3.175 \(\int \frac{A+B x^2}{x^7 \sqrt{a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=177 \[ -\frac{\sqrt{a+b x^2+c x^4} \left (-16 a A c-18 a b B+15 A b^2\right )}{48 a^3 x^2}+\frac{(5 A b-6 a B) \sqrt{a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{7/2}}-\frac{A \sqrt{a+b x^2+c x^4}}{6 a x^6} \]

[Out]

-(A*Sqrt[a + b*x^2 + c*x^4])/(6*a*x^6) + ((5*A*b - 6*a*B)*Sqrt[a + b*x^2 + c*x^4
])/(24*a^2*x^4) - ((15*A*b^2 - 18*a*b*B - 16*a*A*c)*Sqrt[a + b*x^2 + c*x^4])/(48
*a^3*x^2) + ((5*A*b^3 - 6*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*ArcTanh[(2*a + b*x^2
)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(32*a^(7/2))

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Rubi [A]  time = 0.600473, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{\sqrt{a+b x^2+c x^4} \left (-16 a A c-18 a b B+15 A b^2\right )}{48 a^3 x^2}+\frac{(5 A b-6 a B) \sqrt{a+b x^2+c x^4}}{24 a^2 x^4}+\frac{\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{32 a^{7/2}}-\frac{A \sqrt{a+b x^2+c x^4}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^7*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

-(A*Sqrt[a + b*x^2 + c*x^4])/(6*a*x^6) + ((5*A*b - 6*a*B)*Sqrt[a + b*x^2 + c*x^4
])/(24*a^2*x^4) - ((15*A*b^2 - 18*a*b*B - 16*a*A*c)*Sqrt[a + b*x^2 + c*x^4])/(48
*a^3*x^2) + ((5*A*b^3 - 6*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*ArcTanh[(2*a + b*x^2
)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(32*a^(7/2))

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Rubi in Sympy [A]  time = 54.4566, size = 172, normalized size = 0.97 \[ - \frac{A \sqrt{a + b x^{2} + c x^{4}}}{6 a x^{6}} + \frac{\left (5 A b - 6 B a\right ) \sqrt{a + b x^{2} + c x^{4}}}{24 a^{2} x^{4}} - \frac{\sqrt{a + b x^{2} + c x^{4}} \left (- 16 A a c + 15 A b^{2} - 18 B a b\right )}{48 a^{3} x^{2}} + \frac{\left (- 12 A a b c + 5 A b^{3} + 8 B a^{2} c - 6 B a b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{32 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**7/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

-A*sqrt(a + b*x**2 + c*x**4)/(6*a*x**6) + (5*A*b - 6*B*a)*sqrt(a + b*x**2 + c*x*
*4)/(24*a**2*x**4) - sqrt(a + b*x**2 + c*x**4)*(-16*A*a*c + 15*A*b**2 - 18*B*a*b
)/(48*a**3*x**2) + (-12*A*a*b*c + 5*A*b**3 + 8*B*a**2*c - 6*B*a*b**2)*atanh((2*a
+ b*x**2)/(2*sqrt(a)*sqrt(a + b*x**2 + c*x**4)))/(32*a**(7/2))

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Mathematica [A]  time = 0.266458, size = 146, normalized size = 0.82 \[ -\frac{\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )\right )}{32 a^{7/2}}-\frac{\sqrt{a+b x^2+c x^4} \left (8 a^2 A+x^4 \left (-16 a A c-18 a b B+15 A b^2\right )+2 a x^2 (6 a B-5 A b)\right )}{48 a^3 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^7*Sqrt[a + b*x^2 + c*x^4]),x]

[Out]

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

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Maple [A]  time = 0.016, size = 311, normalized size = 1.8 \[ -{\frac{A}{6\,a{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,Ab}{24\,{a}^{2}{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{5\,A{b}^{2}}{16\,{a}^{3}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,A{b}^{3}}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{3\,Abc}{8}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{Ac}{3\,{a}^{2}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{B}{4\,a{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,bB}{8\,{a}^{2}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{b}^{2}B}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{Bc}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^7/(c*x^4+b*x^2+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.358216, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} x^{6} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) + 4 \,{\left ({\left (18 \, B a b - 15 \, A b^{2} + 16 \, A a c\right )} x^{4} - 8 \, A a^{2} - 2 \,{\left (6 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{a}}{192 \, a^{\frac{7}{2}} x^{6}}, -\frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} x^{6} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) - 2 \,{\left ({\left (18 \, B a b - 15 \, A b^{2} + 16 \, A a c\right )} x^{4} - 8 \, A a^{2} - 2 \,{\left (6 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}{96 \, \sqrt{-a} a^{3} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*x^7),x, algorithm="fricas")

[Out]

[1/192*(3*(6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*x^6*log((4*sqrt(c*x^4
+ b*x^2 + a)*(a*b*x^2 + 2*a^2) - ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*sqrt(a)
)/x^4) + 4*((18*B*a*b - 15*A*b^2 + 16*A*a*c)*x^4 - 8*A*a^2 - 2*(6*B*a^2 - 5*A*a*
b)*x^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(a))/(a^(7/2)*x^6), -1/96*(3*(6*B*a*b^2 - 5*
A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*x^6*arctan(1/2*(b*x^2 + 2*a)*sqrt(-a)/(sqrt(c*x
^4 + b*x^2 + a)*a)) - 2*((18*B*a*b - 15*A*b^2 + 16*A*a*c)*x^4 - 8*A*a^2 - 2*(6*B
*a^2 - 5*A*a*b)*x^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^3*x^6)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x^{2}}{x^{7} \sqrt{a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**7/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

Integral((A + B*x**2)/(x**7*sqrt(a + b*x**2 + c*x**4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2} + a} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*x^7),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*x^7), x)