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

Optimal. Leaf size=61 \[ \frac{(b c-a d)^3 \log (a+b x)}{a^2 b^2}-\frac{c^2 \log (x) (b c-3 a d)}{a^2}-\frac{c^3}{a x}+\frac{d^3 x}{b} \]

[Out]

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

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Rubi [A]  time = 0.0490497, antiderivative size = 61, 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} \[ \frac{(b c-a d)^3 \log (a+b x)}{a^2 b^2}-\frac{c^2 \log (x) (b c-3 a d)}{a^2}-\frac{c^3}{a x}+\frac{d^3 x}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^2*(a + b*x)),x]

[Out]

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

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

Mathematica [A]  time = 0.0311796, size = 66, normalized size = 1.08 \[ \frac{b^2 c^2 x \log (x) (3 a d-b c)+a b \left (a d^3 x^2-b c^3\right )+x (b c-a d)^3 \log (a+b x)}{a^2 b^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 102, normalized size = 1.7 \begin{align*}{\frac{{d}^{3}x}{b}}-{\frac{{c}^{3}}{ax}}+3\,{\frac{{c}^{2}\ln \left ( x \right ) d}{a}}-{\frac{{c}^{3}\ln \left ( x \right ) b}{{a}^{2}}}-{\frac{a\ln \left ( bx+a \right ){d}^{3}}{{b}^{2}}}+3\,{\frac{\ln \left ( bx+a \right ) c{d}^{2}}{b}}-3\,{\frac{\ln \left ( bx+a \right ){c}^{2}d}{a}}+{\frac{b\ln \left ( bx+a \right ){c}^{3}}{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04657, size = 120, normalized size = 1.97 \begin{align*} \frac{d^{3} x}{b} - \frac{c^{3}}{a x} - \frac{{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (x\right )}{a^{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^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.54468, size = 198, normalized size = 3.25 \begin{align*} \frac{a^{2} b d^{3} x^{2} - a b^{2} c^{3} +{\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 \log \left (b x + a\right ) -{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d\right )} x \log \left (x\right )}{a^{2} b^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.84652, size = 196, normalized size = 3.21 \begin{align*} \frac{d^{3} x}{b} - \frac{c^{3}}{a x} + \frac{c^{2} \left (3 a d - b c\right ) \log{\left (x + \frac{3 a^{2} b c^{2} d - a b^{2} c^{3} - a b c^{2} \left (3 a d - b c\right )}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{2}} - \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{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} - 3 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15787, size = 123, normalized size = 2.02 \begin{align*} \frac{d^{3} x}{b} - \frac{c^{3}}{a x} - \frac{{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left ({\left | x \right |}\right )}{a^{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^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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