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

Optimal. Leaf size=74 \[ -\frac{(b c-a d)^2 (2 a d+b c) \log (a+b x)}{a^2 b^3}+\frac{c^3 \log (x)}{a^2}+\frac{(b c-a d)^3}{a b^3 (a+b x)}+\frac{d^3 x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0604346, antiderivative size = 74, 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} \[ -\frac{(b c-a d)^2 (2 a d+b c) \log (a+b x)}{a^2 b^3}+\frac{c^3 \log (x)}{a^2}+\frac{(b c-a d)^3}{a b^3 (a+b x)}+\frac{d^3 x}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x*(a + b*x)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0625318, size = 74, normalized size = 1. \[ -\frac{(b c-a d)^2 (2 a d+b c) \log (a+b x)}{a^2 b^3}+\frac{c^3 \log (x)}{a^2}+\frac{(b c-a d)^3}{a b^3 (a+b x)}+\frac{d^3 x}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 128, normalized size = 1.7 \begin{align*}{\frac{{d}^{3}x}{{b}^{2}}}+{\frac{{c}^{3}\ln \left ( x \right ) }{{a}^{2}}}-2\,{\frac{a\ln \left ( bx+a \right ){d}^{3}}{{b}^{3}}}+3\,{\frac{\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{2}}}-{\frac{\ln \left ( bx+a \right ){c}^{3}}{{a}^{2}}}-{\frac{{a}^{2}{d}^{3}}{{b}^{3} \left ( bx+a \right ) }}+3\,{\frac{ac{d}^{2}}{{b}^{2} \left ( bx+a \right ) }}-3\,{\frac{{c}^{2}d}{b \left ( bx+a \right ) }}+{\frac{{c}^{3}}{a \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07033, size = 150, normalized size = 2.03 \begin{align*} \frac{d^{3} x}{b^{2}} + \frac{c^{3} \log \left (x\right )}{a^{2}} + \frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{a b^{4} x + a^{2} b^{3}} - \frac{{\left (b^{3} c^{3} - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.31007, size = 331, normalized size = 4.47 \begin{align*} \frac{a^{2} b^{2} d^{3} x^{2} + a^{3} b d^{3} x + a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} -{\left (a b^{3} c^{3} - 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )} x\right )} \log \left (b x + a\right ) +{\left (b^{4} c^{3} x + a b^{3} c^{3}\right )} \log \left (x\right )}{a^{2} b^{4} x + a^{3} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.02218, size = 153, normalized size = 2.07 \begin{align*} - \frac{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{a^{2} b^{3} + a b^{4} x} + \frac{d^{3} x}{b^{2}} + \frac{c^{3} \log{\left (x \right )}}{a^{2}} - \frac{\left (a d - b c\right )^{2} \left (2 a d + b c\right ) \log{\left (x + \frac{a b^{2} c^{3} + \frac{a \left (a d - b c\right )^{2} \left (2 a d + b c\right )}{b}}{2 a^{3} d^{3} - 3 a^{2} b c d^{2} + 2 b^{3} c^{3}} \right )}}{a^{2} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [B]  time = 1.15609, size = 207, normalized size = 2.8 \begin{align*} b{\left (\frac{c^{3} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{2} b} + \frac{{\left (b x + a\right )} d^{3}}{b^{4}} - \frac{{\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} + \frac{\frac{b^{5} c^{3}}{b x + a} - \frac{3 \, a b^{4} c^{2} d}{b x + a} + \frac{3 \, a^{2} b^{3} c d^{2}}{b x + a} - \frac{a^{3} b^{2} d^{3}}{b x + a}}{a b^{6}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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