3.152 \(\int \frac{A+B x^2}{x^5 \left (b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=138 \[ -\frac{8 c^2 \left (b+2 c x^2\right ) (7 b B-8 A c)}{35 b^5 \sqrt{b x^2+c x^4}}+\frac{2 c (7 b B-8 A c)}{35 b^3 x^2 \sqrt{b x^2+c x^4}}-\frac{7 b B-8 A c}{35 b^2 x^4 \sqrt{b x^2+c x^4}}-\frac{A}{7 b x^6 \sqrt{b x^2+c x^4}} \]

[Out]

-A/(7*b*x^6*Sqrt[b*x^2 + c*x^4]) - (7*b*B - 8*A*c)/(35*b^2*x^4*Sqrt[b*x^2 + c*x^
4]) + (2*c*(7*b*B - 8*A*c))/(35*b^3*x^2*Sqrt[b*x^2 + c*x^4]) - (8*c^2*(7*b*B - 8
*A*c)*(b + 2*c*x^2))/(35*b^5*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.507474, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{8 c^2 \left (b+2 c x^2\right ) (7 b B-8 A c)}{35 b^5 \sqrt{b x^2+c x^4}}+\frac{2 c (7 b B-8 A c)}{35 b^3 x^2 \sqrt{b x^2+c x^4}}-\frac{7 b B-8 A c}{35 b^2 x^4 \sqrt{b x^2+c x^4}}-\frac{A}{7 b x^6 \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^5*(b*x^2 + c*x^4)^(3/2)),x]

[Out]

-A/(7*b*x^6*Sqrt[b*x^2 + c*x^4]) - (7*b*B - 8*A*c)/(35*b^2*x^4*Sqrt[b*x^2 + c*x^
4]) + (2*c*(7*b*B - 8*A*c))/(35*b^3*x^2*Sqrt[b*x^2 + c*x^4]) - (8*c^2*(7*b*B - 8
*A*c)*(b + 2*c*x^2))/(35*b^5*Sqrt[b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 29.1504, size = 133, normalized size = 0.96 \[ - \frac{A}{7 b x^{6} \sqrt{b x^{2} + c x^{4}}} + \frac{8 A c - 7 B b}{35 b^{2} x^{4} \sqrt{b x^{2} + c x^{4}}} - \frac{2 c \left (8 A c - 7 B b\right )}{35 b^{3} x^{2} \sqrt{b x^{2} + c x^{4}}} + \frac{4 c^{2} \left (2 b + 4 c x^{2}\right ) \left (8 A c - 7 B b\right )}{35 b^{5} \sqrt{b x^{2} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**5/(c*x**4+b*x**2)**(3/2),x)

[Out]

-A/(7*b*x**6*sqrt(b*x**2 + c*x**4)) + (8*A*c - 7*B*b)/(35*b**2*x**4*sqrt(b*x**2
+ c*x**4)) - 2*c*(8*A*c - 7*B*b)/(35*b**3*x**2*sqrt(b*x**2 + c*x**4)) + 4*c**2*(
2*b + 4*c*x**2)*(8*A*c - 7*B*b)/(35*b**5*sqrt(b*x**2 + c*x**4))

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Mathematica [A]  time = 0.116338, size = 108, normalized size = 0.78 \[ \frac{A \left (-5 b^4+8 b^3 c x^2-16 b^2 c^2 x^4+64 b c^3 x^6+128 c^4 x^8\right )-7 b B x^2 \left (b^3-2 b^2 c x^2+8 b c^2 x^4+16 c^3 x^6\right )}{35 b^5 x^6 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^5*(b*x^2 + c*x^4)^(3/2)),x]

[Out]

(-7*b*B*x^2*(b^3 - 2*b^2*c*x^2 + 8*b*c^2*x^4 + 16*c^3*x^6) + A*(-5*b^4 + 8*b^3*c
*x^2 - 16*b^2*c^2*x^4 + 64*b*c^3*x^6 + 128*c^4*x^8))/(35*b^5*x^6*Sqrt[x^2*(b + c
*x^2)])

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Maple [A]  time = 0.009, size = 118, normalized size = 0.9 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -128\,A{c}^{4}{x}^{8}+112\,Bb{c}^{3}{x}^{8}-64\,Ab{c}^{3}{x}^{6}+56\,B{b}^{2}{c}^{2}{x}^{6}+16\,A{b}^{2}{c}^{2}{x}^{4}-14\,B{b}^{3}c{x}^{4}-8\,A{b}^{3}c{x}^{2}+7\,B{b}^{4}{x}^{2}+5\,A{b}^{4} \right ) }{35\,{x}^{4}{b}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^5/(c*x^4+b*x^2)^(3/2),x)

[Out]

-1/35*(c*x^2+b)*(-128*A*c^4*x^8+112*B*b*c^3*x^8-64*A*b*c^3*x^6+56*B*b^2*c^2*x^6+
16*A*b^2*c^2*x^4-14*B*b^3*c*x^4-8*A*b^3*c*x^2+7*B*b^4*x^2+5*A*b^4)/x^4/b^5/(c*x^
4+b*x^2)^(3/2)

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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)/((c*x^4 + b*x^2)^(3/2)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303843, size = 163, normalized size = 1.18 \[ -\frac{{\left (16 \,{\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{8} + 8 \,{\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{6} + 5 \, A b^{4} - 2 \,{\left (7 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{4} +{\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{35 \,{\left (b^{5} c x^{10} + b^{6} x^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((c*x^4 + b*x^2)^(3/2)*x^5),x, algorithm="fricas")

[Out]

-1/35*(16*(7*B*b*c^3 - 8*A*c^4)*x^8 + 8*(7*B*b^2*c^2 - 8*A*b*c^3)*x^6 + 5*A*b^4
- 2*(7*B*b^3*c - 8*A*b^2*c^2)*x^4 + (7*B*b^4 - 8*A*b^3*c)*x^2)*sqrt(c*x^4 + b*x^
2)/(b^5*c*x^10 + b^6*x^8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x^{2}}{x^{5} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**5/(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral((A + B*x**2)/(x**5*(x**2*(b + c*x**2))**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((c*x^4 + b*x^2)^(3/2)*x^5),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)/((c*x^4 + b*x^2)^(3/2)*x^5), x)