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

Optimal. Leaf size=93 \[ -\left (\frac{c^3}{a^3}-\frac{d^3}{b^3}\right ) \log (a+b x)+\frac{(b c-a d)^2 (2 a d+b c)}{a^2 b^3 (a+b x)}+\frac{c^3 \log (x)}{a^3}+\frac{(b c-a d)^3}{2 a b^3 (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0747503, antiderivative size = 93, 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} \[ -\left (\frac{c^3}{a^3}-\frac{d^3}{b^3}\right ) \log (a+b x)+\frac{(b c-a d)^2 (2 a d+b c)}{a^2 b^3 (a+b x)}+\frac{c^3 \log (x)}{a^3}+\frac{(b c-a d)^3}{2 a b^3 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x*(a + b*x)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0861169, size = 88, normalized size = 0.95 \[ \frac{\frac{2 \left (a^3 d^3-b^3 c^3\right ) \log (a+b x)+\frac{a (b c-a d)^2 \left (3 a^2 d+a b (3 c+4 d x)+2 b^2 c x\right )}{(a+b x)^2}}{b^3}+2 c^3 \log (x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 150, normalized size = 1.6 \begin{align*}{\frac{{c}^{3}\ln \left ( x \right ) }{{a}^{3}}}+{\frac{\ln \left ( bx+a \right ){d}^{3}}{{b}^{3}}}-{\frac{\ln \left ( bx+a \right ){c}^{3}}{{a}^{3}}}+2\,{\frac{{d}^{3}a}{{b}^{3} \left ( bx+a \right ) }}-3\,{\frac{c{d}^{2}}{{b}^{2} \left ( bx+a \right ) }}+{\frac{{c}^{3}}{{a}^{2} \left ( bx+a \right ) }}-{\frac{{d}^{3}{a}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{3\,c{d}^{2}a}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{c}^{2}d}{2\,b \left ( bx+a \right ) ^{2}}}+{\frac{{c}^{3}}{2\,a \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.30236, size = 193, normalized size = 2.08 \begin{align*} \frac{c^{3} \log \left (x\right )}{a^{3}} + \frac{3 \, a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d - 3 \, a^{3} b c d^{2} + 3 \, a^{4} d^{3} + 2 \,{\left (b^{4} c^{3} - 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )} x}{2 \,{\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} - \frac{{\left (b^{3} c^{3} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.62089, size = 420, normalized size = 4.52 \begin{align*} \frac{3 \, a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d - 3 \, a^{4} b c d^{2} + 3 \, a^{5} d^{3} + 2 \,{\left (a b^{4} c^{3} - 3 \, a^{3} b^{2} c d^{2} + 2 \, a^{4} b d^{3}\right )} x - 2 \,{\left (a^{2} b^{3} c^{3} - a^{5} d^{3} +{\left (b^{5} c^{3} - a^{3} b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{4} c^{3} - a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right ) + 2 \,{\left (b^{5} c^{3} x^{2} + 2 \, a b^{4} c^{3} x + a^{2} b^{3} c^{3}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b^{5} x^{2} + 2 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.67337, size = 209, normalized size = 2.25 \begin{align*} \frac{3 a^{4} d^{3} - 3 a^{3} b c d^{2} - 3 a^{2} b^{2} c^{2} d + 3 a b^{3} c^{3} + x \left (4 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 2 b^{4} c^{3}\right )}{2 a^{4} b^{3} + 4 a^{3} b^{4} x + 2 a^{2} b^{5} x^{2}} + \frac{c^{3} \log{\left (x \right )}}{a^{3}} + \frac{\left (a d - b c\right ) \left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right ) \log{\left (x + \frac{- a b^{2} c^{3} + \frac{a \left (a d - b c\right ) \left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right )}{b}}{a^{3} d^{3} - 2 b^{3} c^{3}} \right )}}{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)**3,x)

[Out]

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

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Giac [A]  time = 1.18579, size = 180, normalized size = 1.94 \begin{align*} \frac{c^{3} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} c^{3} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b^{3}} + \frac{2 \,{\left (a b^{3} c^{3} - 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3}\right )} x + \frac{3 \,{\left (a^{2} b^{3} c^{3} - a^{3} b^{2} c^{2} d - a^{4} b c d^{2} + a^{5} d^{3}\right )}}{b}}{2 \,{\left (b x + a\right )}^{2} a^{3} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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