3.317 \(\int \frac{1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=215 \[ \frac{b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{b^4}{2 a^2 \left (a+b x^2\right ) (b c-a d)^3}+\frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}-\frac{1}{2 a^2 c^3 x^2}-\frac{d^3 (2 b c-a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{d^3}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)^2} \]

[Out]

-1/(2*a^2*c^3*x^2) - b^4/(2*a^2*(b*c - a*d)^3*(a + b*x^2)) - d^3/(4*c^2*(b*c - a
*d)^2*(c + d*x^2)^2) - (d^3*(2*b*c - a*d))/(c^3*(b*c - a*d)^3*(c + d*x^2)) - ((2
*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log[a + b*x^2])/(2*a^3*(b
*c - a*d)^4) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x^2])/(2*c^4
*(b*c - a*d)^4)

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Rubi [A]  time = 0.612914, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{2 a^3 (b c-a d)^4}-\frac{\log (x) (3 a d+2 b c)}{a^3 c^4}-\frac{b^4}{2 a^2 \left (a+b x^2\right ) (b c-a d)^3}+\frac{d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^4}-\frac{1}{2 a^2 c^3 x^2}-\frac{d^3 (2 b c-a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{d^3}{4 c^2 \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

-1/(2*a^2*c^3*x^2) - b^4/(2*a^2*(b*c - a*d)^3*(a + b*x^2)) - d^3/(4*c^2*(b*c - a
*d)^2*(c + d*x^2)^2) - (d^3*(2*b*c - a*d))/(c^3*(b*c - a*d)^3*(c + d*x^2)) - ((2
*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (b^4*(2*b*c - 5*a*d)*Log[a + b*x^2])/(2*a^3*(b
*c - a*d)^4) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x^2])/(2*c^4
*(b*c - a*d)^4)

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Rubi in Sympy [A]  time = 124.796, size = 202, normalized size = 0.94 \[ - \frac{d^{3}}{4 c^{2} \left (c + d x^{2}\right )^{2} \left (a d - b c\right )^{2}} - \frac{d^{3} \left (a d - 2 b c\right )}{c^{3} \left (c + d x^{2}\right ) \left (a d - b c\right )^{3}} + \frac{d^{3} \left (3 a^{2} d^{2} - 10 a b c d + 10 b^{2} c^{2}\right ) \log{\left (c + d x^{2} \right )}}{2 c^{4} \left (a d - b c\right )^{4}} + \frac{b^{4}}{2 a^{2} \left (a + b x^{2}\right ) \left (a d - b c\right )^{3}} - \frac{1}{2 a^{2} c^{3} x^{2}} - \frac{b^{4} \left (5 a d - 2 b c\right ) \log{\left (a + b x^{2} \right )}}{2 a^{3} \left (a d - b c\right )^{4}} - \frac{\left (3 a d + 2 b c\right ) \log{\left (x^{2} \right )}}{2 a^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

-d**3/(4*c**2*(c + d*x**2)**2*(a*d - b*c)**2) - d**3*(a*d - 2*b*c)/(c**3*(c + d*
x**2)*(a*d - b*c)**3) + d**3*(3*a**2*d**2 - 10*a*b*c*d + 10*b**2*c**2)*log(c + d
*x**2)/(2*c**4*(a*d - b*c)**4) + b**4/(2*a**2*(a + b*x**2)*(a*d - b*c)**3) - 1/(
2*a**2*c**3*x**2) - b**4*(5*a*d - 2*b*c)*log(a + b*x**2)/(2*a**3*(a*d - b*c)**4)
- (3*a*d + 2*b*c)*log(x**2)/(2*a**3*c**4)

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Mathematica [A]  time = 0.62441, size = 208, normalized size = 0.97 \[ \frac{1}{4} \left (\frac{2 b^4 (2 b c-5 a d) \log \left (a+b x^2\right )}{a^3 (b c-a d)^4}-\frac{4 \log (x) (3 a d+2 b c)}{a^3 c^4}+\frac{2 b^4}{a^2 \left (a+b x^2\right ) (a d-b c)^3}+\frac{2 d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \log \left (c+d x^2\right )}{c^4 (b c-a d)^4}-\frac{2}{a^2 c^3 x^2}+\frac{4 d^3 (a d-2 b c)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{d^3}{c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(-2/(a^2*c^3*x^2) + (2*b^4)/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - d^3/(c^2*(b*c -
a*d)^2*(c + d*x^2)^2) + (4*d^3*(-2*b*c + a*d))/(c^3*(b*c - a*d)^3*(c + d*x^2))
- (4*(2*b*c + 3*a*d)*Log[x])/(a^3*c^4) + (2*b^4*(2*b*c - 5*a*d)*Log[a + b*x^2])/
(a^3*(b*c - a*d)^4) + (2*d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Log[c + d*x^2
])/(c^4*(b*c - a*d)^4))/4

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Maple [A]  time = 0.04, size = 405, normalized size = 1.9 \[ -{\frac{1}{2\,{a}^{2}{c}^{3}{x}^{2}}}-3\,{\frac{\ln \left ( x \right ) d}{{a}^{2}{c}^{4}}}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}{c}^{3}}}-{\frac{{d}^{5}{a}^{2}}{{c}^{3} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}+3\,{\frac{{d}^{4}ab}{{c}^{2} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}-2\,{\frac{{b}^{2}{d}^{3}}{c \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}-{\frac{{d}^{5}{a}^{2}}{4\,{c}^{2} \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{d}^{4}ab}{2\,c \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{b}^{2}{d}^{3}}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{d}^{5}\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{c}^{4} \left ( ad-bc \right ) ^{4}}}-5\,{\frac{{d}^{4}\ln \left ( d{x}^{2}+c \right ) ab}{{c}^{3} \left ( ad-bc \right ) ^{4}}}+5\,{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ){b}^{2}}{{c}^{2} \left ( ad-bc \right ) ^{4}}}-{\frac{5\,{b}^{4}\ln \left ( b{x}^{2}+a \right ) d}{2\,{a}^{2} \left ( ad-bc \right ) ^{4}}}+{\frac{{b}^{5}\ln \left ( b{x}^{2}+a \right ) c}{{a}^{3} \left ( ad-bc \right ) ^{4}}}+{\frac{{b}^{4}d}{2\,a \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{5}c}{2\,{a}^{2} \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.40466, size = 879, normalized size = 4.09 \[ \frac{{\left (2 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )}} + \frac{{\left (10 \, b^{2} c^{2} d^{3} - 10 \, a b c d^{4} + 3 \, a^{2} d^{5}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )}} - \frac{2 \, a b^{3} c^{5} - 6 \, a^{2} b^{2} c^{4} d + 6 \, a^{3} b c^{3} d^{2} - 2 \, a^{4} c^{2} d^{3} + 2 \,{\left (2 \, b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 7 \, a^{2} b^{2} c d^{4} - 3 \, a^{3} b d^{5}\right )} x^{6} +{\left (8 \, b^{4} c^{4} d - 10 \, a b^{3} c^{3} d^{2} + 15 \, a^{2} b^{2} c^{2} d^{3} + 5 \, a^{3} b c d^{4} - 6 \, a^{4} d^{5}\right )} x^{4} +{\left (4 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d - 6 \, a^{2} b^{2} c^{3} d^{2} + 19 \, a^{3} b c^{2} d^{3} - 9 \, a^{4} c d^{4}\right )} x^{2}}{4 \,{\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{8} +{\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{6} +{\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{4} +{\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x^{2}\right )}} - \frac{{\left (2 \, b c + 3 \, a d\right )} \log \left (x^{2}\right )}{2 \, a^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^3),x, algorithm="maxima")

[Out]

1/2*(2*b^5*c - 5*a*b^4*d)*log(b*x^2 + a)/(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*
b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4) + 1/2*(10*b^2*c^2*d^3 - 10*a*b*c*d^4 + 3*
a^2*d^5)*log(d*x^2 + c)/(b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d^2 - 4*a^3*b*c
^5*d^3 + a^4*c^4*d^4) - 1/4*(2*a*b^3*c^5 - 6*a^2*b^2*c^4*d + 6*a^3*b*c^3*d^2 - 2
*a^4*c^2*d^3 + 2*(2*b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 7*a^2*b^2*c*d^4 - 3*a^3*b*d^
5)*x^6 + (8*b^4*c^4*d - 10*a*b^3*c^3*d^2 + 15*a^2*b^2*c^2*d^3 + 5*a^3*b*c*d^4 -
6*a^4*d^5)*x^4 + (4*b^4*c^5 - 2*a*b^3*c^4*d - 6*a^2*b^2*c^3*d^2 + 19*a^3*b*c^2*d
^3 - 9*a^4*c*d^4)*x^2)/((a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4*d^4
- a^5*b*c^3*d^5)*x^8 + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3
+ a^5*b*c^4*d^4 - a^6*c^3*d^5)*x^6 + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c
^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^4 + (a^3*b^3*c^8 - 3*a^4*b^2*c^7*d +
3*a^5*b*c^6*d^2 - a^6*c^5*d^3)*x^2) - 1/2*(2*b*c + 3*a*d)*log(x^2)/(a^3*c^4)

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Fricas [A]  time = 29.6754, size = 1656, normalized size = 7.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^3),x, algorithm="fricas")

[Out]

-1/4*(2*a^2*b^4*c^7 - 8*a^3*b^3*c^6*d + 12*a^4*b^2*c^5*d^2 - 8*a^5*b*c^4*d^3 + 2
*a^6*c^3*d^4 + 2*(2*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3 + 10*a^3*b^3*c^3*d^4 - 10*
a^4*b^2*c^2*d^5 + 3*a^5*b*c*d^6)*x^6 + (8*a*b^5*c^6*d - 18*a^2*b^4*c^5*d^2 + 25*
a^3*b^3*c^4*d^3 - 10*a^4*b^2*c^3*d^4 - 11*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^4 + (4*
a*b^5*c^7 - 6*a^2*b^4*c^6*d - 4*a^3*b^3*c^5*d^2 + 25*a^4*b^2*c^4*d^3 - 28*a^5*b*
c^3*d^4 + 9*a^6*c^2*d^5)*x^2 - 2*((2*b^6*c^5*d^2 - 5*a*b^5*c^4*d^3)*x^8 + (4*b^6
*c^6*d - 8*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3)*x^6 + (2*b^6*c^7 - a*b^5*c^6*d - 1
0*a^2*b^4*c^5*d^2)*x^4 + (2*a*b^5*c^7 - 5*a^2*b^4*c^6*d)*x^2)*log(b*x^2 + a) - 2
*((10*a^3*b^3*c^2*d^5 - 10*a^4*b^2*c*d^6 + 3*a^5*b*d^7)*x^8 + (20*a^3*b^3*c^3*d^
4 - 10*a^4*b^2*c^2*d^5 - 4*a^5*b*c*d^6 + 3*a^6*d^7)*x^6 + (10*a^3*b^3*c^4*d^3 +
10*a^4*b^2*c^3*d^4 - 17*a^5*b*c^2*d^5 + 6*a^6*c*d^6)*x^4 + (10*a^4*b^2*c^4*d^3 -
10*a^5*b*c^3*d^4 + 3*a^6*c^2*d^5)*x^2)*log(d*x^2 + c) + 4*((2*b^6*c^5*d^2 - 5*a
*b^5*c^4*d^3 + 10*a^3*b^3*c^2*d^5 - 10*a^4*b^2*c*d^6 + 3*a^5*b*d^7)*x^8 + (4*b^6
*c^6*d - 8*a*b^5*c^5*d^2 - 5*a^2*b^4*c^4*d^3 + 20*a^3*b^3*c^3*d^4 - 10*a^4*b^2*c
^2*d^5 - 4*a^5*b*c*d^6 + 3*a^6*d^7)*x^6 + (2*b^6*c^7 - a*b^5*c^6*d - 10*a^2*b^4*
c^5*d^2 + 10*a^3*b^3*c^4*d^3 + 10*a^4*b^2*c^3*d^4 - 17*a^5*b*c^2*d^5 + 6*a^6*c*d
^6)*x^4 + (2*a*b^5*c^7 - 5*a^2*b^4*c^6*d + 10*a^4*b^2*c^4*d^3 - 10*a^5*b*c^3*d^4
+ 3*a^6*c^2*d^5)*x^2)*log(x))/((a^3*b^5*c^8*d^2 - 4*a^4*b^4*c^7*d^3 + 6*a^5*b^3
*c^6*d^4 - 4*a^6*b^2*c^5*d^5 + a^7*b*c^4*d^6)*x^8 + (2*a^3*b^5*c^9*d - 7*a^4*b^4
*c^8*d^2 + 8*a^5*b^3*c^7*d^3 - 2*a^6*b^2*c^6*d^4 - 2*a^7*b*c^5*d^5 + a^8*c^4*d^6
)*x^6 + (a^3*b^5*c^10 - 2*a^4*b^4*c^9*d - 2*a^5*b^3*c^8*d^2 + 8*a^6*b^2*c^7*d^3
- 7*a^7*b*c^6*d^4 + 2*a^8*c^5*d^5)*x^4 + (a^4*b^4*c^10 - 4*a^5*b^3*c^9*d + 6*a^6
*b^2*c^8*d^2 - 4*a^7*b*c^7*d^3 + a^8*c^6*d^4)*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275722, size = 861, normalized size = 4. \[ \frac{{\left (2 \, b^{6} c - 5 \, a b^{5} d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )}} + \frac{{\left (10 \, b^{2} c^{2} d^{4} - 10 \, a b c d^{5} + 3 \, a^{2} d^{6}\right )}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )}} + \frac{10 \, a^{2} b^{3} c^{2} d^{3} x^{4} - 10 \, a^{3} b^{2} c d^{4} x^{4} + 3 \, a^{4} b d^{5} x^{4} - 4 \, b^{5} c^{5} x^{2} + 10 \, a b^{4} c^{4} d x^{2} - 12 \, a^{2} b^{3} c^{3} d^{2} x^{2} + 18 \, a^{3} b^{2} c^{2} d^{3} x^{2} - 12 \, a^{4} b c d^{4} x^{2} + 3 \, a^{5} d^{5} x^{2} - 2 \, a b^{4} c^{5} + 8 \, a^{2} b^{3} c^{4} d - 12 \, a^{3} b^{2} c^{3} d^{2} + 8 \, a^{4} b c^{2} d^{3} - 2 \, a^{5} c d^{4}}{4 \,{\left (a^{2} b^{4} c^{8} - 4 \, a^{3} b^{3} c^{7} d + 6 \, a^{4} b^{2} c^{6} d^{2} - 4 \, a^{5} b c^{5} d^{3} + a^{6} c^{4} d^{4}\right )}{\left (b x^{4} + a x^{2}\right )}} - \frac{30 \, b^{2} c^{2} d^{5} x^{4} - 30 \, a b c d^{6} x^{4} + 9 \, a^{2} d^{7} x^{4} + 68 \, b^{2} c^{3} d^{4} x^{2} - 72 \, a b c^{2} d^{5} x^{2} + 22 \, a^{2} c d^{6} x^{2} + 39 \, b^{2} c^{4} d^{3} - 44 \, a b c^{3} d^{4} + 14 \, a^{2} c^{2} d^{5}}{4 \,{\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )}{\left (d x^{2} + c\right )}^{2}} - \frac{{\left (2 \, b c + 3 \, a d\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^3),x, algorithm="giac")

[Out]

1/2*(2*b^6*c - 5*a*b^5*d)*ln(abs(b*x^2 + a))/(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*
a^5*b^3*c^2*d^2 - 4*a^6*b^2*c*d^3 + a^7*b*d^4) + 1/2*(10*b^2*c^2*d^4 - 10*a*b*c*
d^5 + 3*a^2*d^6)*ln(abs(d*x^2 + c))/(b^4*c^8*d - 4*a*b^3*c^7*d^2 + 6*a^2*b^2*c^6
*d^3 - 4*a^3*b*c^5*d^4 + a^4*c^4*d^5) + 1/4*(10*a^2*b^3*c^2*d^3*x^4 - 10*a^3*b^2
*c*d^4*x^4 + 3*a^4*b*d^5*x^4 - 4*b^5*c^5*x^2 + 10*a*b^4*c^4*d*x^2 - 12*a^2*b^3*c
^3*d^2*x^2 + 18*a^3*b^2*c^2*d^3*x^2 - 12*a^4*b*c*d^4*x^2 + 3*a^5*d^5*x^2 - 2*a*b
^4*c^5 + 8*a^2*b^3*c^4*d - 12*a^3*b^2*c^3*d^2 + 8*a^4*b*c^2*d^3 - 2*a^5*c*d^4)/(
(a^2*b^4*c^8 - 4*a^3*b^3*c^7*d + 6*a^4*b^2*c^6*d^2 - 4*a^5*b*c^5*d^3 + a^6*c^4*d
^4)*(b*x^4 + a*x^2)) - 1/4*(30*b^2*c^2*d^5*x^4 - 30*a*b*c*d^6*x^4 + 9*a^2*d^7*x^
4 + 68*b^2*c^3*d^4*x^2 - 72*a*b*c^2*d^5*x^2 + 22*a^2*c*d^6*x^2 + 39*b^2*c^4*d^3
- 44*a*b*c^3*d^4 + 14*a^2*c^2*d^5)/((b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d^2
- 4*a^3*b*c^5*d^3 + a^4*c^4*d^4)*(d*x^2 + c)^2) - 1/2*(2*b*c + 3*a*d)*ln(x^2)/(
a^3*c^4)