3.313 \(\int \frac{x^3}{(a+b x)^3 (c+d x)^3} \, dx\)
Optimal. Leaf size=155 \[ \frac{a^3}{2 b (a+b x)^2 (b c-a d)^3}-\frac{3 a^2 c}{(a+b x) (b c-a d)^4}-\frac{3 a c^2}{(c+d x) (b c-a d)^4}-\frac{c^3}{2 d (c+d x)^2 (b c-a d)^3}-\frac{3 a c (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac{3 a c (a d+b c) \log (c+d x)}{(b c-a d)^5} \]
[Out]
a^3/(2*b*(b*c - a*d)^3*(a + b*x)^2) - (3*a^2*c)/((b*c - a*d)^4*(a + b*x)) - c^3/(2*d*(b*c - a*d)^3*(c + d*x)^2
) - (3*a*c^2)/((b*c - a*d)^4*(c + d*x)) - (3*a*c*(b*c + a*d)*Log[a + b*x])/(b*c - a*d)^5 + (3*a*c*(b*c + a*d)*
Log[c + d*x])/(b*c - a*d)^5
________________________________________________________________________________________
Rubi [A] time = 0.223893, antiderivative size = 155, 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} \[ \frac{a^3}{2 b (a+b x)^2 (b c-a d)^3}-\frac{3 a^2 c}{(a+b x) (b c-a d)^4}-\frac{3 a c^2}{(c+d x) (b c-a d)^4}-\frac{c^3}{2 d (c+d x)^2 (b c-a d)^3}-\frac{3 a c (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac{3 a c (a d+b c) \log (c+d x)}{(b c-a d)^5} \]
Antiderivative was successfully verified.
[In]
Int[x^3/((a + b*x)^3*(c + d*x)^3),x]
[Out]
a^3/(2*b*(b*c - a*d)^3*(a + b*x)^2) - (3*a^2*c)/((b*c - a*d)^4*(a + b*x)) - c^3/(2*d*(b*c - a*d)^3*(c + d*x)^2
) - (3*a*c^2)/((b*c - a*d)^4*(c + d*x)) - (3*a*c*(b*c + a*d)*Log[a + b*x])/(b*c - a*d)^5 + (3*a*c*(b*c + a*d)*
Log[c + d*x])/(b*c - a*d)^5
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)^3 (c+d x)^3} \, dx &=\int \left (-\frac{a^3}{(b c-a d)^3 (a+b x)^3}+\frac{3 a^2 b c}{(b c-a d)^4 (a+b x)^2}-\frac{3 a b c (b c+a d)}{(b c-a d)^5 (a+b x)}+\frac{c^3}{(b c-a d)^3 (c+d x)^3}+\frac{3 a c^2 d}{(-b c+a d)^4 (c+d x)^2}-\frac{3 a c d (b c+a d)}{(-b c+a d)^5 (c+d x)}\right ) \, dx\\ &=\frac{a^3}{2 b (b c-a d)^3 (a+b x)^2}-\frac{3 a^2 c}{(b c-a d)^4 (a+b x)}-\frac{c^3}{2 d (b c-a d)^3 (c+d x)^2}-\frac{3 a c^2}{(b c-a d)^4 (c+d x)}-\frac{3 a c (b c+a d) \log (a+b x)}{(b c-a d)^5}+\frac{3 a c (b c+a d) \log (c+d x)}{(b c-a d)^5}\\ \end{align*}
Mathematica [A] time = 0.226078, size = 153, normalized size = 0.99 \[ \frac{1}{2} \left (\frac{a^3}{b (a+b x)^2 (b c-a d)^3}-\frac{6 a^2 c}{(a+b x) (b c-a d)^4}-\frac{6 a c^2}{(c+d x) (b c-a d)^4}+\frac{c^3}{d (c+d x)^2 (a d-b c)^3}-\frac{6 a c (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac{6 a c (a d+b c) \log (c+d x)}{(b c-a d)^5}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^3/((a + b*x)^3*(c + d*x)^3),x]
[Out]
(a^3/(b*(b*c - a*d)^3*(a + b*x)^2) - (6*a^2*c)/((b*c - a*d)^4*(a + b*x)) + c^3/(d*(-(b*c) + a*d)^3*(c + d*x)^2
) - (6*a*c^2)/((b*c - a*d)^4*(c + d*x)) - (6*a*c*(b*c + a*d)*Log[a + b*x])/(b*c - a*d)^5 + (6*a*c*(b*c + a*d)*
Log[c + d*x])/(b*c - a*d)^5)/2
________________________________________________________________________________________
Maple [A] time = 0.011, size = 190, normalized size = 1.2 \begin{align*}{\frac{{c}^{3}}{2\, \left ( ad-bc \right ) ^{3}d \left ( dx+c \right ) ^{2}}}-3\,{\frac{{c}^{2}a}{ \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}-3\,{\frac{{a}^{2}c\ln \left ( dx+c \right ) d}{ \left ( ad-bc \right ) ^{5}}}-3\,{\frac{{c}^{2}a\ln \left ( dx+c \right ) b}{ \left ( ad-bc \right ) ^{5}}}-{\frac{{a}^{3}}{2\, \left ( ad-bc \right ) ^{3}b \left ( bx+a \right ) ^{2}}}-3\,{\frac{{a}^{2}c}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }}+3\,{\frac{{a}^{2}c\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{5}}}+3\,{\frac{{c}^{2}a\ln \left ( bx+a \right ) b}{ \left ( ad-bc \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3/(b*x+a)^3/(d*x+c)^3,x)
[Out]
1/2*c^3/(a*d-b*c)^3/d/(d*x+c)^2-3*c^2*a/(a*d-b*c)^4/(d*x+c)-3*c*a^2/(a*d-b*c)^5*ln(d*x+c)*d-3*c^2*a/(a*d-b*c)^
5*ln(d*x+c)*b-1/2/(a*d-b*c)^3*a^3/b/(b*x+a)^2-3*a^2*c/(a*d-b*c)^4/(b*x+a)+3*c*a^2/(a*d-b*c)^5*ln(b*x+a)*d+3*c^
2*a/(a*d-b*c)^5*ln(b*x+a)*b
________________________________________________________________________________________
Maxima [B] time = 1.16674, size = 921, normalized size = 5.94 \begin{align*} -\frac{3 \,{\left (a b c^{2} + a^{2} c d\right )} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac{3 \,{\left (a b c^{2} + a^{2} c d\right )} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac{a^{2} b^{2} c^{4} + 10 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + 6 \,{\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x^{3} +{\left (b^{4} c^{4} + 5 \, a b^{3} c^{3} d + 24 \, a^{2} b^{2} c^{2} d^{2} + 5 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{2} + 2 \,{\left (a b^{3} c^{4} + 8 \, a^{2} b^{2} c^{3} d + 8 \, a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x}{2 \,{\left (a^{2} b^{5} c^{6} d - 4 \, a^{3} b^{4} c^{5} d^{2} + 6 \, a^{4} b^{3} c^{4} d^{3} - 4 \, a^{5} b^{2} c^{3} d^{4} + a^{6} b c^{2} d^{5} +{\left (b^{7} c^{4} d^{3} - 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{4} + 2 \,{\left (b^{7} c^{5} d^{2} - 3 \, a b^{6} c^{4} d^{3} + 2 \, a^{2} b^{5} c^{3} d^{4} + 2 \, a^{3} b^{4} c^{2} d^{5} - 3 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{3} +{\left (b^{7} c^{6} d - 9 \, a^{2} b^{5} c^{4} d^{3} + 16 \, a^{3} b^{4} c^{3} d^{4} - 9 \, a^{4} b^{3} c^{2} d^{5} + a^{6} b d^{7}\right )} x^{2} + 2 \,{\left (a b^{6} c^{6} d - 3 \, a^{2} b^{5} c^{5} d^{2} + 2 \, a^{3} b^{4} c^{4} d^{3} + 2 \, a^{4} b^{3} c^{3} d^{4} - 3 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b 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)^3/(d*x+c)^3,x, algorithm="maxima")
[Out]
-3*(a*b*c^2 + a^2*c*d)*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4
*b*c*d^4 - a^5*d^5) + 3*(a*b*c^2 + a^2*c*d)*log(d*x + c)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^
3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5) - 1/2*(a^2*b^2*c^4 + 10*a^3*b*c^3*d + a^4*c^2*d^2 + 6*(a*b^3*c^2*d^2
+ a^2*b^2*c*d^3)*x^3 + (b^4*c^4 + 5*a*b^3*c^3*d + 24*a^2*b^2*c^2*d^2 + 5*a^3*b*c*d^3 + a^4*d^4)*x^2 + 2*(a*b^3
*c^4 + 8*a^2*b^2*c^3*d + 8*a^3*b*c^2*d^2 + a^4*c*d^3)*x)/(a^2*b^5*c^6*d - 4*a^3*b^4*c^5*d^2 + 6*a^4*b^3*c^4*d^
3 - 4*a^5*b^2*c^3*d^4 + a^6*b*c^2*d^5 + (b^7*c^4*d^3 - 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4*c*d^6 +
a^4*b^3*d^7)*x^4 + 2*(b^7*c^5*d^2 - 3*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4*c^2*d^5 - 3*a^4*b^3*c*d^6
+ a^5*b^2*d^7)*x^3 + (b^7*c^6*d - 9*a^2*b^5*c^4*d^3 + 16*a^3*b^4*c^3*d^4 - 9*a^4*b^3*c^2*d^5 + a^6*b*d^7)*x^2
+ 2*(a*b^6*c^6*d - 3*a^2*b^5*c^5*d^2 + 2*a^3*b^4*c^4*d^3 + 2*a^4*b^3*c^3*d^4 - 3*a^5*b^2*c^2*d^5 + a^6*b*c*d^
6)*x)
________________________________________________________________________________________
Fricas [B] time = 2.7516, size = 1931, normalized size = 12.46 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")
[Out]
-1/2*(a^2*b^3*c^5 + 9*a^3*b^2*c^4*d - 9*a^4*b*c^3*d^2 - a^5*c^2*d^3 + 6*(a*b^4*c^3*d^2 - a^3*b^2*c*d^4)*x^3 +
(b^5*c^5 + 4*a*b^4*c^4*d + 19*a^2*b^3*c^3*d^2 - 19*a^3*b^2*c^2*d^3 - 4*a^4*b*c*d^4 - a^5*d^5)*x^2 + 2*(a*b^4*c
^5 + 7*a^2*b^3*c^4*d - 7*a^4*b*c^2*d^3 - a^5*c*d^4)*x + 6*(a^3*b^2*c^4*d + a^4*b*c^3*d^2 + (a*b^4*c^2*d^3 + a^
2*b^3*c*d^4)*x^4 + 2*(a*b^4*c^3*d^2 + 2*a^2*b^3*c^2*d^3 + a^3*b^2*c*d^4)*x^3 + (a*b^4*c^4*d + 5*a^2*b^3*c^3*d^
2 + 5*a^3*b^2*c^2*d^3 + a^4*b*c*d^4)*x^2 + 2*(a^2*b^3*c^4*d + 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x)*log(b*x +
a) - 6*(a^3*b^2*c^4*d + a^4*b*c^3*d^2 + (a*b^4*c^2*d^3 + a^2*b^3*c*d^4)*x^4 + 2*(a*b^4*c^3*d^2 + 2*a^2*b^3*c^2
*d^3 + a^3*b^2*c*d^4)*x^3 + (a*b^4*c^4*d + 5*a^2*b^3*c^3*d^2 + 5*a^3*b^2*c^2*d^3 + a^4*b*c*d^4)*x^2 + 2*(a^2*b
^3*c^4*d + 2*a^3*b^2*c^3*d^2 + a^4*b*c^2*d^3)*x)*log(d*x + c))/(a^2*b^6*c^7*d - 5*a^3*b^5*c^6*d^2 + 10*a^4*b^4
*c^5*d^3 - 10*a^5*b^3*c^4*d^4 + 5*a^6*b^2*c^3*d^5 - a^7*b*c^2*d^6 + (b^8*c^5*d^3 - 5*a*b^7*c^4*d^4 + 10*a^2*b^
6*c^3*d^5 - 10*a^3*b^5*c^2*d^6 + 5*a^4*b^4*c*d^7 - a^5*b^3*d^8)*x^4 + 2*(b^8*c^6*d^2 - 4*a*b^7*c^5*d^3 + 5*a^2
*b^6*c^4*d^4 - 5*a^4*b^4*c^2*d^6 + 4*a^5*b^3*c*d^7 - a^6*b^2*d^8)*x^3 + (b^8*c^7*d - a*b^7*c^6*d^2 - 9*a^2*b^6
*c^5*d^3 + 25*a^3*b^5*c^4*d^4 - 25*a^4*b^4*c^3*d^5 + 9*a^5*b^3*c^2*d^6 + a^6*b^2*c*d^7 - a^7*b*d^8)*x^2 + 2*(a
*b^7*c^7*d - 4*a^2*b^6*c^6*d^2 + 5*a^3*b^5*c^5*d^3 - 5*a^5*b^3*c^3*d^5 + 4*a^6*b^2*c^2*d^6 - a^7*b*c*d^7)*x)
________________________________________________________________________________________
Sympy [B] time = 3.7776, size = 1112, normalized size = 7.17 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3/(b*x+a)**3/(d*x+c)**3,x)
[Out]
-3*a*c*(a*d + b*c)*log(x + (-3*a**7*c*d**6*(a*d + b*c)/(a*d - b*c)**5 + 18*a**6*b*c**2*d**5*(a*d + b*c)/(a*d -
b*c)**5 - 45*a**5*b**2*c**3*d**4*(a*d + b*c)/(a*d - b*c)**5 + 60*a**4*b**3*c**4*d**3*(a*d + b*c)/(a*d - b*c)*
*5 - 45*a**3*b**4*c**5*d**2*(a*d + b*c)/(a*d - b*c)**5 + 3*a**3*c*d**2 + 18*a**2*b**5*c**6*d*(a*d + b*c)/(a*d
- b*c)**5 + 6*a**2*b*c**2*d - 3*a*b**6*c**7*(a*d + b*c)/(a*d - b*c)**5 + 3*a*b**2*c**3)/(6*a**2*b*c*d**2 + 6*a
*b**2*c**2*d))/(a*d - b*c)**5 + 3*a*c*(a*d + b*c)*log(x + (3*a**7*c*d**6*(a*d + b*c)/(a*d - b*c)**5 - 18*a**6*
b*c**2*d**5*(a*d + b*c)/(a*d - b*c)**5 + 45*a**5*b**2*c**3*d**4*(a*d + b*c)/(a*d - b*c)**5 - 60*a**4*b**3*c**4
*d**3*(a*d + b*c)/(a*d - b*c)**5 + 45*a**3*b**4*c**5*d**2*(a*d + b*c)/(a*d - b*c)**5 + 3*a**3*c*d**2 - 18*a**2
*b**5*c**6*d*(a*d + b*c)/(a*d - b*c)**5 + 6*a**2*b*c**2*d + 3*a*b**6*c**7*(a*d + b*c)/(a*d - b*c)**5 + 3*a*b**
2*c**3)/(6*a**2*b*c*d**2 + 6*a*b**2*c**2*d))/(a*d - b*c)**5 - (a**4*c**2*d**2 + 10*a**3*b*c**3*d + a**2*b**2*c
**4 + x**3*(6*a**2*b**2*c*d**3 + 6*a*b**3*c**2*d**2) + x**2*(a**4*d**4 + 5*a**3*b*c*d**3 + 24*a**2*b**2*c**2*d
**2 + 5*a*b**3*c**3*d + b**4*c**4) + x*(2*a**4*c*d**3 + 16*a**3*b*c**2*d**2 + 16*a**2*b**2*c**3*d + 2*a*b**3*c
**4))/(2*a**6*b*c**2*d**5 - 8*a**5*b**2*c**3*d**4 + 12*a**4*b**3*c**4*d**3 - 8*a**3*b**4*c**5*d**2 + 2*a**2*b*
*5*c**6*d + x**4*(2*a**4*b**3*d**7 - 8*a**3*b**4*c*d**6 + 12*a**2*b**5*c**2*d**5 - 8*a*b**6*c**3*d**4 + 2*b**7
*c**4*d**3) + x**3*(4*a**5*b**2*d**7 - 12*a**4*b**3*c*d**6 + 8*a**3*b**4*c**2*d**5 + 8*a**2*b**5*c**3*d**4 - 1
2*a*b**6*c**4*d**3 + 4*b**7*c**5*d**2) + x**2*(2*a**6*b*d**7 - 18*a**4*b**3*c**2*d**5 + 32*a**3*b**4*c**3*d**4
- 18*a**2*b**5*c**4*d**3 + 2*b**7*c**6*d) + x*(4*a**6*b*c*d**6 - 12*a**5*b**2*c**2*d**5 + 8*a**4*b**3*c**3*d*
*4 + 8*a**3*b**4*c**4*d**3 - 12*a**2*b**5*c**5*d**2 + 4*a*b**6*c**6*d))
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3/(b*x+a)^3/(d*x+c)^3,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError