3.783 \(\int \frac{1}{x^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=304 \[ -\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 (b c-a d)^{7/2}}+\frac{(5 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3 c^{7/2}}-\frac{d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{6 a^2 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac{d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{2 a^2 c^3 \sqrt{c+d x^2} (b c-a d)^3}-\frac{b (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \]

[Out]

-(d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(6*a^2*c^2*(b*c - a*d)^2*(c + d*x^2)^(3
/2)) - (b*(2*b*c - a*d))/(2*a^2*c*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) - 1
/(2*a*c*x^2*(a + b*x^2)*(c + d*x^2)^(3/2)) - (d*(2*b*c - a*d)*(b^2*c^2 - a*b*c*d
+ 5*a^2*d^2))/(2*a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x^2]) + ((4*b*c + 5*a*d)*ArcT
anh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3*c^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)*ArcTanh
[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*a^3*(b*c - a*d)^(7/2))

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Rubi [A]  time = 1.46975, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 (b c-a d)^{7/2}}+\frac{(5 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3 c^{7/2}}-\frac{d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{6 a^2 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac{d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{2 a^2 c^3 \sqrt{c+d x^2} (b c-a d)^3}-\frac{b (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]

[Out]

-(d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(6*a^2*c^2*(b*c - a*d)^2*(c + d*x^2)^(3
/2)) - (b*(2*b*c - a*d))/(2*a^2*c*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) - 1
/(2*a*c*x^2*(a + b*x^2)*(c + d*x^2)^(3/2)) - (d*(2*b*c - a*d)*(b^2*c^2 - a*b*c*d
+ 5*a^2*d^2))/(2*a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x^2]) + ((4*b*c + 5*a*d)*ArcT
anh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3*c^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)*ArcTanh
[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*a^3*(b*c - a*d)^(7/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 6.3297, size = 489, normalized size = 1.61 \[ \frac{1}{4} \left (-\frac{b^{7/2} (4 b c-9 a d) \log \left (\frac{4 a^3 (b c-a d)^2 \left (-i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{7/2} \left (\sqrt{b} x+i \sqrt{a}\right ) (4 b c-9 a d)}\right )}{a^3 (b c-a d)^{7/2}}-\frac{b^{7/2} (4 b c-9 a d) \log \left (\frac{4 a^3 (b c-a d)^2 \left (i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{7/2} \left (\sqrt{b} x-i \sqrt{a}\right ) (4 b c-9 a d)}\right )}{a^3 (b c-a d)^{7/2}}+\frac{2 (5 a d+4 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{a^3 c^{7/2}}-\frac{2 \log (x) (5 a d+4 b c)}{a^3 c^{7/2}}+\frac{2}{3} \sqrt{c+d x^2} \left (\frac{3 b^4}{a^2 \left (a+b x^2\right ) (a d-b c)^3}-\frac{3}{a^2 c^3 x^2}+\frac{12 d^3 (a d-2 b c)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{2 d^3}{c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]

[Out]

((2*Sqrt[c + d*x^2]*(-3/(a^2*c^3*x^2) + (3*b^4)/(a^2*(-(b*c) + a*d)^3*(a + b*x^2
)) - (2*d^3)/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) + (12*d^3*(-2*b*c + a*d))/(c^3*(b
*c - a*d)^3*(c + d*x^2))))/3 - (2*(4*b*c + 5*a*d)*Log[x])/(a^3*c^(7/2)) + (2*(4*
b*c + 5*a*d)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/(a^3*c^(7/2)) - (b^(7/2)*(4*b*c -
9*a*d)*Log[(4*a^3*(b*c - a*d)^2*(Sqrt[b]*c*Sqrt[b*c - a*d] - I*Sqrt[a]*d*Sqrt[b
*c - a*d]*x + b*c*Sqrt[c + d*x^2] - a*d*Sqrt[c + d*x^2]))/(b^(7/2)*(4*b*c - 9*a*
d)*(I*Sqrt[a] + Sqrt[b]*x))])/(a^3*(b*c - a*d)^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)
*Log[(4*a^3*(b*c - a*d)^2*(Sqrt[b]*c*Sqrt[b*c - a*d] + I*Sqrt[a]*d*Sqrt[b*c - a*
d]*x + b*c*Sqrt[c + d*x^2] - a*d*Sqrt[c + d*x^2]))/(b^(7/2)*(4*b*c - 9*a*d)*((-I
)*Sqrt[a] + Sqrt[b]*x))])/(a^3*(b*c - a*d)^(7/2)))/4

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Maple [B]  time = 0.028, size = 2980, normalized size = 9.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^2+a)^2/(d*x^2+c)^(5/2),x)

[Out]

-1/3*b^2/a^3/(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)+b^3/a^3/(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/a^2/c/x^2/(d*x^2+c)^(3/
2)-5/6/a^2*d/c^2/(d*x^2+c)^(3/2)-1/3*b^2/a^3/(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)+b^3/a^3/(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)-2/3*b/a^3/c/(d*x^2+c)^(3/2)-2*b/a^3/c^2/(d*x^2+c)^(1/2)+2*b/a^3/c^(5/2)*ln(
(2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-5/2/a^2*d/c^3/(d*x^2+c)^(1/2)+5/2/a^2*d/c^(7/
2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+1/4*b^2/a^2/(-a*b)^(1/2)/(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)-5/4*b^3/a^2*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)))-1/4*b^2/a^2/(-a*b)^(1/2)/(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)-
5/4*b^3/a^2*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)))-5/1
2*b^2/a^2*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)+5/4*b^3/a^2*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)-b^3/a^3/(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)))-b^3/a^3/(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)))+5/4*b^3/a^2*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)+5/6*b^2
/a/(-a*b)^(1/2)*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*b^3/a/(-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-1/3*b^2/a^2/(-a*b)^(1/2)*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-2/3*b^2/a^2/(-a*b)^(1/2
)*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*b^2/a/(-a*b)^(1/2)*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*b^
2/a/(-a*b)^(1/2)*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*b^3/a/(-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-5/12*b^2/a^2*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*b^2/a/(-a*b)^(1/2)*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+1/3*b^2/a^2/(-a*b)^(1/2)*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+2/3*b^2/a^2/(-a*b)^(1
/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-2/3*b/a^3*d*(-a*b)^(1/2)/(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+b^2/a^3
/(a*d-b*c)^2*(-a*b)^(1/2)/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-1/3*b/a^3*d*(-a*b)^(1/2)/(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
-b^2/a^3/(a*d-b*c)^2*(-a*b)^(1/2)/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+2/3*b/a^3*d*(-a*b)^(1/2)/(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+1/3*b/a^3*d*(-a*b)^(1/2)/(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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{5}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^3),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^3), x)

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Fricas [A]  time = 14.2661, 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)^2*(d*x^2 + c)^(5/2)*x^3),x, algorithm="fricas")

[Out]

[1/24*(3*((4*b^5*c^4*d^2 - 9*a*b^4*c^3*d^3)*x^8 + (8*b^5*c^5*d - 14*a*b^4*c^4*d^
2 - 9*a^2*b^3*c^3*d^3)*x^6 + (4*b^5*c^6 - a*b^4*c^5*d - 18*a^2*b^3*c^4*d^2)*x^4
+ (4*a*b^4*c^6 - 9*a^2*b^3*c^5*d)*x^2)*sqrt(c)*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*(3*a^2*b^3*c^5 - 9*a^3*b^2*c^4*d + 9
*a^4*b*c^3*d^2 - 3*a^5*c^2*d^3 + 3*(2*a*b^4*c^3*d^2 - 3*a^2*b^3*c^2*d^3 + 11*a^3
*b^2*c*d^4 - 5*a^4*b*d^5)*x^6 + (12*a*b^4*c^4*d - 15*a^2*b^3*c^3*d^2 + 35*a^3*b^
2*c^2*d^3 + 13*a^4*b*c*d^4 - 15*a^5*d^5)*x^4 + (6*a*b^4*c^5 - 3*a^2*b^3*c^4*d -
9*a^3*b^2*c^3*d^2 + 41*a^4*b*c^2*d^3 - 20*a^5*c*d^4)*x^2)*sqrt(d*x^2 + c)*sqrt(c
) + 6*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 -
5*a^4*b*d^6)*x^8 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^
3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2
*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^4 + (4*a*b
^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)
*x^2)*log(-((d*x^2 + 2*c)*sqrt(c) + 2*sqrt(d*x^2 + c)*c)/x^2))/(((a^3*b^4*c^6*d^
2 - 3*a^4*b^3*c^5*d^3 + 3*a^5*b^2*c^4*d^4 - a^6*b*c^3*d^5)*x^8 + (2*a^3*b^4*c^7*
d - 5*a^4*b^3*c^6*d^2 + 3*a^5*b^2*c^5*d^3 + a^6*b*c^4*d^4 - a^7*c^3*d^5)*x^6 + (
a^3*b^4*c^8 - a^4*b^3*c^7*d - 3*a^5*b^2*c^6*d^2 + 5*a^6*b*c^5*d^3 - 2*a^7*c^4*d^
4)*x^4 + (a^4*b^3*c^8 - 3*a^5*b^2*c^7*d + 3*a^6*b*c^6*d^2 - a^7*c^5*d^3)*x^2)*sq
rt(c)), 1/24*(3*((4*b^5*c^4*d^2 - 9*a*b^4*c^3*d^3)*x^8 + (8*b^5*c^5*d - 14*a*b^4
*c^4*d^2 - 9*a^2*b^3*c^3*d^3)*x^6 + (4*b^5*c^6 - a*b^4*c^5*d - 18*a^2*b^3*c^4*d^
2)*x^4 + (4*a*b^4*c^6 - 9*a^2*b^3*c^5*d)*x^2)*sqrt(-c)*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)*sq
rt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(3*a^2*b^3*c^5 - 9*a^3*b^2*c
^4*d + 9*a^4*b*c^3*d^2 - 3*a^5*c^2*d^3 + 3*(2*a*b^4*c^3*d^2 - 3*a^2*b^3*c^2*d^3
+ 11*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^6 + (12*a*b^4*c^4*d - 15*a^2*b^3*c^3*d^2 + 3
5*a^3*b^2*c^2*d^3 + 13*a^4*b*c*d^4 - 15*a^5*d^5)*x^4 + (6*a*b^4*c^5 - 3*a^2*b^3*
c^4*d - 9*a^3*b^2*c^3*d^2 + 41*a^4*b*c^2*d^3 - 20*a^5*c*d^4)*x^2)*sqrt(d*x^2 + c
)*sqrt(-c) + 12*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b
^2*c*d^5 - 5*a^4*b*d^6)*x^8 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d
^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6 + a*b^4*c^5*
d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^
4 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^
5*c^2*d^4)*x^2)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/(((a^3*b^4*c^6*d^2 - 3*a^4*b^3
*c^5*d^3 + 3*a^5*b^2*c^4*d^4 - a^6*b*c^3*d^5)*x^8 + (2*a^3*b^4*c^7*d - 5*a^4*b^3
*c^6*d^2 + 3*a^5*b^2*c^5*d^3 + a^6*b*c^4*d^4 - a^7*c^3*d^5)*x^6 + (a^3*b^4*c^8 -
a^4*b^3*c^7*d - 3*a^5*b^2*c^6*d^2 + 5*a^6*b*c^5*d^3 - 2*a^7*c^4*d^4)*x^4 + (a^4
*b^3*c^8 - 3*a^5*b^2*c^7*d + 3*a^6*b*c^6*d^2 - a^7*c^5*d^3)*x^2)*sqrt(-c)), 1/12
*(3*((4*b^5*c^4*d^2 - 9*a*b^4*c^3*d^3)*x^8 + (8*b^5*c^5*d - 14*a*b^4*c^4*d^2 - 9
*a^2*b^3*c^3*d^3)*x^6 + (4*b^5*c^6 - a*b^4*c^5*d - 18*a^2*b^3*c^4*d^2)*x^4 + (4*
a*b^4*c^6 - 9*a^2*b^3*c^5*d)*x^2)*sqrt(c)*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*(3*a^
2*b^3*c^5 - 9*a^3*b^2*c^4*d + 9*a^4*b*c^3*d^2 - 3*a^5*c^2*d^3 + 3*(2*a*b^4*c^3*d
^2 - 3*a^2*b^3*c^2*d^3 + 11*a^3*b^2*c*d^4 - 5*a^4*b*d^5)*x^6 + (12*a*b^4*c^4*d -
15*a^2*b^3*c^3*d^2 + 35*a^3*b^2*c^2*d^3 + 13*a^4*b*c*d^4 - 15*a^5*d^5)*x^4 + (6
*a*b^4*c^5 - 3*a^2*b^3*c^4*d - 9*a^3*b^2*c^3*d^2 + 41*a^4*b*c^2*d^3 - 20*a^5*c*d
^4)*x^2)*sqrt(d*x^2 + c)*sqrt(c) + 3*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b
^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^8 + (8*b^5*c^5*d - 10*a*b^4*c^4*d
^2 - 13*a^2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4
*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d
^4 - 10*a^5*c*d^5)*x^4 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11
*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x^2)*log(-((d*x^2 + 2*c)*sqrt(c) + 2*sqrt(d*x^2
+ c)*c)/x^2))/(((a^3*b^4*c^6*d^2 - 3*a^4*b^3*c^5*d^3 + 3*a^5*b^2*c^4*d^4 - a^6*b
*c^3*d^5)*x^8 + (2*a^3*b^4*c^7*d - 5*a^4*b^3*c^6*d^2 + 3*a^5*b^2*c^5*d^3 + a^6*b
*c^4*d^4 - a^7*c^3*d^5)*x^6 + (a^3*b^4*c^8 - a^4*b^3*c^7*d - 3*a^5*b^2*c^6*d^2 +
5*a^6*b*c^5*d^3 - 2*a^7*c^4*d^4)*x^4 + (a^4*b^3*c^8 - 3*a^5*b^2*c^7*d + 3*a^6*b
*c^6*d^2 - a^7*c^5*d^3)*x^2)*sqrt(c)), 1/12*(3*((4*b^5*c^4*d^2 - 9*a*b^4*c^3*d^3
)*x^8 + (8*b^5*c^5*d - 14*a*b^4*c^4*d^2 - 9*a^2*b^3*c^3*d^3)*x^6 + (4*b^5*c^6 -
a*b^4*c^5*d - 18*a^2*b^3*c^4*d^2)*x^4 + (4*a*b^4*c^6 - 9*a^2*b^3*c^5*d)*x^2)*sqr
t(-c)*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*(3*a^2*b^3*c^5 - 9*a^3*b^2*c^4*d + 9*a^4*
b*c^3*d^2 - 3*a^5*c^2*d^3 + 3*(2*a*b^4*c^3*d^2 - 3*a^2*b^3*c^2*d^3 + 11*a^3*b^2*
c*d^4 - 5*a^4*b*d^5)*x^6 + (12*a*b^4*c^4*d - 15*a^2*b^3*c^3*d^2 + 35*a^3*b^2*c^2
*d^3 + 13*a^4*b*c*d^4 - 15*a^5*d^5)*x^4 + (6*a*b^4*c^5 - 3*a^2*b^3*c^4*d - 9*a^3
*b^2*c^3*d^2 + 41*a^4*b*c^2*d^3 - 20*a^5*c*d^4)*x^2)*sqrt(d*x^2 + c)*sqrt(-c) +
6*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a
^4*b*d^6)*x^8 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^3*b^
2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^3
*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^4 + (4*a*b^4*c
^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x^2
)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/(((a^3*b^4*c^6*d^2 - 3*a^4*b^3*c^5*d^3 + 3*a
^5*b^2*c^4*d^4 - a^6*b*c^3*d^5)*x^8 + (2*a^3*b^4*c^7*d - 5*a^4*b^3*c^6*d^2 + 3*a
^5*b^2*c^5*d^3 + a^6*b*c^4*d^4 - a^7*c^3*d^5)*x^6 + (a^3*b^4*c^8 - a^4*b^3*c^7*d
- 3*a^5*b^2*c^6*d^2 + 5*a^6*b*c^5*d^3 - 2*a^7*c^4*d^4)*x^4 + (a^4*b^3*c^8 - 3*a
^5*b^2*c^7*d + 3*a^6*b*c^6*d^2 - a^7*c^5*d^3)*x^2)*sqrt(-c))]

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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(1/x**3/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.258918, size = 684, normalized size = 2.25 \[ \frac{1}{6} \, d^{3}{\left (\frac{3 \,{\left (4 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{3} c^{3} d^{3} - 3 \, a^{4} b^{2} c^{2} d^{4} + 3 \, a^{5} b c d^{5} - a^{6} d^{6}\right )} \sqrt{-b^{2} c + a b d}} - \frac{3 \,{\left (2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{4} c^{3} - 2 \, \sqrt{d x^{2} + c} b^{4} c^{4} - 3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b^{3} c^{2} d + 4 \, \sqrt{d x^{2} + c} a b^{3} c^{3} d + 3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} b^{2} c d^{2} - 6 \, \sqrt{d x^{2} + c} a^{2} b^{2} c^{2} d^{2} -{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{3} b d^{3} + 4 \, \sqrt{d x^{2} + c} a^{3} b c d^{3} - \sqrt{d x^{2} + c} a^{4} d^{4}\right )}}{{\left (a^{2} b^{3} c^{6} d^{2} - 3 \, a^{3} b^{2} c^{5} d^{3} + 3 \, a^{4} b c^{4} d^{4} - a^{5} c^{3} d^{5}\right )}{\left ({\left (d x^{2} + c\right )}^{2} b - 2 \,{\left (d x^{2} + c\right )} b c + b c^{2} +{\left (d x^{2} + c\right )} a d - a c d\right )}} - \frac{2 \,{\left (12 \,{\left (d x^{2} + c\right )} b c + b c^{2} - 6 \,{\left (d x^{2} + c\right )} a d - a c d\right )}}{{\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}}} - \frac{3 \,{\left (4 \, b c + 5 \, a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c^{3} d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^3),x, algorithm="giac")

[Out]

1/6*d^3*(3*(4*b^5*c - 9*a*b^4*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/
((a^3*b^3*c^3*d^3 - 3*a^4*b^2*c^2*d^4 + 3*a^5*b*c*d^5 - a^6*d^6)*sqrt(-b^2*c + a
*b*d)) - 3*(2*(d*x^2 + c)^(3/2)*b^4*c^3 - 2*sqrt(d*x^2 + c)*b^4*c^4 - 3*(d*x^2 +
c)^(3/2)*a*b^3*c^2*d + 4*sqrt(d*x^2 + c)*a*b^3*c^3*d + 3*(d*x^2 + c)^(3/2)*a^2*
b^2*c*d^2 - 6*sqrt(d*x^2 + c)*a^2*b^2*c^2*d^2 - (d*x^2 + c)^(3/2)*a^3*b*d^3 + 4*
sqrt(d*x^2 + c)*a^3*b*c*d^3 - sqrt(d*x^2 + c)*a^4*d^4)/((a^2*b^3*c^6*d^2 - 3*a^3
*b^2*c^5*d^3 + 3*a^4*b*c^4*d^4 - a^5*c^3*d^5)*((d*x^2 + c)^2*b - 2*(d*x^2 + c)*b
*c + b*c^2 + (d*x^2 + c)*a*d - a*c*d)) - 2*(12*(d*x^2 + c)*b*c + b*c^2 - 6*(d*x^
2 + c)*a*d - a*c*d)/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*(
d*x^2 + c)^(3/2)) - 3*(4*b*c + 5*a*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^3*sqrt
(-c)*c^3*d^3))