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

Optimal. Leaf size=142 \[ \frac{a b}{2 \left (a+b x^2\right ) (b c-a d)^3}+\frac{a d+b c}{2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{c}{4 \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{b (2 a d+b c) \log \left (a+b x^2\right )}{2 (b c-a d)^4}-\frac{b (2 a d+b c) \log \left (c+d x^2\right )}{2 (b c-a d)^4} \]

[Out]

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

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Rubi [A]  time = 0.329991, antiderivative size = 142, 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 \[ \frac{a b}{2 \left (a+b x^2\right ) (b c-a d)^3}+\frac{a d+b c}{2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{c}{4 \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{b (2 a d+b c) \log \left (a+b x^2\right )}{2 (b c-a d)^4}-\frac{b (2 a d+b c) \log \left (c+d x^2\right )}{2 (b c-a d)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 52.1401, size = 121, normalized size = 0.85 \[ - \frac{a b}{2 \left (a + b x^{2}\right ) \left (a d - b c\right )^{3}} + \frac{b \left (2 a d + b c\right ) \log{\left (a + b x^{2} \right )}}{2 \left (a d - b c\right )^{4}} - \frac{b \left (2 a d + b c\right ) \log{\left (c + d x^{2} \right )}}{2 \left (a d - b c\right )^{4}} + \frac{c}{4 \left (c + d x^{2}\right )^{2} \left (a d - b c\right )^{2}} - \frac{a d + b c}{2 \left (c + d x^{2}\right ) \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.161484, size = 121, normalized size = 0.85 \[ \frac{\frac{c (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac{2 a b (b c-a d)}{a+b x^2}+\frac{2 (a d+b c) (b c-a d)}{c+d x^2}+2 b (2 a d+b c) \log \left (a+b x^2\right )-2 b (2 a d+b c) \log \left (c+d x^2\right )}{4 (b c-a d)^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 283, normalized size = 2. \[ -{\frac{{a}^{2}{d}^{2}}{2\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}+{\frac{{b}^{2}{c}^{2}}{2\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) }}+{\frac{{a}^{2}c{d}^{2}}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{ab{c}^{2}d}{2\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{b}^{2}{c}^{3}}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{bd\ln \left ( d{x}^{2}+c \right ) a}{ \left ( ad-bc \right ) ^{4}}}-{\frac{{b}^{2}\ln \left ( d{x}^{2}+c \right ) c}{2\, \left ( ad-bc \right ) ^{4}}}+{\frac{b\ln \left ( b{x}^{2}+a \right ) ad}{ \left ( ad-bc \right ) ^{4}}}+{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) c}{2\, \left ( ad-bc \right ) ^{4}}}-{\frac{b{a}^{2}d}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{a{b}^{2}c}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.38297, size = 560, normalized size = 3.94 \[ \frac{{\left (b^{2} c + 2 \, a b d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} - \frac{{\left (b^{2} c + 2 \, a b d\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}} + \frac{2 \,{\left (b^{2} c d + 2 \, a b d^{2}\right )} x^{4} + 5 \, a b c^{2} + a^{2} c d +{\left (3 \, b^{2} c^{2} + 7 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}}{4 \,{\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} +{\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} +{\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} +{\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.265148, size = 807, normalized size = 5.68 \[ \frac{5 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + 2 \,{\left (b^{3} c^{2} d + a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{4} +{\left (3 \, b^{3} c^{3} + 4 \, a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3}\right )} x^{2} + 2 \,{\left ({\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{6} + a b^{2} c^{3} + 2 \, a^{2} b c^{2} d +{\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left ({\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{6} + a b^{2} c^{3} + 2 \, a^{2} b c^{2} d +{\left (2 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} + 4 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4} +{\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} x^{6} +{\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{4} +{\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 57.4828, size = 780, normalized size = 5.49 \[ - \frac{b \left (2 a d + b c\right ) \log{\left (x^{2} + \frac{- \frac{a^{5} b d^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac{5 a^{4} b^{2} c d^{4} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac{10 a^{3} b^{3} c^{2} d^{3} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac{10 a^{2} b^{4} c^{3} d^{2} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} - \frac{5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d + \frac{b^{6} c^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{2 \left (a d - b c\right )^{4}} + \frac{b \left (2 a d + b c\right ) \log{\left (x^{2} + \frac{\frac{a^{5} b d^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac{5 a^{4} b^{2} c d^{4} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + \frac{10 a^{3} b^{3} c^{2} d^{3} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} - \frac{10 a^{2} b^{4} c^{3} d^{2} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{2} b d^{2} + \frac{5 a b^{5} c^{4} d \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + 3 a b^{2} c d - \frac{b^{6} c^{5} \left (2 a d + b c\right )}{\left (a d - b c\right )^{4}} + b^{3} c^{2}}{4 a b^{2} d^{2} + 2 b^{3} c d} \right )}}{2 \left (a d - b c\right )^{4}} - \frac{a^{2} c d + 5 a b c^{2} + x^{4} \left (4 a b d^{2} + 2 b^{2} c d\right ) + x^{2} \left (2 a^{2} d^{2} + 7 a b c d + 3 b^{2} c^{2}\right )}{4 a^{4} c^{2} d^{3} - 12 a^{3} b c^{3} d^{2} + 12 a^{2} b^{2} c^{4} d - 4 a b^{3} c^{5} + x^{6} \left (4 a^{3} b d^{5} - 12 a^{2} b^{2} c d^{4} + 12 a b^{3} c^{2} d^{3} - 4 b^{4} c^{3} d^{2}\right ) + x^{4} \left (4 a^{4} d^{5} - 4 a^{3} b c d^{4} - 12 a^{2} b^{2} c^{2} d^{3} + 20 a b^{3} c^{3} d^{2} - 8 b^{4} c^{4} d\right ) + x^{2} \left (8 a^{4} c d^{4} - 20 a^{3} b c^{2} d^{3} + 12 a^{2} b^{2} c^{3} d^{2} + 4 a b^{3} c^{4} d - 4 b^{4} c^{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(2*a*d + b*c)*log(x**2 + (-a**5*b*d**5*(2*a*d + b*c)/(a*d - b*c)**4 + 5*a**4*
b**2*c*d**4*(2*a*d + b*c)/(a*d - b*c)**4 - 10*a**3*b**3*c**2*d**3*(2*a*d + b*c)/
(a*d - b*c)**4 + 10*a**2*b**4*c**3*d**2*(2*a*d + b*c)/(a*d - b*c)**4 + 2*a**2*b*
d**2 - 5*a*b**5*c**4*d*(2*a*d + b*c)/(a*d - b*c)**4 + 3*a*b**2*c*d + b**6*c**5*(
2*a*d + b*c)/(a*d - b*c)**4 + b**3*c**2)/(4*a*b**2*d**2 + 2*b**3*c*d))/(2*(a*d -
b*c)**4) + b*(2*a*d + b*c)*log(x**2 + (a**5*b*d**5*(2*a*d + b*c)/(a*d - b*c)**4
- 5*a**4*b**2*c*d**4*(2*a*d + b*c)/(a*d - b*c)**4 + 10*a**3*b**3*c**2*d**3*(2*a
*d + b*c)/(a*d - b*c)**4 - 10*a**2*b**4*c**3*d**2*(2*a*d + b*c)/(a*d - b*c)**4 +
2*a**2*b*d**2 + 5*a*b**5*c**4*d*(2*a*d + b*c)/(a*d - b*c)**4 + 3*a*b**2*c*d - b
**6*c**5*(2*a*d + b*c)/(a*d - b*c)**4 + b**3*c**2)/(4*a*b**2*d**2 + 2*b**3*c*d))
/(2*(a*d - b*c)**4) - (a**2*c*d + 5*a*b*c**2 + x**4*(4*a*b*d**2 + 2*b**2*c*d) +
x**2*(2*a**2*d**2 + 7*a*b*c*d + 3*b**2*c**2))/(4*a**4*c**2*d**3 - 12*a**3*b*c**3
*d**2 + 12*a**2*b**2*c**4*d - 4*a*b**3*c**5 + x**6*(4*a**3*b*d**5 - 12*a**2*b**2
*c*d**4 + 12*a*b**3*c**2*d**3 - 4*b**4*c**3*d**2) + x**4*(4*a**4*d**5 - 4*a**3*b
*c*d**4 - 12*a**2*b**2*c**2*d**3 + 20*a*b**3*c**3*d**2 - 8*b**4*c**4*d) + x**2*(
8*a**4*c*d**4 - 20*a**3*b*c**2*d**3 + 12*a**2*b**2*c**3*d**2 + 4*a*b**3*c**4*d -
4*b**4*c**5))

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GIAC/XCAS [A]  time = 0.239851, size = 360, normalized size = 2.54 \[ \frac{\frac{2 \, a b^{5}}{{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )}{\left (b x^{2} + a\right )}} - \frac{2 \,{\left (b^{4} c + 2 \, a b^{3} d\right )}{\rm ln}\left ({\left | \frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac{3 \, b^{3} c d^{2} + 2 \, a b^{2} d^{3} + \frac{2 \,{\left (2 \, b^{5} c^{2} d - a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )}}{{\left (b x^{2} + a\right )} b}}{{\left (b c - a d\right )}^{4}{\left (\frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d\right )}^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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