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

Optimal. Leaf size=155 \[ \frac{a^3}{2 b (a+b x)^2 (b c-a d)^3}-\frac{3 a^2 c}{(a+b x) (b c-a d)^4}-\frac{c^3}{2 d (c+d x)^2 (b c-a d)^3}-\frac{3 a c^2}{(c+d x) (b c-a d)^4}-\frac{3 a c (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac{3 a c (a d+b c) \log (c+d x)}{(b c-a d)^5} \]

[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 = 0.357196, 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 \[ \frac{a^3}{2 b (a+b x)^2 (b c-a d)^3}-\frac{3 a^2 c}{(a+b x) (b c-a d)^4}-\frac{c^3}{2 d (c+d x)^2 (b c-a d)^3}-\frac{3 a c^2}{(c+d x) (b c-a d)^4}-\frac{3 a c (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac{3 a c (a d+b c) \log (c+d x)}{(b c-a d)^5} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 78.8252, size = 138, normalized size = 0.89 \[ - \frac{a^{3}}{2 b \left (a + b x\right )^{2} \left (a d - b c\right )^{3}} - \frac{3 a^{2} c}{\left (a + b x\right ) \left (a d - b c\right )^{4}} - \frac{3 a c^{2}}{\left (c + d x\right ) \left (a d - b c\right )^{4}} + \frac{3 a c \left (a d + b c\right ) \log{\left (a + b x \right )}}{\left (a d - b c\right )^{5}} - \frac{3 a c \left (a d + b c\right ) \log{\left (c + d x \right )}}{\left (a d - b c\right )^{5}} + \frac{c^{3}}{2 d \left (c + d x\right )^{2} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x+a)**3/(d*x+c)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.54028, size = 153, normalized size = 0.99 \[ \frac{1}{2} \left (\frac{a^3}{b (a+b x)^2 (b c-a d)^3}-\frac{6 a^2 c}{(a+b x) (b c-a d)^4}+\frac{c^3}{d (c+d x)^2 (a d-b c)^3}-\frac{6 a c^2}{(c+d x) (b c-a d)^4}-\frac{6 a c (a d+b c) \log (a+b x)}{(b c-a d)^5}+\frac{6 a c (a d+b c) \log (c+d x)}{(b c-a d)^5}\right ) \]

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 = 0.022, size = 190, normalized size = 1.2 \[{\frac{{c}^{3}}{2\, \left ( ad-bc \right ) ^{3}d \left ( dx+c \right ) ^{2}}}-3\,{\frac{{c}^{2}a}{ \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}-3\,{\frac{{a}^{2}c\ln \left ( dx+c \right ) d}{ \left ( ad-bc \right ) ^{5}}}-3\,{\frac{{c}^{2}a\ln \left ( dx+c \right ) b}{ \left ( ad-bc \right ) ^{5}}}-{\frac{{a}^{3}}{2\, \left ( ad-bc \right ) ^{3}b \left ( bx+a \right ) ^{2}}}-3\,{\frac{{a}^{2}c}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }}+3\,{\frac{{a}^{2}c\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{5}}}+3\,{\frac{{c}^{2}a\ln \left ( bx+a \right ) b}{ \left ( ad-bc \right ) ^{5}}} \]

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 [A]  time = 1.40328, size = 921, normalized size = 5.94 \[ -\frac{3 \,{\left (a b c^{2} + a^{2} c d\right )} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac{3 \,{\left (a b c^{2} + a^{2} c d\right )} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac{a^{2} b^{2} c^{4} + 10 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + 6 \,{\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x^{3} +{\left (b^{4} c^{4} + 5 \, a b^{3} c^{3} d + 24 \, a^{2} b^{2} c^{2} d^{2} + 5 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x^{2} + 2 \,{\left (a b^{3} c^{4} + 8 \, a^{2} b^{2} c^{3} d + 8 \, a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x}{2 \,{\left (a^{2} b^{5} c^{6} d - 4 \, a^{3} b^{4} c^{5} d^{2} + 6 \, a^{4} b^{3} c^{4} d^{3} - 4 \, a^{5} b^{2} c^{3} d^{4} + a^{6} b c^{2} d^{5} +{\left (b^{7} c^{4} d^{3} - 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{4} + 2 \,{\left (b^{7} c^{5} d^{2} - 3 \, a b^{6} c^{4} d^{3} + 2 \, a^{2} b^{5} c^{3} d^{4} + 2 \, a^{3} b^{4} c^{2} d^{5} - 3 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{3} +{\left (b^{7} c^{6} d - 9 \, a^{2} b^{5} c^{4} d^{3} + 16 \, a^{3} b^{4} c^{3} d^{4} - 9 \, a^{4} b^{3} c^{2} d^{5} + a^{6} b d^{7}\right )} x^{2} + 2 \,{\left (a b^{6} c^{6} d - 3 \, a^{2} b^{5} c^{5} d^{2} + 2 \, a^{3} b^{4} c^{4} d^{3} + 2 \, a^{4} b^{3} c^{3} d^{4} - 3 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6}\right )} x\right )}} \]

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 [A]  time = 0.232716, size = 1338, normalized size = 8.63 \[ \text{result too large to display} \]

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 [A]  time = 19.8449, size = 1112, normalized size = 7.17 \[ \text{result too large to display} \]

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 - 12*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/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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