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

Optimal. Leaf size=112 \[ -\frac{(b c-a d)^2 (a d+2 b c)}{a^3 b^2 (a+b x)}-\frac{(b c-a d)^3}{2 a^2 b^2 (a+b x)^2}-\frac{3 c^2 \log (x) (b c-a d)}{a^4}+\frac{3 c^2 (b c-a d) \log (a+b x)}{a^4}-\frac{c^3}{a^3 x} \]

[Out]

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

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Rubi [A]  time = 0.0915244, antiderivative size = 112, 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} \[ -\frac{(b c-a d)^2 (a d+2 b c)}{a^3 b^2 (a+b x)}-\frac{(b c-a d)^3}{2 a^2 b^2 (a+b x)^2}-\frac{3 c^2 \log (x) (b c-a d)}{a^4}+\frac{3 c^2 (b c-a d) \log (a+b x)}{a^4}-\frac{c^3}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^2*(a + b*x)^3),x]

[Out]

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

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)^3} \, dx &=\int \left (\frac{c^3}{a^3 x^2}+\frac{3 c^2 (-b c+a d)}{a^4 x}-\frac{(-b c+a d)^3}{a^2 b (a+b x)^3}+\frac{(-b c+a d)^2 (2 b c+a d)}{a^3 b (a+b x)^2}-\frac{3 b c^2 (-b c+a d)}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac{c^3}{a^3 x}-\frac{(b c-a d)^3}{2 a^2 b^2 (a+b x)^2}-\frac{(b c-a d)^2 (2 b c+a d)}{a^3 b^2 (a+b x)}-\frac{3 c^2 (b c-a d) \log (x)}{a^4}+\frac{3 c^2 (b c-a d) \log (a+b x)}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.0973096, size = 106, normalized size = 0.95 \[ \frac{\frac{a^2 (a d-b c)^3}{b^2 (a+b x)^2}-\frac{2 a (b c-a d)^2 (a d+2 b c)}{b^2 (a+b x)}+6 c^2 \log (x) (a d-b c)+6 c^2 (b c-a d) \log (a+b x)-\frac{2 a c^3}{x}}{2 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a*c^3)/x + (a^2*(-(b*c) + a*d)^3)/(b^2*(a + b*x)^2) - (2*a*(b*c - a*d)^2*(2*b*c + a*d))/(b^2*(a + b*x)) +
6*c^2*(-(b*c) + a*d)*Log[x] + 6*c^2*(b*c - a*d)*Log[a + b*x])/(2*a^4)

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Maple [A]  time = 0.009, size = 176, normalized size = 1.6 \begin{align*} -{\frac{{c}^{3}}{{a}^{3}x}}+3\,{\frac{{c}^{2}\ln \left ( x \right ) d}{{a}^{3}}}-3\,{\frac{{c}^{3}\ln \left ( x \right ) b}{{a}^{4}}}-{\frac{{d}^{3}}{{b}^{2} \left ( bx+a \right ) }}+3\,{\frac{{c}^{2}d}{{a}^{2} \left ( bx+a \right ) }}-2\,{\frac{{c}^{3}b}{{a}^{3} \left ( bx+a \right ) }}+{\frac{{d}^{3}a}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{3\,c{d}^{2}}{2\,b \left ( bx+a \right ) ^{2}}}+{\frac{3\,{c}^{2}d}{2\,a \left ( bx+a \right ) ^{2}}}-{\frac{{c}^{3}b}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}}-3\,{\frac{{c}^{2}\ln \left ( bx+a \right ) d}{{a}^{3}}}+3\,{\frac{{c}^{3}\ln \left ( bx+a \right ) b}{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17447, size = 221, normalized size = 1.97 \begin{align*} -\frac{2 \, a^{2} b^{2} c^{3} + 2 \,{\left (3 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{3} b d^{3}\right )} x^{2} +{\left (9 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x}{2 \,{\left (a^{3} b^{4} x^{3} + 2 \, a^{4} b^{3} x^{2} + a^{5} b^{2} x\right )}} + \frac{3 \,{\left (b c^{3} - a c^{2} d\right )} \log \left (b x + a\right )}{a^{4}} - \frac{3 \,{\left (b c^{3} - a c^{2} d\right )} \log \left (x\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a^2*b^2*c^3 + 2*(3*b^4*c^3 - 3*a*b^3*c^2*d + a^3*b*d^3)*x^2 + (9*a*b^3*c^3 - 9*a^2*b^2*c^2*d + 3*a^3*b
*c*d^2 + a^4*d^3)*x)/(a^3*b^4*x^3 + 2*a^4*b^3*x^2 + a^5*b^2*x) + 3*(b*c^3 - a*c^2*d)*log(b*x + a)/a^4 - 3*(b*c
^3 - a*c^2*d)*log(x)/a^4

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Fricas [B]  time = 2.54154, size = 551, normalized size = 4.92 \begin{align*} -\frac{2 \, a^{3} b^{2} c^{3} + 2 \,{\left (3 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{4} b d^{3}\right )} x^{2} +{\left (9 \, a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} + a^{5} d^{3}\right )} x - 6 \,{\left ({\left (b^{5} c^{3} - a b^{4} c^{2} d\right )} x^{3} + 2 \,{\left (a b^{4} c^{3} - a^{2} b^{3} c^{2} d\right )} x^{2} +{\left (a^{2} b^{3} c^{3} - a^{3} b^{2} c^{2} d\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left ({\left (b^{5} c^{3} - a b^{4} c^{2} d\right )} x^{3} + 2 \,{\left (a b^{4} c^{3} - a^{2} b^{3} c^{2} d\right )} x^{2} +{\left (a^{2} b^{3} c^{3} - a^{3} b^{2} c^{2} d\right )} x\right )} \log \left (x\right )}{2 \,{\left (a^{4} b^{4} x^{3} + 2 \, a^{5} b^{3} x^{2} + a^{6} b^{2} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a^3*b^2*c^3 + 2*(3*a*b^4*c^3 - 3*a^2*b^3*c^2*d + a^4*b*d^3)*x^2 + (9*a^2*b^3*c^3 - 9*a^3*b^2*c^2*d + 3
*a^4*b*c*d^2 + a^5*d^3)*x - 6*((b^5*c^3 - a*b^4*c^2*d)*x^3 + 2*(a*b^4*c^3 - a^2*b^3*c^2*d)*x^2 + (a^2*b^3*c^3
- a^3*b^2*c^2*d)*x)*log(b*x + a) + 6*((b^5*c^3 - a*b^4*c^2*d)*x^3 + 2*(a*b^4*c^3 - a^2*b^3*c^2*d)*x^2 + (a^2*b
^3*c^3 - a^3*b^2*c^2*d)*x)*log(x))/(a^4*b^4*x^3 + 2*a^5*b^3*x^2 + a^6*b^2*x)

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Sympy [B]  time = 1.8856, size = 262, normalized size = 2.34 \begin{align*} - \frac{2 a^{2} b^{2} c^{3} + x^{2} \left (2 a^{3} b d^{3} - 6 a b^{3} c^{2} d + 6 b^{4} c^{3}\right ) + x \left (a^{4} d^{3} + 3 a^{3} b c d^{2} - 9 a^{2} b^{2} c^{2} d + 9 a b^{3} c^{3}\right )}{2 a^{5} b^{2} x + 4 a^{4} b^{3} x^{2} + 2 a^{3} b^{4} x^{3}} + \frac{3 c^{2} \left (a d - b c\right ) \log{\left (x + \frac{3 a^{2} c^{2} d - 3 a b c^{3} - 3 a c^{2} \left (a d - b c\right )}{6 a b c^{2} d - 6 b^{2} c^{3}} \right )}}{a^{4}} - \frac{3 c^{2} \left (a d - b c\right ) \log{\left (x + \frac{3 a^{2} c^{2} d - 3 a b c^{3} + 3 a c^{2} \left (a d - b c\right )}{6 a b c^{2} d - 6 b^{2} c^{3}} \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a**2*b**2*c**3 + x**2*(2*a**3*b*d**3 - 6*a*b**3*c**2*d + 6*b**4*c**3) + x*(a**4*d**3 + 3*a**3*b*c*d**2 - 9
*a**2*b**2*c**2*d + 9*a*b**3*c**3))/(2*a**5*b**2*x + 4*a**4*b**3*x**2 + 2*a**3*b**4*x**3) + 3*c**2*(a*d - b*c)
*log(x + (3*a**2*c**2*d - 3*a*b*c**3 - 3*a*c**2*(a*d - b*c))/(6*a*b*c**2*d - 6*b**2*c**3))/a**4 - 3*c**2*(a*d
- b*c)*log(x + (3*a**2*c**2*d - 3*a*b*c**3 + 3*a*c**2*(a*d - b*c))/(6*a*b*c**2*d - 6*b**2*c**3))/a**4

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Giac [A]  time = 1.22487, size = 217, normalized size = 1.94 \begin{align*} -\frac{3 \,{\left (b c^{3} - a c^{2} d\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{3 \,{\left (b^{2} c^{3} - a b c^{2} d\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{2 \, a^{3} b^{2} c^{3} + 2 \,{\left (3 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{4} b d^{3}\right )} x^{2} +{\left (9 \, a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} + a^{5} d^{3}\right )} x}{2 \,{\left (b x + a\right )}^{2} a^{4} 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)^3,x, algorithm="giac")

[Out]

-3*(b*c^3 - a*c^2*d)*log(abs(x))/a^4 + 3*(b^2*c^3 - a*b*c^2*d)*log(abs(b*x + a))/(a^4*b) - 1/2*(2*a^3*b^2*c^3
+ 2*(3*a*b^4*c^3 - 3*a^2*b^3*c^2*d + a^4*b*d^3)*x^2 + (9*a^2*b^3*c^3 - 9*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 + a^5*d
^3)*x)/((b*x + a)^2*a^4*b^2*x)