3.3.77 \(\int \frac {(c+d x)^3}{x^4 (a+b x)^2} \, dx\) [277]

Optimal. Leaf size=132 \[ -\frac {c^3}{3 a^2 x^3}+\frac {c^2 (2 b c-3 a d)}{2 a^3 x^2}-\frac {3 c (b c-a d)^2}{a^4 x}-\frac {(b c-a d)^3}{a^4 (a+b x)}-\frac {(b c-a d)^2 (4 b c-a d) \log (x)}{a^5}+\frac {(b c-a d)^2 (4 b c-a d) \log (a+b x)}{a^5} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 132, 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)^2 (4 b c-a d)}{a^5}+\frac {(b c-a d)^2 (4 b c-a d) \log (a+b x)}{a^5}-\frac {3 c (b c-a d)^2}{a^4 x}-\frac {(b c-a d)^3}{a^4 (a+b x)}+\frac {c^2 (2 b c-3 a d)}{2 a^3 x^2}-\frac {c^3}{3 a^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^4*(a + b*x)^2),x]

[Out]

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

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

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Mathematica [A]
time = 0.05, size = 126, normalized size = 0.95 \begin {gather*} -\frac {\frac {2 a^3 c^3}{x^3}+\frac {3 a^2 c^2 (-2 b c+3 a d)}{x^2}+\frac {18 a c (b c-a d)^2}{x}-\frac {6 a (-b c+a d)^3}{a+b x}+6 (b c-a d)^2 (4 b c-a d) \log (x)-6 (b c-a d)^2 (4 b c-a d) \log (a+b x)}{6 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(x^4*(a + b*x)^2),x]

[Out]

-1/6*((2*a^3*c^3)/x^3 + (3*a^2*c^2*(-2*b*c + 3*a*d))/x^2 + (18*a*c*(b*c - a*d)^2)/x - (6*a*(-(b*c) + a*d)^3)/(
a + b*x) + 6*(b*c - a*d)^2*(4*b*c - a*d)*Log[x] - 6*(b*c - a*d)^2*(4*b*c - a*d)*Log[a + b*x])/a^5

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Maple [A]
time = 0.08, size = 199, normalized size = 1.51

method	result	size
default	\(-\frac {\left (a^{3} d^{3}-6 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{5}}+\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{a^{4} \left (b x +a \right )}-\frac {c^{3}}{3 a^{2} x^{3}}+\frac {\left (a^{3} d^{3}-6 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{5}}-\frac {c^{2} \left (3 a d -2 b c \right )}{2 a^{3} x^{2}}-\frac {3 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{a^{4} x}\)	\(199\)
norman	\(\frac {\frac {b \left (-a^{3} d^{3}+6 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) x^{4}}{a^{5}}-\frac {c^{3}}{3 a}-\frac {c \left (6 a^{2} d^{2}-9 a b c d +4 b^{2} c^{2}\right ) x^{2}}{2 a^{3}}-\frac {c^{2} \left (9 a d -4 b c \right ) x}{6 a^{2}}}{x^{3} \left (b x +a \right )}+\frac {\left (a^{3} d^{3}-6 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{5}}-\frac {\left (a^{3} d^{3}-6 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{5}}\)	\(206\)
risch	\(\frac {\frac {\left (a^{3} d^{3}-6 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{3}}{a^{4}}-\frac {c \left (6 a^{2} d^{2}-9 a b c d +4 b^{2} c^{2}\right ) x^{2}}{2 a^{3}}-\frac {c^{2} \left (9 a d -4 b c \right ) x}{6 a^{2}}-\frac {c^{3}}{3 a}}{x^{3} \left (b x +a \right )}-\frac {\ln \left (b x +a \right ) d^{3}}{a^{2}}+\frac {6 \ln \left (b x +a \right ) b c \,d^{2}}{a^{3}}-\frac {9 \ln \left (b x +a \right ) b^{2} c^{2} d}{a^{4}}+\frac {4 \ln \left (b x +a \right ) b^{3} c^{3}}{a^{5}}+\frac {\ln \left (-x \right ) d^{3}}{a^{2}}-\frac {6 \ln \left (-x \right ) b c \,d^{2}}{a^{3}}+\frac {9 \ln \left (-x \right ) b^{2} c^{2} d}{a^{4}}-\frac {4 \ln \left (-x \right ) b^{3} c^{3}}{a^{5}}\)	\(236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/x^4/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-(a^3*d^3-6*a^2*b*c*d^2+9*a*b^2*c^2*d-4*b^3*c^3)/a^5*ln(b*x+a)+(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/a
^4/(b*x+a)-1/3*c^3/a^2/x^3+(a^3*d^3-6*a^2*b*c*d^2+9*a*b^2*c^2*d-4*b^3*c^3)/a^5*ln(x)-1/2*c^2*(3*a*d-2*b*c)/a^3
/x^2-3*c*(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^4/x

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Maxima [A]
time = 0.31, size = 219, normalized size = 1.66 \begin {gather*} -\frac {2 \, a^{3} c^{3} + 6 \, {\left (4 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} + 3 \, {\left (4 \, a b^{2} c^{3} - 9 \, a^{2} b c^{2} d + 6 \, a^{3} c d^{2}\right )} x^{2} - {\left (4 \, a^{2} b c^{3} - 9 \, a^{3} c^{2} d\right )} x}{6 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} + \frac {{\left (4 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{5}} - \frac {{\left (4 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x\right )}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/6*(2*a^3*c^3 + 6*(4*b^3*c^3 - 9*a*b^2*c^2*d + 6*a^2*b*c*d^2 - a^3*d^3)*x^3 + 3*(4*a*b^2*c^3 - 9*a^2*b*c^2*d
+ 6*a^3*c*d^2)*x^2 - (4*a^2*b*c^3 - 9*a^3*c^2*d)*x)/(a^4*b*x^4 + a^5*x^3) + (4*b^3*c^3 - 9*a*b^2*c^2*d + 6*a^
2*b*c*d^2 - a^3*d^3)*log(b*x + a)/a^5 - (4*b^3*c^3 - 9*a*b^2*c^2*d + 6*a^2*b*c*d^2 - a^3*d^3)*log(x)/a^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (128) = 256\).
time = 1.45, size = 322, normalized size = 2.44 \begin {gather*} -\frac {2 \, a^{4} c^{3} + 6 \, {\left (4 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3} + 3 \, {\left (4 \, a^{2} b^{2} c^{3} - 9 \, a^{3} b c^{2} d + 6 \, a^{4} c d^{2}\right )} x^{2} - {\left (4 \, a^{3} b c^{3} - 9 \, a^{4} c^{2} d\right )} x - 6 \, {\left ({\left (4 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} + {\left (4 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) + 6 \, {\left ({\left (4 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} + {\left (4 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \, {\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^4/(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/6*(2*a^4*c^3 + 6*(4*a*b^3*c^3 - 9*a^2*b^2*c^2*d + 6*a^3*b*c*d^2 - a^4*d^3)*x^3 + 3*(4*a^2*b^2*c^3 - 9*a^3*b
*c^2*d + 6*a^4*c*d^2)*x^2 - (4*a^3*b*c^3 - 9*a^4*c^2*d)*x - 6*((4*b^4*c^3 - 9*a*b^3*c^2*d + 6*a^2*b^2*c*d^2 -
a^3*b*d^3)*x^4 + (4*a*b^3*c^3 - 9*a^2*b^2*c^2*d + 6*a^3*b*c*d^2 - a^4*d^3)*x^3)*log(b*x + a) + 6*((4*b^4*c^3 -
9*a*b^3*c^2*d + 6*a^2*b^2*c*d^2 - a^3*b*d^3)*x^4 + (4*a*b^3*c^3 - 9*a^2*b^2*c^2*d + 6*a^3*b*c*d^2 - a^4*d^3)*
x^3)*log(x))/(a^5*b*x^4 + a^6*x^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (117) = 234\).
time = 0.79, size = 386, normalized size = 2.92 \begin {gather*} \frac {- 2 a^{3} c^{3} + x^{3} \cdot \left (6 a^{3} d^{3} - 36 a^{2} b c d^{2} + 54 a b^{2} c^{2} d - 24 b^{3} c^{3}\right ) + x^{2} \left (- 18 a^{3} c d^{2} + 27 a^{2} b c^{2} d - 12 a b^{2} c^{3}\right ) + x \left (- 9 a^{3} c^{2} d + 4 a^{2} b c^{3}\right )}{6 a^{5} x^{3} + 6 a^{4} b x^{4}} + \frac {\left (a d - 4 b c\right ) \left (a d - b c\right )^{2} \log {\left (x + \frac {a^{4} d^{3} - 6 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - 4 a b^{3} c^{3} - a \left (a d - 4 b c\right ) \left (a d - b c\right )^{2}}{2 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 18 a b^{3} c^{2} d - 8 b^{4} c^{3}} \right )}}{a^{5}} - \frac {\left (a d - 4 b c\right ) \left (a d - b c\right )^{2} \log {\left (x + \frac {a^{4} d^{3} - 6 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - 4 a b^{3} c^{3} + a \left (a d - 4 b c\right ) \left (a d - b c\right )^{2}}{2 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 18 a b^{3} c^{2} d - 8 b^{4} c^{3}} \right )}}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/x**4/(b*x+a)**2,x)

[Out]

(-2*a**3*c**3 + x**3*(6*a**3*d**3 - 36*a**2*b*c*d**2 + 54*a*b**2*c**2*d - 24*b**3*c**3) + x**2*(-18*a**3*c*d**
2 + 27*a**2*b*c**2*d - 12*a*b**2*c**3) + x*(-9*a**3*c**2*d + 4*a**2*b*c**3))/(6*a**5*x**3 + 6*a**4*b*x**4) + (
a*d - 4*b*c)*(a*d - b*c)**2*log(x + (a**4*d**3 - 6*a**3*b*c*d**2 + 9*a**2*b**2*c**2*d - 4*a*b**3*c**3 - a*(a*d
- 4*b*c)*(a*d - b*c)**2)/(2*a**3*b*d**3 - 12*a**2*b**2*c*d**2 + 18*a*b**3*c**2*d - 8*b**4*c**3))/a**5 - (a*d
- 4*b*c)*(a*d - b*c)**2*log(x + (a**4*d**3 - 6*a**3*b*c*d**2 + 9*a**2*b**2*c**2*d - 4*a*b**3*c**3 + a*(a*d - 4
*b*c)*(a*d - b*c)**2)/(2*a**3*b*d**3 - 12*a**2*b**2*c*d**2 + 18*a*b**3*c**2*d - 8*b**4*c**3))/a**5

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (128) = 256\).
time = 1.70, size = 280, normalized size = 2.12 \begin {gather*} -\frac {{\left (4 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{5} b} - \frac {\frac {b^{7} c^{3}}{b x + a} - \frac {3 \, a b^{6} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{5} c d^{2}}{b x + a} - \frac {a^{3} b^{4} d^{3}}{b x + a}}{a^{4} b^{4}} + \frac {26 \, b^{3} c^{3} - 45 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - \frac {3 \, {\left (20 \, a b^{4} c^{3} - 33 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2}\right )}}{{\left (b x + a\right )} b} + \frac {18 \, {\left (2 \, a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + a^{4} b^{3} c d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{6 \, a^{5} {\left (\frac {a}{b x + a} - 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/x^4/(b*x+a)^2,x, algorithm="giac")

[Out]

-(4*b^4*c^3 - 9*a*b^3*c^2*d + 6*a^2*b^2*c*d^2 - a^3*b*d^3)*log(abs(-a/(b*x + a) + 1))/(a^5*b) - (b^7*c^3/(b*x
+ a) - 3*a*b^6*c^2*d/(b*x + a) + 3*a^2*b^5*c*d^2/(b*x + a) - a^3*b^4*d^3/(b*x + a))/(a^4*b^4) + 1/6*(26*b^3*c^
3 - 45*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 3*(20*a*b^4*c^3 - 33*a^2*b^3*c^2*d + 12*a^3*b^2*c*d^2)/((b*x + a)*b) + 1
8*(2*a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + a^4*b^3*c*d^2)/((b*x + a)^2*b^2))/(a^5*(a/(b*x + a) - 1)^3)

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Mupad [B]
time = 0.44, size = 209, normalized size = 1.58 \begin {gather*} -\frac {\frac {c^3}{3\,a}-\frac {x^3\,\left (a^3\,d^3-6\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d-4\,b^3\,c^3\right )}{a^4}+\frac {c^2\,x\,\left (9\,a\,d-4\,b\,c\right )}{6\,a^2}+\frac {c\,x^2\,\left (6\,a^2\,d^2-9\,a\,b\,c\,d+4\,b^2\,c^2\right )}{2\,a^3}}{b\,x^4+a\,x^3}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (a\,d-b\,c\right )}^2\,\left (a\,d-4\,b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (a^3\,d^3-6\,a^2\,b\,c\,d^2+9\,a\,b^2\,c^2\,d-4\,b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d-4\,b\,c\right )}{a^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^3/(x^4*(a + b*x)^2),x)

[Out]

- (c^3/(3*a) - (x^3*(a^3*d^3 - 4*b^3*c^3 + 9*a*b^2*c^2*d - 6*a^2*b*c*d^2))/a^4 + (c^2*x*(9*a*d - 4*b*c))/(6*a^
2) + (c*x^2*(6*a^2*d^2 + 4*b^2*c^2 - 9*a*b*c*d))/(2*a^3))/(a*x^3 + b*x^4) - (2*atanh(((a*d - b*c)^2*(a*d - 4*b
*c)*(a + 2*b*x))/(a*(a^3*d^3 - 4*b^3*c^3 + 9*a*b^2*c^2*d - 6*a^2*b*c*d^2)))*(a*d - b*c)^2*(a*d - 4*b*c))/a^5

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