^{3}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ) d{c}^{2}}{2\,{b}^{2}}}+{\frac{{a}^{3}{d}^{3}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,{a}^{2}c{d}^{2}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,ad{c}^{2}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{c}^{3}}{2\,b \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35585, size = 167, normalized size = 1.9 \[ -\frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{2 \,{\left (b^{5} x^{2} + a b^{4}\right )}} + \frac{b d^{3} x^{4} + 2 \,{\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x^{2}}{4 \, b^{3}} + \frac{3 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216234, size = 244, normalized size = 2.77 \[ \frac{b^{3} d^{3} x^{6} - 2 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} + 3 \,{\left (2 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 2 \,{\left (3 \, a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{2} + 6 \,{\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{5} x^{2} + a b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.27337, size = 112, normalized size = 1.27 \[ \frac{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{2 a b^{4} + 2 b^{5} x^{2}} + \frac{d^{3} x^{4}}{4 b^{2}} - \frac{x^{2} \left (2 a d^{3} - 3 b c d^{2}\right )}{2 b^{3}} + \frac{3 d \left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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GIAC/XCAS [A]  time = 0.236639, size = 247, normalized size = 2.81 \[ \frac{{\left (d^{3} + \frac{6 \,{\left (b^{2} c d^{2} - a b d^{3}\right )}}{{\left (b x^{2} + a\right )} b}\right )}{\left (b x^{2} + a\right )}^{2}}{4 \, b^{4}} - \frac{3 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}{\rm ln}\left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{2 \, b^{4}} - \frac{\frac{b^{5} c^{3}}{b x^{2} + a} - \frac{3 \, a b^{4} c^{2} d}{b x^{2} + a} + \frac{3 \, a^{2} b^{3} c d^{2}}{b x^{2} + a} - \frac{a^{3} b^{2} d^{3}}{b x^{2} + a}}{2 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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