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

Optimal. Leaf size=128 \[ \frac{c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^3}-\frac{a^3 \log (a+b x)}{b (b c-a d)^3}+\frac{c^2 (2 b c-3 a d)}{d^3 (c+d x) (b c-a d)^2}-\frac{c^3}{2 d^3 (c+d x)^2 (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.106833, antiderivative size = 128, 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} \[ \frac{c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^3}-\frac{a^3 \log (a+b x)}{b (b c-a d)^3}+\frac{c^2 (2 b c-3 a d)}{d^3 (c+d x) (b c-a d)^2}-\frac{c^3}{2 d^3 (c+d x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((a + b*x)*(c + d*x)^3),x]

[Out]

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

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

Mathematica [A]  time = 0.0596631, size = 134, normalized size = 1.05 \[ -\frac{\left (-3 a^2 c d^2+3 a b c^2 d-b^2 c^3\right ) \log (c+d x)}{d^3 (b c-a d)^3}-\frac{a^3 \log (a+b x)}{b (b c-a d)^3}+\frac{2 b c^3-3 a c^2 d}{d^3 (c+d x) (a d-b c)^2}+\frac{c^3}{2 d^3 (c+d x)^2 (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 180, normalized size = 1.4 \begin{align*} -3\,{\frac{{c}^{2}a}{{d}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+2\,{\frac{{c}^{3}b}{{d}^{3} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+{\frac{{c}^{3}}{2\,{d}^{3} \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}-3\,{\frac{c\ln \left ( dx+c \right ){a}^{2}}{d \left ( ad-bc \right ) ^{3}}}+3\,{\frac{{c}^{2}\ln \left ( dx+c \right ) ab}{ \left ( ad-bc \right ) ^{3}{d}^{2}}}-{\frac{{c}^{3}\ln \left ( dx+c \right ){b}^{2}}{ \left ( ad-bc \right ) ^{3}{d}^{3}}}+{\frac{{a}^{3}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.16668, size = 350, normalized size = 2.73 \begin{align*} -\frac{a^{3} \log \left (b x + a\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac{{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}} + \frac{3 \, b c^{4} - 5 \, a c^{3} d + 2 \,{\left (2 \, b c^{3} d - 3 \, a c^{2} d^{2}\right )} x}{2 \,{\left (b^{2} c^{4} d^{3} - 2 \, a b c^{3} d^{4} + a^{2} c^{2} d^{5} +{\left (b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d^{4} - 2 \, a b c^{2} d^{5} + a^{2} c d^{6}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.34761, size = 736, normalized size = 5.75 \begin{align*} \frac{3 \, b^{3} c^{5} - 8 \, a b^{2} c^{4} d + 5 \, a^{2} b c^{3} d^{2} + 2 \,{\left (2 \, b^{3} c^{4} d - 5 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3}\right )} x - 2 \,{\left (a^{3} d^{5} x^{2} + 2 \, a^{3} c d^{4} x + a^{3} c^{2} d^{3}\right )} \log \left (b x + a\right ) + 2 \,{\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} +{\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x^{2} + 2 \,{\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (b^{4} c^{5} d^{3} - 3 \, a b^{3} c^{4} d^{4} + 3 \, a^{2} b^{2} c^{3} d^{5} - a^{3} b c^{2} d^{6} +{\left (b^{4} c^{3} d^{5} - 3 \, a b^{3} c^{2} d^{6} + 3 \, a^{2} b^{2} c d^{7} - a^{3} b d^{8}\right )} x^{2} + 2 \,{\left (b^{4} c^{4} d^{4} - 3 \, a b^{3} c^{3} d^{5} + 3 \, a^{2} b^{2} c^{2} d^{6} - a^{3} b c d^{7}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 3.53052, size = 653, normalized size = 5.1 \begin{align*} \frac{a^{3} \log{\left (x + \frac{\frac{a^{7} d^{6}}{b \left (a d - b c\right )^{3}} - \frac{4 a^{6} c d^{5}}{\left (a d - b c\right )^{3}} + \frac{6 a^{5} b c^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a^{4} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{3}} + \frac{a^{3} b^{3} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + 4 a^{3} c d^{2} - 3 a^{2} b c^{2} d + a b^{2} c^{3}}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{b \left (a d - b c\right )^{3}} - \frac{c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log{\left (x + \frac{- \frac{a^{4} c d^{3} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b c^{2} d^{2} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + 4 a^{3} c d^{2} - \frac{6 a^{2} b^{2} c^{3} d \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} - 3 a^{2} b c^{2} d + \frac{4 a b^{3} c^{4} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{3}} + a b^{2} c^{3} - \frac{b^{4} c^{5} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{d \left (a d - b c\right )^{3}}}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{d^{3} \left (a d - b c\right )^{3}} - \frac{5 a c^{3} d - 3 b c^{4} + x \left (6 a c^{2} d^{2} - 4 b c^{3} d\right )}{2 a^{2} c^{2} d^{5} - 4 a b c^{3} d^{4} + 2 b^{2} c^{4} d^{3} + x^{2} \left (2 a^{2} d^{7} - 4 a b c d^{6} + 2 b^{2} c^{2} d^{5}\right ) + x \left (4 a^{2} c d^{6} - 8 a b c^{2} d^{5} + 4 b^{2} c^{3} d^{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*log(x + (a**7*d**6/(b*(a*d - b*c)**3) - 4*a**6*c*d**5/(a*d - b*c)**3 + 6*a**5*b*c**2*d**4/(a*d - b*c)**3
- 4*a**4*b**2*c**3*d**3/(a*d - b*c)**3 + a**3*b**3*c**4*d**2/(a*d - b*c)**3 + 4*a**3*c*d**2 - 3*a**2*b*c**2*d
+ a*b**2*c**3)/(a**3*d**3 + 3*a**2*b*c*d**2 - 3*a*b**2*c**2*d + b**3*c**3))/(b*(a*d - b*c)**3) - c*(3*a**2*d**
2 - 3*a*b*c*d + b**2*c**2)*log(x + (-a**4*c*d**3*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)/(a*d - b*c)**3 + 4*a**3
*b*c**2*d**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)/(a*d - b*c)**3 + 4*a**3*c*d**2 - 6*a**2*b**2*c**3*d*(3*a**2
*d**2 - 3*a*b*c*d + b**2*c**2)/(a*d - b*c)**3 - 3*a**2*b*c**2*d + 4*a*b**3*c**4*(3*a**2*d**2 - 3*a*b*c*d + b**
2*c**2)/(a*d - b*c)**3 + a*b**2*c**3 - b**4*c**5*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)/(d*(a*d - b*c)**3))/(a*
*3*d**3 + 3*a**2*b*c*d**2 - 3*a*b**2*c**2*d + b**3*c**3))/(d**3*(a*d - b*c)**3) - (5*a*c**3*d - 3*b*c**4 + x*(
6*a*c**2*d**2 - 4*b*c**3*d))/(2*a**2*c**2*d**5 - 4*a*b*c**3*d**4 + 2*b**2*c**4*d**3 + x**2*(2*a**2*d**7 - 4*a*
b*c*d**6 + 2*b**2*c**2*d**5) + x*(4*a**2*c*d**6 - 8*a*b*c**2*d**5 + 4*b**2*c**3*d**4))

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Giac [A]  time = 3.04267, size = 292, normalized size = 2.28 \begin{align*} -\frac{a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} + \frac{{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}} + \frac{2 \,{\left (2 \, b^{2} c^{4} - 5 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2}\right )} x + \frac{3 \, b^{2} c^{5} - 8 \, a b c^{4} d + 5 \, a^{2} c^{3} d^{2}}{d}}{2 \,{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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