3.3.29 \(\int \frac {(c+d x)^3}{x^4 (a+b x)} \, dx\) [229]

Optimal. Leaf size=103 \[ -\frac {c^3}{3 a x^3}+\frac {c^2 (b c-3 a d)}{2 a^2 x^2}-\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x}-\frac {(b c-a d)^3 \log (x)}{a^4}+\frac {(b c-a d)^3 \log (a+b x)}{a^4} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 103, 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 {\log (x) (b c-a d)^3}{a^4}+\frac {(b c-a d)^3 \log (a+b x)}{a^4}+\frac {c^2 (b c-3 a d)}{2 a^2 x^2}-\frac {c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3 x}-\frac {c^3}{3 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^4*(a + b*x)),x]

[Out]

-1/3*c^3/(a*x^3) + (c^2*(b*c - 3*a*d))/(2*a^2*x^2) - (c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2))/(a^3*x) - ((b*c - a
*d)^3*Log[x])/a^4 + ((b*c - a*d)^3*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)^3}{x^4 (a+b x)} \, dx &=\int \left (\frac {c^3}{a x^4}+\frac {c^2 (-b c+3 a d)}{a^2 x^3}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x^2}+\frac {(-b c+a d)^3}{a^4 x}-\frac {b (-b c+a d)^3}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac {c^3}{3 a x^3}+\frac {c^2 (b c-3 a d)}{2 a^2 x^2}-\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x}-\frac {(b c-a d)^3 \log (x)}{a^4}+\frac {(b c-a d)^3 \log (a+b x)}{a^4}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 93, normalized size = 0.90 \begin {gather*} -\frac {\frac {a c \left (6 b^2 c^2 x^2-3 a b c x (c+6 d x)+a^2 \left (2 c^2+9 c d x+18 d^2 x^2\right )\right )}{x^3}+6 (b c-a d)^3 \log (x)-6 (b c-a d)^3 \log (a+b x)}{6 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(x^4*(a + b*x)),x]

[Out]

-1/6*((a*c*(6*b^2*c^2*x^2 - 3*a*b*c*x*(c + 6*d*x) + a^2*(2*c^2 + 9*c*d*x + 18*d^2*x^2)))/x^3 + 6*(b*c - a*d)^3
*Log[x] - 6*(b*c - a*d)^3*Log[a + b*x])/a^4

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Maple [A]
time = 0.08, size = 153, normalized size = 1.49

method	result	size
default	\(-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{4}}-\frac {c^{3}}{3 a \,x^{3}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (x \right )}{a^{4}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{a^{3} x}-\frac {c^{2} \left (3 a d -b c \right )}{2 a^{2} x^{2}}\)	\(153\)
norman	\(\frac {-\frac {c^{3}}{3 a}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{2}}{a^{3}}-\frac {c^{2} \left (3 a d -b c \right ) x}{2 a^{2}}}{x^{3}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (x \right )}{a^{4}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{4}}\)	\(153\)
risch	\(\frac {-\frac {c^{3}}{3 a}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{2}}{a^{3}}-\frac {c^{2} \left (3 a d -b c \right ) x}{2 a^{2}}}{x^{3}}-\frac {\ln \left (b x +a \right ) d^{3}}{a}+\frac {3 \ln \left (b x +a \right ) b c \,d^{2}}{a^{2}}-\frac {3 \ln \left (b x +a \right ) b^{2} c^{2} d}{a^{3}}+\frac {\ln \left (b x +a \right ) b^{3} c^{3}}{a^{4}}+\frac {\ln \left (-x \right ) d^{3}}{a}-\frac {3 \ln \left (-x \right ) b c \,d^{2}}{a^{2}}+\frac {3 \ln \left (-x \right ) b^{2} c^{2} d}{a^{3}}-\frac {\ln \left (-x \right ) b^{3} c^{3}}{a^{4}}\)	\(184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/x^4/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.33, size = 156, normalized size = 1.51 \begin {gather*} \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{4}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x\right )}{a^{4}} - \frac {2 \, a^{2} c^{3} + 6 \, {\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{2} - 3 \, {\left (a b c^{3} - 3 \, a^{2} c^{2} 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)^3/x^4/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.04, size = 161, normalized size = 1.56 \begin {gather*} -\frac {2 \, a^{3} c^{3} - 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (b x + a\right ) + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (x\right ) + 6 \, {\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 3 \, {\left (a^{2} b c^{3} - 3 \, a^{3} c^{2} 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)^3/x^4/(b*x+a),x, algorithm="fricas")

[Out]

-1/6*(2*a^3*c^3 - 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^3*log(b*x + a) + 6*(b^3*c^3 - 3*a*b^
2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^3*log(x) + 6*(a*b^2*c^3 - 3*a^2*b*c^2*d + 3*a^3*c*d^2)*x^2 - 3*(a^2*b*c^3
- 3*a^3*c^2*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. 289 vs. \(2 (94) = 188\).
time = 0.61, size = 289, normalized size = 2.81 \begin {gather*} \frac {- 2 a^{2} c^{3} + x^{2} \left (- 18 a^{2} c d^{2} + 18 a b c^{2} d - 6 b^{2} c^{3}\right ) + x \left (- 9 a^{2} c^{2} d + 3 a b c^{3}\right )}{6 a^{3} x^{3}} + \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3} - a \left (a d - b c\right )^{3}}{2 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 6 a b^{3} c^{2} d - 2 b^{4} c^{3}} \right )}}{a^{4}} - \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3} + a \left (a d - b c\right )^{3}}{2 a^{3} b d^{3} - 6 a^{2} b^{2} c d^{2} + 6 a b^{3} c^{2} d - 2 b^{4} c^{3}} \right )}}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/x**4/(b*x+a),x)

[Out]

(-2*a**2*c**3 + x**2*(-18*a**2*c*d**2 + 18*a*b*c**2*d - 6*b**2*c**3) + x*(-9*a**2*c**2*d + 3*a*b*c**3))/(6*a**
3*x**3) + (a*d - b*c)**3*log(x + (a**4*d**3 - 3*a**3*b*c*d**2 + 3*a**2*b**2*c**2*d - a*b**3*c**3 - a*(a*d - b*
c)**3)/(2*a**3*b*d**3 - 6*a**2*b**2*c*d**2 + 6*a*b**3*c**2*d - 2*b**4*c**3))/a**4 - (a*d - b*c)**3*log(x + (a*
*4*d**3 - 3*a**3*b*c*d**2 + 3*a**2*b**2*c**2*d - a*b**3*c**3 + a*(a*d - b*c)**3)/(2*a**3*b*d**3 - 6*a**2*b**2*
c*d**2 + 6*a*b**3*c**2*d - 2*b**4*c**3))/a**4

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Giac [A]
time = 1.59, size = 169, normalized size = 1.64 \begin {gather*} -\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, a^{3} c^{3} + 6 \, {\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 3 \, {\left (a^{2} b c^{3} - 3 \, a^{3} c^{2} 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)^3/x^4/(b*x+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.39, size = 137, normalized size = 1.33 \begin {gather*} -\frac {\frac {c^3}{3\,a}+\frac {c^2\,x\,\left (3\,a\,d-b\,c\right )}{2\,a^2}+\frac {c\,x^2\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{a^3}}{x^3}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (a\,d-b\,c\right )}^3\,\left (a+2\,b\,x\right )}{a\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(x^4*(a + b*x)),x)

[Out]

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

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