3.97 \(\int \frac{\left (A+B x^2\right ) \sqrt{b x^2+c x^4}}{x^9} \, dx\)

Optimal. Leaf size=96 \[ \frac{2 c \left (b x^2+c x^4\right )^{3/2} (7 b B-4 A c)}{105 b^3 x^6}-\frac{\left (b x^2+c x^4\right )^{3/2} (7 b B-4 A c)}{35 b^2 x^8}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}} \]

[Out]

-(A*(b*x^2 + c*x^4)^(3/2))/(7*b*x^10) - ((7*b*B - 4*A*c)*(b*x^2 + c*x^4)^(3/2))/
(35*b^2*x^8) + (2*c*(7*b*B - 4*A*c)*(b*x^2 + c*x^4)^(3/2))/(105*b^3*x^6)

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Rubi [A]  time = 0.408782, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 c \left (b x^2+c x^4\right )^{3/2} (7 b B-4 A c)}{105 b^3 x^6}-\frac{\left (b x^2+c x^4\right )^{3/2} (7 b B-4 A c)}{35 b^2 x^8}-\frac{A \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^9,x]

[Out]

-(A*(b*x^2 + c*x^4)^(3/2))/(7*b*x^10) - ((7*b*B - 4*A*c)*(b*x^2 + c*x^4)^(3/2))/
(35*b^2*x^8) + (2*c*(7*b*B - 4*A*c)*(b*x^2 + c*x^4)^(3/2))/(105*b^3*x^6)

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Rubi in Sympy [A]  time = 23.8281, size = 88, normalized size = 0.92 \[ - \frac{A \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{7 b x^{10}} + \frac{\left (4 A c - 7 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{35 b^{2} x^{8}} - \frac{2 c \left (4 A c - 7 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{105 b^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**(1/2)/x**9,x)

[Out]

-A*(b*x**2 + c*x**4)**(3/2)/(7*b*x**10) + (4*A*c - 7*B*b)*(b*x**2 + c*x**4)**(3/
2)/(35*b**2*x**8) - 2*c*(4*A*c - 7*B*b)*(b*x**2 + c*x**4)**(3/2)/(105*b**3*x**6)

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Mathematica [A]  time = 0.0811512, size = 66, normalized size = 0.69 \[ -\frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (A \left (15 b^2-12 b c x^2+8 c^2 x^4\right )+7 b B x^2 \left (3 b-2 c x^2\right )\right )}{105 b^3 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^9,x]

[Out]

-((x^2*(b + c*x^2))^(3/2)*(7*b*B*x^2*(3*b - 2*c*x^2) + A*(15*b^2 - 12*b*c*x^2 +
8*c^2*x^4)))/(105*b^3*x^10)

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Maple [A]  time = 0.01, size = 70, normalized size = 0.7 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( 8\,A{c}^{2}{x}^{4}-14\,B{x}^{4}bc-12\,Abc{x}^{2}+21\,B{b}^{2}{x}^{2}+15\,{b}^{2}A \right ) }{105\,{x}^{8}{b}^{3}}\sqrt{c{x}^{4}+b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(c*x^4+b*x^2)^(1/2)/x^9,x)

[Out]

-1/105*(c*x^2+b)*(8*A*c^2*x^4-14*B*b*c*x^4-12*A*b*c*x^2+21*B*b^2*x^2+15*A*b^2)*(
c*x^4+b*x^2)^(1/2)/x^8/b^3

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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(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265416, size = 115, normalized size = 1.2 \[ \frac{{\left (2 \,{\left (7 \, B b c^{2} - 4 \, A c^{3}\right )} x^{6} -{\left (7 \, B b^{2} c - 4 \, A b c^{2}\right )} x^{4} - 15 \, A b^{3} - 3 \,{\left (7 \, B b^{3} + A b^{2} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{105 \, b^{3} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^9,x, algorithm="fricas")

[Out]

1/105*(2*(7*B*b*c^2 - 4*A*c^3)*x^6 - (7*B*b^2*c - 4*A*b*c^2)*x^4 - 15*A*b^3 - 3*
(7*B*b^3 + A*b^2*c)*x^2)*sqrt(c*x^4 + b*x^2)/(b^3*x^8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(c*x**4+b*x**2)**(1/2)/x**9,x)

[Out]

Integral(sqrt(x**2*(b + c*x**2))*(A + B*x**2)/x**9, x)

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GIAC/XCAS [A]  time = 0.562984, size = 419, normalized size = 4.36 \[ \frac{4 \,{\left (105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{10} B c^{\frac{5}{2}}{\rm sign}\left (x\right ) - 175 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} B b c^{\frac{5}{2}}{\rm sign}\left (x\right ) + 280 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{8} A c^{\frac{7}{2}}{\rm sign}\left (x\right ) + 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} B b^{2} c^{\frac{5}{2}}{\rm sign}\left (x\right ) + 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{6} A b c^{\frac{7}{2}}{\rm sign}\left (x\right ) - 42 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} B b^{3} c^{\frac{5}{2}}{\rm sign}\left (x\right ) + 84 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} A b^{2} c^{\frac{7}{2}}{\rm sign}\left (x\right ) + 49 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} B b^{4} c^{\frac{5}{2}}{\rm sign}\left (x\right ) - 28 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} A b^{3} c^{\frac{7}{2}}{\rm sign}\left (x\right ) - 7 \, B b^{5} c^{\frac{5}{2}}{\rm sign}\left (x\right ) + 4 \, A b^{4} c^{\frac{7}{2}}{\rm sign}\left (x\right )\right )}}{105 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^9,x, algorithm="giac")

[Out]

4/105*(105*(sqrt(c)*x - sqrt(c*x^2 + b))^10*B*c^(5/2)*sign(x) - 175*(sqrt(c)*x -
sqrt(c*x^2 + b))^8*B*b*c^(5/2)*sign(x) + 280*(sqrt(c)*x - sqrt(c*x^2 + b))^8*A*
c^(7/2)*sign(x) + 70*(sqrt(c)*x - sqrt(c*x^2 + b))^6*B*b^2*c^(5/2)*sign(x) + 140
*(sqrt(c)*x - sqrt(c*x^2 + b))^6*A*b*c^(7/2)*sign(x) - 42*(sqrt(c)*x - sqrt(c*x^
2 + b))^4*B*b^3*c^(5/2)*sign(x) + 84*(sqrt(c)*x - sqrt(c*x^2 + b))^4*A*b^2*c^(7/
2)*sign(x) + 49*(sqrt(c)*x - sqrt(c*x^2 + b))^2*B*b^4*c^(5/2)*sign(x) - 28*(sqrt
(c)*x - sqrt(c*x^2 + b))^2*A*b^3*c^(7/2)*sign(x) - 7*B*b^5*c^(5/2)*sign(x) + 4*A
*b^4*c^(7/2)*sign(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^7