3.310 \(\int \frac{x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=207 \[ \frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} \sqrt{d} (b c-a d)^4}+\frac{3 x (3 a d+b c)}{8 \left (c+d x^2\right ) (b c-a d)^3}+\frac{x (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac{3 \sqrt{a} \sqrt{b} (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 (b c-a d)^4} \]

[Out]

((b*c + 2*a*d)*x)/(4*b*(b*c - a*d)^2*(c + d*x^2)^2) + (a*x)/(2*b*(b*c - a*d)*(a
+ b*x^2)*(c + d*x^2)^2) + (3*(b*c + 3*a*d)*x)/(8*(b*c - a*d)^3*(c + d*x^2)) - (3
*Sqrt[a]*Sqrt[b]*(b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*(b*c - a*d)^4) + (3
*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[c]*Sqrt[d]
*(b*c - a*d)^4)

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Rubi [A]  time = 0.629957, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} \sqrt{d} (b c-a d)^4}+\frac{3 x (3 a d+b c)}{8 \left (c+d x^2\right ) (b c-a d)^3}+\frac{x (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac{3 \sqrt{a} \sqrt{b} (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

((b*c + 2*a*d)*x)/(4*b*(b*c - a*d)^2*(c + d*x^2)^2) + (a*x)/(2*b*(b*c - a*d)*(a
+ b*x^2)*(c + d*x^2)^2) + (3*(b*c + 3*a*d)*x)/(8*(b*c - a*d)^3*(c + d*x^2)) - (3
*Sqrt[a]*Sqrt[b]*(b*c + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*(b*c - a*d)^4) + (3
*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*Sqrt[c]*Sqrt[d]
*(b*c - a*d)^4)

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Rubi in Sympy [A]  time = 119.91, size = 187, normalized size = 0.9 \[ - \frac{3 \sqrt{a} \sqrt{b} \left (a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \left (a d - b c\right )^{4}} - \frac{a x}{2 b \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} - \frac{3 x \left (3 a d + b c\right )}{8 \left (c + d x^{2}\right ) \left (a d - b c\right )^{3}} + \frac{3 \left (a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 \sqrt{c} \sqrt{d} \left (a d - b c\right )^{4}} + \frac{x \left (2 a d + b c\right )}{4 b \left (c + d x^{2}\right )^{2} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

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

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Mathematica [A]  time = 0.611585, size = 166, normalized size = 0.8 \[ \frac{\frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} \sqrt{d}}+\frac{2 c x (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac{4 a b x (b c-a d)}{a+b x^2}+\frac{x (5 a d+3 b c) (b c-a d)}{c+d x^2}-12 \sqrt{a} \sqrt{b} (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

((4*a*b*(b*c - a*d)*x)/(a + b*x^2) + (2*c*(b*c - a*d)^2*x)/(c + d*x^2)^2 + ((b*c
- a*d)*(3*b*c + 5*a*d)*x)/(c + d*x^2) - 12*Sqrt[a]*Sqrt[b]*(b*c + a*d)*ArcTan[(
Sqrt[b]*x)/Sqrt[a]] + (3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt
[c]])/(Sqrt[c]*Sqrt[d]))/(8*(b*c - a*d)^4)

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Maple [B]  time = 0.023, size = 388, normalized size = 1.9 \[ -{\frac{5\,{x}^{3}{a}^{2}{d}^{3}}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{x}^{3}abc{d}^{2}}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{x}^{3}{b}^{2}{c}^{2}d}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{3\,x{a}^{2}c{d}^{2}}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{xab{c}^{2}d}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,x{b}^{2}{c}^{3}}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{a}^{2}{d}^{2}}{8\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{9\,cabd}{4\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,{b}^{2}{c}^{2}}{8\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}bxd}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{a{b}^{2}xc}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,{a}^{2}bd}{2\, \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,a{b}^{2}c}{2\, \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

-5/8/(a*d-b*c)^4/(d*x^2+c)^2*x^3*a^2*d^3+1/4/(a*d-b*c)^4/(d*x^2+c)^2*x^3*a*b*c*d
^2+3/8/(a*d-b*c)^4/(d*x^2+c)^2*x^3*b^2*c^2*d-3/8/(a*d-b*c)^4/(d*x^2+c)^2*x*a^2*c
*d^2-1/4/(a*d-b*c)^4/(d*x^2+c)^2*x*a*b*c^2*d+5/8/(a*d-b*c)^4/(d*x^2+c)^2*x*b^2*c
^3+3/8/(a*d-b*c)^4/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2*d^2+9/4/(a*d-b*c)^4/(
c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*c*a*b*d+3/8/(a*d-b*c)^4/(c*d)^(1/2)*arctan(x*
d/(c*d)^(1/2))*b^2*c^2-1/2*a^2*b/(a*d-b*c)^4*x/(b*x^2+a)*d+1/2*a*b^2/(a*d-b*c)^4
*x/(b*x^2+a)*c-3/2*a^2*b/(a*d-b*c)^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d-3/2*a
*b^2/(a*d-b*c)^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.10676, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="fricas")

[Out]

[1/16*(12*((b^2*c*d^2 + a*b*d^3)*x^6 + a*b*c^3 + a^2*c^2*d + (2*b^2*c^2*d + 3*a*
b*c*d^2 + a^2*d^3)*x^4 + (b^2*c^3 + 3*a*b*c^2*d + 2*a^2*c*d^2)*x^2)*sqrt(-a*b)*s
qrt(-c*d)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 3*(a*b^2*c^4 + 6*a^2*b
*c^3*d + a^3*c^2*d^2 + (b^3*c^2*d^2 + 6*a*b^2*c*d^3 + a^2*b*d^4)*x^6 + (2*b^3*c^
3*d + 13*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^4 + 8*a*b^2*c^3*d
+ 13*a^2*b*c^2*d^2 + 2*a^3*c*d^3)*x^2)*log((2*c*d*x + (d*x^2 - c)*sqrt(-c*d))/(
d*x^2 + c)) + 2*(3*(b^3*c^2*d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x^5 + (5*b^3*c^3 +
9*a*b^2*c^2*d - 9*a^2*b*c*d^2 - 5*a^3*d^3)*x^3 + 3*(3*a*b^2*c^3 - 2*a^2*b*c^2*d
- a^3*c*d^2)*x)*sqrt(-c*d))/((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)*sqrt(-c*d)), 1/8*(6*((b^2*c*d^2 + a*b*d^3)*x^6 + a*b*c^3 + a^2*c^2*d + (2
*b^2*c^2*d + 3*a*b*c*d^2 + a^2*d^3)*x^4 + (b^2*c^3 + 3*a*b*c^2*d + 2*a^2*c*d^2)*
x^2)*sqrt(-a*b)*sqrt(c*d)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 3*(a*b
^2*c^4 + 6*a^2*b*c^3*d + a^3*c^2*d^2 + (b^3*c^2*d^2 + 6*a*b^2*c*d^3 + a^2*b*d^4)
*x^6 + (2*b^3*c^3*d + 13*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^4
+ 8*a*b^2*c^3*d + 13*a^2*b*c^2*d^2 + 2*a^3*c*d^3)*x^2)*arctan(sqrt(c*d)*x/c) +
(3*(b^3*c^2*d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x^5 + (5*b^3*c^3 + 9*a*b^2*c^2*d -
9*a^2*b*c*d^2 - 5*a^3*d^3)*x^3 + 3*(3*a*b^2*c^3 - 2*a^2*b*c^2*d - a^3*c*d^2)*x)*
sqrt(c*d))/((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)*sqrt(c*d))
, -1/16*(24*((b^2*c*d^2 + a*b*d^3)*x^6 + a*b*c^3 + a^2*c^2*d + (2*b^2*c^2*d + 3*
a*b*c*d^2 + a^2*d^3)*x^4 + (b^2*c^3 + 3*a*b*c^2*d + 2*a^2*c*d^2)*x^2)*sqrt(a*b)*
sqrt(-c*d)*arctan(b*x/sqrt(a*b)) - 3*(a*b^2*c^4 + 6*a^2*b*c^3*d + a^3*c^2*d^2 +
(b^3*c^2*d^2 + 6*a*b^2*c*d^3 + a^2*b*d^4)*x^6 + (2*b^3*c^3*d + 13*a*b^2*c^2*d^2
+ 8*a^2*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^4 + 8*a*b^2*c^3*d + 13*a^2*b*c^2*d^2 + 2
*a^3*c*d^3)*x^2)*log((2*c*d*x + (d*x^2 - c)*sqrt(-c*d))/(d*x^2 + c)) - 2*(3*(b^3
*c^2*d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x^5 + (5*b^3*c^3 + 9*a*b^2*c^2*d - 9*a^2*b
*c*d^2 - 5*a^3*d^3)*x^3 + 3*(3*a*b^2*c^3 - 2*a^2*b*c^2*d - a^3*c*d^2)*x)*sqrt(-c
*d))/((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)*sqrt(-c*d)), -1/
8*(12*((b^2*c*d^2 + a*b*d^3)*x^6 + a*b*c^3 + a^2*c^2*d + (2*b^2*c^2*d + 3*a*b*c*
d^2 + a^2*d^3)*x^4 + (b^2*c^3 + 3*a*b*c^2*d + 2*a^2*c*d^2)*x^2)*sqrt(a*b)*sqrt(c
*d)*arctan(b*x/sqrt(a*b)) - 3*(a*b^2*c^4 + 6*a^2*b*c^3*d + a^3*c^2*d^2 + (b^3*c^
2*d^2 + 6*a*b^2*c*d^3 + a^2*b*d^4)*x^6 + (2*b^3*c^3*d + 13*a*b^2*c^2*d^2 + 8*a^2
*b*c*d^3 + a^3*d^4)*x^4 + (b^3*c^4 + 8*a*b^2*c^3*d + 13*a^2*b*c^2*d^2 + 2*a^3*c*
d^3)*x^2)*arctan(sqrt(c*d)*x/c) - (3*(b^3*c^2*d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x
^5 + (5*b^3*c^3 + 9*a*b^2*c^2*d - 9*a^2*b*c*d^2 - 5*a^3*d^3)*x^3 + 3*(3*a*b^2*c^
3 - 2*a^2*b*c^2*d - a^3*c*d^2)*x)*sqrt(c*d))/((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)*sqrt(c*d))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.247166, size = 406, normalized size = 1.96 \[ \frac{a b x}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (b x^{2} + a\right )}} - \frac{3 \,{\left (a b^{2} c + a^{2} b d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{a b}} + \frac{3 \,{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{c d}} + \frac{3 \, b c d x^{3} + 5 \, a d^{2} x^{3} + 5 \, b c^{2} x + 3 \, a c d x}{8 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (d x^{2} + c\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="giac")

[Out]

1/2*a*b*x/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x^2 + a)) - 3/
2*(a*b^2*c + a^2*b*d)*arctan(b*x/sqrt(a*b))/((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)*sqrt(a*b)) + 3/8*(b^2*c^2 + 6*a*b*c*d + a^2
*d^2)*arctan(d*x/sqrt(c*d))/((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)*sqrt(c*d)) + 1/8*(3*b*c*d*x^3 + 5*a*d^2*x^3 + 5*b*c^2*x + 3
*a*c*d*x)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(d*x^2 + c)^2)