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

Optimal. Leaf size=90 \[ \frac{c (b c-2 a d)}{2 a^2 x^2}-\frac{(b c-a d)^2}{a^3 x}-\frac{b \log (x) (b c-a d)^2}{a^4}+\frac{b (b c-a d)^2 \log (a+b x)}{a^4}-\frac{c^2}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0588979, antiderivative size = 90, 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} \[ \frac{c (b c-2 a d)}{2 a^2 x^2}-\frac{(b c-a d)^2}{a^3 x}-\frac{b \log (x) (b c-a d)^2}{a^4}+\frac{b (b c-a d)^2 \log (a+b x)}{a^4}-\frac{c^2}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x^4*(a + b*x)),x]

[Out]

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

Mathematica [A]  time = 0.0440564, size = 99, normalized size = 1.1 \[ \frac{3 a^2 b c x (c+4 d x)-2 a^3 \left (c^2+3 c d x+3 d^2 x^2\right )-6 a b^2 c^2 x^2-6 b x^3 \log (x) (b c-a d)^2+6 b x^3 (b c-a d)^2 \log (a+b x)}{6 a^4 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 153, normalized size = 1.7 \begin{align*} -{\frac{{c}^{2}}{3\,a{x}^{3}}}-{\frac{{d}^{2}}{ax}}+2\,{\frac{bcd}{{a}^{2}x}}-{\frac{{b}^{2}{c}^{2}}{{a}^{3}x}}-{\frac{cd}{a{x}^{2}}}+{\frac{{c}^{2}b}{2\,{a}^{2}{x}^{2}}}-{\frac{b\ln \left ( x \right ){d}^{2}}{{a}^{2}}}+2\,{\frac{{b}^{2}\ln \left ( x \right ) cd}{{a}^{3}}}-{\frac{{b}^{3}\ln \left ( x \right ){c}^{2}}{{a}^{4}}}+{\frac{b\ln \left ( bx+a \right ){d}^{2}}{{a}^{2}}}-2\,{\frac{{b}^{2}\ln \left ( bx+a \right ) cd}{{a}^{3}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ){c}^{2}}{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06754, size = 170, normalized size = 1.89 \begin{align*} \frac{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (b x + a\right )}{a^{4}} - \frac{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (x\right )}{a^{4}} - \frac{2 \, a^{2} c^{2} + 6 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} - 3 \,{\left (a b c^{2} - 2 \, a^{2} c d\right )} x}{6 \, a^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.19309, size = 284, normalized size = 3.16 \begin{align*} -\frac{2 \, a^{3} c^{2} - 6 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} \log \left (b x + a\right ) + 6 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} \log \left (x\right ) + 6 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 3 \,{\left (a^{2} b c^{2} - 2 \, a^{3} c d\right )} x}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.20386, size = 240, normalized size = 2.67 \begin{align*} - \frac{2 a^{2} c^{2} + x^{2} \left (6 a^{2} d^{2} - 12 a b c d + 6 b^{2} c^{2}\right ) + x \left (6 a^{2} c d - 3 a b c^{2}\right )}{6 a^{3} x^{3}} - \frac{b \left (a d - b c\right )^{2} \log{\left (x + \frac{a^{3} b d^{2} - 2 a^{2} b^{2} c d + a b^{3} c^{2} - a b \left (a d - b c\right )^{2}}{2 a^{2} b^{2} d^{2} - 4 a b^{3} c d + 2 b^{4} c^{2}} \right )}}{a^{4}} + \frac{b \left (a d - b c\right )^{2} \log{\left (x + \frac{a^{3} b d^{2} - 2 a^{2} b^{2} c d + a b^{3} c^{2} + a b \left (a d - b c\right )^{2}}{2 a^{2} b^{2} d^{2} - 4 a b^{3} c d + 2 b^{4} c^{2}} \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22518, size = 186, normalized size = 2.07 \begin{align*} -\frac{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{2 \, a^{3} c^{2} + 6 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 3 \,{\left (a^{2} b c^{2} - 2 \, a^{3} c d\right )} x}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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