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

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

[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 = 0.097313, 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, Rules used = {88} \[ -\frac{a^2 (b c-a d)^2}{b^5 (a+b x)}+\frac{d x^2 (b c-a d)}{b^3}+\frac{x (b c-3 a d) (b c-a d)}{b^4}-\frac{2 a (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac{d^2 x^3}{3 b^2} \]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{x^2 (c+d x)^2}{(a+b x)^2} \, dx &=\int \left (\frac{(b c-3 a d) (b c-a d)}{b^4}+\frac{2 d (b c-a d) x}{b^3}+\frac{d^2 x^2}{b^2}+\frac{a^2 (-b c+a d)^2}{b^4 (a+b x)^2}+\frac{2 a (b c-2 a d) (-b c+a d)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{(b c-3 a d) (b c-a d) x}{b^4}+\frac{d (b c-a d) x^2}{b^3}+\frac{d^2 x^3}{3 b^2}-\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}\\ \end{align*}

Mathematica [A]  time = 0.0744315, 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 = 0.008, size = 164, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{3}}{3\,{b}^{2}}}-{\frac{a{x}^{2}{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}}}-{\frac{{a}^{4}{d}^{2}}{{b}^{5} \left ( bx+a \right ) }}+2\,{\frac{{a}^{3}cd}{{b}^{4} \left ( bx+a \right ) }}-{\frac{{a}^{2}{c}^{2}}{{b}^{3} \left ( bx+a \right ) }}-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}}} \end{align*}

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

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Maxima [A]  time = 1.00557, size = 186, normalized size = 1.79 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d*x+c)^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 = 2.21397, size = 416, normalized size = 4. \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d*x+c)^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 = 0.878595, size = 122, normalized size = 1.17 \begin{align*} - \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}} \end{align*}

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 [A]  time = 1.14805, size = 254, normalized size = 2.44 \begin{align*} \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 )} \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d*x+c)^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)*log(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