3.1253 \(\int \frac{(c+d x)^2}{(a+b x)^3} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 d (b c-a d)}{b^3 (a+b x)}-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac{d^2 \log (a+b x)}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0350432, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ -\frac{2 d (b c-a d)}{b^3 (a+b x)}-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac{d^2 \log (a+b x)}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a + b*x)^3,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0234555, size = 49, normalized size = 0.83 \[ \frac{2 d^2 \log (a+b x)-\frac{(b c-a d) (3 a d+b (c+4 d x))}{(a+b x)^2}}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a + b*x)^3,x]

[Out]

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

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Maple [A]  time = 0.005, size = 92, normalized size = 1.6 \begin{align*} 2\,{\frac{a{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}-2\,{\frac{cd}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{a}^{2}{d}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{acd}{{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{c}^{2}}{2\,b \left ( bx+a \right ) ^{2}}}+{\frac{{d}^{2}\ln \left ( bx+a \right ) }{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.964279, size = 107, normalized size = 1.81 \begin{align*} -\frac{b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac{d^{2} \log \left (b x + a\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.88601, size = 207, normalized size = 3.51 \begin{align*} -\frac{b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x - 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.630731, size = 80, normalized size = 1.36 \begin{align*} \frac{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2} + x \left (4 a b d^{2} - 4 b^{2} c d\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac{d^{2} \log{\left (a + b x \right )}}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.0557, size = 92, normalized size = 1.56 \begin{align*} \frac{d^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac{4 \,{\left (b c d - a d^{2}\right )} x + \frac{b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}}{b}}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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