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

Optimal. Leaf size=153 \[ \frac{\sqrt{a+b x^2+c x^4} \left (-16 a B c-2 c x^2 (5 b B-6 A c)-18 A b c+15 b^2 B\right )}{48 c^3}-\frac{\left (8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{7/2}}+\frac{B x^4 \sqrt{a+b x^2+c x^4}}{6 c} \]

[Out]

(B*x^4*Sqrt[a + b*x^2 + c*x^4])/(6*c) + ((15*b^2*B - 18*A*b*c - 16*a*B*c - 2*c*(
5*b*B - 6*A*c)*x^2)*Sqrt[a + b*x^2 + c*x^4])/(48*c^3) - ((5*b^3*B - 6*A*b^2*c -
12*a*b*B*c + 8*a*A*c^2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]
)])/(32*c^(7/2))

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Rubi [A]  time = 0.507291, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{\sqrt{a+b x^2+c x^4} \left (-16 a B c-2 c x^2 (5 b B-6 A c)-18 A b c+15 b^2 B\right )}{48 c^3}-\frac{\left (8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{7/2}}+\frac{B x^4 \sqrt{a+b x^2+c x^4}}{6 c} \]

Antiderivative was successfully verified.

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

[Out]

(B*x^4*Sqrt[a + b*x^2 + c*x^4])/(6*c) + ((15*b^2*B - 18*A*b*c - 16*a*B*c - 2*c*(
5*b*B - 6*A*c)*x^2)*Sqrt[a + b*x^2 + c*x^4])/(48*c^3) - ((5*b^3*B - 6*A*b^2*c -
12*a*b*B*c + 8*a*A*c^2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]
)])/(32*c^(7/2))

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Rubi in Sympy [A]  time = 35.4782, size = 153, normalized size = 1. \[ \frac{B x^{4} \sqrt{a + b x^{2} + c x^{4}}}{6 c} - \frac{\sqrt{a + b x^{2} + c x^{4}} \left (4 B a c + \frac{3 b \left (6 A c - 5 B b\right )}{4} - \frac{c x^{2} \left (6 A c - 5 B b\right )}{2}\right )}{12 c^{3}} - \frac{\left (8 A a c^{2} - 6 A b^{2} c - 12 B a b c + 5 B b^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{32 c^{\frac{7}{2}}} \]

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)**(1/2),x)

[Out]

B*x**4*sqrt(a + b*x**2 + c*x**4)/(6*c) - sqrt(a + b*x**2 + c*x**4)*(4*B*a*c + 3*
b*(6*A*c - 5*B*b)/4 - c*x**2*(6*A*c - 5*B*b)/2)/(12*c**3) - (8*A*a*c**2 - 6*A*b*
*2*c - 12*B*a*b*c + 5*B*b**3)*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 +
c*x**4)))/(32*c**(7/2))

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Mathematica [A]  time = 0.163228, size = 137, normalized size = 0.9 \[ \frac{2 \sqrt{c} \sqrt{a+b x^2+c x^4} \left (4 c \left (-4 a B+3 A c x^2+2 B c x^4\right )-2 b c \left (9 A+5 B x^2\right )+15 b^2 B\right )-3 \left (8 a A c^2-12 a b B c-6 A b^2 c+5 b^3 B\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{96 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]*(15*b^2*B - 2*b*c*(9*A + 5*B*x^2) + 4*c*(-4*a
*B + 3*A*c*x^2 + 2*B*c*x^4)) - 3*(5*b^3*B - 6*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*
Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])/(96*c^(7/2))

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Maple [B]  time = 0.015, size = 286, normalized size = 1.9 \[{\frac{A{x}^{2}}{4\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,Ab}{8\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,A{b}^{2}}{16}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{Aa}{4}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{B{x}^{4}}{6\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{5\,bB{x}^{2}}{24\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,{b}^{2}B}{16\,{c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{5\,{b}^{3}B}{32}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{3\,abB}{8}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{Ba}{3\,{c}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}} \]

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)^(1/2),x)

[Out]

1/4*A*x^2/c*(c*x^4+b*x^2+a)^(1/2)-3/8*A*b/c^2*(c*x^4+b*x^2+a)^(1/2)+3/16*A*b^2/c
^(5/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-1/4*A*a/c^(3/2)*ln((1/2*b
+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+1/6*B*x^4*(c*x^4+b*x^2+a)^(1/2)/c-5/24*B*
b/c^2*x^2*(c*x^4+b*x^2+a)^(1/2)+5/16*B*b^2/c^3*(c*x^4+b*x^2+a)^(1/2)-5/32*B*b^3/
c^(7/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+3/8*B*b/c^(5/2)*a*ln((1/
2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-1/3*B/c^2*a*(c*x^4+b*x^2+a)^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.35913, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, B c^{2} x^{4} + 15 \, B b^{2} - 2 \,{\left (5 \, B b c - 6 \, A c^{2}\right )} x^{2} - 2 \,{\left (8 \, B a + 9 \, A b\right )} c\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{c} + 3 \,{\left (5 \, B b^{3} + 8 \, A a c^{2} - 6 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} \log \left (4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} + b c\right )} -{\left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{192 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (8 \, B c^{2} x^{4} + 15 \, B b^{2} - 2 \,{\left (5 \, B b c - 6 \, A c^{2}\right )} x^{2} - 2 \,{\left (8 \, B a + 9 \, A b\right )} c\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c} - 3 \,{\left (5 \, B b^{3} + 8 \, A a c^{2} - 6 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} \arctan \left (\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{2} + a} c}\right )}{96 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/192*(4*(8*B*c^2*x^4 + 15*B*b^2 - 2*(5*B*b*c - 6*A*c^2)*x^2 - 2*(8*B*a + 9*A*b
)*c)*sqrt(c*x^4 + b*x^2 + a)*sqrt(c) + 3*(5*B*b^3 + 8*A*a*c^2 - 6*(2*B*a*b + A*b
^2)*c)*log(4*sqrt(c*x^4 + b*x^2 + a)*(2*c^2*x^2 + b*c) - (8*c^2*x^4 + 8*b*c*x^2
+ b^2 + 4*a*c)*sqrt(c)))/c^(7/2), 1/96*(2*(8*B*c^2*x^4 + 15*B*b^2 - 2*(5*B*b*c -
6*A*c^2)*x^2 - 2*(8*B*a + 9*A*b)*c)*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c) - 3*(5*B*b
^3 + 8*A*a*c^2 - 6*(2*B*a*b + A*b^2)*c)*arctan(1/2*(2*c*x^2 + b)*sqrt(-c)/(sqrt(
c*x^4 + b*x^2 + a)*c)))/(sqrt(-c)*c^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5} \left (A + B x^{2}\right )}{\sqrt{a + b x^{2} + c x^{4}}}\, dx \]

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)**(1/2),x)

[Out]

Integral(x**5*(A + B*x**2)/sqrt(a + b*x**2 + c*x**4), x)

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GIAC/XCAS [A]  time = 0.305565, size = 203, normalized size = 1.33 \[ \frac{1}{48} \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \,{\left (\frac{4 \, B x^{2}}{c} - \frac{5 \, B b c^{2} - 6 \, A c^{3}}{c^{4}}\right )} x^{2} + \frac{15 \, B b^{2} c - 16 \, B a c^{2} - 18 \, A b c^{2}}{c^{4}}\right )} + \frac{{\left (5 \, B b^{3} c - 12 \, B a b c^{2} - 6 \, A b^{2} c^{2} + 8 \, A a c^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{32 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*sqrt(c*x^4 + b*x^2 + a)*(2*(4*B*x^2/c - (5*B*b*c^2 - 6*A*c^3)/c^4)*x^2 + (1
5*B*b^2*c - 16*B*a*c^2 - 18*A*b*c^2)/c^4) + 1/32*(5*B*b^3*c - 12*B*a*b*c^2 - 6*A
*b^2*c^2 + 8*A*a*c^3)*ln(abs(-2*(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c)
- b))/c^(9/2)