3.267 \(\int \frac{(c+d x)^2}{x^4 (a+b x)^2} \, dx\)

Optimal. Leaf size=132 \[ \frac{c (b c-a d)}{a^3 x^2}-\frac{(b c-a d) (3 b c-a d)}{a^4 x}-\frac{b (b c-a d)^2}{a^4 (a+b x)}-\frac{2 b \log (x) (b c-a d) (2 b c-a d)}{a^5}+\frac{2 b (b c-a d) (2 b c-a d) \log (a+b x)}{a^5}-\frac{c^2}{3 a^2 x^3} \]

[Out]

-c^2/(3*a^2*x^3) + (c*(b*c - a*d))/(a^3*x^2) - ((b*c - a*d)*(3*b*c - a*d))/(a^4*x) - (b*(b*c - a*d)^2)/(a^4*(a
+ b*x)) - (2*b*(b*c - a*d)*(2*b*c - a*d)*Log[x])/a^5 + (2*b*(b*c - a*d)*(2*b*c - a*d)*Log[a + b*x])/a^5

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Rubi [A]  time = 0.117689, antiderivative size = 132, 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} \[ \frac{c (b c-a d)}{a^3 x^2}-\frac{(b c-a d) (3 b c-a d)}{a^4 x}-\frac{b (b c-a d)^2}{a^4 (a+b x)}-\frac{2 b \log (x) (b c-a d) (2 b c-a d)}{a^5}+\frac{2 b (b c-a d) (2 b c-a d) \log (a+b x)}{a^5}-\frac{c^2}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x^4*(a + b*x)^2),x]

[Out]

-c^2/(3*a^2*x^3) + (c*(b*c - a*d))/(a^3*x^2) - ((b*c - a*d)*(3*b*c - a*d))/(a^4*x) - (b*(b*c - a*d)^2)/(a^4*(a
+ b*x)) - (2*b*(b*c - a*d)*(2*b*c - a*d)*Log[x])/a^5 + (2*b*(b*c - a*d)*(2*b*c - a*d)*Log[a + b*x])/a^5

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)^2}{x^4 (a+b x)^2} \, dx &=\int \left (\frac{c^2}{a^2 x^4}+\frac{2 c (-b c+a d)}{a^3 x^3}+\frac{(b c-a d) (3 b c-a d)}{a^4 x^2}+\frac{2 b (b c-a d) (-2 b c+a d)}{a^5 x}+\frac{b^2 (-b c+a d)^2}{a^4 (a+b x)^2}+\frac{2 b^2 (b c-a d) (2 b c-a d)}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{c^2}{3 a^2 x^3}+\frac{c (b c-a d)}{a^3 x^2}-\frac{(b c-a d) (3 b c-a d)}{a^4 x}-\frac{b (b c-a d)^2}{a^4 (a+b x)}-\frac{2 b (b c-a d) (2 b c-a d) \log (x)}{a^5}+\frac{2 b (b c-a d) (2 b c-a d) \log (a+b x)}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.135075, size = 142, normalized size = 1.08 \[ -\frac{\frac{3 a \left (a^2 d^2-4 a b c d+3 b^2 c^2\right )}{x}+6 b \log (x) \left (a^2 d^2-3 a b c d+2 b^2 c^2\right )-6 b \left (a^2 d^2-3 a b c d+2 b^2 c^2\right ) \log (a+b x)+\frac{3 a^2 c (a d-b c)}{x^2}+\frac{a^3 c^2}{x^3}+\frac{3 a b (b c-a d)^2}{a+b x}}{3 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(x^4*(a + b*x)^2),x]

[Out]

-((a^3*c^2)/x^3 + (3*a^2*c*(-(b*c) + a*d))/x^2 + (3*a*(3*b^2*c^2 - 4*a*b*c*d + a^2*d^2))/x + (3*a*b*(b*c - a*d
)^2)/(a + b*x) + 6*b*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Log[x] - 6*b*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Log[a +
b*x])/(3*a^5)

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Maple [A]  time = 0.01, size = 205, normalized size = 1.6 \begin{align*} -{\frac{{c}^{2}}{3\,{a}^{2}{x}^{3}}}-{\frac{{d}^{2}}{{a}^{2}x}}+4\,{\frac{cdb}{{a}^{3}x}}-3\,{\frac{{b}^{2}{c}^{2}}{{a}^{4}x}}-2\,{\frac{b\ln \left ( x \right ){d}^{2}}{{a}^{3}}}+6\,{\frac{{b}^{2}\ln \left ( x \right ) cd}{{a}^{4}}}-4\,{\frac{{b}^{3}\ln \left ( x \right ){c}^{2}}{{a}^{5}}}-{\frac{cd}{{a}^{2}{x}^{2}}}+{\frac{{c}^{2}b}{{a}^{3}{x}^{2}}}-{\frac{{d}^{2}b}{{a}^{2} \left ( bx+a \right ) }}+2\,{\frac{cd{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}-{\frac{{c}^{2}{b}^{3}}{{a}^{4} \left ( bx+a \right ) }}+2\,{\frac{b\ln \left ( bx+a \right ){d}^{2}}{{a}^{3}}}-6\,{\frac{{b}^{2}\ln \left ( bx+a \right ) cd}{{a}^{4}}}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ){c}^{2}}{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/x^4/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.0758, size = 239, normalized size = 1.81 \begin{align*} -\frac{a^{3} c^{2} + 6 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + 3 \,{\left (2 \, a b^{2} c^{2} - 3 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} -{\left (2 \, a^{2} b c^{2} - 3 \, a^{3} c d\right )} x}{3 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac{2 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (b x + a\right )}{a^{5}} - \frac{2 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (x\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/x^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.21337, size = 516, normalized size = 3.91 \begin{align*} -\frac{a^{4} c^{2} + 6 \,{\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3} + 3 \,{\left (2 \, a^{2} b^{2} c^{2} - 3 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2} -{\left (2 \, a^{3} b c^{2} - 3 \, a^{4} c d\right )} x - 6 \,{\left ({\left (2 \, b^{4} c^{2} - 3 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} +{\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3}\right )} \log \left (b x + a\right ) + 6 \,{\left ({\left (2 \, b^{4} c^{2} - 3 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} +{\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/x^4/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.55844, size = 326, normalized size = 2.47 \begin{align*} - \frac{a^{3} c^{2} + x^{3} \left (6 a^{2} b d^{2} - 18 a b^{2} c d + 12 b^{3} c^{2}\right ) + x^{2} \left (3 a^{3} d^{2} - 9 a^{2} b c d + 6 a b^{2} c^{2}\right ) + x \left (3 a^{3} c d - 2 a^{2} b c^{2}\right )}{3 a^{5} x^{3} + 3 a^{4} b x^{4}} - \frac{2 b \left (a d - 2 b c\right ) \left (a d - b c\right ) \log{\left (x + \frac{2 a^{3} b d^{2} - 6 a^{2} b^{2} c d + 4 a b^{3} c^{2} - 2 a b \left (a d - 2 b c\right ) \left (a d - b c\right )}{4 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 8 b^{4} c^{2}} \right )}}{a^{5}} + \frac{2 b \left (a d - 2 b c\right ) \left (a d - b c\right ) \log{\left (x + \frac{2 a^{3} b d^{2} - 6 a^{2} b^{2} c d + 4 a b^{3} c^{2} + 2 a b \left (a d - 2 b c\right ) \left (a d - b c\right )}{4 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 8 b^{4} c^{2}} \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/x**4/(b*x+a)**2,x)

[Out]

-(a**3*c**2 + x**3*(6*a**2*b*d**2 - 18*a*b**2*c*d + 12*b**3*c**2) + x**2*(3*a**3*d**2 - 9*a**2*b*c*d + 6*a*b**
2*c**2) + x*(3*a**3*c*d - 2*a**2*b*c**2))/(3*a**5*x**3 + 3*a**4*b*x**4) - 2*b*(a*d - 2*b*c)*(a*d - b*c)*log(x
+ (2*a**3*b*d**2 - 6*a**2*b**2*c*d + 4*a*b**3*c**2 - 2*a*b*(a*d - 2*b*c)*(a*d - b*c))/(4*a**2*b**2*d**2 - 12*a
*b**3*c*d + 8*b**4*c**2))/a**5 + 2*b*(a*d - 2*b*c)*(a*d - b*c)*log(x + (2*a**3*b*d**2 - 6*a**2*b**2*c*d + 4*a*
b**3*c**2 + 2*a*b*(a*d - 2*b*c)*(a*d - b*c))/(4*a**2*b**2*d**2 - 12*a*b**3*c*d + 8*b**4*c**2))/a**5

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Giac [A]  time = 1.37907, size = 319, normalized size = 2.42 \begin{align*} -\frac{2 \,{\left (2 \, b^{4} c^{2} - 3 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{5} b} - \frac{\frac{b^{7} c^{2}}{b x + a} - \frac{2 \, a b^{6} c d}{b x + a} + \frac{a^{2} b^{5} d^{2}}{b x + a}}{a^{4} b^{4}} + \frac{13 \, b^{3} c^{2} - 15 \, a b^{2} c d + 3 \, a^{2} b d^{2} - \frac{3 \,{\left (10 \, a b^{4} c^{2} - 11 \, a^{2} b^{3} c d + 2 \, a^{3} b^{2} d^{2}\right )}}{{\left (b x + a\right )} b} + \frac{3 \,{\left (6 \, a^{2} b^{5} c^{2} - 6 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{3 \, a^{5}{\left (\frac{a}{b x + a} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/x^4/(b*x+a)^2,x, algorithm="giac")

[Out]

-2*(2*b^4*c^2 - 3*a*b^3*c*d + a^2*b^2*d^2)*log(abs(-a/(b*x + a) + 1))/(a^5*b) - (b^7*c^2/(b*x + a) - 2*a*b^6*c
*d/(b*x + a) + a^2*b^5*d^2/(b*x + a))/(a^4*b^4) + 1/3*(13*b^3*c^2 - 15*a*b^2*c*d + 3*a^2*b*d^2 - 3*(10*a*b^4*c
^2 - 11*a^2*b^3*c*d + 2*a^3*b^2*d^2)/((b*x + a)*b) + 3*(6*a^2*b^5*c^2 - 6*a^3*b^4*c*d + a^4*b^3*d^2)/((b*x + a
)^2*b^2))/(a^5*(a/(b*x + a) - 1)^3)