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

Optimal. Leaf size=103 \[ \frac {\log (x) (b c-a d) (3 b c-a d)}{a^4}-\frac {(b c-a d) (3 b c-a d) \log (a+b x)}{a^4}+\frac {2 c (b c-a d)}{a^3 x}+\frac {(b c-a d)^2}{a^3 (a+b x)}-\frac {c^2}{2 a^2 x^2} \]

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Rubi [A]  time = 0.08, antiderivative size = 103, 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 {2 c (b c-a d)}{a^3 x}+\frac {(b c-a d)^2}{a^3 (a+b x)}+\frac {\log (x) (b c-a d) (3 b c-a d)}{a^4}-\frac {(b c-a d) (3 b c-a d) \log (a+b x)}{a^4}-\frac {c^2}{2 a^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x^3*(a + b*x)^2),x]

[Out]

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

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Mathematica [A]  time = 0.13, size = 109, normalized size = 1.06 \begin {gather*} -\frac {-2 \log (x) \left (a^2 d^2-4 a b c d+3 b^2 c^2\right )+2 \left (a^2 d^2-4 a b c d+3 b^2 c^2\right ) \log (a+b x)+\frac {a^2 c^2}{x^2}+\frac {4 a c (a d-b c)}{x}-\frac {2 a (b c-a d)^2}{a+b x}}{2 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^2}{x^3 (a+b x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.93, size = 208, normalized size = 2.02 \begin {gather*} -\frac {a^{3} c^{2} - 2 \, {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - {\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d\right )} x + 2 \, {\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.02, size = 166, normalized size = 1.61 \begin {gather*} \frac {{\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{4} b} + \frac {\frac {b^{5} c^{2}}{b x + a} - \frac {2 \, a b^{4} c d}{b x + a} + \frac {a^{2} b^{3} d^{2}}{b x + a}}{a^{3} b^{3}} + \frac {5 \, b^{2} c^{2} - 4 \, a b c d - \frac {2 \, {\left (3 \, a b^{3} c^{2} - 2 \, a^{2} b^{2} c d\right )}}{{\left (b x + a\right )} b}}{2 \, a^{4} {\left (\frac {a}{b x + a} - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 158, normalized size = 1.53 \begin {gather*} \frac {d^{2}}{\left (b x +a \right ) a}-\frac {2 b c d}{\left (b x +a \right ) a^{2}}+\frac {d^{2} \ln \relax (x )}{a^{2}}-\frac {d^{2} \ln \left (b x +a \right )}{a^{2}}+\frac {b^{2} c^{2}}{\left (b x +a \right ) a^{3}}-\frac {4 b c d \ln \relax (x )}{a^{3}}+\frac {4 b c d \ln \left (b x +a \right )}{a^{3}}+\frac {3 b^{2} c^{2} \ln \relax (x )}{a^{4}}-\frac {3 b^{2} c^{2} \ln \left (b x +a \right )}{a^{4}}-\frac {2 c d}{a^{2} x}+\frac {2 b \,c^{2}}{a^{3} x}-\frac {c^{2}}{2 a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.21, size = 135, normalized size = 1.31 \begin {gather*} -\frac {a^{2} c^{2} - 2 \, {\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} x^{2} - {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x}{2 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} - \frac {{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{4}} + \frac {{\left (3 \, b^{2} c^{2} - 4 \, a b c d + a^{2} d^{2}\right )} \log \relax (x)}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.15, size = 143, normalized size = 1.39 \begin {gather*} -\frac {\frac {c^2}{2\,a}-\frac {x^2\,\left (a^2\,d^2-4\,a\,b\,c\,d+3\,b^2\,c^2\right )}{a^3}+\frac {c\,x\,\left (4\,a\,d-3\,b\,c\right )}{2\,a^2}}{b\,x^3+a\,x^2}-\frac {2\,\mathrm {atanh}\left (\frac {\left (a\,d-b\,c\right )\,\left (a\,d-3\,b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (a^2\,d^2-4\,a\,b\,c\,d+3\,b^2\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d-3\,b\,c\right )}{a^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/(x^3*(a + b*x)^2),x)

[Out]

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

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sympy [B]  time = 1.00, size = 262, normalized size = 2.54 \begin {gather*} \frac {- a^{2} c^{2} + x^{2} \left (2 a^{2} d^{2} - 8 a b c d + 6 b^{2} c^{2}\right ) + x \left (- 4 a^{2} c d + 3 a b c^{2}\right )}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} + \frac {\left (a d - 3 b c\right ) \left (a d - b c\right ) \log {\left (x + \frac {a^{3} d^{2} - 4 a^{2} b c d + 3 a b^{2} c^{2} - a \left (a d - 3 b c\right ) \left (a d - b c\right )}{2 a^{2} b d^{2} - 8 a b^{2} c d + 6 b^{3} c^{2}} \right )}}{a^{4}} - \frac {\left (a d - 3 b c\right ) \left (a d - b c\right ) \log {\left (x + \frac {a^{3} d^{2} - 4 a^{2} b c d + 3 a b^{2} c^{2} + a \left (a d - 3 b c\right ) \left (a d - b c\right )}{2 a^{2} b d^{2} - 8 a b^{2} c d + 6 b^{3} c^{2}} \right )}}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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