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

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

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 132, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.17, size = 142, normalized size = 1.08 \[ -\frac {\frac {a^3 c^2}{x^3}+\frac {3 a \left (a^2 d^2-4 a b c d+3 b^2 c^2\right )}{x}+6 b \log (x) \left (a^2 d^2-3 a b c d+2 b^2 c^2\right )-6 b \left (a^2 d^2-3 a b c d+2 b^2 c^2\right ) \log (a+b x)+\frac {3 a^2 c (a d-b c)}{x^2}+\frac {3 a b (b c-a d)^2}{a+b x}}{3 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*((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])/a^5

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fricas [A]  time = 0.93, size = 253, normalized size = 1.92 \[ -\frac {a^{4} c^{2} + 6 \, {\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3} + 3 \, {\left (2 \, a^{2} b^{2} c^{2} - 3 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2} - {\left (2 \, a^{3} b c^{2} - 3 \, a^{4} c d\right )} x - 6 \, {\left ({\left (2 \, b^{4} c^{2} - 3 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + {\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3}\right )} \log \left (b x + a\right ) + 6 \, {\left ({\left (2 \, b^{4} c^{2} - 3 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} + {\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3}\right )} \log \relax (x)}{3 \, {\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \]

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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giac [A]  time = 1.14, size = 236, normalized size = 1.79 \[ -\frac {2 \, {\left (2 \, b^{4} c^{2} - 3 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{5} b} - \frac {\frac {b^{7} c^{2}}{b x + a} - \frac {2 \, a b^{6} c d}{b x + a} + \frac {a^{2} b^{5} d^{2}}{b x + a}}{a^{4} b^{4}} + \frac {13 \, b^{3} c^{2} - 15 \, a b^{2} c d + 3 \, a^{2} b d^{2} - \frac {3 \, {\left (10 \, a b^{4} c^{2} - 11 \, a^{2} b^{3} c d + 2 \, a^{3} b^{2} d^{2}\right )}}{{\left (b x + a\right )} b} + \frac {3 \, {\left (6 \, a^{2} b^{5} c^{2} - 6 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{3 \, a^{5} {\left (\frac {a}{b x + a} - 1\right )}^{3}} \]

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)

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maple [A]  time = 0.01, size = 205, normalized size = 1.55 \[ -\frac {b \,d^{2}}{\left (b x +a \right ) a^{2}}+\frac {2 b^{2} c d}{\left (b x +a \right ) a^{3}}-\frac {2 b \,d^{2} \ln \relax (x )}{a^{3}}+\frac {2 b \,d^{2} \ln \left (b x +a \right )}{a^{3}}-\frac {b^{3} c^{2}}{\left (b x +a \right ) a^{4}}+\frac {6 b^{2} c d \ln \relax (x )}{a^{4}}-\frac {6 b^{2} c d \ln \left (b x +a \right )}{a^{4}}-\frac {4 b^{3} c^{2} \ln \relax (x )}{a^{5}}+\frac {4 b^{3} c^{2} \ln \left (b x +a \right )}{a^{5}}-\frac {d^{2}}{a^{2} x}+\frac {4 b c d}{a^{3} x}-\frac {3 b^{2} c^{2}}{a^{4} x}-\frac {c d}{a^{2} x^{2}}+\frac {b \,c^{2}}{a^{3} x^{2}}-\frac {c^{2}}{3 a^{2} x^{3}} \]

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 = 1.08, size = 177, normalized size = 1.34 \[ -\frac {a^{3} c^{2} + 6 \, {\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + 3 \, {\left (2 \, a b^{2} c^{2} - 3 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - {\left (2 \, a^{2} b c^{2} - 3 \, a^{3} c d\right )} x}{3 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac {2 \, {\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (b x + a\right )}{a^{5}} - \frac {2 \, {\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \relax (x)}{a^{5}} \]

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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mupad [B]  time = 0.44, size = 180, normalized size = 1.36 \[ \frac {4\,b\,\mathrm {atanh}\left (\frac {2\,b\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (2\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+4\,b^3\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )}{a^5}-\frac {\frac {c^2}{3\,a}+\frac {x^2\,\left (a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{a^3}+\frac {2\,b\,x^3\,\left (a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2\right )}{a^4}+\frac {c\,x\,\left (3\,a\,d-2\,b\,c\right )}{3\,a^2}}{b\,x^4+a\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/(x^4*(a + b*x)^2),x)

[Out]

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

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sympy [B]  time = 1.18, size = 326, normalized size = 2.47 \[ \frac {- a^{3} c^{2} + x^{3} \left (- 6 a^{2} b d^{2} + 18 a b^{2} c d - 12 b^{3} c^{2}\right ) + x^{2} \left (- 3 a^{3} d^{2} + 9 a^{2} b c d - 6 a b^{2} c^{2}\right ) + x \left (- 3 a^{3} c d + 2 a^{2} b c^{2}\right )}{3 a^{5} x^{3} + 3 a^{4} b x^{4}} - \frac {2 b \left (a d - 2 b c\right ) \left (a d - b c\right ) \log {\left (x + \frac {2 a^{3} b d^{2} - 6 a^{2} b^{2} c d + 4 a b^{3} c^{2} - 2 a b \left (a d - 2 b c\right ) \left (a d - b c\right )}{4 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 8 b^{4} c^{2}} \right )}}{a^{5}} + \frac {2 b \left (a d - 2 b c\right ) \left (a d - b c\right ) \log {\left (x + \frac {2 a^{3} b d^{2} - 6 a^{2} b^{2} c d + 4 a b^{3} c^{2} + 2 a b \left (a d - 2 b c\right ) \left (a d - b c\right )}{4 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 8 b^{4} c^{2}} \right )}}{a^{5}} \]

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 - 1
2*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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