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

Optimal. Leaf size=85 \[ \frac{c \log (x) \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{(b c-a d)^3 \log (a+b x)}{a^3 b}-\frac{c^3}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0541983, antiderivative size = 85, 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} \[ \frac{c \log (x) \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{(b c-a d)^3 \log (a+b x)}{a^3 b}-\frac{c^3}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^3*(a + b*x)),x]

[Out]

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

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

Mathematica [A]  time = 0.0810459, size = 78, normalized size = 0.92 \[ -\frac{-2 c \log (x) \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )+\frac{a c^2 (a (c+6 d x)-2 b c x)}{x^2}+\frac{2 (b c-a d)^3 \log (a+b x)}{b}}{2 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 132, normalized size = 1.6 \begin{align*} -{\frac{{c}^{3}}{2\,a{x}^{2}}}+3\,{\frac{c\ln \left ( x \right ){d}^{2}}{a}}-3\,{\frac{{c}^{2}\ln \left ( x \right ) bd}{{a}^{2}}}+{\frac{{c}^{3}\ln \left ( x \right ){b}^{2}}{{a}^{3}}}-3\,{\frac{{c}^{2}d}{ax}}+{\frac{b{c}^{3}}{{a}^{2}x}}+{\frac{\ln \left ( bx+a \right ){d}^{3}}{b}}-3\,{\frac{\ln \left ( bx+a \right ) c{d}^{2}}{a}}+3\,{\frac{b\ln \left ( bx+a \right ){c}^{2}d}{{a}^{2}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ){c}^{3}}{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0478, size = 151, normalized size = 1.78 \begin{align*} \frac{{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left (x\right )}{a^{3}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{3} b} - \frac{a c^{3} - 2 \,{\left (b c^{3} - 3 \, a c^{2} d\right )} x}{2 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.5604, size = 262, normalized size = 3.08 \begin{align*} -\frac{a^{2} b c^{3} + 2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2} \log \left (b x + a\right ) - 2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2}\right )} x^{2} \log \left (x\right ) - 2 \,{\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d\right )} x}{2 \, a^{3} b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.75313, size = 257, normalized size = 3.02 \begin{align*} - \frac{a c^{3} + x \left (6 a c^{2} d - 2 b c^{3}\right )}{2 a^{2} x^{2}} + \frac{c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log{\left (x + \frac{- 3 a^{3} c d^{2} + 3 a^{2} b c^{2} d - a b^{2} c^{3} + a c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{3}} + \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{- 3 a^{3} c d^{2} + 3 a^{2} b c^{2} d - a b^{2} c^{3} + \frac{a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{3} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18194, size = 161, normalized size = 1.89 \begin{align*} \frac{{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac{a^{2} c^{3} - 2 \,{\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x}{2 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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