3.223 \(\int \frac{x^2}{(a+b x) (c+d x)^3} \, dx\)

Optimal. Leaf size=100 \[ \frac{a^2 \log (a+b x)}{(b c-a d)^3}-\frac{a^2 \log (c+d x)}{(b c-a d)^3}+\frac{c^2}{2 d^2 (c+d x)^2 (b c-a d)}-\frac{c (b c-2 a d)}{d^2 (c+d x) (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.19257, antiderivative size = 100, 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 \[ \frac{a^2 \log (a+b x)}{(b c-a d)^3}-\frac{a^2 \log (c+d x)}{(b c-a d)^3}+\frac{c^2}{2 d^2 (c+d x)^2 (b c-a d)}-\frac{c (b c-2 a d)}{d^2 (c+d x) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((a + b*x)*(c + d*x)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 47.9046, size = 85, normalized size = 0.85 \[ - \frac{a^{2} \log{\left (a + b x \right )}}{\left (a d - b c\right )^{3}} + \frac{a^{2} \log{\left (c + d x \right )}}{\left (a d - b c\right )^{3}} - \frac{c^{2}}{2 d^{2} \left (c + d x\right )^{2} \left (a d - b c\right )} + \frac{c \left (2 a d - b c\right )}{d^{2} \left (c + d x\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x+a)/(d*x+c)**3,x)

[Out]

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

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Mathematica [A]  time = 0.105363, size = 99, normalized size = 0.99 \[ \frac{-2 a^2 d^2 (c+d x)^2 \log (a+b x)+2 a^2 d^2 (c+d x)^2 \log (c+d x)+c (b c-a d) (b c (c+2 d x)-a d (3 c+4 d x))}{2 d^2 (c+d x)^2 (a d-b c)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/((a + b*x)*(c + d*x)^3),x]

[Out]

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

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Maple [A]  time = 0.014, size = 118, normalized size = 1.2 \[{\frac{{a}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}}-{\frac{{c}^{2}}{2\,{d}^{2} \left ( ad-bc \right ) \left ( dx+c \right ) ^{2}}}+2\,{\frac{ac}{d \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-{\frac{{c}^{2}b}{ \left ( ad-bc \right ) ^{2}{d}^{2} \left ( dx+c \right ) }}-{\frac{{a}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x+a)/(d*x+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.36138, size = 304, normalized size = 3.04 \[ \frac{a^{2} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{a^{2} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{b c^{3} - 3 \, a c^{2} d + 2 \,{\left (b c^{2} d - 2 \, a c d^{2}\right )} x}{2 \,{\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4} +{\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )} x^{2} + 2 \,{\left (b^{2} c^{3} d^{3} - 2 \, a b c^{2} d^{4} + a^{2} c d^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x + a)*(d*x + c)^3),x, algorithm="maxima")

[Out]

a^2*log(b*x + a)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - a^2*log(d
*x + c)/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - 1/2*(b*c^3 - 3*a*c
^2*d + 2*(b*c^2*d - 2*a*c*d^2)*x)/(b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4 + (
b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6)*x^2 + 2*(b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*
c*d^5)*x)

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Fricas [A]  time = 0.207824, size = 375, normalized size = 3.75 \[ -\frac{b^{2} c^{4} - 4 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + 2 \,{\left (b^{2} c^{3} d - 3 \, a b c^{2} d^{2} + 2 \, a^{2} c d^{3}\right )} x - 2 \,{\left (a^{2} d^{4} x^{2} + 2 \, a^{2} c d^{3} x + a^{2} c^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left (a^{2} d^{4} x^{2} + 2 \, a^{2} c d^{3} x + a^{2} c^{2} d^{2}\right )} \log \left (d x + c\right )}{2 \,{\left (b^{3} c^{5} d^{2} - 3 \, a b^{2} c^{4} d^{3} + 3 \, a^{2} b c^{3} d^{4} - a^{3} c^{2} d^{5} +{\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} x^{2} + 2 \,{\left (b^{3} c^{4} d^{3} - 3 \, a b^{2} c^{3} d^{4} + 3 \, a^{2} b c^{2} d^{5} - a^{3} c d^{6}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x + a)*(d*x + c)^3),x, algorithm="fricas")

[Out]

-1/2*(b^2*c^4 - 4*a*b*c^3*d + 3*a^2*c^2*d^2 + 2*(b^2*c^3*d - 3*a*b*c^2*d^2 + 2*a
^2*c*d^3)*x - 2*(a^2*d^4*x^2 + 2*a^2*c*d^3*x + a^2*c^2*d^2)*log(b*x + a) + 2*(a^
2*d^4*x^2 + 2*a^2*c*d^3*x + a^2*c^2*d^2)*log(d*x + c))/(b^3*c^5*d^2 - 3*a*b^2*c^
4*d^3 + 3*a^2*b*c^3*d^4 - a^3*c^2*d^5 + (b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b
*c*d^6 - a^3*d^7)*x^2 + 2*(b^3*c^4*d^3 - 3*a*b^2*c^3*d^4 + 3*a^2*b*c^2*d^5 - a^3
*c*d^6)*x)

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Sympy [A]  time = 7.90104, size = 408, normalized size = 4.08 \[ \frac{a^{2} \log{\left (x + \frac{- \frac{a^{6} d^{4}}{\left (a d - b c\right )^{3}} + \frac{4 a^{5} b c d^{3}}{\left (a d - b c\right )^{3}} - \frac{6 a^{4} b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b^{3} c^{3} d}{\left (a d - b c\right )^{3}} + a^{3} d - \frac{a^{2} b^{4} c^{4}}{\left (a d - b c\right )^{3}} + a^{2} b c}{2 a^{2} b d} \right )}}{\left (a d - b c\right )^{3}} - \frac{a^{2} \log{\left (x + \frac{\frac{a^{6} d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a^{5} b c d^{3}}{\left (a d - b c\right )^{3}} + \frac{6 a^{4} b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b^{3} c^{3} d}{\left (a d - b c\right )^{3}} + a^{3} d + \frac{a^{2} b^{4} c^{4}}{\left (a d - b c\right )^{3}} + a^{2} b c}{2 a^{2} b d} \right )}}{\left (a d - b c\right )^{3}} + \frac{3 a c^{2} d - b c^{3} + x \left (4 a c d^{2} - 2 b c^{2} d\right )}{2 a^{2} c^{2} d^{4} - 4 a b c^{3} d^{3} + 2 b^{2} c^{4} d^{2} + x^{2} \left (2 a^{2} d^{6} - 4 a b c d^{5} + 2 b^{2} c^{2} d^{4}\right ) + x \left (4 a^{2} c d^{5} - 8 a b c^{2} d^{4} + 4 b^{2} c^{3} d^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x+a)/(d*x+c)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.273103, size = 254, normalized size = 2.54 \[ \frac{a^{2} b{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{a^{2} d{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}} - \frac{b^{2} c^{4} - 4 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} + 2 \,{\left (b^{2} c^{3} d - 3 \, a b c^{2} d^{2} + 2 \, a^{2} c d^{3}\right )} x}{2 \,{\left (b c - a d\right )}^{3}{\left (d x + c\right )}^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x + a)*(d*x + c)^3),x, algorithm="giac")

[Out]

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