3.230 \(\int \frac{x^2 (c+d x)^2}{(a+b x)^2} \, dx\)

Optimal. Leaf size=104 \[ -\frac{a^2 (b c-a d)^2}{b^5 (a+b x)}-\frac{2 a (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac{x (b c-3 a d) (b c-a d)}{b^4}+\frac{d x^2 (b c-a d)}{b^3}+\frac{d^2 x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.242164, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{a^2 (b c-a d)^2}{b^5 (a+b x)}-\frac{2 a (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac{x (b c-3 a d) (b c-a d)}{b^4}+\frac{d x^2 (b c-a d)}{b^3}+\frac{d^2 x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \left (a d - b c\right )^{2}}{b^{5} \left (a + b x\right )} - \frac{2 a \left (a d - b c\right ) \left (2 a d - b c\right ) \log{\left (a + b x \right )}}{b^{5}} + \left (a d - b c\right ) \left (3 a d - b c\right ) \int \frac{1}{b^{4}}\, dx + \frac{d^{2} x^{3}}{3 b^{2}} - \frac{2 d \left (a d - b c\right ) \int x\, dx}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.14349, size = 114, normalized size = 1.1 \[ \frac{3 b x \left (3 a^2 d^2-4 a b c d+b^2 c^2\right )-6 a \left (2 a^2 d^2-3 a b c d+b^2 c^2\right ) \log (a+b x)-\frac{3 a^2 (b c-a d)^2}{a+b x}+3 b^2 d x^2 (b c-a d)+b^3 d^2 x^3}{3 b^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 164, normalized size = 1.6 \[{\frac{{d}^{2}{x}^{3}}{3\,{b}^{2}}}-{\frac{{x}^{2}a{d}^{2}}{{b}^{3}}}+{\frac{c{x}^{2}d}{{b}^{2}}}+3\,{\frac{{a}^{2}{d}^{2}x}{{b}^{4}}}-4\,{\frac{acdx}{{b}^{3}}}+{\frac{{c}^{2}x}{{b}^{2}}}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ){d}^{2}}{{b}^{5}}}+6\,{\frac{{a}^{2}\ln \left ( bx+a \right ) cd}{{b}^{4}}}-2\,{\frac{a\ln \left ( bx+a \right ){c}^{2}}{{b}^{3}}}-{\frac{{a}^{4}{d}^{2}}{ \left ( bx+a \right ){b}^{5}}}+2\,{\frac{{a}^{3}cd}{ \left ( bx+a \right ){b}^{4}}}-{\frac{{a}^{2}{c}^{2}}{ \left ( bx+a \right ){b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(d*x+c)^2/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.35202, size = 186, normalized size = 1.79 \[ -\frac{a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}}{b^{6} x + a b^{5}} + \frac{b^{2} d^{2} x^{3} + 3 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2} + 3 \,{\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x}{3 \, b^{4}} - \frac{2 \,{\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} \log \left (b x + a\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216282, size = 273, normalized size = 2.62 \[ \frac{b^{4} d^{2} x^{4} - 3 \, a^{2} b^{2} c^{2} + 6 \, a^{3} b c d - 3 \, a^{4} d^{2} +{\left (3 \, b^{4} c d - 2 \, a b^{3} d^{2}\right )} x^{3} + 3 \,{\left (b^{4} c^{2} - 3 \, a b^{3} c d + 2 \, a^{2} b^{2} d^{2}\right )} x^{2} + 3 \,{\left (a b^{3} c^{2} - 4 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x - 6 \,{\left (a^{2} b^{2} c^{2} - 3 \, a^{3} b c d + 2 \, a^{4} d^{2} +{\left (a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{3 \,{\left (b^{6} x + a b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.79086, size = 122, normalized size = 1.17 \[ - \frac{2 a \left (a d - b c\right ) \left (2 a d - b c\right ) \log{\left (a + b x \right )}}{b^{5}} - \frac{a^{4} d^{2} - 2 a^{3} b c d + a^{2} b^{2} c^{2}}{a b^{5} + b^{6} x} + \frac{d^{2} x^{3}}{3 b^{2}} - \frac{x^{2} \left (a d^{2} - b c d\right )}{b^{3}} + \frac{x \left (3 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282545, size = 254, normalized size = 2.44 \[ \frac{{\left (d^{2} + \frac{3 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )}}{{\left (b x + a\right )} b} + \frac{3 \,{\left (b^{4} c^{2} - 6 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )}{\left (b x + a\right )}^{3}}{3 \, b^{5}} + \frac{2 \,{\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{5}} - \frac{\frac{a^{2} b^{5} c^{2}}{b x + a} - \frac{2 \, a^{3} b^{4} c d}{b x + a} + \frac{a^{4} b^{3} d^{2}}{b x + a}}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(d^2 + 3*(b^2*c*d - 2*a*b*d^2)/((b*x + a)*b) + 3*(b^4*c^2 - 6*a*b^3*c*d + 6*
a^2*b^2*d^2)/((b*x + a)^2*b^2))*(b*x + a)^3/b^5 + 2*(a*b^2*c^2 - 3*a^2*b*c*d + 2
*a^3*d^2)*ln(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^5 - (a^2*b^5*c^2/(b*x + a) - 2
*a^3*b^4*c*d/(b*x + a) + a^4*b^3*d^2/(b*x + a))/b^8