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

Optimal. Leaf size=64 \[ \frac{d^2 x (3 b c-a d)}{b^2}-\frac{(b c-a d)^3 \log (a+b x)}{a b^3}+\frac{c^3 \log (x)}{a}+\frac{d^3 x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0458137, antiderivative size = 64, 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 = {72} \[ \frac{d^2 x (3 b c-a d)}{b^2}-\frac{(b c-a d)^3 \log (a+b x)}{a b^3}+\frac{c^3 \log (x)}{a}+\frac{d^3 x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x*(a + b*x)),x]

[Out]

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0259397, size = 59, normalized size = 0.92 \[ \frac{a b d^2 x (-2 a d+6 b c+b d x)-2 (b c-a d)^3 \log (a+b x)+2 b^3 c^3 \log (x)}{2 a b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 103, normalized size = 1.6 \begin{align*}{\frac{{d}^{3}{x}^{2}}{2\,b}}-{\frac{{d}^{3}ax}{{b}^{2}}}+3\,{\frac{{d}^{2}xc}{b}}+{\frac{{c}^{3}\ln \left ( x \right ) }{a}}+{\frac{{a}^{2}\ln \left ( bx+a \right ){d}^{3}}{{b}^{3}}}-3\,{\frac{a\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{2}}}+3\,{\frac{\ln \left ( bx+a \right ){c}^{2}d}{b}}-{\frac{\ln \left ( bx+a \right ){c}^{3}}{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0186, size = 123, normalized size = 1.92 \begin{align*} \frac{c^{3} \log \left (x\right )}{a} + \frac{b d^{3} x^{2} + 2 \,{\left (3 \, b c d^{2} - a d^{3}\right )} x}{2 \, b^{2}} - \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 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.32918, size = 204, normalized size = 3.19 \begin{align*} \frac{a b^{2} d^{3} x^{2} + 2 \, b^{3} c^{3} \log \left (x\right ) + 2 \,{\left (3 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x - 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 )} \log \left (b x + a\right )}{2 \, a b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.48075, size = 112, normalized size = 1.75 \begin{align*} \frac{d^{3} x^{2}}{2 b} - \frac{x \left (a d^{3} - 3 b c d^{2}\right )}{b^{2}} + \frac{c^{3} \log{\left (x \right )}}{a} + \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{- a b^{2} c^{3} + \frac{a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18915, size = 123, normalized size = 1.92 \begin{align*} \frac{c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac{b d^{3} x^{2} + 6 \, b c d^{2} x - 2 \, a d^{3} x}{2 \, b^{2}} - \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 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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