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

Optimal. Leaf size=73 \[ -\frac {2 c \log (x) (b c-a d)}{a^3}+\frac {2 c (b c-a d) \log (a+b x)}{a^3}-\frac {(b c-a d)^2}{a^2 b (a+b x)}-\frac {c^2}{a^2 x} \]

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Rubi [A]  time = 0.05, antiderivative size = 73, 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} \begin {gather*} -\frac {(b c-a d)^2}{a^2 b (a+b x)}-\frac {2 c \log (x) (b c-a d)}{a^3}+\frac {2 c (b c-a d) \log (a+b x)}{a^3}-\frac {c^2}{a^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x^2*(a + b*x)^2),x]

[Out]

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

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

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Mathematica [A]  time = 0.09, size = 67, normalized size = 0.92 \begin {gather*} \frac {-\frac {a (b c-a d)^2}{b (a+b x)}+2 c \log (x) (a d-b c)+2 c (b c-a d) \log (a+b x)-\frac {a c^2}{x}}{a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^2}{x^2 (a+b x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.73, size = 149, normalized size = 2.04 \begin {gather*} -\frac {a^{2} b c^{2} + {\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x - 2 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{2} + {\left (a b^{2} c^{2} - a^{2} b c d\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{2} + {\left (a b^{2} c^{2} - a^{2} b c d\right )} x\right )} \log \relax (x)}{a^{3} b^{2} x^{2} + a^{4} b x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.14, size = 111, normalized size = 1.52 \begin {gather*} \frac {b c^{2}}{a^{3} {\left (\frac {a}{b x + a} - 1\right )}} - \frac {2 \, {\left (b^{2} c^{2} - a b c d\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{3} b} - \frac {\frac {b^{3} c^{2}}{b x + a} - \frac {2 \, a b^{2} c d}{b x + a} + \frac {a^{2} b d^{2}}{b x + a}}{a^{2} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 106, normalized size = 1.45 \begin {gather*} \frac {2 c d}{\left (b x +a \right ) a}-\frac {b \,c^{2}}{\left (b x +a \right ) a^{2}}+\frac {2 c d \ln \relax (x )}{a^{2}}-\frac {2 c d \ln \left (b x +a \right )}{a^{2}}-\frac {2 b \,c^{2} \ln \relax (x )}{a^{3}}+\frac {2 b \,c^{2} \ln \left (b x +a \right )}{a^{3}}-\frac {d^{2}}{\left (b x +a \right ) b}-\frac {c^{2}}{a^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.99, size = 93, normalized size = 1.27 \begin {gather*} -\frac {a b c^{2} + {\left (2 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{a^{2} b^{2} x^{2} + a^{3} b x} + \frac {2 \, {\left (b c^{2} - a c d\right )} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, {\left (b c^{2} - a c d\right )} \log \relax (x)}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.39, size = 101, normalized size = 1.38 \begin {gather*} \frac {4\,c\,\mathrm {atanh}\left (\frac {2\,c\,\left (a\,d-b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (2\,b\,c^2-2\,a\,c\,d\right )}\right )\,\left (a\,d-b\,c\right )}{a^3}-\frac {\frac {c^2}{a}+\frac {x\,\left (a^2\,d^2-2\,a\,b\,c\,d+2\,b^2\,c^2\right )}{a^2\,b}}{b\,x^2+a\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/(x^2*(a + b*x)^2),x)

[Out]

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

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sympy [B]  time = 0.84, size = 173, normalized size = 2.37 \begin {gather*} \frac {- a b c^{2} + x \left (- a^{2} d^{2} + 2 a b c d - 2 b^{2} c^{2}\right )}{a^{3} b x + a^{2} b^{2} x^{2}} + \frac {2 c \left (a d - b c\right ) \log {\left (x + \frac {2 a^{2} c d - 2 a b c^{2} - 2 a c \left (a d - b c\right )}{4 a b c d - 4 b^{2} c^{2}} \right )}}{a^{3}} - \frac {2 c \left (a d - b c\right ) \log {\left (x + \frac {2 a^{2} c d - 2 a b c^{2} + 2 a c \left (a d - b c\right )}{4 a b c d - 4 b^{2} c^{2}} \right )}}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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