3.113 \(\int \frac{x^5 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=147 \[ -\frac{x^2 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (4 a A c^2-6 a b B c+b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c^2} \]

[Out]

-(x^2*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(2*c*(b^2 - 4*a*c)*(a +
b*x^2 + c*x^4)) + ((b^3*B - 6*a*b*B*c + 4*a*A*c^2)*ArcTanh[(b + 2*c*x^2)/Sqrt[b
^2 - 4*a*c]])/(2*c^2*(b^2 - 4*a*c)^(3/2)) + (B*Log[a + b*x^2 + c*x^4])/(4*c^2)

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Rubi [A]  time = 0.371795, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{x^2 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (4 a A c^2-6 a b B c+b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{B \log \left (a+b x^2+c x^4\right )}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]

[Out]

-(x^2*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x^2))/(2*c*(b^2 - 4*a*c)*(a +
b*x^2 + c*x^4)) + ((b^3*B - 6*a*b*B*c + 4*a*A*c^2)*ArcTanh[(b + 2*c*x^2)/Sqrt[b
^2 - 4*a*c]])/(2*c^2*(b^2 - 4*a*c)^(3/2)) + (B*Log[a + b*x^2 + c*x^4])/(4*c^2)

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Rubi in Sympy [A]  time = 49.5501, size = 138, normalized size = 0.94 \[ \frac{B \log{\left (a + b x^{2} + c x^{4} \right )}}{4 c^{2}} + \frac{x^{2} \left (a \left (2 A c - B b\right ) - x^{2} \left (- A b c - 2 B a c + B b^{2}\right )\right )}{2 c \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} + \frac{\left (4 A a c^{2} - 6 B a b c + B b^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c^{2} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(B*x**2+A)/(c*x**4+b*x**2+a)**2,x)

[Out]

B*log(a + b*x**2 + c*x**4)/(4*c**2) + x**2*(a*(2*A*c - B*b) - x**2*(-A*b*c - 2*B
*a*c + B*b**2))/(2*c*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)) + (4*A*a*c**2 - 6*B*
a*b*c + B*b**3)*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(2*c**2*(-4*a*c + b**2
)**(3/2))

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Mathematica [A]  time = 0.337398, size = 160, normalized size = 1.09 \[ \frac{-\frac{2 \left (2 a^2 B c+a \left (b c \left (A+3 B x^2\right )-2 A c^2 x^2+b^2 (-B)\right )+b^2 x^2 (A c-b B)\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{2 \left (4 a A c^2-6 a b B c+b^3 B\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+B \log \left (a+b x^2+c x^4\right )}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]

[Out]

((-2*(2*a^2*B*c + b^2*(-(b*B) + A*c)*x^2 + a*(-(b^2*B) - 2*A*c^2*x^2 + b*c*(A +
3*B*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (2*(b^3*B - 6*a*b*B*c + 4*a*A*
c^2)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + B*Log[a +
b*x^2 + c*x^4])/(4*c^2)

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Maple [B]  time = 0.022, size = 542, normalized size = 3.7 \[{\frac{1}{2\,c{x}^{4}+2\,b{x}^{2}+2\,a} \left ( -{\frac{ \left ( 2\,aA{c}^{2}-A{b}^{2}c-3\,abBc+{b}^{3}B \right ){x}^{2}}{ \left ( 4\,ac-{b}^{2} \right ){c}^{2}}}+{\frac{a \left ( Abc+2\,aBc-{b}^{2}B \right ) }{ \left ( 4\,ac-{b}^{2} \right ){c}^{2}}} \right ) }+{\frac{\ln \left ( c \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) \right ) aB}{ \left ( 4\,ac-{b}^{2} \right ) c}}-{\frac{\ln \left ( c \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) \right ){b}^{2}B}{ \left ( 16\,ac-4\,{b}^{2} \right ){c}^{2}}}+2\,{\frac{aAc}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}\arctan \left ({\frac{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ){x}^{2}+c \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}} \right ) }-3\,{\frac{abB}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}\arctan \left ({\frac{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ){x}^{2}+c \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}} \right ) }+{\frac{{b}^{3}B}{2\,c}\arctan \left ({(2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ){x}^{2}+c \left ( 4\,ac-{b}^{2} \right ) b){\frac{1}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}}} \right ){\frac{1}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(B*x^2+A)/(c*x^4+b*x^2+a)^2,x)

[Out]

1/2*(-1/c^2*(2*A*a*c^2-A*b^2*c-3*B*a*b*c+B*b^3)/(4*a*c-b^2)*x^2+a*(A*b*c+2*B*a*c
-B*b^2)/(4*a*c-b^2)/c^2)/(c*x^4+b*x^2+a)+1/(4*a*c-b^2)/c*ln(c*(4*a*c-b^2)*(c*x^4
+b*x^2+a))*a*B-1/4/(4*a*c-b^2)/c^2*ln(c*(4*a*c-b^2)*(c*x^4+b*x^2+a))*b^2*B+2/(64
*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b^2)*x^
2+c*(4*a*c-b^2)*b)/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))*a*A*c
-3/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b
^2)*x^2+c*(4*a*c-b^2)*b)/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))
*a*b*B+1/2/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*
(4*a*c-b^2)*x^2+c*(4*a*c-b^2)*b)/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2
)^(1/2))*b^3/c*B

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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((B*x^2 + A)*x^5/(c*x^4 + b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284112, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (B a b^{3} - 6 \, B a^{2} b c + 4 \, A a^{2} c^{2} +{\left (B b^{3} c - 6 \, B a b c^{2} + 4 \, A a c^{3}\right )} x^{4} +{\left (B b^{4} - 6 \, B a b^{2} c + 4 \, A a b c^{2}\right )} x^{2}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} -{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left (2 \, B a b^{2} + 2 \,{\left (B b^{3} + 2 \, A a c^{2} -{\left (3 \, B a b + A b^{2}\right )} c\right )} x^{2} - 2 \,{\left (2 \, B a^{2} + A a b\right )} c +{\left ({\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{4} + B a b^{2} - 4 \, B a^{2} c +{\left (B b^{3} - 4 \, B a b c\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{4 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{2 \,{\left (B a b^{3} - 6 \, B a^{2} b c + 4 \, A a^{2} c^{2} +{\left (B b^{3} c - 6 \, B a b c^{2} + 4 \, A a c^{3}\right )} x^{4} +{\left (B b^{4} - 6 \, B a b^{2} c + 4 \, A a b c^{2}\right )} x^{2}\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, B a b^{2} + 2 \,{\left (B b^{3} + 2 \, A a c^{2} -{\left (3 \, B a b + A b^{2}\right )} c\right )} x^{2} - 2 \,{\left (2 \, B a^{2} + A a b\right )} c +{\left ({\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{4} + B a b^{2} - 4 \, B a^{2} c +{\left (B b^{3} - 4 \, B a b c\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^5/(c*x^4 + b*x^2 + a)^2,x, algorithm="fricas")

[Out]

[-1/4*((B*a*b^3 - 6*B*a^2*b*c + 4*A*a^2*c^2 + (B*b^3*c - 6*B*a*b*c^2 + 4*A*a*c^3
)*x^4 + (B*b^4 - 6*B*a*b^2*c + 4*A*a*b*c^2)*x^2)*log(-(b^3 - 4*a*b*c + 2*(b^2*c
- 4*a*c^2)*x^2 - (2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4
+ b*x^2 + a)) - (2*B*a*b^2 + 2*(B*b^3 + 2*A*a*c^2 - (3*B*a*b + A*b^2)*c)*x^2 -
2*(2*B*a^2 + A*a*b)*c + ((B*b^2*c - 4*B*a*c^2)*x^4 + B*a*b^2 - 4*B*a^2*c + (B*b^
3 - 4*B*a*b*c)*x^2)*log(c*x^4 + b*x^2 + a))*sqrt(b^2 - 4*a*c))/((a*b^2*c^2 - 4*a
^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^4 + (b^3*c^2 - 4*a*b*c^3)*x^2)*sqrt(b^2 - 4*a*c))
, -1/4*(2*(B*a*b^3 - 6*B*a^2*b*c + 4*A*a^2*c^2 + (B*b^3*c - 6*B*a*b*c^2 + 4*A*a*
c^3)*x^4 + (B*b^4 - 6*B*a*b^2*c + 4*A*a*b*c^2)*x^2)*arctan(-(2*c*x^2 + b)*sqrt(-
b^2 + 4*a*c)/(b^2 - 4*a*c)) - (2*B*a*b^2 + 2*(B*b^3 + 2*A*a*c^2 - (3*B*a*b + A*b
^2)*c)*x^2 - 2*(2*B*a^2 + A*a*b)*c + ((B*b^2*c - 4*B*a*c^2)*x^4 + B*a*b^2 - 4*B*
a^2*c + (B*b^3 - 4*B*a*b*c)*x^2)*log(c*x^4 + b*x^2 + a))*sqrt(-b^2 + 4*a*c))/((a
*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^4 + (b^3*c^2 - 4*a*b*c^3)*x^2)*sqrt
(-b^2 + 4*a*c))]

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Sympy [A]  time = 54.662, size = 916, normalized size = 6.23 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(B*x**2+A)/(c*x**4+b*x**2+a)**2,x)

[Out]

(B/(4*c**2) - sqrt(-(4*a*c - b**2)**3)*(-4*A*a*c**2 + 6*B*a*b*c - B*b**3)/(4*c**
2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x**2 + (-2*A*a*b
*c + 8*B*a**2*c - B*a*b**2 - 32*a**2*c**3*(B/(4*c**2) - sqrt(-(4*a*c - b**2)**3)
*(-4*A*a*c**2 + 6*B*a*b*c - B*b**3)/(4*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 +
12*a*b**4*c - b**6))) + 16*a*b**2*c**2*(B/(4*c**2) - sqrt(-(4*a*c - b**2)**3)*(-
4*A*a*c**2 + 6*B*a*b*c - B*b**3)/(4*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*
a*b**4*c - b**6))) - 2*b**4*c*(B/(4*c**2) - sqrt(-(4*a*c - b**2)**3)*(-4*A*a*c**
2 + 6*B*a*b*c - B*b**3)/(4*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6))))/(-4*A*a*c**2 + 6*B*a*b*c - B*b**3)) + (B/(4*c**2) + sqrt(-(4*a*c - b*
*2)**3)*(-4*A*a*c**2 + 6*B*a*b*c - B*b**3)/(4*c**2*(64*a**3*c**3 - 48*a**2*b**2*
c**2 + 12*a*b**4*c - b**6)))*log(x**2 + (-2*A*a*b*c + 8*B*a**2*c - B*a*b**2 - 32
*a**2*c**3*(B/(4*c**2) + sqrt(-(4*a*c - b**2)**3)*(-4*A*a*c**2 + 6*B*a*b*c - B*b
**3)/(4*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 16*a*b*
*2*c**2*(B/(4*c**2) + sqrt(-(4*a*c - b**2)**3)*(-4*A*a*c**2 + 6*B*a*b*c - B*b**3
)/(4*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 2*b**4*c*(
B/(4*c**2) + sqrt(-(4*a*c - b**2)**3)*(-4*A*a*c**2 + 6*B*a*b*c - B*b**3)/(4*c**2
*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))))/(-4*A*a*c**2 + 6*B*a
*b*c - B*b**3)) + (A*a*b*c + 2*B*a**2*c - B*a*b**2 + x**2*(-2*A*a*c**2 + A*b**2*
c + 3*B*a*b*c - B*b**3))/(8*a**2*c**3 - 2*a*b**2*c**2 + x**4*(8*a*c**4 - 2*b**2*
c**3) + x**2*(8*a*b*c**3 - 2*b**3*c**2))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^5/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError