3.157 \(\int \frac{c+d x^2+e x^4+f x^6}{x^8 \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=140 \[ \frac{\sqrt{a+b x^2} (6 b c-7 a d)}{35 a^2 x^5}-\frac{\sqrt{a+b x^2} \left (35 a^2 e-28 a b d+24 b^2 c\right )}{105 a^3 x^3}+\frac{\sqrt{a+b x^2} \left (-105 a^3 f+70 a^2 b e-56 a b^2 d+48 b^3 c\right )}{105 a^4 x}-\frac{c \sqrt{a+b x^2}}{7 a x^7} \]

[Out]

-(c*Sqrt[a + b*x^2])/(7*a*x^7) + ((6*b*c - 7*a*d)*Sqrt[a + b*x^2])/(35*a^2*x^5)
- ((24*b^2*c - 28*a*b*d + 35*a^2*e)*Sqrt[a + b*x^2])/(105*a^3*x^3) + ((48*b^3*c
- 56*a*b^2*d + 70*a^2*b*e - 105*a^3*f)*Sqrt[a + b*x^2])/(105*a^4*x)

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Rubi [A]  time = 0.335471, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ \frac{\sqrt{a+b x^2} (6 b c-7 a d)}{35 a^2 x^5}-\frac{\sqrt{a+b x^2} \left (35 a^2 e-28 a b d+24 b^2 c\right )}{105 a^3 x^3}+\frac{\sqrt{a+b x^2} \left (-105 a^3 f+70 a^2 b e-56 a b^2 d+48 b^3 c\right )}{105 a^4 x}-\frac{c \sqrt{a+b x^2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*Sqrt[a + b*x^2]),x]

[Out]

-(c*Sqrt[a + b*x^2])/(7*a*x^7) + ((6*b*c - 7*a*d)*Sqrt[a + b*x^2])/(35*a^2*x^5)
- ((24*b^2*c - 28*a*b*d + 35*a^2*e)*Sqrt[a + b*x^2])/(105*a^3*x^3) + ((48*b^3*c
- 56*a*b^2*d + 70*a^2*b*e - 105*a^3*f)*Sqrt[a + b*x^2])/(105*a^4*x)

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Rubi in Sympy [A]  time = 94.4329, size = 163, normalized size = 1.16 \[ - \frac{f \sqrt{a + b x^{2}}}{2 b x^{3}} - \frac{c \sqrt{a + b x^{2}}}{7 a x^{7}} - \frac{\sqrt{a + b x^{2}} \left (7 a d - 6 b c\right )}{35 a^{2} x^{5}} + \frac{\sqrt{a + b x^{2}} \left (105 a^{3} f - 70 a^{2} b e + 56 a b^{2} d - 48 b^{3} c\right )}{210 a^{3} b x^{3}} - \frac{\sqrt{a + b x^{2}} \left (105 a^{3} f - 70 a^{2} b e + 56 a b^{2} d - 48 b^{3} c\right )}{105 a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**6+e*x**4+d*x**2+c)/x**8/(b*x**2+a)**(1/2),x)

[Out]

-f*sqrt(a + b*x**2)/(2*b*x**3) - c*sqrt(a + b*x**2)/(7*a*x**7) - sqrt(a + b*x**2
)*(7*a*d - 6*b*c)/(35*a**2*x**5) + sqrt(a + b*x**2)*(105*a**3*f - 70*a**2*b*e +
56*a*b**2*d - 48*b**3*c)/(210*a**3*b*x**3) - sqrt(a + b*x**2)*(105*a**3*f - 70*a
**2*b*e + 56*a*b**2*d - 48*b**3*c)/(105*a**4*x)

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Mathematica [A]  time = 0.139509, size = 103, normalized size = 0.74 \[ \frac{\sqrt{a+b x^2} \left (-a^3 \left (15 c+21 d x^2+35 x^4 \left (e+3 f x^2\right )\right )+2 a^2 b x^2 \left (9 c+14 d x^2+35 e x^4\right )-8 a b^2 x^4 \left (3 c+7 d x^2\right )+48 b^3 c x^6\right )}{105 a^4 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*Sqrt[a + b*x^2]),x]

[Out]

(Sqrt[a + b*x^2]*(48*b^3*c*x^6 - 8*a*b^2*x^4*(3*c + 7*d*x^2) + 2*a^2*b*x^2*(9*c
+ 14*d*x^2 + 35*e*x^4) - a^3*(15*c + 21*d*x^2 + 35*x^4*(e + 3*f*x^2))))/(105*a^4
*x^7)

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Maple [A]  time = 0.009, size = 111, normalized size = 0.8 \[ -{\frac{105\,{a}^{3}f{x}^{6}-70\,{a}^{2}be{x}^{6}+56\,a{b}^{2}d{x}^{6}-48\,{b}^{3}c{x}^{6}+35\,{a}^{3}e{x}^{4}-28\,{a}^{2}bd{x}^{4}+24\,a{b}^{2}c{x}^{4}+21\,{a}^{3}d{x}^{2}-18\,{a}^{2}bc{x}^{2}+15\,c{a}^{3}}{105\,{x}^{7}{a}^{4}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a)^(1/2),x)

[Out]

-1/105*(b*x^2+a)^(1/2)*(105*a^3*f*x^6-70*a^2*b*e*x^6+56*a*b^2*d*x^6-48*b^3*c*x^6
+35*a^3*e*x^4-28*a^2*b*d*x^4+24*a*b^2*c*x^4+21*a^3*d*x^2-18*a^2*b*c*x^2+15*a^3*c
)/x^7/a^4

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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((f*x^6 + e*x^4 + d*x^2 + c)/(sqrt(b*x^2 + a)*x^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.416038, size = 135, normalized size = 0.96 \[ \frac{{\left ({\left (48 \, b^{3} c - 56 \, a b^{2} d + 70 \, a^{2} b e - 105 \, a^{3} f\right )} x^{6} -{\left (24 \, a b^{2} c - 28 \, a^{2} b d + 35 \, a^{3} e\right )} x^{4} - 15 \, a^{3} c + 3 \,{\left (6 \, a^{2} b c - 7 \, a^{3} d\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, a^{4} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)/(sqrt(b*x^2 + a)*x^8),x, algorithm="fricas")

[Out]

1/105*((48*b^3*c - 56*a*b^2*d + 70*a^2*b*e - 105*a^3*f)*x^6 - (24*a*b^2*c - 28*a
^2*b*d + 35*a^3*e)*x^4 - 15*a^3*c + 3*(6*a^2*b*c - 7*a^3*d)*x^2)*sqrt(b*x^2 + a)
/(a^4*x^7)

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Sympy [A]  time = 7.21836, size = 891, normalized size = 6.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**6+e*x**4+d*x**2+c)/x**8/(b*x**2+a)**(1/2),x)

[Out]

-5*a**6*b**(19/2)*c*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**
8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 9*a**5*b**(21/2)*c*x**2*sqrt(a
/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 +
35*a**4*b**12*x**12) - 5*a**4*b**(23/2)*c*x**4*sqrt(a/(b*x**2) + 1)/(35*a**7*b*
*9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 3*
a**4*b**(9/2)*d*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15
*a**3*b**6*x**8) + 5*a**3*b**(25/2)*c*x**6*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x*
*6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 2*a**3*
b**(11/2)*d*x**2*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 1
5*a**3*b**6*x**8) + 30*a**2*b**(27/2)*c*x**8*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*
x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 3*a**
2*b**(13/2)*d*x**4*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 +
15*a**3*b**6*x**8) + 40*a*b**(29/2)*c*x**10*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*
x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 12*a*
b**(15/2)*d*x**6*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 1
5*a**3*b**6*x**8) + 16*b**(31/2)*c*x**12*sqrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6
+ 105*a**6*b**10*x**8 + 105*a**5*b**11*x**10 + 35*a**4*b**12*x**12) - 8*b**(17/
2)*d*x**8*sqrt(a/(b*x**2) + 1)/(15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*
b**6*x**8) - sqrt(b)*e*sqrt(a/(b*x**2) + 1)/(3*a*x**2) - sqrt(b)*f*sqrt(a/(b*x**
2) + 1)/a + 2*b**(3/2)*e*sqrt(a/(b*x**2) + 1)/(3*a**2)

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GIAC/XCAS [A]  time = 0.230394, size = 748, normalized size = 5.34 \[ \frac{2 \,{\left (105 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} \sqrt{b} f - 630 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a \sqrt{b} f + 210 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} b^{\frac{3}{2}} e + 560 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} b^{\frac{5}{2}} d + 1575 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{2} \sqrt{b} f - 910 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a b^{\frac{3}{2}} e + 1680 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} b^{\frac{7}{2}} c - 1400 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a b^{\frac{5}{2}} d - 2100 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{3} \sqrt{b} f + 1540 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{2} b^{\frac{3}{2}} e - 1008 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a b^{\frac{7}{2}} c + 1176 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{2} b^{\frac{5}{2}} d + 1575 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{4} \sqrt{b} f - 1260 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{3} b^{\frac{3}{2}} e + 336 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{2} b^{\frac{7}{2}} c - 392 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{3} b^{\frac{5}{2}} d - 630 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{5} \sqrt{b} f + 490 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{4} b^{\frac{3}{2}} e - 48 \, a^{3} b^{\frac{7}{2}} c + 56 \, a^{4} b^{\frac{5}{2}} d + 105 \, a^{6} \sqrt{b} f - 70 \, a^{5} b^{\frac{3}{2}} e\right )}}{105 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)/(sqrt(b*x^2 + a)*x^8),x, algorithm="giac")

[Out]

2/105*(105*(sqrt(b)*x - sqrt(b*x^2 + a))^12*sqrt(b)*f - 630*(sqrt(b)*x - sqrt(b*
x^2 + a))^10*a*sqrt(b)*f + 210*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(3/2)*e + 560*
(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(5/2)*d + 1575*(sqrt(b)*x - sqrt(b*x^2 + a))^8
*a^2*sqrt(b)*f - 910*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(3/2)*e + 1680*(sqrt(b)
*x - sqrt(b*x^2 + a))^6*b^(7/2)*c - 1400*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(5/
2)*d - 2100*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*sqrt(b)*f + 1540*(sqrt(b)*x - sq
rt(b*x^2 + a))^6*a^2*b^(3/2)*e - 1008*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(7/2)*
c + 1176*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(5/2)*d + 1575*(sqrt(b)*x - sqrt(
b*x^2 + a))^4*a^4*sqrt(b)*f - 1260*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*b^(3/2)*e
+ 336*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(7/2)*c - 392*(sqrt(b)*x - sqrt(b*x
^2 + a))^2*a^3*b^(5/2)*d - 630*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*sqrt(b)*f + 4
90*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*b^(3/2)*e - 48*a^3*b^(7/2)*c + 56*a^4*b^(
5/2)*d + 105*a^6*sqrt(b)*f - 70*a^5*b^(3/2)*e)/((sqrt(b)*x - sqrt(b*x^2 + a))^2
- a)^7