3.308 \(\int \frac{(c+d x)^3}{x^4 (a+b x)^3} \, dx\)

Optimal. Leaf size=193 \[ -\frac{\log (x) (b c-a d) \left (a^2 d^2-8 a b c d+10 b^2 c^2\right )}{a^6}+\frac{(b c-a d) \left (a^2 d^2-8 a b c d+10 b^2 c^2\right ) \log (a+b x)}{a^6}+\frac{3 c^2 (b c-a d)}{2 a^4 x^2}-\frac{3 c (b c-a d) (2 b c-a d)}{a^5 x}-\frac{(b c-a d)^2 (4 b c-a d)}{a^5 (a+b x)}-\frac{(b c-a d)^3}{2 a^4 (a+b x)^2}-\frac{c^3}{3 a^3 x^3} \]

[Out]

-c^3/(3*a^3*x^3) + (3*c^2*(b*c - a*d))/(2*a^4*x^2) - (3*c*(b*c - a*d)*(2*b*c - a*d))/(a^5*x) - (b*c - a*d)^3/(
2*a^4*(a + b*x)^2) - ((b*c - a*d)^2*(4*b*c - a*d))/(a^5*(a + b*x)) - ((b*c - a*d)*(10*b^2*c^2 - 8*a*b*c*d + a^
2*d^2)*Log[x])/a^6 + ((b*c - a*d)*(10*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*Log[a + b*x])/a^6

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Rubi [A]  time = 0.180577, antiderivative size = 193, normalized size of antiderivative = 1., 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} \[ -\frac{\log (x) (b c-a d) \left (a^2 d^2-8 a b c d+10 b^2 c^2\right )}{a^6}+\frac{(b c-a d) \left (a^2 d^2-8 a b c d+10 b^2 c^2\right ) \log (a+b x)}{a^6}+\frac{3 c^2 (b c-a d)}{2 a^4 x^2}-\frac{3 c (b c-a d) (2 b c-a d)}{a^5 x}-\frac{(b c-a d)^2 (4 b c-a d)}{a^5 (a+b x)}-\frac{(b c-a d)^3}{2 a^4 (a+b x)^2}-\frac{c^3}{3 a^3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(x^4*(a + b*x)^3),x]

[Out]

-c^3/(3*a^3*x^3) + (3*c^2*(b*c - a*d))/(2*a^4*x^2) - (3*c*(b*c - a*d)*(2*b*c - a*d))/(a^5*x) - (b*c - a*d)^3/(
2*a^4*(a + b*x)^2) - ((b*c - a*d)^2*(4*b*c - a*d))/(a^5*(a + b*x)) - ((b*c - a*d)*(10*b^2*c^2 - 8*a*b*c*d + a^
2*d^2)*Log[x])/a^6 + ((b*c - a*d)*(10*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*Log[a + b*x])/a^6

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)^3}{x^4 (a+b x)^3} \, dx &=\int \left (\frac{c^3}{a^3 x^4}+\frac{3 c^2 (-b c+a d)}{a^4 x^3}+\frac{3 c (b c-a d) (2 b c-a d)}{a^5 x^2}+\frac{(b c-a d) \left (-10 b^2 c^2+8 a b c d-a^2 d^2\right )}{a^6 x}-\frac{b (-b c+a d)^3}{a^4 (a+b x)^3}-\frac{b (-4 b c+a d) (-b c+a d)^2}{a^5 (a+b x)^2}+\frac{b (b c-a d) \left (10 b^2 c^2-8 a b c d+a^2 d^2\right )}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac{c^3}{3 a^3 x^3}+\frac{3 c^2 (b c-a d)}{2 a^4 x^2}-\frac{3 c (b c-a d) (2 b c-a d)}{a^5 x}-\frac{(b c-a d)^3}{2 a^4 (a+b x)^2}-\frac{(b c-a d)^2 (4 b c-a d)}{a^5 (a+b x)}-\frac{(b c-a d) \left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) \log (x)}{a^6}+\frac{(b c-a d) \left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) \log (a+b x)}{a^6}\\ \end{align*}

Mathematica [A]  time = 0.13604, size = 202, normalized size = 1.05 \[ \frac{-\frac{18 a c \left (a^2 d^2-3 a b c d+2 b^2 c^2\right )}{x}+6 \log (x) \left (-9 a^2 b c d^2+a^3 d^3+18 a b^2 c^2 d-10 b^3 c^3\right )+6 \left (9 a^2 b c d^2-a^3 d^3-18 a b^2 c^2 d+10 b^3 c^3\right ) \log (a+b x)-\frac{9 a^2 c^2 (a d-b c)}{x^2}+\frac{3 a^2 (a d-b c)^3}{(a+b x)^2}-\frac{2 a^3 c^3}{x^3}+\frac{6 a (b c-a d)^2 (a d-4 b c)}{a+b x}}{6 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 326, normalized size = 1.7 \begin{align*} -{\frac{{c}^{3}}{3\,{a}^{3}{x}^{3}}}+{\frac{\ln \left ( x \right ){d}^{3}}{{a}^{3}}}-9\,{\frac{\ln \left ( x \right ) cb{d}^{2}}{{a}^{4}}}+18\,{\frac{\ln \left ( x \right ){b}^{2}{c}^{2}d}{{a}^{5}}}-10\,{\frac{\ln \left ( x \right ){b}^{3}{c}^{3}}{{a}^{6}}}-3\,{\frac{c{d}^{2}}{{a}^{3}x}}+9\,{\frac{{c}^{2}db}{{a}^{4}x}}-6\,{\frac{{c}^{3}{b}^{2}}{{a}^{5}x}}-{\frac{3\,{c}^{2}d}{2\,{a}^{3}{x}^{2}}}+{\frac{3\,{c}^{3}b}{2\,{a}^{4}{x}^{2}}}-{\frac{\ln \left ( bx+a \right ){d}^{3}}{{a}^{3}}}+9\,{\frac{\ln \left ( bx+a \right ) cb{d}^{2}}{{a}^{4}}}-18\,{\frac{\ln \left ( bx+a \right ){b}^{2}{c}^{2}d}{{a}^{5}}}+10\,{\frac{\ln \left ( bx+a \right ){b}^{3}{c}^{3}}{{a}^{6}}}+{\frac{{d}^{3}}{{a}^{2} \left ( bx+a \right ) }}-6\,{\frac{c{d}^{2}b}{{a}^{3} \left ( bx+a \right ) }}+9\,{\frac{{c}^{2}d{b}^{2}}{{a}^{4} \left ( bx+a \right ) }}-4\,{\frac{{b}^{3}{c}^{3}}{{a}^{5} \left ( bx+a \right ) }}+{\frac{{d}^{3}}{2\,a \left ( bx+a \right ) ^{2}}}-{\frac{3\,c{d}^{2}b}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}}+{\frac{3\,{c}^{2}d{b}^{2}}{2\,{a}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{{b}^{3}{c}^{3}}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*c^3/a^3/x^3+1/a^3*ln(x)*d^3-9/a^4*ln(x)*c*b*d^2+18/a^5*ln(x)*b^2*c^2*d-10/a^6*ln(x)*b^3*c^3-3*c/a^3/x*d^2
+9*c^2/a^4/x*b*d-6*c^3/a^5/x*b^2-3/2*c^2/a^3/x^2*d+3/2*c^3/a^4/x^2*b-1/a^3*ln(b*x+a)*d^3+9/a^4*ln(b*x+a)*c*b*d
^2-18/a^5*ln(b*x+a)*b^2*c^2*d+10/a^6*ln(b*x+a)*b^3*c^3+1/a^2/(b*x+a)*d^3-6/a^3/(b*x+a)*c*b*d^2+9/a^4/(b*x+a)*b
^2*c^2*d-4/a^5/(b*x+a)*b^3*c^3+1/2/a/(b*x+a)^2*d^3-3/2/a^2/(b*x+a)^2*c*b*d^2+3/2/a^3/(b*x+a)^2*b^2*c^2*d-1/2/a
^4/(b*x+a)^2*b^3*c^3

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Maxima [A]  time = 1.09856, size = 378, normalized size = 1.96 \begin{align*} -\frac{2 \, a^{4} c^{3} + 6 \,{\left (10 \, b^{4} c^{3} - 18 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} + 9 \,{\left (10 \, a b^{3} c^{3} - 18 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3} + 2 \,{\left (10 \, a^{2} b^{2} c^{3} - 18 \, a^{3} b c^{2} d + 9 \, a^{4} c d^{2}\right )} x^{2} -{\left (5 \, a^{3} b c^{3} - 9 \, a^{4} c^{2} d\right )} x}{6 \,{\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} + \frac{{\left (10 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{6}} - \frac{{\left (10 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x\right )}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*a^4*c^3 + 6*(10*b^4*c^3 - 18*a*b^3*c^2*d + 9*a^2*b^2*c*d^2 - a^3*b*d^3)*x^4 + 9*(10*a*b^3*c^3 - 18*a^2
*b^2*c^2*d + 9*a^3*b*c*d^2 - a^4*d^3)*x^3 + 2*(10*a^2*b^2*c^3 - 18*a^3*b*c^2*d + 9*a^4*c*d^2)*x^2 - (5*a^3*b*c
^3 - 9*a^4*c^2*d)*x)/(a^5*b^2*x^5 + 2*a^6*b*x^4 + a^7*x^3) + (10*b^3*c^3 - 18*a*b^2*c^2*d + 9*a^2*b*c*d^2 - a^
3*d^3)*log(b*x + a)/a^6 - (10*b^3*c^3 - 18*a*b^2*c^2*d + 9*a^2*b*c*d^2 - a^3*d^3)*log(x)/a^6

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Fricas [B]  time = 2.79524, size = 986, normalized size = 5.11 \begin{align*} -\frac{2 \, a^{5} c^{3} + 6 \,{\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + 9 \,{\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 2 \,{\left (10 \, a^{3} b^{2} c^{3} - 18 \, a^{4} b c^{2} d + 9 \, a^{5} c d^{2}\right )} x^{2} -{\left (5 \, a^{4} b c^{3} - 9 \, a^{5} c^{2} d\right )} x - 6 \,{\left ({\left (10 \, b^{5} c^{3} - 18 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 2 \,{\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} +{\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) + 6 \,{\left ({\left (10 \, b^{5} c^{3} - 18 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 2 \,{\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} +{\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*a^5*c^3 + 6*(10*a*b^4*c^3 - 18*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - a^4*b*d^3)*x^4 + 9*(10*a^2*b^3*c^3 -
18*a^3*b^2*c^2*d + 9*a^4*b*c*d^2 - a^5*d^3)*x^3 + 2*(10*a^3*b^2*c^3 - 18*a^4*b*c^2*d + 9*a^5*c*d^2)*x^2 - (5*a
^4*b*c^3 - 9*a^5*c^2*d)*x - 6*((10*b^5*c^3 - 18*a*b^4*c^2*d + 9*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5 + 2*(10*a*b^4
*c^3 - 18*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - a^4*b*d^3)*x^4 + (10*a^2*b^3*c^3 - 18*a^3*b^2*c^2*d + 9*a^4*b*c*d^
2 - a^5*d^3)*x^3)*log(b*x + a) + 6*((10*b^5*c^3 - 18*a*b^4*c^2*d + 9*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^5 + 2*(10*
a*b^4*c^3 - 18*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - a^4*b*d^3)*x^4 + (10*a^2*b^3*c^3 - 18*a^3*b^2*c^2*d + 9*a^4*b
*c*d^2 - a^5*d^3)*x^3)*log(x))/(a^6*b^2*x^5 + 2*a^7*b*x^4 + a^8*x^3)

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Sympy [B]  time = 2.57849, size = 505, normalized size = 2.62 \begin{align*} \frac{- 2 a^{4} c^{3} + x^{4} \left (6 a^{3} b d^{3} - 54 a^{2} b^{2} c d^{2} + 108 a b^{3} c^{2} d - 60 b^{4} c^{3}\right ) + x^{3} \left (9 a^{4} d^{3} - 81 a^{3} b c d^{2} + 162 a^{2} b^{2} c^{2} d - 90 a b^{3} c^{3}\right ) + x^{2} \left (- 18 a^{4} c d^{2} + 36 a^{3} b c^{2} d - 20 a^{2} b^{2} c^{3}\right ) + x \left (- 9 a^{4} c^{2} d + 5 a^{3} b c^{3}\right )}{6 a^{7} x^{3} + 12 a^{6} b x^{4} + 6 a^{5} b^{2} x^{5}} + \frac{\left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right ) \log{\left (x + \frac{a^{4} d^{3} - 9 a^{3} b c d^{2} + 18 a^{2} b^{2} c^{2} d - 10 a b^{3} c^{3} - a \left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right )}{2 a^{3} b d^{3} - 18 a^{2} b^{2} c d^{2} + 36 a b^{3} c^{2} d - 20 b^{4} c^{3}} \right )}}{a^{6}} - \frac{\left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right ) \log{\left (x + \frac{a^{4} d^{3} - 9 a^{3} b c d^{2} + 18 a^{2} b^{2} c^{2} d - 10 a b^{3} c^{3} + a \left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right )}{2 a^{3} b d^{3} - 18 a^{2} b^{2} c d^{2} + 36 a b^{3} c^{2} d - 20 b^{4} c^{3}} \right )}}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*a**4*c**3 + x**4*(6*a**3*b*d**3 - 54*a**2*b**2*c*d**2 + 108*a*b**3*c**2*d - 60*b**4*c**3) + x**3*(9*a**4*d
**3 - 81*a**3*b*c*d**2 + 162*a**2*b**2*c**2*d - 90*a*b**3*c**3) + x**2*(-18*a**4*c*d**2 + 36*a**3*b*c**2*d - 2
0*a**2*b**2*c**3) + x*(-9*a**4*c**2*d + 5*a**3*b*c**3))/(6*a**7*x**3 + 12*a**6*b*x**4 + 6*a**5*b**2*x**5) + (a
*d - b*c)*(a**2*d**2 - 8*a*b*c*d + 10*b**2*c**2)*log(x + (a**4*d**3 - 9*a**3*b*c*d**2 + 18*a**2*b**2*c**2*d -
10*a*b**3*c**3 - a*(a*d - b*c)*(a**2*d**2 - 8*a*b*c*d + 10*b**2*c**2))/(2*a**3*b*d**3 - 18*a**2*b**2*c*d**2 +
36*a*b**3*c**2*d - 20*b**4*c**3))/a**6 - (a*d - b*c)*(a**2*d**2 - 8*a*b*c*d + 10*b**2*c**2)*log(x + (a**4*d**3
- 9*a**3*b*c*d**2 + 18*a**2*b**2*c**2*d - 10*a*b**3*c**3 + a*(a*d - b*c)*(a**2*d**2 - 8*a*b*c*d + 10*b**2*c**
2))/(2*a**3*b*d**3 - 18*a**2*b**2*c*d**2 + 36*a*b**3*c**2*d - 20*b**4*c**3))/a**6

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Giac [A]  time = 1.21521, size = 374, normalized size = 1.94 \begin{align*} -\frac{{\left (10 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac{{\left (10 \, b^{4} c^{3} - 18 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac{2 \, a^{5} c^{3} + 6 \,{\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + 9 \,{\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 2 \,{\left (10 \, a^{3} b^{2} c^{3} - 18 \, a^{4} b c^{2} d + 9 \, a^{5} c d^{2}\right )} x^{2} -{\left (5 \, a^{4} b c^{3} - 9 \, a^{5} c^{2} d\right )} x}{6 \,{\left (b x + a\right )}^{2} a^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(10*b^3*c^3 - 18*a*b^2*c^2*d + 9*a^2*b*c*d^2 - a^3*d^3)*log(abs(x))/a^6 + (10*b^4*c^3 - 18*a*b^3*c^2*d + 9*a^
2*b^2*c*d^2 - a^3*b*d^3)*log(abs(b*x + a))/(a^6*b) - 1/6*(2*a^5*c^3 + 6*(10*a*b^4*c^3 - 18*a^2*b^3*c^2*d + 9*a
^3*b^2*c*d^2 - a^4*b*d^3)*x^4 + 9*(10*a^2*b^3*c^3 - 18*a^3*b^2*c^2*d + 9*a^4*b*c*d^2 - a^5*d^3)*x^3 + 2*(10*a^
3*b^2*c^3 - 18*a^4*b*c^2*d + 9*a^5*c*d^2)*x^2 - (5*a^4*b*c^3 - 9*a^5*c^2*d)*x)/((b*x + a)^2*a^6*x^3)