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

Optimal. Leaf size=67 \[ \frac{c (b c-2 a d)}{a^2 x}+\frac{\log (x) (b c-a d)^2}{a^3}-\frac{(b c-a d)^2 \log (a+b x)}{a^3}-\frac{c^2}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0505042, antiderivative size = 67, 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} \[ \frac{c (b c-2 a d)}{a^2 x}+\frac{\log (x) (b c-a d)^2}{a^3}-\frac{(b c-a d)^2 \log (a+b x)}{a^3}-\frac{c^2}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x^3*(a + b*x)),x]

[Out]

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

Mathematica [A]  time = 0.0526087, size = 60, normalized size = 0.9 \[ -\frac{\frac{a c (a c+4 a d x-2 b c x)}{x^2}-2 \log (x) (b c-a d)^2+2 (b c-a d)^2 \log (a+b x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 110, normalized size = 1.6 \begin{align*} -{\frac{{c}^{2}}{2\,a{x}^{2}}}+{\frac{\ln \left ( x \right ){d}^{2}}{a}}-2\,{\frac{b\ln \left ( x \right ) cd}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}{c}^{2}}{{a}^{3}}}-2\,{\frac{cd}{ax}}+{\frac{{c}^{2}b}{{a}^{2}x}}-{\frac{\ln \left ( bx+a \right ){d}^{2}}{a}}+2\,{\frac{\ln \left ( bx+a \right ) bcd}{{a}^{2}}}-{\frac{\ln \left ( bx+a \right ){b}^{2}{c}^{2}}{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05051, size = 119, normalized size = 1.78 \begin{align*} -\frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{3}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x\right )}{a^{3}} - \frac{a c^{2} - 2 \,{\left (b c^{2} - 2 \, a c d\right )} x}{2 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.24603, size = 208, normalized size = 3.1 \begin{align*} -\frac{a^{2} c^{2} + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (b x + a\right ) - 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (x\right ) - 2 \,{\left (a b c^{2} - 2 \, a^{2} c d\right )} x}{2 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.973504, size = 187, normalized size = 2.79 \begin{align*} - \frac{a c^{2} + x \left (4 a c d - 2 b c^{2}\right )}{2 a^{2} x^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (x + \frac{a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2} - a \left (a d - b c\right )^{2}}{2 a^{2} b d^{2} - 4 a b^{2} c d + 2 b^{3} c^{2}} \right )}}{a^{3}} - \frac{\left (a d - b c\right )^{2} \log{\left (x + \frac{a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2} + a \left (a d - b c\right )^{2}}{2 a^{2} b d^{2} - 4 a b^{2} c d + 2 b^{3} c^{2}} \right )}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20935, size = 136, normalized size = 2.03 \begin{align*} \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac{a^{2} c^{2} - 2 \,{\left (a b c^{2} - 2 \, a^{2} c d\right )} x}{2 \, a^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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