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

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

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Rubi [A]  time = 0.06, antiderivative size = 90, 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 {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} \end {gather*}

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*}

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Mathematica [A]  time = 0.06, size = 99, normalized size = 1.10 \begin {gather*} \frac {-2 a^3 \left (c^2+3 c d x+3 d^2 x^2\right )+3 a^2 b c x (c+4 d x)-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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^2}{x^4 (a+b x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.28, size = 131, normalized size = 1.46 \begin {gather*} -\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 \relax (x) + 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 {gather*}

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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giac [A]  time = 0.86, size = 138, normalized size = 1.53 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.01, size = 153, normalized size = 1.70 \begin {gather*} -\frac {b \,d^{2} \ln \relax (x )}{a^{2}}+\frac {b \,d^{2} \ln \left (b x +a \right )}{a^{2}}+\frac {2 b^{2} c d \ln \relax (x )}{a^{3}}-\frac {2 b^{2} c d \ln \left (b x +a \right )}{a^{3}}-\frac {b^{3} c^{2} \ln \relax (x )}{a^{4}}+\frac {b^{3} c^{2} \ln \left (b x +a \right )}{a^{4}}-\frac {d^{2}}{a x}+\frac {2 b c d}{a^{2} x}-\frac {b^{2} c^{2}}{a^{3} x}-\frac {c d}{a \,x^{2}}+\frac {b \,c^{2}}{2 a^{2} x^{2}}-\frac {c^{2}}{3 a \,x^{3}} \end {gather*}

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 = 1.01, size = 126, normalized size = 1.40 \begin {gather*} \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 \relax (x)}{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 {gather*}

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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mupad [B]  time = 0.12, size = 123, normalized size = 1.37 \begin {gather*} \frac {2\,b\,\mathrm {atanh}\left (\frac {b\,{\left (a\,d-b\,c\right )}^2\,\left (a+2\,b\,x\right )}{a\,\left (a^2\,b\,d^2-2\,a\,b^2\,c\,d+b^3\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{a^4}-\frac {\frac {c^2}{3\,a}+\frac {x^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{a^3}+\frac {c\,x\,\left (2\,a\,d-b\,c\right )}{2\,a^2}}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/(x^4*(a + b*x)),x)

[Out]

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

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sympy [B]  time = 0.94, size = 240, normalized size = 2.67 \begin {gather*} \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 {gather*}

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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