3.778 \(\int \frac{x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=163 \[ -\frac{d x (2 a d+13 b c)}{6 c \sqrt{c+d x^2} (b c-a d)^3}-\frac{x}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{5 d x}{6 \left (c+d x^2\right )^{3/2} (b c-a d)^2}+\frac{b (4 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} (b c-a d)^{7/2}} \]

[Out]

(-5*d*x)/(6*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - x/(2*(b*c - a*d)*(a + b*x^2)*(c +
d*x^2)^(3/2)) - (d*(13*b*c + 2*a*d)*x)/(6*c*(b*c - a*d)^3*Sqrt[c + d*x^2]) + (b
*(b*c + 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]
*(b*c - a*d)^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.444429, antiderivative size = 163, 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 \[ -\frac{d x (2 a d+13 b c)}{6 c \sqrt{c+d x^2} (b c-a d)^3}-\frac{x}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{5 d x}{6 \left (c+d x^2\right )^{3/2} (b c-a d)^2}+\frac{b (4 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} (b c-a d)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]

[Out]

(-5*d*x)/(6*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - x/(2*(b*c - a*d)*(a + b*x^2)*(c +
d*x^2)^(3/2)) - (d*(13*b*c + 2*a*d)*x)/(6*c*(b*c - a*d)^3*Sqrt[c + d*x^2]) + (b
*(b*c + 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]
*(b*c - a*d)^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 79.0366, size = 143, normalized size = 0.88 \[ - \frac{5 d x}{6 \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{x}{2 \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{d x \left (2 a d + 13 b c\right )}{6 c \sqrt{c + d x^{2}} \left (a d - b c\right )^{3}} - \frac{b \left (4 a d + b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 \sqrt{a} \left (a d - b c\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

-5*d*x/(6*(c + d*x**2)**(3/2)*(a*d - b*c)**2) + x/(2*(a + b*x**2)*(c + d*x**2)**
(3/2)*(a*d - b*c)) + d*x*(2*a*d + 13*b*c)/(6*c*sqrt(c + d*x**2)*(a*d - b*c)**3)
- b*(4*a*d + b*c)*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/(2*sqrt(a)
*(a*d - b*c)**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.383999, size = 163, normalized size = 1. \[ \sqrt{c+d x^2} \left (-\frac{b^2 x}{2 \left (a+b x^2\right ) (b c-a d)^3}-\frac{d x (a d+5 b c)}{3 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{d x}{3 \left (c+d x^2\right )^2 (b c-a d)^2}\right )+\frac{b (4 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 \sqrt{a} (b c-a d)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/((a + b*x^2)^2*(c + d*x^2)^(5/2)),x]

[Out]

Sqrt[c + d*x^2]*(-(b^2*x)/(2*(b*c - a*d)^3*(a + b*x^2)) - (d*x)/(3*(b*c - a*d)^2
*(c + d*x^2)^2) - (d*(5*b*c + a*d)*x)/(3*c*(b*c - a*d)^3*(c + d*x^2))) + (b*(b*c
+ 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*Sqrt[a]*(b*c
- a*d)^(7/2))

_______________________________________________________________________________________

Maple [B]  time = 0.024, size = 2369, normalized size = 14.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^2+a)^2/(d*x^2+c)^(5/2),x)

[Out]

-1/4/(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+5/12/b*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*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)))-1/4/(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-1/4/(-a*b)^(1/2)*b/(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)))+1/12/(-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)-1/12/(-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/b*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)+5/12/b*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+5/6/b*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+5/4*d^2*a/(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*d^2*a/(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/b*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+5/6/b*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+1/4/(-a*b)^(1/2)*b/(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)))-5/12/b*d^2*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)^(3/2)*x-5/6/b*d^2*a/(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-5/12/b*d^2*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)^(3/2
)*x-5/6/b*d^2*a/(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*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)))+1/4/(-a*b)^(1/2)*b/(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)-1/4
/(-a*b)^(1/2)*b/(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)+1/4/b/(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)+5/4*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)+1/4/b/(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)-5/4*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)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)),x, algorithm="maxima")

[Out]

integrate(x^2/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)), x)

_______________________________________________________________________________________

Fricas [A]  time = 1.18564, 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^2/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)),x, algorithm="fricas")

[Out]

[-1/24*(4*((13*b^2*c*d^2 + 2*a*b*d^3)*x^5 + 2*(9*b^2*c^2*d + 5*a*b*c*d^2 + a^2*d
^3)*x^3 + 3*(b^2*c^3 + 4*a*b*c^2*d)*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c) + 3*
(a*b^2*c^4 + 4*a^2*b*c^3*d + (b^3*c^2*d^2 + 4*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d +
9*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3)*x^4 + (b^3*c^4 + 6*a*b^2*c^3*d + 8*a^2*b*c^2*d^
2)*x^2)*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
*b^3*c^6 - 3*a^2*b^2*c^5*d + 3*a^3*b*c^4*d^2 - a^4*c^3*d^3 + (b^4*c^4*d^2 - 3*a*
b^3*c^3*d^3 + 3*a^2*b^2*c^2*d^4 - a^3*b*c*d^5)*x^6 + (2*b^4*c^5*d - 5*a*b^3*c^4*
d^2 + 3*a^2*b^2*c^3*d^3 + a^3*b*c^2*d^4 - a^4*c*d^5)*x^4 + (b^4*c^6 - a*b^3*c^5*
d - 3*a^2*b^2*c^4*d^2 + 5*a^3*b*c^3*d^3 - 2*a^4*c^2*d^4)*x^2)*sqrt(-a*b*c + a^2*
d)), -1/12*(2*((13*b^2*c*d^2 + 2*a*b*d^3)*x^5 + 2*(9*b^2*c^2*d + 5*a*b*c*d^2 + a
^2*d^3)*x^3 + 3*(b^2*c^3 + 4*a*b*c^2*d)*x)*sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c) -
3*(a*b^2*c^4 + 4*a^2*b*c^3*d + (b^3*c^2*d^2 + 4*a*b^2*c*d^3)*x^6 + (2*b^3*c^3*d
+ 9*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3)*x^4 + (b^3*c^4 + 6*a*b^2*c^3*d + 8*a^2*b*c^2
*d^2)*x^2)*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*b^3*c^6 - 3*a^2*b^2*c^5*d + 3*a^3*b*c^4*d^2 - a^4*c^3*d^3 + (b^4*c
^4*d^2 - 3*a*b^3*c^3*d^3 + 3*a^2*b^2*c^2*d^4 - a^3*b*c*d^5)*x^6 + (2*b^4*c^5*d -
5*a*b^3*c^4*d^2 + 3*a^2*b^2*c^3*d^3 + a^3*b*c^2*d^4 - a^4*c*d^5)*x^4 + (b^4*c^6
- a*b^3*c^5*d - 3*a^2*b^2*c^4*d^2 + 5*a^3*b*c^3*d^3 - 2*a^4*c^2*d^4)*x^2)*sqrt(
a*b*c - a^2*d))]

_______________________________________________________________________________________

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**2/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 4.49879, size = 803, normalized size = 4.93 \[ -\frac{{\left (\frac{{\left (5 \, b^{4} c^{4} d^{3} - 14 \, a b^{3} c^{3} d^{4} + 12 \, a^{2} b^{2} c^{2} d^{5} - 2 \, a^{3} b c d^{6} - a^{4} d^{7}\right )} x^{2}}{b^{6} c^{7} d - 6 \, a b^{5} c^{6} d^{2} + 15 \, a^{2} b^{4} c^{5} d^{3} - 20 \, a^{3} b^{3} c^{4} d^{4} + 15 \, a^{4} b^{2} c^{3} d^{5} - 6 \, a^{5} b c^{2} d^{6} + a^{6} c d^{7}} + \frac{6 \,{\left (b^{4} c^{5} d^{2} - 3 \, a b^{3} c^{4} d^{3} + 3 \, a^{2} b^{2} c^{3} d^{4} - a^{3} b c^{2} d^{5}\right )}}{b^{6} c^{7} d - 6 \, a b^{5} c^{6} d^{2} + 15 \, a^{2} b^{4} c^{5} d^{3} - 20 \, a^{3} b^{3} c^{4} d^{4} + 15 \, a^{4} b^{2} c^{3} d^{5} - 6 \, a^{5} b c^{2} d^{6} + a^{6} c d^{7}}\right )} x}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}}} - \frac{{\left (b^{2} c \sqrt{d} + 4 \, a b d^{\frac{3}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{a b c d - a^{2} d^{2}}} + \frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c \sqrt{d} - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b d^{\frac{3}{2}} - b^{2} c^{2} \sqrt{d}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)),x, algorithm="giac")

[Out]

-1/3*((5*b^4*c^4*d^3 - 14*a*b^3*c^3*d^4 + 12*a^2*b^2*c^2*d^5 - 2*a^3*b*c*d^6 - a
^4*d^7)*x^2/(b^6*c^7*d - 6*a*b^5*c^6*d^2 + 15*a^2*b^4*c^5*d^3 - 20*a^3*b^3*c^4*d
^4 + 15*a^4*b^2*c^3*d^5 - 6*a^5*b*c^2*d^6 + a^6*c*d^7) + 6*(b^4*c^5*d^2 - 3*a*b^
3*c^4*d^3 + 3*a^2*b^2*c^3*d^4 - a^3*b*c^2*d^5)/(b^6*c^7*d - 6*a*b^5*c^6*d^2 + 15
*a^2*b^4*c^5*d^3 - 20*a^3*b^3*c^4*d^4 + 15*a^4*b^2*c^3*d^5 - 6*a^5*b*c^2*d^6 + a
^6*c*d^7))*x/(d*x^2 + c)^(3/2) - 1/2*(b^2*c*sqrt(d) + 4*a*b*d^(3/2))*arctan(1/2*
((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/((b^3
*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a*b*c*d - a^2*d^2)) + ((sqr
t(d)*x - sqrt(d*x^2 + c))^2*b^2*c*sqrt(d) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*
b*d^(3/2) - b^2*c^2*sqrt(d))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3
)*((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4
*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2))