3.787 \(\int \left (a+\frac{b}{x^2}\right ) \left (c+\frac{d}{x^2}\right )^{3/2} x^6 \, dx\)

Optimal. Leaf size=53 \[ \frac{x^5 \left (c+\frac{d}{x^2}\right )^{5/2} (7 b c-2 a d)}{35 c^2}+\frac{a x^7 \left (c+\frac{d}{x^2}\right )^{5/2}}{7 c} \]

[Out]

((7*b*c - 2*a*d)*(c + d/x^2)^(5/2)*x^5)/(35*c^2) + (a*(c + d/x^2)^(5/2)*x^7)/(7*
c)

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Rubi [A]  time = 0.103252, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^5 \left (c+\frac{d}{x^2}\right )^{5/2} (7 b c-2 a d)}{35 c^2}+\frac{a x^7 \left (c+\frac{d}{x^2}\right )^{5/2}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x^6,x]

[Out]

((7*b*c - 2*a*d)*(c + d/x^2)^(5/2)*x^5)/(35*c^2) + (a*(c + d/x^2)^(5/2)*x^7)/(7*
c)

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Rubi in Sympy [A]  time = 8.2219, size = 46, normalized size = 0.87 \[ \frac{a x^{7} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{7 c} - \frac{x^{5} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (2 a d - 7 b c\right )}{35 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**6,x)

[Out]

a*x**7*(c + d/x**2)**(5/2)/(7*c) - x**5*(c + d/x**2)**(5/2)*(2*a*d - 7*b*c)/(35*
c**2)

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Mathematica [A]  time = 0.0485513, size = 44, normalized size = 0.83 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right )^2 \left (5 a c x^2-2 a d+7 b c\right )}{35 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^6,x]

[Out]

(Sqrt[c + d/x^2]*x*(d + c*x^2)^2*(7*b*c - 2*a*d + 5*a*c*x^2))/(35*c^2)

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Maple [A]  time = 0.008, size = 45, normalized size = 0.9 \[{\frac{{x}^{3} \left ( 5\,a{x}^{2}c-2\,ad+7\,bc \right ) \left ( c{x}^{2}+d \right ) }{35\,{c}^{2}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*(c+d/x^2)^(3/2)*x^6,x)

[Out]

1/35*((c*x^2+d)/x^2)^(3/2)*x^3*(5*a*c*x^2-2*a*d+7*b*c)*(c*x^2+d)/c^2

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Maxima [A]  time = 1.44227, size = 74, normalized size = 1.4 \[ \frac{b{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} x^{5}}{5 \, c} + \frac{{\left (5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} x^{7} - 7 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} d x^{5}\right )} a}{35 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x^6,x, algorithm="maxima")

[Out]

1/5*b*(c + d/x^2)^(5/2)*x^5/c + 1/35*(5*(c + d/x^2)^(7/2)*x^7 - 7*(c + d/x^2)^(5
/2)*d*x^5)*a/c^2

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Fricas [A]  time = 0.222626, size = 108, normalized size = 2.04 \[ \frac{{\left (5 \, a c^{3} x^{7} +{\left (7 \, b c^{3} + 8 \, a c^{2} d\right )} x^{5} +{\left (14 \, b c^{2} d + a c d^{2}\right )} x^{3} +{\left (7 \, b c d^{2} - 2 \, a d^{3}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{35 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x^6,x, algorithm="fricas")

[Out]

1/35*(5*a*c^3*x^7 + (7*b*c^3 + 8*a*c^2*d)*x^5 + (14*b*c^2*d + a*c*d^2)*x^3 + (7*
b*c*d^2 - 2*a*d^3)*x)*sqrt((c*x^2 + d)/x^2)/c^2

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Sympy [A]  time = 11.1271, size = 498, normalized size = 9.4 \[ \frac{15 a c^{6} d^{\frac{9}{2}} x^{10} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{33 a c^{5} d^{\frac{11}{2}} x^{8} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{17 a c^{4} d^{\frac{13}{2}} x^{6} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{3 a c^{3} d^{\frac{15}{2}} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{12 a c^{2} d^{\frac{17}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{8 a c d^{\frac{19}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac{a d^{\frac{3}{2}} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{a d^{\frac{5}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c} - \frac{2 a d^{\frac{7}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c^{2}} + \frac{b c \sqrt{d} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{2 b d^{\frac{3}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{b d^{\frac{5}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**6,x)

[Out]

15*a*c**6*d**(9/2)*x**10*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*
x**2 + 105*c**3*d**6) + 33*a*c**5*d**(11/2)*x**8*sqrt(c*x**2/d + 1)/(105*c**5*d*
*4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) + 17*a*c**4*d**(13/2)*x**6*sqrt(c*
x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) + 3*a*c**3
*d**(15/2)*x**4*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 + 210*c**4*d**5*x**2 + 10
5*c**3*d**6) + 12*a*c**2*d**(17/2)*x**2*sqrt(c*x**2/d + 1)/(105*c**5*d**4*x**4 +
210*c**4*d**5*x**2 + 105*c**3*d**6) + 8*a*c*d**(19/2)*sqrt(c*x**2/d + 1)/(105*c
**5*d**4*x**4 + 210*c**4*d**5*x**2 + 105*c**3*d**6) + a*d**(3/2)*x**4*sqrt(c*x**
2/d + 1)/5 + a*d**(5/2)*x**2*sqrt(c*x**2/d + 1)/(15*c) - 2*a*d**(7/2)*sqrt(c*x**
2/d + 1)/(15*c**2) + b*c*sqrt(d)*x**4*sqrt(c*x**2/d + 1)/5 + 2*b*d**(3/2)*x**2*s
qrt(c*x**2/d + 1)/5 + b*d**(5/2)*sqrt(c*x**2/d + 1)/(5*c)

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GIAC/XCAS [A]  time = 0.215442, size = 203, normalized size = 3.83 \[ \frac{35 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} b d{\rm sign}\left (x\right ) + 7 \,{\left (3 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d\right )} b{\rm sign}\left (x\right ) + \frac{7 \,{\left (3 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d\right )} a d{\rm sign}\left (x\right )}{c} + \frac{{\left (15 \,{\left (c x^{2} + d\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} d + 35 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d^{2}\right )} a{\rm sign}\left (x\right )}{c}}{105 \, c} - \frac{{\left (7 \, b c d^{\frac{5}{2}} - 2 \, a d^{\frac{7}{2}}\right )}{\rm sign}\left (x\right )}{35 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*(c + d/x^2)^(3/2)*x^6,x, algorithm="giac")

[Out]

1/105*(35*(c*x^2 + d)^(3/2)*b*d*sign(x) + 7*(3*(c*x^2 + d)^(5/2) - 5*(c*x^2 + d)
^(3/2)*d)*b*sign(x) + 7*(3*(c*x^2 + d)^(5/2) - 5*(c*x^2 + d)^(3/2)*d)*a*d*sign(x
)/c + (15*(c*x^2 + d)^(7/2) - 42*(c*x^2 + d)^(5/2)*d + 35*(c*x^2 + d)^(3/2)*d^2)
*a*sign(x)/c)/c - 1/35*(7*b*c*d^(5/2) - 2*a*d^(7/2))*sign(x)/c^2