3.3.31 \(\int \frac {(c+d x)^3}{x^6 (a+b x)} \, dx\) [231]

Optimal. Leaf size=150 \[ -\frac {c^3}{5 a x^5}+\frac {c^2 (b c-3 a d)}{4 a^2 x^4}-\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^3}+\frac {(b c-a d)^3}{2 a^4 x^2}-\frac {b (b c-a d)^3}{a^5 x}-\frac {b^2 (b c-a d)^3 \log (x)}{a^6}+\frac {b^2 (b c-a d)^3 \log (a+b x)}{a^6} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 150, 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^2 \log (x) (b c-a d)^3}{a^6}+\frac {b^2 (b c-a d)^3 \log (a+b x)}{a^6}-\frac {b (b c-a d)^3}{a^5 x}+\frac {(b c-a d)^3}{2 a^4 x^2}+\frac {c^2 (b c-3 a d)}{4 a^2 x^4}-\frac {c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 a^3 x^3}-\frac {c^3}{5 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^6*(a + b*x)),x]

[Out]

-1/5*c^3/(a*x^5) + (c^2*(b*c - 3*a*d))/(4*a^2*x^4) - (c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2))/(3*a^3*x^3) + (b*c
- a*d)^3/(2*a^4*x^2) - (b*(b*c - a*d)^3)/(a^5*x) - (b^2*(b*c - a*d)^3*Log[x])/a^6 + (b^2*(b*c - a*d)^3*Log[a +
b*x])/a^6

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^6 (a+b x)} \, dx &=\int \left (\frac {c^3}{a x^6}+\frac {c^2 (-b c+3 a d)}{a^2 x^5}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x^4}+\frac {(-b c+a d)^3}{a^4 x^3}-\frac {b (-b c+a d)^3}{a^5 x^2}+\frac {b^2 (-b c+a d)^3}{a^6 x}-\frac {b^3 (-b c+a d)^3}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac {c^3}{5 a x^5}+\frac {c^2 (b c-3 a d)}{4 a^2 x^4}-\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^3}+\frac {(b c-a d)^3}{2 a^4 x^2}-\frac {b (b c-a d)^3}{a^5 x}-\frac {b^2 (b c-a d)^3 \log (x)}{a^6}+\frac {b^2 (b c-a d)^3 \log (a+b x)}{a^6}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 188, normalized size = 1.25 \begin {gather*} \frac {-60 a b^4 c^3 x^4+30 a^2 b^3 c^2 x^3 (c+6 d x)-10 a^3 b^2 c x^2 \left (2 c^2+9 c d x+18 d^2 x^2\right )+15 a^4 b x \left (c^3+4 c^2 d x+6 c d^2 x^2+4 d^3 x^3\right )-3 a^5 \left (4 c^3+15 c^2 d x+20 c d^2 x^2+10 d^3 x^3\right )-60 b^2 (b c-a d)^3 x^5 \log (x)+60 b^2 (b c-a d)^3 x^5 \log (a+b x)}{60 a^6 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(x^6*(a + b*x)),x]

[Out]

(-60*a*b^4*c^3*x^4 + 30*a^2*b^3*c^2*x^3*(c + 6*d*x) - 10*a^3*b^2*c*x^2*(2*c^2 + 9*c*d*x + 18*d^2*x^2) + 15*a^4
*b*x*(c^3 + 4*c^2*d*x + 6*c*d^2*x^2 + 4*d^3*x^3) - 3*a^5*(4*c^3 + 15*c^2*d*x + 20*c*d^2*x^2 + 10*d^3*x^3) - 60
*b^2*(b*c - a*d)^3*x^5*Log[x] + 60*b^2*(b*c - a*d)^3*x^5*Log[a + b*x])/(60*a^6*x^5)

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Maple [A]
time = 0.08, size = 247, normalized size = 1.65

method	result	size
default	\(-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \ln \left (b x +a \right )}{a^{6}}-\frac {c^{3}}{5 a \,x^{5}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{2 a^{4} x^{2}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right )}{3 a^{3} x^{3}}-\frac {c^{2} \left (3 a d -b c \right )}{4 a^{2} x^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \ln \left (x \right )}{a^{6}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}{a^{5} x}\)	\(247\)
norman	\(\frac {\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b \,x^{4}}{a^{5}}-\frac {c^{3}}{5 a}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{3}}{2 a^{4}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{2}}{3 a^{3}}-\frac {c^{2} \left (3 a d -b c \right ) x}{4 a^{2}}}{x^{5}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \ln \left (x \right )}{a^{6}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \ln \left (b x +a \right )}{a^{6}}\)	\(247\)
risch	\(\frac {\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b \,x^{4}}{a^{5}}-\frac {c^{3}}{5 a}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{3}}{2 a^{4}}-\frac {c \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{2}}{3 a^{3}}-\frac {c^{2} \left (3 a d -b c \right ) x}{4 a^{2}}}{x^{5}}+\frac {b^{2} \ln \left (-x \right ) d^{3}}{a^{3}}-\frac {3 b^{3} \ln \left (-x \right ) c \,d^{2}}{a^{4}}+\frac {3 b^{4} \ln \left (-x \right ) c^{2} d}{a^{5}}-\frac {b^{5} \ln \left (-x \right ) c^{3}}{a^{6}}-\frac {b^{2} \ln \left (b x +a \right ) d^{3}}{a^{3}}+\frac {3 b^{3} \ln \left (b x +a \right ) c \,d^{2}}{a^{4}}-\frac {3 b^{4} \ln \left (b x +a \right ) c^{2} d}{a^{5}}+\frac {b^{5} \ln \left (b x +a \right ) c^{3}}{a^{6}}\)	\(282\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]
time = 0.28, size = 261, normalized size = 1.74 \begin {gather*} \frac {{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (b x + a\right )}{a^{6}} - \frac {{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (x\right )}{a^{6}} - \frac {12 \, a^{4} c^{3} + 60 \, {\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 )} x^{4} - 30 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3} + 20 \, {\left (a^{2} b^{2} c^{3} - 3 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 15 \, {\left (a^{3} b c^{3} - 3 \, a^{4} c^{2} d\right )} x}{60 \, a^{5} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^6/(b*x+a),x, algorithm="maxima")

[Out]

(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(b*x + a)/a^6 - (b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*
b^3*c*d^2 - a^3*b^2*d^3)*log(x)/a^6 - 1/60*(12*a^4*c^3 + 60*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b
*d^3)*x^4 - 30*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*x^3 + 20*(a^2*b^2*c^3 - 3*a^3*b*c^2*d +
3*a^4*c*d^2)*x^2 - 15*(a^3*b*c^3 - 3*a^4*c^2*d)*x)/(a^5*x^5)

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Fricas [A]
time = 1.29, size = 266, normalized size = 1.77 \begin {gather*} -\frac {12 \, a^{5} c^{3} - 60 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} \log \left (b x + a\right ) + 60 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} \log \left (x\right ) + 60 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} - 30 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 20 \, {\left (a^{3} b^{2} c^{3} - 3 \, a^{4} b c^{2} d + 3 \, a^{5} c d^{2}\right )} x^{2} - 15 \, {\left (a^{4} b c^{3} - 3 \, a^{5} c^{2} d\right )} x}{60 \, a^{6} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^6/(b*x+a),x, algorithm="fricas")

[Out]

-1/60*(12*a^5*c^3 - 60*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5*log(b*x + a) + 60*(b^5*c^
3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5*log(x) + 60*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*
d^2 - a^4*b*d^3)*x^4 - 30*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*x^3 + 20*(a^3*b^2*c^3 - 3*
a^4*b*c^2*d + 3*a^5*c*d^2)*x^2 - 15*(a^4*b*c^3 - 3*a^5*c^2*d)*x)/(a^6*x^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (136) = 272\).
time = 0.78, size = 418, normalized size = 2.79 \begin {gather*} \frac {- 12 a^{4} c^{3} + x^{4} \cdot \left (60 a^{3} b d^{3} - 180 a^{2} b^{2} c d^{2} + 180 a b^{3} c^{2} d - 60 b^{4} c^{3}\right ) + x^{3} \left (- 30 a^{4} d^{3} + 90 a^{3} b c d^{2} - 90 a^{2} b^{2} c^{2} d + 30 a b^{3} c^{3}\right ) + x^{2} \left (- 60 a^{4} c d^{2} + 60 a^{3} b c^{2} d - 20 a^{2} b^{2} c^{3}\right ) + x \left (- 45 a^{4} c^{2} d + 15 a^{3} b c^{3}\right )}{60 a^{5} x^{5}} + \frac {b^{2} \left (a d - b c\right )^{3} \log {\left (x + \frac {a^{4} b^{2} d^{3} - 3 a^{3} b^{3} c d^{2} + 3 a^{2} b^{4} c^{2} d - a b^{5} c^{3} - a b^{2} \left (a d - b c\right )^{3}}{2 a^{3} b^{3} d^{3} - 6 a^{2} b^{4} c d^{2} + 6 a b^{5} c^{2} d - 2 b^{6} c^{3}} \right )}}{a^{6}} - \frac {b^{2} \left (a d - b c\right )^{3} \log {\left (x + \frac {a^{4} b^{2} d^{3} - 3 a^{3} b^{3} c d^{2} + 3 a^{2} b^{4} c^{2} d - a b^{5} c^{3} + a b^{2} \left (a d - b c\right )^{3}}{2 a^{3} b^{3} d^{3} - 6 a^{2} b^{4} c d^{2} + 6 a b^{5} c^{2} d - 2 b^{6} c^{3}} \right )}}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/x**6/(b*x+a),x)

[Out]

(-12*a**4*c**3 + x**4*(60*a**3*b*d**3 - 180*a**2*b**2*c*d**2 + 180*a*b**3*c**2*d - 60*b**4*c**3) + x**3*(-30*a
**4*d**3 + 90*a**3*b*c*d**2 - 90*a**2*b**2*c**2*d + 30*a*b**3*c**3) + x**2*(-60*a**4*c*d**2 + 60*a**3*b*c**2*d
- 20*a**2*b**2*c**3) + x*(-45*a**4*c**2*d + 15*a**3*b*c**3))/(60*a**5*x**5) + b**2*(a*d - b*c)**3*log(x + (a*
*4*b**2*d**3 - 3*a**3*b**3*c*d**2 + 3*a**2*b**4*c**2*d - a*b**5*c**3 - a*b**2*(a*d - b*c)**3)/(2*a**3*b**3*d**
3 - 6*a**2*b**4*c*d**2 + 6*a*b**5*c**2*d - 2*b**6*c**3))/a**6 - b**2*(a*d - b*c)**3*log(x + (a**4*b**2*d**3 -
3*a**3*b**3*c*d**2 + 3*a**2*b**4*c**2*d - a*b**5*c**3 + a*b**2*(a*d - b*c)**3)/(2*a**3*b**3*d**3 - 6*a**2*b**4
*c*d**2 + 6*a*b**5*c**2*d - 2*b**6*c**3))/a**6

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Giac [A]
time = 1.42, size = 271, normalized size = 1.81 \begin {gather*} -\frac {{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac {12 \, a^{5} c^{3} + 60 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} - 30 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 20 \, {\left (a^{3} b^{2} c^{3} - 3 \, a^{4} b c^{2} d + 3 \, a^{5} c d^{2}\right )} x^{2} - 15 \, {\left (a^{4} b c^{3} - 3 \, a^{5} c^{2} d\right )} x}{60 \, a^{6} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^6/(b*x+a),x, algorithm="giac")

[Out]

-(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(abs(x))/a^6 + (b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*
b^4*c*d^2 - a^3*b^3*d^3)*log(abs(b*x + a))/(a^6*b) - 1/60*(12*a^5*c^3 + 60*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^
3*b^2*c*d^2 - a^4*b*d^3)*x^4 - 30*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*x^3 + 20*(a^3*b^2*
c^3 - 3*a^4*b*c^2*d + 3*a^5*c*d^2)*x^2 - 15*(a^4*b*c^3 - 3*a^5*c^2*d)*x)/(a^6*x^5)

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Mupad [B]
time = 0.42, size = 238, normalized size = 1.59 \begin {gather*} \frac {2\,b^2\,\mathrm {atanh}\left (\frac {b^2\,{\left (a\,d-b\,c\right )}^3\,\left (a+2\,b\,x\right )}{a\,\left (-a^3\,b^2\,d^3+3\,a^2\,b^3\,c\,d^2-3\,a\,b^4\,c^2\,d+b^5\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^6}-\frac {\frac {c^3}{5\,a}+\frac {x^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,a^4}+\frac {c^2\,x\,\left (3\,a\,d-b\,c\right )}{4\,a^2}+\frac {c\,x^2\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{3\,a^3}-\frac {b\,x^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{a^5}}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(x^6*(a + b*x)),x)

[Out]

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

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