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

Optimal. Leaf size=77 \[ \frac{a (b c-a d)^2}{b^4 (a+b x)}+\frac{2 d x (b c-a d)}{b^3}+\frac{(b c-3 a d) (b c-a d) \log (a+b x)}{b^4}+\frac{d^2 x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0651912, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{a (b c-a d)^2}{b^4 (a+b x)}+\frac{2 d x (b c-a d)}{b^3}+\frac{(b c-3 a d) (b c-a d) \log (a+b x)}{b^4}+\frac{d^2 x^2}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x)^2)/(a + b*x)^2,x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0486992, size = 81, normalized size = 1.05 \[ \frac{2 \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (a+b x)+\frac{2 a (b c-a d)^2}{a+b x}+4 b d x (b c-a d)+b^2 d^2 x^2}{2 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 124, normalized size = 1.6 \begin{align*}{\frac{{d}^{2}{x}^{2}}{2\,{b}^{2}}}-2\,{\frac{a{d}^{2}x}{{b}^{3}}}+2\,{\frac{cdx}{{b}^{2}}}+{\frac{{a}^{3}{d}^{2}}{{b}^{4} \left ( bx+a \right ) }}-2\,{\frac{{a}^{2}cd}{{b}^{3} \left ( bx+a \right ) }}+{\frac{a{c}^{2}}{{b}^{2} \left ( bx+a \right ) }}+3\,{\frac{{a}^{2}\ln \left ( bx+a \right ){d}^{2}}{{b}^{4}}}-4\,{\frac{\ln \left ( bx+a \right ) acd}{{b}^{3}}}+{\frac{\ln \left ( bx+a \right ){c}^{2}}{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d*x+c)^2/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.07295, size = 134, normalized size = 1.74 \begin{align*} \frac{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}}{b^{5} x + a b^{4}} + \frac{b d^{2} x^{2} + 4 \,{\left (b c d - a d^{2}\right )} x}{2 \, b^{3}} + \frac{{\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (b x + a\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.14776, size = 316, normalized size = 4.1 \begin{align*} \frac{b^{3} d^{2} x^{3} + 2 \, a b^{2} c^{2} - 4 \, a^{2} b c d + 2 \, a^{3} d^{2} +{\left (4 \, b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{2} + 4 \,{\left (a b^{2} c d - a^{2} b d^{2}\right )} x + 2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} b c d + 3 \, a^{3} d^{2} +{\left (b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x + a b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.766265, size = 90, normalized size = 1.17 \begin{align*} \frac{a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}}{a b^{4} + b^{5} x} + \frac{d^{2} x^{2}}{2 b^{2}} - \frac{x \left (2 a d^{2} - 2 b c d\right )}{b^{3}} + \frac{\left (a d - b c\right ) \left (3 a d - b c\right ) \log{\left (a + b x \right )}}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18661, size = 201, normalized size = 2.61 \begin{align*} \frac{\frac{{\left (d^{2} + \frac{2 \,{\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )}}{{\left (b x + a\right )} b}\right )}{\left (b x + a\right )}^{2}}{b^{3}} - \frac{2 \,{\left (b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} + \frac{2 \,{\left (\frac{a b^{4} c^{2}}{b x + a} - \frac{2 \, a^{2} b^{3} c d}{b x + a} + \frac{a^{3} b^{2} d^{2}}{b x + a}\right )}}{b^{5}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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