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

Optimal. Leaf size=133 \[ -\frac{8 c^2 \sqrt{b x^2+c x^4} (7 b B-6 A c)}{105 b^4 x^2}+\frac{4 c \sqrt{b x^2+c x^4} (7 b B-6 A c)}{105 b^3 x^4}-\frac{\sqrt{b x^2+c x^4} (7 b B-6 A c)}{35 b^2 x^6}-\frac{A \sqrt{b x^2+c x^4}}{7 b x^8} \]

[Out]

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

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Rubi [A]  time = 0.503336, antiderivative size = 133, 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 \[ -\frac{8 c^2 \sqrt{b x^2+c x^4} (7 b B-6 A c)}{105 b^4 x^2}+\frac{4 c \sqrt{b x^2+c x^4} (7 b B-6 A c)}{105 b^3 x^4}-\frac{\sqrt{b x^2+c x^4} (7 b B-6 A c)}{35 b^2 x^6}-\frac{A \sqrt{b x^2+c x^4}}{7 b x^8} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.3254, size = 126, normalized size = 0.95 \[ - \frac{A \sqrt{b x^{2} + c x^{4}}}{7 b x^{8}} + \frac{\left (6 A c - 7 B b\right ) \sqrt{b x^{2} + c x^{4}}}{35 b^{2} x^{6}} - \frac{4 c \left (6 A c - 7 B b\right ) \sqrt{b x^{2} + c x^{4}}}{105 b^{3} x^{4}} + \frac{8 c^{2} \left (6 A c - 7 B b\right ) \sqrt{b x^{2} + c x^{4}}}{105 b^{4} x^{2}} \]

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)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0937102, size = 89, normalized size = 0.67 \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )} \left (3 A \left (5 b^3-6 b^2 c x^2+8 b c^2 x^4-16 c^3 x^6\right )+7 b B x^2 \left (3 b^2-4 b c x^2+8 c^2 x^4\right )\right )}{105 b^4 x^8} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[x^2*(b + c*x^2)]*(7*b*B*x^2*(3*b^2 - 4*b*c*x^2 + 8*c^2*x^4) + 3*A*(5*b^3
- 6*b^2*c*x^2 + 8*b*c^2*x^4 - 16*c^3*x^6)))/(105*b^4*x^8)

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Maple [A]  time = 0.009, size = 94, normalized size = 0.7 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -48\,A{c}^{3}{x}^{6}+56\,B{x}^{6}b{c}^{2}+24\,Ab{c}^{2}{x}^{4}-28\,B{x}^{4}{b}^{2}c-18\,A{b}^{2}c{x}^{2}+21\,B{x}^{2}{b}^{3}+15\,A{b}^{3} \right ) }{105\,{x}^{6}{b}^{4}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(c*x^2+b)*(-48*A*c^3*x^6+56*B*b*c^2*x^6+24*A*b*c^2*x^4-28*B*b^2*c*x^4-18*
A*b^2*c*x^2+21*B*b^3*x^2+15*A*b^3)/x^6/b^4/(c*x^4+b*x^2)^(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)/(sqrt(c*x^4 + b*x^2)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258011, size = 116, normalized size = 0.87 \[ -\frac{{\left (8 \,{\left (7 \, B b c^{2} - 6 \, A c^{3}\right )} x^{6} - 4 \,{\left (7 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{4} + 15 \, A b^{3} + 3 \,{\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{105 \, b^{4} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.225244, size = 140, normalized size = 1.05 \[ -\frac{21 \, B b{\left (c + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} + 15 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{7}{2}} - 70 \, B b{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} c - 63 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} c + 105 \, B b \sqrt{c + \frac{b}{x^{2}}} c^{2} + 105 \, A{\left (c + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} c^{2} - 105 \, A \sqrt{c + \frac{b}{x^{2}}} c^{3}}{105 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(21*B*b*(c + b/x^2)^(5/2) + 15*A*(c + b/x^2)^(7/2) - 70*B*b*(c + b/x^2)^(
3/2)*c - 63*A*(c + b/x^2)^(5/2)*c + 105*B*b*sqrt(c + b/x^2)*c^2 + 105*A*(c + b/x
^2)^(3/2)*c^2 - 105*A*sqrt(c + b/x^2)*c^3)/b^4