3.3.7 \(\int \frac {(c+d x)^2}{x^5 (a+b x)} \, dx\)

Optimal. Leaf size=114 \[ \frac {b^2 \log (x) (b c-a d)^2}{a^5}-\frac {b^2 (b c-a d)^2 \log (a+b x)}{a^5}+\frac {b (b c-a d)^2}{a^4 x}-\frac {(b c-a d)^2}{2 a^3 x^2}+\frac {c (b c-2 a d)}{3 a^2 x^3}-\frac {c^2}{4 a x^4} \]

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Rubi [A]  time = 0.08, antiderivative size = 114, 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^2 \log (x) (b c-a d)^2}{a^5}-\frac {b^2 (b c-a d)^2 \log (a+b x)}{a^5}+\frac {c (b c-2 a d)}{3 a^2 x^3}-\frac {(b c-a d)^2}{2 a^3 x^2}+\frac {b (b c-a d)^2}{a^4 x}-\frac {c^2}{4 a x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x^5*(a + b*x)),x]

[Out]

-c^2/(4*a*x^4) + (c*(b*c - 2*a*d))/(3*a^2*x^3) - (b*c - a*d)^2/(2*a^3*x^2) + (b*(b*c - a*d)^2)/(a^4*x) + (b^2*
(b*c - a*d)^2*Log[x])/a^5 - (b^2*(b*c - a*d)^2*Log[a + b*x])/a^5

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^5 (a+b x)} \, dx &=\int \left (\frac {c^2}{a x^5}+\frac {c (-b c+2 a d)}{a^2 x^4}+\frac {(-b c+a d)^2}{a^3 x^3}-\frac {b (-b c+a d)^2}{a^4 x^2}+\frac {b^2 (-b c+a d)^2}{a^5 x}-\frac {b^3 (-b c+a d)^2}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac {c^2}{4 a x^4}+\frac {c (b c-2 a d)}{3 a^2 x^3}-\frac {(b c-a d)^2}{2 a^3 x^2}+\frac {b (b c-a d)^2}{a^4 x}+\frac {b^2 (b c-a d)^2 \log (x)}{a^5}-\frac {b^2 (b c-a d)^2 \log (a+b x)}{a^5}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 127, normalized size = 1.11 \begin {gather*} \frac {\frac {a \left (-\left (a^3 \left (3 c^2+8 c d x+6 d^2 x^2\right )\right )+4 a^2 b x \left (c^2+3 c d x+3 d^2 x^2\right )-6 a b^2 c x^2 (c+4 d x)+12 b^3 c^2 x^3\right )}{x^4}+12 b^2 \log (x) (b c-a d)^2-12 b^2 (b c-a d)^2 \log (a+b x)}{12 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(x^5*(a + b*x)),x]

[Out]

((a*(12*b^3*c^2*x^3 - 6*a*b^2*c*x^2*(c + 4*d*x) + 4*a^2*b*x*(c^2 + 3*c*d*x + 3*d^2*x^2) - a^3*(3*c^2 + 8*c*d*x
+ 6*d^2*x^2)))/x^4 + 12*b^2*(b*c - a*d)^2*Log[x] - 12*b^2*(b*c - a*d)^2*Log[a + b*x])/(12*a^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^2}{x^5 (a+b x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^2/(x^5*(a + b*x)),x]

[Out]

IntegrateAlgebraic[(c + d*x)^2/(x^5*(a + b*x)), x]

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fricas [A]  time = 1.29, size = 169, normalized size = 1.48 \begin {gather*} -\frac {3 \, a^{4} c^{2} + 12 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} \log \left (b x + a\right ) - 12 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} \log \relax (x) - 12 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3} + 6 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2} - 4 \, {\left (a^{3} b c^{2} - 2 \, a^{4} c d\right )} x}{12 \, a^{5} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/x^5/(b*x+a),x, algorithm="fricas")

[Out]

-1/12*(3*a^4*c^2 + 12*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^4*log(b*x + a) - 12*(b^4*c^2 - 2*a*b^3*c*d + a^2
*b^2*d^2)*x^4*log(x) - 12*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^3 + 6*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2
)*x^2 - 4*(a^3*b*c^2 - 2*a^4*c*d)*x)/(a^5*x^4)

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giac [A]  time = 0.90, size = 174, normalized size = 1.53 \begin {gather*} \frac {{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac {3 \, a^{4} c^{2} - 12 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{3} + 6 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} x^{2} - 4 \, {\left (a^{3} b c^{2} - 2 \, a^{4} c d\right )} x}{12 \, a^{5} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/x^5/(b*x+a),x, algorithm="giac")

[Out]

(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*log(abs(x))/a^5 - (b^5*c^2 - 2*a*b^4*c*d + a^2*b^3*d^2)*log(abs(b*x + a)
)/(a^5*b) - 1/12*(3*a^4*c^2 - 12*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^3 + 6*(a^2*b^2*c^2 - 2*a^3*b*c*d +
a^4*d^2)*x^2 - 4*(a^3*b*c^2 - 2*a^4*c*d)*x)/(a^5*x^4)

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maple [A]  time = 0.01, size = 193, normalized size = 1.69 \begin {gather*} \frac {b^{2} d^{2} \ln \relax (x )}{a^{3}}-\frac {b^{2} d^{2} \ln \left (b x +a \right )}{a^{3}}-\frac {2 b^{3} c d \ln \relax (x )}{a^{4}}+\frac {2 b^{3} c d \ln \left (b x +a \right )}{a^{4}}+\frac {b^{4} c^{2} \ln \relax (x )}{a^{5}}-\frac {b^{4} c^{2} \ln \left (b x +a \right )}{a^{5}}+\frac {b \,d^{2}}{a^{2} x}-\frac {2 b^{2} c d}{a^{3} x}+\frac {b^{3} c^{2}}{a^{4} x}-\frac {d^{2}}{2 a \,x^{2}}+\frac {b c d}{a^{2} x^{2}}-\frac {b^{2} c^{2}}{2 a^{3} x^{2}}-\frac {2 c d}{3 a \,x^{3}}+\frac {b \,c^{2}}{3 a^{2} x^{3}}-\frac {c^{2}}{4 a \,x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/x^5/(b*x+a),x)

[Out]

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

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maxima [A]  time = 1.06, size = 164, normalized size = 1.44 \begin {gather*} -\frac {{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{5}} + \frac {{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \relax (x)}{a^{5}} - \frac {3 \, a^{3} c^{2} - 12 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + 6 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 4 \, {\left (a^{2} b c^{2} - 2 \, a^{3} c d\right )} x}{12 \, a^{4} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/x^5/(b*x+a),x, algorithm="maxima")

[Out]

-(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*log(b*x + a)/a^5 + (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*log(x)/a^5 - 1
/12*(3*a^3*c^2 - 12*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^3 + 6*(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*x^2 - 4*(a
^2*b*c^2 - 2*a^3*c*d)*x)/(a^4*x^4)

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mupad [B]  time = 0.42, size = 160, normalized size = 1.40 \begin {gather*} -\frac {\frac {c^2}{4\,a}+\frac {x^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,a^3}-\frac {b\,x^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{a^4}+\frac {c\,x\,\left (2\,a\,d-b\,c\right )}{3\,a^2}}{x^4}-\frac {2\,b^2\,\mathrm {atanh}\left (\frac {b^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+2\,b\,x\right )}{a\,\left (a^2\,b^2\,d^2-2\,a\,b^3\,c\,d+b^4\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{a^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/(x^5*(a + b*x)),x)

[Out]

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

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sympy [B]  time = 1.08, size = 287, normalized size = 2.52 \begin {gather*} \frac {- 3 a^{3} c^{2} + x^{3} \left (12 a^{2} b d^{2} - 24 a b^{2} c d + 12 b^{3} c^{2}\right ) + x^{2} \left (- 6 a^{3} d^{2} + 12 a^{2} b c d - 6 a b^{2} c^{2}\right ) + x \left (- 8 a^{3} c d + 4 a^{2} b c^{2}\right )}{12 a^{4} x^{4}} + \frac {b^{2} \left (a d - b c\right )^{2} \log {\left (x + \frac {a^{3} b^{2} d^{2} - 2 a^{2} b^{3} c d + a b^{4} c^{2} - a b^{2} \left (a d - b c\right )^{2}}{2 a^{2} b^{3} d^{2} - 4 a b^{4} c d + 2 b^{5} c^{2}} \right )}}{a^{5}} - \frac {b^{2} \left (a d - b c\right )^{2} \log {\left (x + \frac {a^{3} b^{2} d^{2} - 2 a^{2} b^{3} c d + a b^{4} c^{2} + a b^{2} \left (a d - b c\right )^{2}}{2 a^{2} b^{3} d^{2} - 4 a b^{4} c d + 2 b^{5} c^{2}} \right )}}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/x**5/(b*x+a),x)

[Out]

(-3*a**3*c**2 + x**3*(12*a**2*b*d**2 - 24*a*b**2*c*d + 12*b**3*c**2) + x**2*(-6*a**3*d**2 + 12*a**2*b*c*d - 6*
a*b**2*c**2) + x*(-8*a**3*c*d + 4*a**2*b*c**2))/(12*a**4*x**4) + b**2*(a*d - b*c)**2*log(x + (a**3*b**2*d**2 -
2*a**2*b**3*c*d + a*b**4*c**2 - a*b**2*(a*d - b*c)**2)/(2*a**2*b**3*d**2 - 4*a*b**4*c*d + 2*b**5*c**2))/a**5
- b**2*(a*d - b*c)**2*log(x + (a**3*b**2*d**2 - 2*a**2*b**3*c*d + a*b**4*c**2 + a*b**2*(a*d - b*c)**2)/(2*a**2
*b**3*d**2 - 4*a*b**4*c*d + 2*b**5*c**2))/a**5

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