3.730 \(\int \frac{1}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=245 \[ \frac{b^4 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} (b c-a d)^{5/2}}+\frac{\sqrt{c+d x^2} (b c-2 a d) \left (-8 a^2 d^2+8 a b c d+3 b^2 c^2\right )}{3 a^2 c^4 x (b c-a d)^2}-\frac{\sqrt{c+d x^2} \left (8 a^2 d^2-12 a b c d+b^2 c^2\right )}{3 a c^3 x^3 (b c-a d)^2}-\frac{d (3 b c-2 a d)}{c^2 x^3 \sqrt{c+d x^2} (b c-a d)^2}-\frac{d}{3 c x^3 \left (c+d x^2\right )^{3/2} (b c-a d)} \]
[Out]
-d/(3*c*(b*c - a*d)*x^3*(c + d*x^2)^(3/2)) - (d*(3*b*c - 2*a*d))/(c^2*(b*c - a*d
)^2*x^3*Sqrt[c + d*x^2]) - ((b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*Sqrt[c + d*x^2])/
(3*a*c^3*(b*c - a*d)^2*x^3) + ((b*c - 2*a*d)*(3*b^2*c^2 + 8*a*b*c*d - 8*a^2*d^2)
*Sqrt[c + d*x^2])/(3*a^2*c^4*(b*c - a*d)^2*x) + (b^4*ArcTan[(Sqrt[b*c - a*d]*x)/
(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(5/2)*(b*c - a*d)^(5/2))
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Rubi [A] time = 1.08825, antiderivative size = 245, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25
\[ \frac{b^4 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} (b c-a d)^{5/2}}+\frac{\sqrt{c+d x^2} (b c-2 a d) \left (-8 a^2 d^2+8 a b c d+3 b^2 c^2\right )}{3 a^2 c^4 x (b c-a d)^2}-\frac{\sqrt{c+d x^2} \left (8 a^2 d^2-12 a b c d+b^2 c^2\right )}{3 a c^3 x^3 (b c-a d)^2}-\frac{d (3 b c-2 a d)}{c^2 x^3 \sqrt{c+d x^2} (b c-a d)^2}-\frac{d}{3 c x^3 \left (c+d x^2\right )^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(a + b*x^2)*(c + d*x^2)^(5/2)),x]
[Out]
-d/(3*c*(b*c - a*d)*x^3*(c + d*x^2)^(3/2)) - (d*(3*b*c - 2*a*d))/(c^2*(b*c - a*d
)^2*x^3*Sqrt[c + d*x^2]) - ((b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*Sqrt[c + d*x^2])/
(3*a*c^3*(b*c - a*d)^2*x^3) + ((b*c - 2*a*d)*(3*b^2*c^2 + 8*a*b*c*d - 8*a^2*d^2)
*Sqrt[c + d*x^2])/(3*a^2*c^4*(b*c - a*d)^2*x) + (b^4*ArcTan[(Sqrt[b*c - a*d]*x)/
(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(5/2)*(b*c - a*d)^(5/2))
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Rubi in Sympy [A] time = 174.2, size = 224, normalized size = 0.91 \[ \frac{d}{3 c x^{3} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{d \left (2 a d - 3 b c\right )}{c^{2} x^{3} \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} - \frac{\sqrt{c + d x^{2}} \left (8 a^{2} d^{2} - 12 a b c d + b^{2} c^{2}\right )}{3 a c^{3} x^{3} \left (a d - b c\right )^{2}} + \frac{\sqrt{c + d x^{2}} \left (2 a d - b c\right ) \left (8 a^{2} d^{2} - 8 a b c d - 3 b^{2} c^{2}\right )}{3 a^{2} c^{4} x \left (a d - b c\right )^{2}} + \frac{b^{4} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{a^{\frac{5}{2}} \left (a d - b c\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(b*x**2+a)/(d*x**2+c)**(5/2),x)
[Out]
d/(3*c*x**3*(c + d*x**2)**(3/2)*(a*d - b*c)) + d*(2*a*d - 3*b*c)/(c**2*x**3*sqrt
(c + d*x**2)*(a*d - b*c)**2) - sqrt(c + d*x**2)*(8*a**2*d**2 - 12*a*b*c*d + b**2
*c**2)/(3*a*c**3*x**3*(a*d - b*c)**2) + sqrt(c + d*x**2)*(2*a*d - b*c)*(8*a**2*d
**2 - 8*a*b*c*d - 3*b**2*c**2)/(3*a**2*c**4*x*(a*d - b*c)**2) + b**4*atanh(x*sqr
t(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/(a**(5/2)*(a*d - b*c)**(5/2))
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Mathematica [A] time = 0.706967, size = 160, normalized size = 0.65 \[ \frac{b^4 \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2} (b c-a d)^{5/2}}+\frac{\sqrt{c+d x^2} \left (\frac{x^2 (8 a d+3 b c)}{a^2}+\frac{d^3 x^4 (8 a d-11 b c)}{\left (c+d x^2\right ) (b c-a d)^2}-\frac{c d^3 x^4}{\left (c+d x^2\right )^2 (b c-a d)}-\frac{c}{a}\right )}{3 c^4 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*(a + b*x^2)*(c + d*x^2)^(5/2)),x]
[Out]
(Sqrt[c + d*x^2]*(-(c/a) + ((3*b*c + 8*a*d)*x^2)/a^2 - (c*d^3*x^4)/((b*c - a*d)*
(c + d*x^2)^2) + (d^3*(-11*b*c + 8*a*d)*x^4)/((b*c - a*d)^2*(c + d*x^2))))/(3*c^
4*x^3) + (b^4*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(a^(5/2)*(b
*c - a*d)^(5/2))
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Maple [B] time = 0.027, size = 1285, normalized size = 5.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(b*x^2+a)/(d*x^2+c)^(5/2),x)
[Out]
-1/3/a/c/x^3/(d*x^2+c)^(3/2)+2/a*d/c^2/x/(d*x^2+c)^(3/2)+8/3/a*d^2/c^3*x/(d*x^2+
c)^(3/2)+16/3/a*d^2/c^4*x/(d*x^2+c)^(1/2)+b/a^2/c/x/(d*x^2+c)^(3/2)+4/3*b/a^2*d/
c^2*x/(d*x^2+c)^(3/2)+8/3*b/a^2*d/c^3*x/(d*x^2+c)^(1/2)-1/6*b^3/a^2/(-a*b)^(1/2)
/(a*d-b*c)/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*
d-b*c)/b)^(3/2)+1/6*b^2/a^2*d/(a*d-b*c)/c/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(
1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x+1/3*b^2/a^2*d/(a*d-b*c)/c^2/((x
-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2
)*x+1/2*b^4/a^2/(-a*b)^(1/2)/(a*d-b*c)^2/((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1
/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1/2*b^3/a^2/(a*d-b*c)^2/c/((x-1/b*
(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d
-1/2*b^4/a^2/(-a*b)^(1/2)/(a*d-b*c)^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*
d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2
))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)
^(1/2)))+1/6*b^3/a^2/(-a*b)^(1/2)/(a*d-b*c)/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)
^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)+1/6*b^2/a^2*d/(a*d-b*c)/c/((x+1
/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*
x+1/3*b^2/a^2*d/(a*d-b*c)/c^2/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/
b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-1/2*b^4/a^2/(-a*b)^(1/2)/(a*d-b*c)^2/((x+1/
b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1
/2*b^3/a^2/(a*d-b*c)^2/c/((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a
*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*d+1/2*b^4/a^2/(-a*b)^(1/2)/(a*d-b*c)^2/(-(a*d-b*
c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-
b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-
(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{5}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x^4),x, algorithm="maxima")
[Out]
integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x^4), x)
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Fricas [A] time = 1.02971, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x^4),x, algorithm="fricas")
[Out]
[-1/12*(4*(a*b^2*c^5 - 2*a^2*b*c^4*d + a^3*c^3*d^2 - (3*b^3*c^3*d^2 + 2*a*b^2*c^
2*d^3 - 24*a^2*b*c*d^4 + 16*a^3*d^5)*x^6 - 3*(2*b^3*c^4*d + a*b^2*c^3*d^2 - 12*a
^2*b*c^2*d^3 + 8*a^3*c*d^4)*x^4 - 3*(b^3*c^5 - 3*a^2*b*c^3*d^2 + 2*a^3*c^2*d^3)*
x^2)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c) - 3*(b^4*c^4*d^2*x^7 + 2*b^4*c^5*d*x^5
+ b^4*c^6*x^3)*log((((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b
*c^2 - 4*a^2*c*d)*x^2)*sqrt(-a*b*c + a^2*d) + 4*((a*b^2*c^2 - 3*a^2*b*c*d + 2*a^
3*d^2)*x^3 - (a^2*b*c^2 - a^3*c*d)*x)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^
2)))/(((a^2*b^2*c^6*d^2 - 2*a^3*b*c^5*d^3 + a^4*c^4*d^4)*x^7 + 2*(a^2*b^2*c^7*d
- 2*a^3*b*c^6*d^2 + a^4*c^5*d^3)*x^5 + (a^2*b^2*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d^
2)*x^3)*sqrt(-a*b*c + a^2*d)), -1/6*(2*(a*b^2*c^5 - 2*a^2*b*c^4*d + a^3*c^3*d^2
- (3*b^3*c^3*d^2 + 2*a*b^2*c^2*d^3 - 24*a^2*b*c*d^4 + 16*a^3*d^5)*x^6 - 3*(2*b^3
*c^4*d + a*b^2*c^3*d^2 - 12*a^2*b*c^2*d^3 + 8*a^3*c*d^4)*x^4 - 3*(b^3*c^5 - 3*a^
2*b*c^3*d^2 + 2*a^3*c^2*d^3)*x^2)*sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c) - 3*(b^4*c
^4*d^2*x^7 + 2*b^4*c^5*d*x^5 + b^4*c^6*x^3)*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)
/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x)))/(((a^2*b^2*c^6*d^2 - 2*a^3*b*c^5*d^3
+ a^4*c^4*d^4)*x^7 + 2*(a^2*b^2*c^7*d - 2*a^3*b*c^6*d^2 + a^4*c^5*d^3)*x^5 + (a^
2*b^2*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d^2)*x^3)*sqrt(a*b*c - a^2*d))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(b*x**2+a)/(d*x**2+c)**(5/2),x)
[Out]
Integral(1/(x**4*(a + b*x**2)*(c + d*x**2)**(5/2)), x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^(5/2)*x^4),x, algorithm="giac")
[Out]
Exception raised: TypeError