3.776 \(\int \frac{x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=174 \[ \frac{x (11 a d+4 b c)}{6 \sqrt{c+d x^2} (b c-a d)^3}+\frac{x (3 a d+2 b c)}{6 b \left (c+d x^2\right )^{3/2} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{\sqrt{a} (2 a d+3 b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 (b c-a d)^{7/2}} \]
[Out]
((2*b*c + 3*a*d)*x)/(6*b*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + (a*x)/(2*b*(b*c - a*
d)*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((4*b*c + 11*a*d)*x)/(6*(b*c - a*d)^3*Sqrt[c
+ d*x^2]) - (Sqrt[a]*(3*b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c
+ d*x^2])])/(2*(b*c - a*d)^(7/2))
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Rubi [A] time = 0.578457, antiderivative size = 174, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208
\[ \frac{x (11 a d+4 b c)}{6 \sqrt{c+d x^2} (b c-a d)^3}+\frac{x (3 a d+2 b c)}{6 b \left (c+d x^2\right )^{3/2} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{\sqrt{a} (2 a d+3 b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 (b c-a d)^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[x^4/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]
[Out]
((2*b*c + 3*a*d)*x)/(6*b*(b*c - a*d)^2*(c + d*x^2)^(3/2)) + (a*x)/(2*b*(b*c - a*
d)*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((4*b*c + 11*a*d)*x)/(6*(b*c - a*d)^3*Sqrt[c
+ d*x^2]) - (Sqrt[a]*(3*b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c
+ d*x^2])])/(2*(b*c - a*d)^(7/2))
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Rubi in Sympy [A] time = 88.8406, size = 151, normalized size = 0.87 \[ \frac{\sqrt{a} \left (2 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 \left (a d - b c\right )^{\frac{7}{2}}} - \frac{a x}{2 b \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{x \left (11 a d + 4 b c\right )}{6 \sqrt{c + d x^{2}} \left (a d - b c\right )^{3}} + \frac{x \left (3 a d + 2 b c\right )}{6 b \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)
[Out]
sqrt(a)*(2*a*d + 3*b*c)*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/(2*(
a*d - b*c)**(7/2)) - a*x/(2*b*(a + b*x**2)*(c + d*x**2)**(3/2)*(a*d - b*c)) - x*
(11*a*d + 4*b*c)/(6*sqrt(c + d*x**2)*(a*d - b*c)**3) + x*(3*a*d + 2*b*c)/(6*b*(c
+ d*x**2)**(3/2)*(a*d - b*c)**2)
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Mathematica [A] time = 0.450762, size = 158, normalized size = 0.91 \[ \sqrt{c+d x^2} \left (\frac{a b x}{2 \left (a+b x^2\right ) (b c-a d)^3}+\frac{2 x (2 a d+b c)}{3 \left (c+d x^2\right ) (b c-a d)^3}+\frac{c x}{3 \left (c+d x^2\right )^2 (b c-a d)^2}\right )-\frac{\sqrt{a} (2 a d+3 b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 (b c-a d)^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]
[Out]
Sqrt[c + d*x^2]*((a*b*x)/(2*(b*c - a*d)^3*(a + b*x^2)) + (c*x)/(3*(b*c - a*d)^2*
(c + d*x^2)^2) + (2*(b*c + 2*a*d)*x)/(3*(b*c - a*d)^3*(c + d*x^2))) - (Sqrt[a]*(
3*b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*(b*c -
a*d)^(7/2))
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Maple [B] time = 0.036, size = 2463, normalized size = 14.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x^2+a)^2/(d*x^2+c)^(5/2),x)
[Out]
3/4*a/(-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)+1/3/b^2*x/c/(d*x^2+c)^(3/2)+2/3/b^2/c^2*x/(d
*x^2+c)^(1/2)-3/4*a/(-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)+3/4*a/(-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)))-1/4/b*a/(-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)-3/4*a/(-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)))-1/4*a/b^2/(a*d-b*c)/(x-1/b*(-a*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)^(3/2)-1/4*a/b^2/(a*d-b*c)/(x+1
/b*(-a*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)^(3/2)+1/4/b*a/(-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)+5/12*a^2/b^2*d^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)^(3/2)*x+5/6*a^2/b^2*d^2/(a*d-b*c)^2/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-5/4*a^2/b*d^2/(a*d-b*c)^3
/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+5/4*a/b*d*(-a*b)^(1/2)/(a*d-b*c)^3/(-(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)))+3/4/b*a/(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-7/12*a/b^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-7
/6*a/b^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+5/12*a^2/b^2*d^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)^(3/2)*x+5/6*a^2/b^
2*d^2/(a*d-b*c)^2/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-5/4*a^2/b*d^2/(a*d-b*c)^3/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+5/12*a/b^2*d*(-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)^(3/2)-5/4*a/b*d*(-a*b)^(1/2)/(a*d-b*c)^3/((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)-5/12*a/b^2*d*
(-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)^(3/2)+5/4*a/b*d*(-a*b)^(1/2)/(a*d-b*c)^3/((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)-5/4*a/b*d*
(-a*b)^(1/2)/(a*d-b*c)^3/(-(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)))-7/12
*a/b^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-7/6*a/b^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+3/4/b*a/(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
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)),x, algorithm="maxima")
[Out]
integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)), x)
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Fricas [A] time = 1.03612, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)),x, algorithm="fricas")
[Out]
[-1/24*(3*((3*b^2*c*d^2 + 2*a*b*d^3)*x^6 + 3*a*b*c^3 + 2*a^2*c^2*d + (6*b^2*c^2*
d + 7*a*b*c*d^2 + 2*a^2*d^3)*x^4 + (3*b^2*c^3 + 8*a*b*c^2*d + 4*a^2*c*d^2)*x^2)*
sqrt(-a/(b*c - a*d))*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 + 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^3 - (a*b*c^2
- a^2*c*d)*x)*sqrt(d*x^2 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2))
- 4*((4*b^2*c*d + 11*a*b*d^2)*x^5 + 2*(3*b^2*c^2 + 8*a*b*c*d + 4*a^2*d^2)*x^3 +
3*(3*a*b*c^2 + 2*a^2*c*d)*x)*sqrt(d*x^2 + c))/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*
a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 -
a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d
^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3
- 2*a^4*c*d^4)*x^2), -1/12*(3*((3*b^2*c*d^2 + 2*a*b*d^3)*x^6 + 3*a*b*c^3 + 2*a^
2*c^2*d + (6*b^2*c^2*d + 7*a*b*c*d^2 + 2*a^2*d^3)*x^4 + (3*b^2*c^3 + 8*a*b*c^2*d
+ 4*a^2*c*d^2)*x^2)*sqrt(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)/(s
qrt(d*x^2 + c)*(b*c - a*d)*x*sqrt(a/(b*c - a*d)))) - 2*((4*b^2*c*d + 11*a*b*d^2)
*x^5 + 2*(3*b^2*c^2 + 8*a*b*c*d + 4*a^2*d^2)*x^3 + 3*(3*a*b*c^2 + 2*a^2*c*d)*x)*
sqrt(d*x^2 + c))/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 +
(b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d
- 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 -
a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x^2)]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 5.01781, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)),x, algorithm="giac")
[Out]
sage0*x