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

Optimal. Leaf size=140 \[ \frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 (b c-a d)^{7/2}}-\frac{5 b d}{2 \sqrt{c+d x^2} (b c-a d)^3}-\frac{1}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{5 d}{6 \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

(-5*d)/(6*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - 1/(2*(b*c - a*d)*(a + b*x^2)*(c + d
*x^2)^(3/2)) - (5*b*d)/(2*(b*c - a*d)^3*Sqrt[c + d*x^2]) + (5*b^(3/2)*d*ArcTanh[
(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*(b*c - a*d)^(7/2))

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Rubi [A]  time = 0.273968, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 (b c-a d)^{7/2}}-\frac{5 b d}{2 \sqrt{c+d x^2} (b c-a d)^3}-\frac{1}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{5 d}{6 \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-5*d)/(6*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - 1/(2*(b*c - a*d)*(a + b*x^2)*(c + d
*x^2)^(3/2)) - (5*b*d)/(2*(b*c - a*d)^3*Sqrt[c + d*x^2]) + (5*b^(3/2)*d*ArcTanh[
(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*(b*c - a*d)^(7/2))

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Rubi in Sympy [A]  time = 38.6956, size = 122, normalized size = 0.87 \[ \frac{5 b^{\frac{3}{2}} d \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 \left (a d - b c\right )^{\frac{7}{2}}} + \frac{5 b d}{2 \sqrt{c + d x^{2}} \left (a d - b c\right )^{3}} - \frac{5 d}{6 \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{1}{2 \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*b**(3/2)*d*atan(sqrt(b)*sqrt(c + d*x**2)/sqrt(a*d - b*c))/(2*(a*d - b*c)**(7/2
)) + 5*b*d/(2*sqrt(c + d*x**2)*(a*d - b*c)**3) - 5*d/(6*(c + d*x**2)**(3/2)*(a*d
 - b*c)**2) + 1/(2*(a + b*x**2)*(c + d*x**2)**(3/2)*(a*d - b*c))

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Mathematica [A]  time = 0.353241, size = 137, normalized size = 0.98 \[ \frac{2 a^2 d^2-2 a b d \left (7 c+5 d x^2\right )+b^2 \left (-\left (3 c^2+20 c d x^2+15 d^2 x^4\right )\right )}{6 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)^3}+\frac{5 b^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 (b c-a d)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*d^2 - 2*a*b*d*(7*c + 5*d*x^2) - b^2*(3*c^2 + 20*c*d*x^2 + 15*d^2*x^4))/(6
*(b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^(3/2)) + (5*b^(3/2)*d*ArcTanh[(Sqrt[b]*Sq
rt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*(b*c - a*d)^(7/2))

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Maple [B]  time = 0.02, size = 1639, normalized size = 11.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.335559, 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/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)),x, algorithm="fricas")

[Out]

[-1/24*(15*(b^2*d^3*x^6 + a*b*c^2*d + (2*b^2*c*d^2 + a*b*d^3)*x^4 + (b^2*c^2*d +
 2*a*b*c*d^2)*x^2)*sqrt(b/(b*c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d
+ a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 +
 (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*
x^2 + a^2)) + 4*(15*b^2*d^2*x^4 + 3*b^2*c^2 + 14*a*b*c*d - 2*a^2*d^2 + 10*(2*b^2
*c*d + a*b*d^2)*x^2)*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*(15*(b^2*d^3*x^6 + a*b*c^2*d + (2*b^2*c*d^2 + a*b*d^3)*x^4 +
(b^2*c^2*d + 2*a*b*c*d^2)*x^2)*sqrt(-b/(b*c - a*d))*arctan(-1/2*(b*d*x^2 + 2*b*c
 - a*d)/(sqrt(d*x^2 + c)*(b*c - a*d)*sqrt(-b/(b*c - a*d)))) + 2*(15*b^2*d^2*x^4
+ 3*b^2*c^2 + 14*a*b*c*d - 2*a^2*d^2 + 10*(2*b^2*c*d + a*b*d^2)*x^2)*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/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228184, size = 301, normalized size = 2.15 \[ -\frac{1}{6} \,{\left (\frac{15 \, b^{2} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} + \frac{3 \, \sqrt{d x^{2} + c} b^{2}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} + \frac{2 \,{\left (6 \,{\left (d x^{2} + c\right )} b + b c - a d\right )}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\right )} d \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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