3.809 \(\int \frac{a+\frac{b}{x^2}}{\left (c+\frac{d}{x^2}\right )^{3/2} x^7} \, dx\)

Optimal. Leaf size=100 \[ -\frac{c^2 (b c-a d)}{d^4 \sqrt{c+\frac{d}{x^2}}}+\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-a d)}{3 d^4}-\frac{c \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^4} \]

[Out]

-((c^2*(b*c - a*d))/(d^4*Sqrt[c + d/x^2])) - (c*(3*b*c - 2*a*d)*Sqrt[c + d/x^2])
/d^4 + ((3*b*c - a*d)*(c + d/x^2)^(3/2))/(3*d^4) - (b*(c + d/x^2)^(5/2))/(5*d^4)

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Rubi [A]  time = 0.225897, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{c^2 (b c-a d)}{d^4 \sqrt{c+\frac{d}{x^2}}}+\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (3 b c-a d)}{3 d^4}-\frac{c \sqrt{c+\frac{d}{x^2}} (3 b c-2 a d)}{d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^7),x]

[Out]

-((c^2*(b*c - a*d))/(d^4*Sqrt[c + d/x^2])) - (c*(3*b*c - 2*a*d)*Sqrt[c + d/x^2])
/d^4 + ((3*b*c - a*d)*(c + d/x^2)^(3/2))/(3*d^4) - (b*(c + d/x^2)^(5/2))/(5*d^4)

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Rubi in Sympy [A]  time = 22.8028, size = 88, normalized size = 0.88 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5 d^{4}} + \frac{c^{2} \left (a d - b c\right )}{d^{4} \sqrt{c + \frac{d}{x^{2}}}} + \frac{c \sqrt{c + \frac{d}{x^{2}}} \left (2 a d - 3 b c\right )}{d^{4}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d - 3 b c\right )}{3 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**7,x)

[Out]

-b*(c + d/x**2)**(5/2)/(5*d**4) + c**2*(a*d - b*c)/(d**4*sqrt(c + d/x**2)) + c*s
qrt(c + d/x**2)*(2*a*d - 3*b*c)/d**4 - (c + d/x**2)**(3/2)*(a*d - 3*b*c)/(3*d**4
)

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Mathematica [A]  time = 0.100012, size = 81, normalized size = 0.81 \[ \frac{-5 a d x^2 \left (-8 c^2 x^4-4 c d x^2+d^2\right )-3 b \left (16 c^3 x^6+8 c^2 d x^4-2 c d^2 x^2+d^3\right )}{15 d^4 x^6 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^7),x]

[Out]

(-5*a*d*x^2*(d^2 - 4*c*d*x^2 - 8*c^2*x^4) - 3*b*(d^3 - 2*c*d^2*x^2 + 8*c^2*d*x^4
+ 16*c^3*x^6))/(15*d^4*Sqrt[c + d/x^2]*x^6)

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Maple [A]  time = 0.011, size = 94, normalized size = 0.9 \[{\frac{ \left ( 40\,a{c}^{2}d{x}^{6}-48\,b{c}^{3}{x}^{6}+20\,ac{d}^{2}{x}^{4}-24\,b{c}^{2}d{x}^{4}-5\,a{d}^{3}{x}^{2}+6\,bc{d}^{2}{x}^{2}-3\,b{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{15\,{d}^{4}{x}^{8}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/(c+d/x^2)^(3/2)/x^7,x)

[Out]

1/15*(40*a*c^2*d*x^6-48*b*c^3*x^6+20*a*c*d^2*x^4-24*b*c^2*d*x^4-5*a*d^3*x^2+6*b*
c*d^2*x^2-3*b*d^3)*(c*x^2+d)/((c*x^2+d)/x^2)^(3/2)/d^4/x^8

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Maxima [A]  time = 1.41474, size = 157, normalized size = 1.57 \[ -\frac{1}{5} \, b{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}}}{d^{4}} - \frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c}{d^{4}} + \frac{15 \, \sqrt{c + \frac{d}{x^{2}}} c^{2}}{d^{4}} + \frac{5 \, c^{3}}{\sqrt{c + \frac{d}{x^{2}}} d^{4}}\right )} - \frac{1}{3} \, a{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}{d^{3}} - \frac{6 \, \sqrt{c + \frac{d}{x^{2}}} c}{d^{3}} - \frac{3 \, c^{2}}{\sqrt{c + \frac{d}{x^{2}}} d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^7),x, algorithm="maxima")

[Out]

-1/5*b*((c + d/x^2)^(5/2)/d^4 - 5*(c + d/x^2)^(3/2)*c/d^4 + 15*sqrt(c + d/x^2)*c
^2/d^4 + 5*c^3/(sqrt(c + d/x^2)*d^4)) - 1/3*a*((c + d/x^2)^(3/2)/d^3 - 6*sqrt(c
+ d/x^2)*c/d^3 - 3*c^2/(sqrt(c + d/x^2)*d^3))

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Fricas [A]  time = 0.247626, size = 132, normalized size = 1.32 \[ -\frac{{\left (8 \,{\left (6 \, b c^{3} - 5 \, a c^{2} d\right )} x^{6} + 4 \,{\left (6 \, b c^{2} d - 5 \, a c d^{2}\right )} x^{4} + 3 \, b d^{3} -{\left (6 \, b c d^{2} - 5 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \,{\left (c d^{4} x^{6} + d^{5} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^7),x, algorithm="fricas")

[Out]

-1/15*(8*(6*b*c^3 - 5*a*c^2*d)*x^6 + 4*(6*b*c^2*d - 5*a*c*d^2)*x^4 + 3*b*d^3 - (
6*b*c*d^2 - 5*a*d^3)*x^2)*sqrt((c*x^2 + d)/x^2)/(c*d^4*x^6 + d^5*x^4)

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Sympy [A]  time = 56.2123, size = 2273, normalized size = 22.73 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**7,x)

[Out]

a*(8*c**(9/2)*d**(7/2)*x**6*sqrt(c*x**2/d + 1)/(3*c**(7/2)*d**6*x**7 + 6*c**(5/2
)*d**7*x**5 + 3*c**(3/2)*d**8*x**3) + 12*c**(7/2)*d**(9/2)*x**4*sqrt(c*x**2/d +
1)/(3*c**(7/2)*d**6*x**7 + 6*c**(5/2)*d**7*x**5 + 3*c**(3/2)*d**8*x**3) + 3*c**(
5/2)*d**(11/2)*x**2*sqrt(c*x**2/d + 1)/(3*c**(7/2)*d**6*x**7 + 6*c**(5/2)*d**7*x
**5 + 3*c**(3/2)*d**8*x**3) - c**(3/2)*d**(13/2)*sqrt(c*x**2/d + 1)/(3*c**(7/2)*
d**6*x**7 + 6*c**(5/2)*d**7*x**5 + 3*c**(3/2)*d**8*x**3) - 8*c**5*d**3*x**7/(3*c
**(7/2)*d**6*x**7 + 6*c**(5/2)*d**7*x**5 + 3*c**(3/2)*d**8*x**3) - 16*c**4*d**4*
x**5/(3*c**(7/2)*d**6*x**7 + 6*c**(5/2)*d**7*x**5 + 3*c**(3/2)*d**8*x**3) - 8*c*
*3*d**5*x**3/(3*c**(7/2)*d**6*x**7 + 6*c**(5/2)*d**7*x**5 + 3*c**(3/2)*d**8*x**3
)) + b*(-16*c**(21/2)*d**(23/2)*x**16*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**1
7 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x*
*11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) -
88*c**(19/2)*d**(25/2)*x**14*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 + 30*c
**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75
*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) - 198*c**
(17/2)*d**(27/2)*x**12*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2
)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/
2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) - 231*c**(15/2)*
d**(29/2)*x**10*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16
*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**1
9*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) - 145*c**(13/2)*d**(31/
2)*x**8*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 +
75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 +
30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) - 46*c**(11/2)*d**(33/2)*x**6*s
qrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(1
3/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7
/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) - 8*c**(9/2)*d**(35/2)*x**4*sqrt(c*x**2/
d + 1)/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*
x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x
**7 + 5*c**(5/2)*d**21*x**5) - 3*c**(7/2)*d**(37/2)*x**2*sqrt(c*x**2/d + 1)/(5*c
**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100
*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**
(5/2)*d**21*x**5) - c**(5/2)*d**(39/2)*sqrt(c*x**2/d + 1)/(5*c**(17/2)*d**15*x**
17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x
**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5)
+ 16*c**11*d**11*x**17/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*
c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*
c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) + 96*c**10*d**12*x**15/(5*c**(17/2)
*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/
2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d*
*21*x**5) + 240*c**9*d**13*x**13/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x
**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*
x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) + 320*c**8*d**14*x**11/(5
*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 1
00*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c
**(5/2)*d**21*x**5) + 240*c**7*d**15*x**9/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2
)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/
2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5) + 96*c**6*d**16*
x**7/(5*c**(17/2)*d**15*x**17 + 30*c**(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x*
*13 + 100*c**(11/2)*d**18*x**11 + 75*c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**
7 + 5*c**(5/2)*d**21*x**5) + 16*c**5*d**17*x**5/(5*c**(17/2)*d**15*x**17 + 30*c*
*(15/2)*d**16*x**15 + 75*c**(13/2)*d**17*x**13 + 100*c**(11/2)*d**18*x**11 + 75*
c**(9/2)*d**19*x**9 + 30*c**(7/2)*d**20*x**7 + 5*c**(5/2)*d**21*x**5))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^7),x, algorithm="giac")

[Out]

integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^7), x)