3.3.20 \(\int \frac {(c+d x)^2}{x^4 (a+b x)} \, dx\) [220]

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

[Out]

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

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Rubi [A]
time = 0.04, 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 = {90} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(x^4*(a + b*x)),x]

[Out]

-1/3*c^2/(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 90

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 = 0.04, size = 99, normalized size = 1.10 \begin {gather*} \frac {-6 a b^2 c^2 x^2+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 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} \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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Maple [A]
time = 0.08, size = 121, normalized size = 1.34

method	result	size
default	\(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \ln \left (b x +a \right )}{a^{4}}-\frac {c^{2}}{3 a \,x^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{a^{3} x}-\frac {c \left (2 a d -b c \right )}{2 a^{2} x^{2}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \ln \left (x \right )}{a^{4}}\)	\(121\)
norman	\(\frac {-\frac {c^{2}}{3 a}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{2}}{a^{3}}-\frac {c \left (2 a d -b c \right ) x}{2 a^{2}}}{x^{3}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \ln \left (b x +a \right )}{a^{4}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \ln \left (x \right )}{a^{4}}\)	\(121\)
risch	\(\frac {-\frac {c^{2}}{3 a}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{2}}{a^{3}}-\frac {c \left (2 a d -b c \right ) x}{2 a^{2}}}{x^{3}}+\frac {b \ln \left (-b x -a \right ) d^{2}}{a^{2}}-\frac {2 b^{2} \ln \left (-b x -a \right ) c d}{a^{3}}+\frac {b^{3} \ln \left (-b x -a \right ) c^{2}}{a^{4}}-\frac {b \ln \left (x \right ) d^{2}}{a^{2}}+\frac {2 b^{2} \ln \left (x \right ) c d}{a^{3}}-\frac {b^{3} \ln \left (x \right ) c^{2}}{a^{4}}\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, 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 \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 {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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Fricas [A]
time = 1.16, 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 \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 {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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (78) = 156\).
time = 0.40, 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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Giac [A]
time = 1.43, 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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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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