3.315 \(\int \frac{x}{(a+b x)^3 (c+d x)^3} \, dx\)
Optimal. Leaf size=157 \[ -\frac{b (2 a d+b c)}{(a+b x) (b c-a d)^4}+\frac{a b}{2 (a+b x)^2 (b c-a d)^3}-\frac{d (a d+2 b c)}{(c+d x) (b c-a d)^4}-\frac{c d}{2 (c+d x)^2 (b c-a d)^3}-\frac{3 b d (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac{3 b d (a d+b c) \log (c+d x)}{(b c-a d)^5} \]
[Out]
(a*b)/(2*(b*c - a*d)^3*(a + b*x)^2) - (b*(b*c + 2*a*d))/((b*c - a*d)^4*(a + b*x)) - (c*d)/(2*(b*c - a*d)^3*(c
+ d*x)^2) - (d*(2*b*c + a*d))/((b*c - a*d)^4*(c + d*x)) - (3*b*d*(b*c + a*d)*Log[a + b*x])/(b*c - a*d)^5 + (3*
b*d*(b*c + a*d)*Log[c + d*x])/(b*c - a*d)^5
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Rubi [A] time = 0.147473, antiderivative size = 157, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used =
{77} \[ -\frac{b (2 a d+b c)}{(a+b x) (b c-a d)^4}+\frac{a b}{2 (a+b x)^2 (b c-a d)^3}-\frac{d (a d+2 b c)}{(c+d x) (b c-a d)^4}-\frac{c d}{2 (c+d x)^2 (b c-a d)^3}-\frac{3 b d (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac{3 b d (a d+b c) \log (c+d x)}{(b c-a d)^5} \]
Antiderivative was successfully verified.
[In]
Int[x/((a + b*x)^3*(c + d*x)^3),x]
[Out]
(a*b)/(2*(b*c - a*d)^3*(a + b*x)^2) - (b*(b*c + 2*a*d))/((b*c - a*d)^4*(a + b*x)) - (c*d)/(2*(b*c - a*d)^3*(c
+ d*x)^2) - (d*(2*b*c + a*d))/((b*c - a*d)^4*(c + d*x)) - (3*b*d*(b*c + a*d)*Log[a + b*x])/(b*c - a*d)^5 + (3*
b*d*(b*c + a*d)*Log[c + d*x])/(b*c - a*d)^5
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps
\begin{align*} \int \frac{x}{(a+b x)^3 (c+d x)^3} \, dx &=\int \left (-\frac{a b^2}{(b c-a d)^3 (a+b x)^3}+\frac{b^2 (b c+2 a d)}{(b c-a d)^4 (a+b x)^2}-\frac{3 b^2 d (b c+a d)}{(b c-a d)^5 (a+b x)}+\frac{c d^2}{(b c-a d)^3 (c+d x)^3}+\frac{d^2 (2 b c+a d)}{(b c-a d)^4 (c+d x)^2}+\frac{3 b d^2 (b c+a d)}{(b c-a d)^5 (c+d x)}\right ) \, dx\\ &=\frac{a b}{2 (b c-a d)^3 (a+b x)^2}-\frac{b (b c+2 a d)}{(b c-a d)^4 (a+b x)}-\frac{c d}{2 (b c-a d)^3 (c+d x)^2}-\frac{d (2 b c+a d)}{(b c-a d)^4 (c+d x)}-\frac{3 b d (b c+a d) \log (a+b x)}{(b c-a d)^5}+\frac{3 b d (b c+a d) \log (c+d x)}{(b c-a d)^5}\\ \end{align*}
Mathematica [A] time = 0.148688, size = 142, normalized size = 0.9 \[ \frac{\frac{a b (b c-a d)^2}{(a+b x)^2}-\frac{c d (b c-a d)^2}{(c+d x)^2}-\frac{2 b (2 a d+b c) (b c-a d)}{a+b x}+\frac{2 d (a d-b c) (a d+2 b c)}{c+d x}-6 b d (a d+b c) \log (a+b x)+6 b d (a d+b c) \log (c+d x)}{2 (b c-a d)^5} \]
Antiderivative was successfully verified.
[In]
Integrate[x/((a + b*x)^3*(c + d*x)^3),x]
[Out]
((a*b*(b*c - a*d)^2)/(a + b*x)^2 - (2*b*(b*c - a*d)*(b*c + 2*a*d))/(a + b*x) - (c*d*(b*c - a*d)^2)/(c + d*x)^2
+ (2*d*(-(b*c) + a*d)*(2*b*c + a*d))/(c + d*x) - 6*b*d*(b*c + a*d)*Log[a + b*x] + 6*b*d*(b*c + a*d)*Log[c + d
*x])/(2*(b*c - a*d)^5)
________________________________________________________________________________________
Maple [A] time = 0.01, size = 226, normalized size = 1.4 \begin{align*} -{\frac{a{d}^{2}}{ \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}-2\,{\frac{bdc}{ \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+{\frac{cd}{2\, \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}-3\,{\frac{{d}^{2}b\ln \left ( dx+c \right ) a}{ \left ( ad-bc \right ) ^{5}}}-3\,{\frac{{b}^{2}d\ln \left ( dx+c \right ) c}{ \left ( ad-bc \right ) ^{5}}}-{\frac{ab}{2\, \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}-2\,{\frac{abd}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }}-{\frac{{b}^{2}c}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }}+3\,{\frac{{d}^{2}b\ln \left ( bx+a \right ) a}{ \left ( ad-bc \right ) ^{5}}}+3\,{\frac{{b}^{2}d\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(b*x+a)^3/(d*x+c)^3,x)
[Out]
-d^2/(a*d-b*c)^4/(d*x+c)*a-2*d/(a*d-b*c)^4/(d*x+c)*b*c+1/2*c*d/(a*d-b*c)^3/(d*x+c)^2-3*d^2*b/(a*d-b*c)^5*ln(d*
x+c)*a-3*d*b^2/(a*d-b*c)^5*ln(d*x+c)*c-1/2*b/(a*d-b*c)^3*a/(b*x+a)^2-2*b/(a*d-b*c)^4/(b*x+a)*a*d-b^2/(a*d-b*c)
^4/(b*x+a)*c+3*d^2*b/(a*d-b*c)^5*ln(b*x+a)*a+3*d*b^2/(a*d-b*c)^5*ln(b*x+a)*c
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Maxima [B] time = 1.54957, size = 846, normalized size = 5.39 \begin{align*} -\frac{3 \,{\left (b^{2} c d + a b d^{2}\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 (b^{2} c d + a b d^{2}\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 b^{2} c^{3} + 10 \, a^{2} b c^{2} d + a^{3} c d^{2} + 6 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + 9 \,{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 2 \,{\left (b^{3} c^{3} + 8 \, a b^{2} c^{2} d + 8 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x}{2 \,{\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \,{\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} +{\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \,{\left (a b^{5} c^{6} - 3 \, 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} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(b*x+a)^3/(d*x+c)^3,x, algorithm="maxima")
[Out]
-3*(b^2*c*d + a*b*d^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) + 3*(b^2*c*d + a*b*d^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*b^2*c^3 + 10*a^2*b*c^2*d + a^3*c*d^2 + 6*(b^3*c*d^2 + a*b^2*
d^3)*x^3 + 9*(b^3*c^2*d + 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + 2*(b^3*c^3 + 8*a*b^2*c^2*d + 8*a^2*b*c*d^2 + a^3*d^
3)*x)/(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*
b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*
b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3
*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*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 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x)
________________________________________________________________________________________
Fricas [B] time = 2.66382, size = 1747, normalized size = 11.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(b*x+a)^3/(d*x+c)^3,x, algorithm="fricas")
[Out]
-1/2*(a*b^3*c^4 + 9*a^2*b^2*c^3*d - 9*a^3*b*c^2*d^2 - a^4*c*d^3 + 6*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 9*(b^4*c
^3*d + a*b^3*c^2*d^2 - a^2*b^2*c*d^3 - a^3*b*d^4)*x^2 + 2*(b^4*c^4 + 7*a*b^3*c^3*d - 7*a^3*b*c*d^3 - a^4*d^4)*
x + 6*(a^2*b^2*c^3*d + a^3*b*c^2*d^2 + (b^4*c*d^3 + a*b^3*d^4)*x^4 + 2*(b^4*c^2*d^2 + 2*a*b^3*c*d^3 + a^2*b^2*
d^4)*x^3 + (b^4*c^3*d + 5*a*b^3*c^2*d^2 + 5*a^2*b^2*c*d^3 + a^3*b*d^4)*x^2 + 2*(a*b^3*c^3*d + 2*a^2*b^2*c^2*d^
2 + a^3*b*c*d^3)*x)*log(b*x + a) - 6*(a^2*b^2*c^3*d + a^3*b*c^2*d^2 + (b^4*c*d^3 + a*b^3*d^4)*x^4 + 2*(b^4*c^2
*d^2 + 2*a*b^3*c*d^3 + a^2*b^2*d^4)*x^3 + (b^4*c^3*d + 5*a*b^3*c^2*d^2 + 5*a^2*b^2*c*d^3 + a^3*b*d^4)*x^2 + 2*
(a*b^3*c^3*d + 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x)*log(d*x + c))/(a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c
^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5 + (b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*
d^4 - 10*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^4 + 2*(b^7*c^6*d - 4*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4
*d^3 - 5*a^4*b^3*c^2*d^5 + 4*a^5*b^2*c*d^6 - a^6*b*d^7)*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 25*
a^3*b^4*c^4*d^3 - 25*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 - a^7*d^7)*x^2 + 2*(a*b^6*c^7 - 4*a^2*b
^5*c^6*d + 5*a^3*b^4*c^5*d^2 - 5*a^5*b^2*c^3*d^4 + 4*a^6*b*c^2*d^5 - a^7*c*d^6)*x)
________________________________________________________________________________________
Sympy [B] time = 3.60663, size = 1047, normalized size = 6.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(b*x+a)**3/(d*x+c)**3,x)
[Out]
-3*b*d*(a*d + b*c)*log(x + (-3*a**6*b*d**7*(a*d + b*c)/(a*d - b*c)**5 + 18*a**5*b**2*c*d**6*(a*d + b*c)/(a*d -
b*c)**5 - 45*a**4*b**3*c**2*d**5*(a*d + b*c)/(a*d - b*c)**5 + 60*a**3*b**4*c**3*d**4*(a*d + b*c)/(a*d - b*c)*
*5 - 45*a**2*b**5*c**4*d**3*(a*d + b*c)/(a*d - b*c)**5 + 3*a**2*b*d**3 + 18*a*b**6*c**5*d**2*(a*d + b*c)/(a*d
- b*c)**5 + 6*a*b**2*c*d**2 - 3*b**7*c**6*d*(a*d + b*c)/(a*d - b*c)**5 + 3*b**3*c**2*d)/(6*a*b**2*d**3 + 6*b**
3*c*d**2))/(a*d - b*c)**5 + 3*b*d*(a*d + b*c)*log(x + (3*a**6*b*d**7*(a*d + b*c)/(a*d - b*c)**5 - 18*a**5*b**2
*c*d**6*(a*d + b*c)/(a*d - b*c)**5 + 45*a**4*b**3*c**2*d**5*(a*d + b*c)/(a*d - b*c)**5 - 60*a**3*b**4*c**3*d**
4*(a*d + b*c)/(a*d - b*c)**5 + 45*a**2*b**5*c**4*d**3*(a*d + b*c)/(a*d - b*c)**5 + 3*a**2*b*d**3 - 18*a*b**6*c
**5*d**2*(a*d + b*c)/(a*d - b*c)**5 + 6*a*b**2*c*d**2 + 3*b**7*c**6*d*(a*d + b*c)/(a*d - b*c)**5 + 3*b**3*c**2
*d)/(6*a*b**2*d**3 + 6*b**3*c*d**2))/(a*d - b*c)**5 - (a**3*c*d**2 + 10*a**2*b*c**2*d + a*b**2*c**3 + x**3*(6*
a*b**2*d**3 + 6*b**3*c*d**2) + x**2*(9*a**2*b*d**3 + 18*a*b**2*c*d**2 + 9*b**3*c**2*d) + x*(2*a**3*d**3 + 16*a
**2*b*c*d**2 + 16*a*b**2*c**2*d + 2*b**3*c**3))/(2*a**6*c**2*d**4 - 8*a**5*b*c**3*d**3 + 12*a**4*b**2*c**4*d**
2 - 8*a**3*b**3*c**5*d + 2*a**2*b**4*c**6 + x**4*(2*a**4*b**2*d**6 - 8*a**3*b**3*c*d**5 + 12*a**2*b**4*c**2*d*
*4 - 8*a*b**5*c**3*d**3 + 2*b**6*c**4*d**2) + x**3*(4*a**5*b*d**6 - 12*a**4*b**2*c*d**5 + 8*a**3*b**3*c**2*d**
4 + 8*a**2*b**4*c**3*d**3 - 12*a*b**5*c**4*d**2 + 4*b**6*c**5*d) + x**2*(2*a**6*d**6 - 18*a**4*b**2*c**2*d**4
+ 32*a**3*b**3*c**3*d**3 - 18*a**2*b**4*c**4*d**2 + 2*b**6*c**6) + x*(4*a**6*c*d**5 - 12*a**5*b*c**2*d**4 + 8*
a**4*b**2*c**3*d**3 + 8*a**3*b**3*c**4*d**2 - 12*a**2*b**4*c**5*d + 4*a*b**5*c**6))
________________________________________________________________________________________
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/(b*x+a)^3/(d*x+c)^3,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError