3.312 \(\int \frac{x^4}{(a+b x)^3 (c+d x)^3} \, dx\)
Optimal. Leaf size=169 \[ -\frac{a^4}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac{a^3 (4 b c-a d)}{b^2 (a+b x) (b c-a d)^4}+\frac{6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}-\frac{c^3 (b c-4 a d)}{d^2 (c+d x) (b c-a d)^4}+\frac{c^4}{2 d^2 (c+d x)^2 (b c-a d)^3} \]
[Out]
-a^4/(2*b^2*(b*c - a*d)^3*(a + b*x)^2) + (a^3*(4*b*c - a*d))/(b^2*(b*c - a*d)^4*(a + b*x)) + c^4/(2*d^2*(b*c -
a*d)^3*(c + d*x)^2) - (c^3*(b*c - 4*a*d))/(d^2*(b*c - a*d)^4*(c + d*x)) + (6*a^2*c^2*Log[a + b*x])/(b*c - a*d
)^5 - (6*a^2*c^2*Log[c + d*x])/(b*c - a*d)^5
________________________________________________________________________________________
Rubi [A] time = 0.191926, antiderivative size = 169, 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} \[ -\frac{a^4}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac{a^3 (4 b c-a d)}{b^2 (a+b x) (b c-a d)^4}+\frac{6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}-\frac{c^3 (b c-4 a d)}{d^2 (c+d x) (b c-a d)^4}+\frac{c^4}{2 d^2 (c+d x)^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
[In]
Int[x^4/((a + b*x)^3*(c + d*x)^3),x]
[Out]
-a^4/(2*b^2*(b*c - a*d)^3*(a + b*x)^2) + (a^3*(4*b*c - a*d))/(b^2*(b*c - a*d)^4*(a + b*x)) + c^4/(2*d^2*(b*c -
a*d)^3*(c + d*x)^2) - (c^3*(b*c - 4*a*d))/(d^2*(b*c - a*d)^4*(c + d*x)) + (6*a^2*c^2*Log[a + b*x])/(b*c - a*d
)^5 - (6*a^2*c^2*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^4}{(a+b x)^3 (c+d x)^3} \, dx &=\int \left (\frac{a^4}{b (b c-a d)^3 (a+b x)^3}+\frac{a^3 (-4 b c+a d)}{b (b c-a d)^4 (a+b x)^2}+\frac{6 a^2 b c^2}{(b c-a d)^5 (a+b x)}+\frac{c^4}{d (-b c+a d)^3 (c+d x)^3}+\frac{c^3 (b c-4 a d)}{d (-b c+a d)^4 (c+d x)^2}+\frac{6 a^2 c^2 d}{(-b c+a d)^5 (c+d x)}\right ) \, dx\\ &=-\frac{a^4}{2 b^2 (b c-a d)^3 (a+b x)^2}+\frac{a^3 (4 b c-a d)}{b^2 (b c-a d)^4 (a+b x)}+\frac{c^4}{2 d^2 (b c-a d)^3 (c+d x)^2}-\frac{c^3 (b c-4 a d)}{d^2 (b c-a d)^4 (c+d x)}+\frac{6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}\\ \end{align*}
Mathematica [A] time = 0.249633, size = 171, normalized size = 1.01 \[ -\frac{a^4}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac{4 a^3 b c-a^4 d}{b^2 (a+b x) (b c-a d)^4}+\frac{6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}-\frac{c^3 (b c-4 a d)}{d^2 (c+d x) (a d-b c)^4}-\frac{c^4}{2 d^2 (c+d x)^2 (a d-b c)^3} \]
Antiderivative was successfully verified.
[In]
Integrate[x^4/((a + b*x)^3*(c + d*x)^3),x]
[Out]
-a^4/(2*b^2*(b*c - a*d)^3*(a + b*x)^2) + (4*a^3*b*c - a^4*d)/(b^2*(b*c - a*d)^4*(a + b*x)) - c^4/(2*d^2*(-(b*c
) + a*d)^3*(c + d*x)^2) - (c^3*(b*c - 4*a*d))/(d^2*(-(b*c) + a*d)^4*(c + d*x)) + (6*a^2*c^2*Log[a + b*x])/(b*c
- a*d)^5 - (6*a^2*c^2*Log[c + d*x])/(b*c - a*d)^5
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Maple [A] time = 0.012, size = 204, normalized size = 1.2 \begin{align*} -{\frac{{c}^{4}}{2\,{d}^{2} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}+6\,{\frac{{a}^{2}{c}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{5}}}+4\,{\frac{{c}^{3}a}{ \left ( ad-bc \right ) ^{4}d \left ( dx+c \right ) }}-{\frac{{c}^{4}b}{ \left ( ad-bc \right ) ^{4}{d}^{2} \left ( dx+c \right ) }}-{\frac{d{a}^{4}}{ \left ( ad-bc \right ) ^{4}{b}^{2} \left ( bx+a \right ) }}+4\,{\frac{{a}^{3}c}{ \left ( ad-bc \right ) ^{4}b \left ( bx+a \right ) }}+{\frac{{a}^{4}}{2\,{b}^{2} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}-6\,{\frac{{a}^{2}{c}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4/(b*x+a)^3/(d*x+c)^3,x)
[Out]
-1/2*c^4/d^2/(a*d-b*c)^3/(d*x+c)^2+6*c^2*a^2/(a*d-b*c)^5*ln(d*x+c)+4*c^3/(a*d-b*c)^4/d/(d*x+c)*a-c^4/(a*d-b*c)
^4/d^2/(d*x+c)*b-a^4/(a*d-b*c)^4/b^2/(b*x+a)*d+4*a^3/(a*d-b*c)^4/b/(b*x+a)*c+1/2/b^2/(a*d-b*c)^3*a^4/(b*x+a)^2
-6*c^2*a^2/(a*d-b*c)^5*ln(b*x+a)
________________________________________________________________________________________
Maxima [B] time = 1.16345, size = 999, normalized size = 5.91 \begin{align*} \frac{6 \, a^{2} c^{2} \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{6 \, a^{2} c^{2} \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^{3} c^{5} - 7 \, a^{3} b^{2} c^{4} d - 7 \, a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3} + 2 \,{\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x^{3} +{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 16 \, a^{2} b^{3} c^{3} d^{2} - 16 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} x^{2} + 2 \,{\left (a b^{4} c^{5} - 6 \, a^{2} b^{3} c^{4} d - 8 \, a^{3} b^{2} c^{3} d^{2} - 6 \, a^{4} b c^{2} d^{3} + a^{5} c d^{4}\right )} x}{2 \,{\left (a^{2} b^{6} c^{6} d^{2} - 4 \, a^{3} b^{5} c^{5} d^{3} + 6 \, a^{4} b^{4} c^{4} d^{4} - 4 \, a^{5} b^{3} c^{3} d^{5} + a^{6} b^{2} c^{2} d^{6} +{\left (b^{8} c^{4} d^{4} - 4 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} - 4 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 2 \,{\left (b^{8} c^{5} d^{3} - 3 \, a b^{7} c^{4} d^{4} + 2 \, a^{2} b^{6} c^{3} d^{5} + 2 \, a^{3} b^{5} c^{2} d^{6} - 3 \, a^{4} b^{4} c d^{7} + a^{5} b^{3} d^{8}\right )} x^{3} +{\left (b^{8} c^{6} d^{2} - 9 \, a^{2} b^{6} c^{4} d^{4} + 16 \, a^{3} b^{5} c^{3} d^{5} - 9 \, a^{4} b^{4} c^{2} d^{6} + a^{6} b^{2} d^{8}\right )} x^{2} + 2 \,{\left (a b^{7} c^{6} d^{2} - 3 \, a^{2} b^{6} c^{5} d^{3} + 2 \, a^{3} b^{5} c^{4} d^{4} + 2 \, a^{4} b^{4} c^{3} d^{5} - 3 \, a^{5} b^{3} c^{2} d^{6} + a^{6} b^{2} c d^{7}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x+a)^3/(d*x+c)^3,x, algorithm="maxima")
[Out]
6*a^2*c^2*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) - 6*a^2*c^2*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^3*c^5 - 7*a^3*b^2*c^4*d - 7*a^4*b*c^3*d^2 + a^5*c^2*d^3 + 2*(b^5*c^4*d - 4*a*b^4*
c^3*d^2 - 4*a^3*b^2*c*d^4 + a^4*b*d^5)*x^3 + (b^5*c^5 - 3*a*b^4*c^4*d - 16*a^2*b^3*c^3*d^2 - 16*a^3*b^2*c^2*d^
3 - 3*a^4*b*c*d^4 + a^5*d^5)*x^2 + 2*(a*b^4*c^5 - 6*a^2*b^3*c^4*d - 8*a^3*b^2*c^3*d^2 - 6*a^4*b*c^2*d^3 + a^5*
c*d^4)*x)/(a^2*b^6*c^6*d^2 - 4*a^3*b^5*c^5*d^3 + 6*a^4*b^4*c^4*d^4 - 4*a^5*b^3*c^3*d^5 + a^6*b^2*c^2*d^6 + (b^
8*c^4*d^4 - 4*a*b^7*c^3*d^5 + 6*a^2*b^6*c^2*d^6 - 4*a^3*b^5*c*d^7 + a^4*b^4*d^8)*x^4 + 2*(b^8*c^5*d^3 - 3*a*b^
7*c^4*d^4 + 2*a^2*b^6*c^3*d^5 + 2*a^3*b^5*c^2*d^6 - 3*a^4*b^4*c*d^7 + a^5*b^3*d^8)*x^3 + (b^8*c^6*d^2 - 9*a^2*
b^6*c^4*d^4 + 16*a^3*b^5*c^3*d^5 - 9*a^4*b^4*c^2*d^6 + a^6*b^2*d^8)*x^2 + 2*(a*b^7*c^6*d^2 - 3*a^2*b^6*c^5*d^3
+ 2*a^3*b^5*c^4*d^4 + 2*a^4*b^4*c^3*d^5 - 3*a^5*b^3*c^2*d^6 + a^6*b^2*c*d^7)*x)
________________________________________________________________________________________
Fricas [B] time = 2.86662, size = 1895, normalized size = 11.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4/(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")
[Out]
-1/2*(a^2*b^4*c^6 - 8*a^3*b^3*c^5*d + 8*a^5*b*c^3*d^3 - a^6*c^2*d^4 + 2*(b^6*c^5*d - 5*a*b^5*c^4*d^2 + 4*a^2*b
^4*c^3*d^3 - 4*a^3*b^3*c^2*d^4 + 5*a^4*b^2*c*d^5 - a^5*b*d^6)*x^3 + (b^6*c^6 - 4*a*b^5*c^5*d - 13*a^2*b^4*c^4*
d^2 + 13*a^4*b^2*c^2*d^4 + 4*a^5*b*c*d^5 - a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 7*a^2*b^4*c^5*d - 2*a^3*b^3*c^4*d^2 +
2*a^4*b^2*c^3*d^3 + 7*a^5*b*c^2*d^4 - a^6*c*d^5)*x - 12*(a^2*b^4*c^2*d^4*x^4 + a^4*b^2*c^4*d^2 + 2*(a^2*b^4*c
^3*d^3 + a^3*b^3*c^2*d^4)*x^3 + (a^2*b^4*c^4*d^2 + 4*a^3*b^3*c^3*d^3 + a^4*b^2*c^2*d^4)*x^2 + 2*(a^3*b^3*c^4*d
^2 + a^4*b^2*c^3*d^3)*x)*log(b*x + a) + 12*(a^2*b^4*c^2*d^4*x^4 + a^4*b^2*c^4*d^2 + 2*(a^2*b^4*c^3*d^3 + a^3*b
^3*c^2*d^4)*x^3 + (a^2*b^4*c^4*d^2 + 4*a^3*b^3*c^3*d^3 + a^4*b^2*c^2*d^4)*x^2 + 2*(a^3*b^3*c^4*d^2 + a^4*b^2*c
^3*d^3)*x)*log(d*x + c))/(a^2*b^7*c^7*d^2 - 5*a^3*b^6*c^6*d^3 + 10*a^4*b^5*c^5*d^4 - 10*a^5*b^4*c^4*d^5 + 5*a^
6*b^3*c^3*d^6 - a^7*b^2*c^2*d^7 + (b^9*c^5*d^4 - 5*a*b^8*c^4*d^5 + 10*a^2*b^7*c^3*d^6 - 10*a^3*b^6*c^2*d^7 + 5
*a^4*b^5*c*d^8 - a^5*b^4*d^9)*x^4 + 2*(b^9*c^6*d^3 - 4*a*b^8*c^5*d^4 + 5*a^2*b^7*c^4*d^5 - 5*a^4*b^5*c^2*d^7 +
4*a^5*b^4*c*d^8 - a^6*b^3*d^9)*x^3 + (b^9*c^7*d^2 - a*b^8*c^6*d^3 - 9*a^2*b^7*c^5*d^4 + 25*a^3*b^6*c^4*d^5 -
25*a^4*b^5*c^3*d^6 + 9*a^5*b^4*c^2*d^7 + a^6*b^3*c*d^8 - a^7*b^2*d^9)*x^2 + 2*(a*b^8*c^7*d^2 - 4*a^2*b^7*c^6*d
^3 + 5*a^3*b^6*c^5*d^4 - 5*a^5*b^4*c^3*d^6 + 4*a^6*b^3*c^2*d^7 - a^7*b^2*c*d^8)*x)
________________________________________________________________________________________
Sympy [B] time = 5.37933, size = 1046, normalized size = 6.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**4/(b*x+a)**3/(d*x+c)**3,x)
[Out]
6*a**2*c**2*log(x + (-6*a**8*c**2*d**6/(a*d - b*c)**5 + 36*a**7*b*c**3*d**5/(a*d - b*c)**5 - 90*a**6*b**2*c**4
*d**4/(a*d - b*c)**5 + 120*a**5*b**3*c**5*d**3/(a*d - b*c)**5 - 90*a**4*b**4*c**6*d**2/(a*d - b*c)**5 + 36*a**
3*b**5*c**7*d/(a*d - b*c)**5 + 6*a**3*c**2*d - 6*a**2*b**6*c**8/(a*d - b*c)**5 + 6*a**2*b*c**3)/(12*a**2*b*c**
2*d))/(a*d - b*c)**5 - 6*a**2*c**2*log(x + (6*a**8*c**2*d**6/(a*d - b*c)**5 - 36*a**7*b*c**3*d**5/(a*d - b*c)*
*5 + 90*a**6*b**2*c**4*d**4/(a*d - b*c)**5 - 120*a**5*b**3*c**5*d**3/(a*d - b*c)**5 + 90*a**4*b**4*c**6*d**2/(
a*d - b*c)**5 - 36*a**3*b**5*c**7*d/(a*d - b*c)**5 + 6*a**3*c**2*d + 6*a**2*b**6*c**8/(a*d - b*c)**5 + 6*a**2*
b*c**3)/(12*a**2*b*c**2*d))/(a*d - b*c)**5 - (a**5*c**2*d**3 - 7*a**4*b*c**3*d**2 - 7*a**3*b**2*c**4*d + a**2*
b**3*c**5 + x**3*(2*a**4*b*d**5 - 8*a**3*b**2*c*d**4 - 8*a*b**4*c**3*d**2 + 2*b**5*c**4*d) + x**2*(a**5*d**5 -
3*a**4*b*c*d**4 - 16*a**3*b**2*c**2*d**3 - 16*a**2*b**3*c**3*d**2 - 3*a*b**4*c**4*d + b**5*c**5) + x*(2*a**5*
c*d**4 - 12*a**4*b*c**2*d**3 - 16*a**3*b**2*c**3*d**2 - 12*a**2*b**3*c**4*d + 2*a*b**4*c**5))/(2*a**6*b**2*c**
2*d**6 - 8*a**5*b**3*c**3*d**5 + 12*a**4*b**4*c**4*d**4 - 8*a**3*b**5*c**5*d**3 + 2*a**2*b**6*c**6*d**2 + x**4
*(2*a**4*b**4*d**8 - 8*a**3*b**5*c*d**7 + 12*a**2*b**6*c**2*d**6 - 8*a*b**7*c**3*d**5 + 2*b**8*c**4*d**4) + x*
*3*(4*a**5*b**3*d**8 - 12*a**4*b**4*c*d**7 + 8*a**3*b**5*c**2*d**6 + 8*a**2*b**6*c**3*d**5 - 12*a*b**7*c**4*d*
*4 + 4*b**8*c**5*d**3) + x**2*(2*a**6*b**2*d**8 - 18*a**4*b**4*c**2*d**6 + 32*a**3*b**5*c**3*d**5 - 18*a**2*b*
*6*c**4*d**4 + 2*b**8*c**6*d**2) + x*(4*a**6*b**2*c*d**7 - 12*a**5*b**3*c**2*d**6 + 8*a**4*b**4*c**3*d**5 + 8*
a**3*b**5*c**4*d**4 - 12*a**2*b**6*c**5*d**3 + 4*a*b**7*c**6*d**2))
________________________________________________________________________________________
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^4/(b*x+a)^3/(d*x+c)^3,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError