3.251 \(\int \frac{x^4}{(a+b x)^2 (c+d x)^2} \, dx\)

Optimal. Leaf size=124 \[ -\frac{a^4}{b^3 (a+b x) (b c-a d)^2}-\frac{2 a^3 (2 b c-a d) \log (a+b x)}{b^3 (b c-a d)^3}-\frac{c^4}{d^3 (c+d x) (b c-a d)^2}-\frac{2 c^3 (b c-2 a d) \log (c+d x)}{d^3 (b c-a d)^3}+\frac{x}{b^2 d^2} \]

[Out]

x/(b^2*d^2) - a^4/(b^3*(b*c - a*d)^2*(a + b*x)) - c^4/(d^3*(b*c - a*d)^2*(c + d*
x)) - (2*a^3*(2*b*c - a*d)*Log[a + b*x])/(b^3*(b*c - a*d)^3) - (2*c^3*(b*c - 2*a
*d)*Log[c + d*x])/(d^3*(b*c - a*d)^3)

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Rubi [A]  time = 0.284685, antiderivative size = 124, 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 \[ -\frac{a^4}{b^3 (a+b x) (b c-a d)^2}-\frac{2 a^3 (2 b c-a d) \log (a+b x)}{b^3 (b c-a d)^3}-\frac{c^4}{d^3 (c+d x) (b c-a d)^2}-\frac{2 c^3 (b c-2 a d) \log (c+d x)}{d^3 (b c-a d)^3}+\frac{x}{b^2 d^2} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x)^2*(c + d*x)^2),x]

[Out]

x/(b^2*d^2) - a^4/(b^3*(b*c - a*d)^2*(a + b*x)) - c^4/(d^3*(b*c - a*d)^2*(c + d*
x)) - (2*a^3*(2*b*c - a*d)*Log[a + b*x])/(b^3*(b*c - a*d)^3) - (2*c^3*(b*c - 2*a
*d)*Log[c + d*x])/(d^3*(b*c - a*d)^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{4}}{b^{3} \left (a + b x\right ) \left (a d - b c\right )^{2}} - \frac{2 a^{3} \left (a d - 2 b c\right ) \log{\left (a + b x \right )}}{b^{3} \left (a d - b c\right )^{3}} - \frac{c^{4}}{d^{3} \left (c + d x\right ) \left (a d - b c\right )^{2}} - \frac{2 c^{3} \left (2 a d - b c\right ) \log{\left (c + d x \right )}}{d^{3} \left (a d - b c\right )^{3}} + \frac{\int \frac{1}{b^{2}}\, dx}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x+a)**2/(d*x+c)**2,x)

[Out]

-a**4/(b**3*(a + b*x)*(a*d - b*c)**2) - 2*a**3*(a*d - 2*b*c)*log(a + b*x)/(b**3*
(a*d - b*c)**3) - c**4/(d**3*(c + d*x)*(a*d - b*c)**2) - 2*c**3*(2*a*d - b*c)*lo
g(c + d*x)/(d**3*(a*d - b*c)**3) + Integral(b**(-2), x)/d**2

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Mathematica [A]  time = 0.245627, size = 123, normalized size = 0.99 \[ -\frac{a^4}{b^3 (a+b x) (b c-a d)^2}+\frac{2 a^3 (a d-2 b c) \log (a+b x)}{b^3 (b c-a d)^3}-\frac{c^4}{d^3 (c+d x) (b c-a d)^2}+\frac{2 c^3 (b c-2 a d) \log (c+d x)}{d^3 (a d-b c)^3}+\frac{x}{b^2 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((a + b*x)^2*(c + d*x)^2),x]

[Out]

x/(b^2*d^2) - a^4/(b^3*(b*c - a*d)^2*(a + b*x)) - c^4/(d^3*(b*c - a*d)^2*(c + d*
x)) + (2*a^3*(-2*b*c + a*d)*Log[a + b*x])/(b^3*(b*c - a*d)^3) + (2*c^3*(b*c - 2*
a*d)*Log[c + d*x])/(d^3*(-(b*c) + a*d)^3)

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Maple [A]  time = 0.023, size = 160, normalized size = 1.3 \[{\frac{x}{{b}^{2}{d}^{2}}}-{\frac{{c}^{4}}{{d}^{3} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-4\,{\frac{{c}^{3}\ln \left ( dx+c \right ) a}{{d}^{2} \left ( ad-bc \right ) ^{3}}}+2\,{\frac{{c}^{4}\ln \left ( dx+c \right ) b}{{d}^{3} \left ( ad-bc \right ) ^{3}}}-{\frac{{a}^{4}}{{b}^{3} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}-2\,{\frac{{a}^{4}\ln \left ( bx+a \right ) d}{{b}^{3} \left ( ad-bc \right ) ^{3}}}+4\,{\frac{{a}^{3}\ln \left ( bx+a \right ) c}{{b}^{2} \left ( ad-bc \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a)^2/(d*x+c)^2,x)

[Out]

x/b^2/d^2-1/d^3*c^4/(a*d-b*c)^2/(d*x+c)-4/d^2*c^3/(a*d-b*c)^3*ln(d*x+c)*a+2/d^3*
c^4/(a*d-b*c)^3*ln(d*x+c)*b-1/b^3*a^4/(a*d-b*c)^2/(b*x+a)-2/b^3*a^4/(a*d-b*c)^3*
ln(b*x+a)*d+4/b^2*a^3/(a*d-b*c)^3*ln(b*x+a)*c

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Maxima [A]  time = 1.38025, size = 398, normalized size = 3.21 \[ -\frac{2 \,{\left (2 \, a^{3} b c - a^{4} d\right )} \log \left (b x + a\right )}{b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}} - \frac{2 \,{\left (b c^{4} - 2 \, a c^{3} d\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}} - \frac{a b^{3} c^{4} + a^{4} c d^{3} +{\left (b^{4} c^{4} + a^{4} d^{4}\right )} x}{a b^{5} c^{3} d^{3} - 2 \, a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} +{\left (b^{6} c^{2} d^{4} - 2 \, a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{2} +{\left (b^{6} c^{3} d^{3} - a b^{5} c^{2} d^{4} - a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x} + \frac{x}{b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x + a)^2*(d*x + c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.231653, size = 725, normalized size = 5.85 \[ -\frac{a b^{4} c^{5} - a^{2} b^{3} c^{4} d + a^{4} b c^{2} d^{3} - a^{5} c d^{4} -{\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{3} -{\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{2} +{\left (b^{5} c^{5} - 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - 3 \, a^{3} b^{2} c^{2} d^{3} + 2 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} x + 2 \,{\left (2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4} +{\left (2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{2} +{\left (2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x\right )} \log \left (b x + a\right ) + 2 \,{\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d +{\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2}\right )} x^{2} +{\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{6} c^{4} d^{3} - 3 \, a^{2} b^{5} c^{3} d^{4} + 3 \, a^{3} b^{4} c^{2} d^{5} - a^{4} b^{3} c d^{6} +{\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{2} +{\left (b^{7} c^{4} d^{3} - 2 \, a b^{6} c^{3} d^{4} + 2 \, a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x + a)^2*(d*x + c)^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 22.1918, size = 694, normalized size = 5.6 \[ - \frac{2 a^{3} \left (a d - 2 b c\right ) \log{\left (x + \frac{\frac{2 a^{7} d^{6} \left (a d - 2 b c\right )}{b \left (a d - b c\right )^{3}} - \frac{8 a^{6} c d^{5} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{3}} + \frac{12 a^{5} b c^{2} d^{4} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{3}} - \frac{8 a^{4} b^{2} c^{3} d^{3} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{3}} + 2 a^{4} c d^{3} + \frac{2 a^{3} b^{3} c^{4} d^{2} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{3}} - 4 a^{3} b c^{2} d^{2} - 4 a^{2} b^{2} c^{3} d + 2 a b^{3} c^{4}}{2 a^{4} d^{4} - 4 a^{3} b c d^{3} - 4 a b^{3} c^{3} d + 2 b^{4} c^{4}} \right )}}{b^{3} \left (a d - b c\right )^{3}} - \frac{2 c^{3} \left (2 a d - b c\right ) \log{\left (x + \frac{\frac{2 a^{4} b^{2} c^{3} d^{3} \left (2 a d - b c\right )}{\left (a d - b c\right )^{3}} + 2 a^{4} c d^{3} - \frac{8 a^{3} b^{3} c^{4} d^{2} \left (2 a d - b c\right )}{\left (a d - b c\right )^{3}} - 4 a^{3} b c^{2} d^{2} + \frac{12 a^{2} b^{4} c^{5} d \left (2 a d - b c\right )}{\left (a d - b c\right )^{3}} - 4 a^{2} b^{2} c^{3} d - \frac{8 a b^{5} c^{6} \left (2 a d - b c\right )}{\left (a d - b c\right )^{3}} + 2 a b^{3} c^{4} + \frac{2 b^{6} c^{7} \left (2 a d - b c\right )}{d \left (a d - b c\right )^{3}}}{2 a^{4} d^{4} - 4 a^{3} b c d^{3} - 4 a b^{3} c^{3} d + 2 b^{4} c^{4}} \right )}}{d^{3} \left (a d - b c\right )^{3}} - \frac{a^{4} c d^{3} + a b^{3} c^{4} + x \left (a^{4} d^{4} + b^{4} c^{4}\right )}{a^{3} b^{3} c d^{5} - 2 a^{2} b^{4} c^{2} d^{4} + a b^{5} c^{3} d^{3} + x^{2} \left (a^{2} b^{4} d^{6} - 2 a b^{5} c d^{5} + b^{6} c^{2} d^{4}\right ) + x \left (a^{3} b^{3} d^{6} - a^{2} b^{4} c d^{5} - a b^{5} c^{2} d^{4} + b^{6} c^{3} d^{3}\right )} + \frac{x}{b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x+a)**2/(d*x+c)**2,x)

[Out]

-2*a**3*(a*d - 2*b*c)*log(x + (2*a**7*d**6*(a*d - 2*b*c)/(b*(a*d - b*c)**3) - 8*
a**6*c*d**5*(a*d - 2*b*c)/(a*d - b*c)**3 + 12*a**5*b*c**2*d**4*(a*d - 2*b*c)/(a*
d - b*c)**3 - 8*a**4*b**2*c**3*d**3*(a*d - 2*b*c)/(a*d - b*c)**3 + 2*a**4*c*d**3
+ 2*a**3*b**3*c**4*d**2*(a*d - 2*b*c)/(a*d - b*c)**3 - 4*a**3*b*c**2*d**2 - 4*a
**2*b**2*c**3*d + 2*a*b**3*c**4)/(2*a**4*d**4 - 4*a**3*b*c*d**3 - 4*a*b**3*c**3*
d + 2*b**4*c**4))/(b**3*(a*d - b*c)**3) - 2*c**3*(2*a*d - b*c)*log(x + (2*a**4*b
**2*c**3*d**3*(2*a*d - b*c)/(a*d - b*c)**3 + 2*a**4*c*d**3 - 8*a**3*b**3*c**4*d*
*2*(2*a*d - b*c)/(a*d - b*c)**3 - 4*a**3*b*c**2*d**2 + 12*a**2*b**4*c**5*d*(2*a*
d - b*c)/(a*d - b*c)**3 - 4*a**2*b**2*c**3*d - 8*a*b**5*c**6*(2*a*d - b*c)/(a*d
- b*c)**3 + 2*a*b**3*c**4 + 2*b**6*c**7*(2*a*d - b*c)/(d*(a*d - b*c)**3))/(2*a**
4*d**4 - 4*a**3*b*c*d**3 - 4*a*b**3*c**3*d + 2*b**4*c**4))/(d**3*(a*d - b*c)**3)
- (a**4*c*d**3 + a*b**3*c**4 + x*(a**4*d**4 + b**4*c**4))/(a**3*b**3*c*d**5 - 2
*a**2*b**4*c**2*d**4 + a*b**5*c**3*d**3 + x**2*(a**2*b**4*d**6 - 2*a*b**5*c*d**5
+ b**6*c**2*d**4) + x*(a**3*b**3*d**6 - a**2*b**4*c*d**5 - a*b**5*c**2*d**4 + b
**6*c**3*d**3)) + x/(b**2*d**2)

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GIAC/XCAS [A]  time = 0.27945, size = 421, normalized size = 3.4 \[ -\frac{a^{4} b^{3}}{{\left (b^{8} c^{2} - 2 \, a b^{7} c d + a^{2} b^{6} d^{2}\right )}{\left (b x + a\right )}} - \frac{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d\right )}{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} d^{3} - 3 \, a b^{3} c^{2} d^{4} + 3 \, a^{2} b^{2} c d^{5} - a^{3} b d^{6}} + \frac{2 \,{\left (b c + a d\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3} d^{3}} + \frac{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3} + \frac{2 \, b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{{\left (b c - a d\right )}{\left (b x + a\right )} b}\right )}{\left (b x + a\right )}}{{\left (b c - a d\right )}^{2} b^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x + a)^2*(d*x + c)^2),x, algorithm="giac")

[Out]

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