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

Optimal. Leaf size=241 \[ -\frac{b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 (b c-a d)^{5/2}}+\frac{(3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3 c^{5/2}}-\frac{d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{2 a^2 c^2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{b (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \sqrt{c+d x^2}} \]

[Out]

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

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Rubi [A]  time = 1.02358, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 (b c-a d)^{5/2}}+\frac{(3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3 c^{5/2}}-\frac{d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{2 a^2 c^2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{b (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}-\frac{1}{2 a c x^2 \left (a+b x^2\right ) \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 123.639, size = 216, normalized size = 0.9 \[ - \frac{1}{2 a c x^{2} \left (a + b x^{2}\right ) \sqrt{c + d x^{2}}} - \frac{b \left (a d - 2 b c\right )}{2 a^{2} c \left (a + b x^{2}\right ) \sqrt{c + d x^{2}} \left (a d - b c\right )} - \frac{d \left (3 a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right )}{2 a^{2} c^{2} \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} - \frac{b^{\frac{5}{2}} \left (7 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 a^{3} \left (a d - b c\right )^{\frac{5}{2}}} + \frac{\left (3 a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 a^{3} c^{\frac{5}{2}}} \]

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)**(3/2),x)

[Out]

-1/(2*a*c*x**2*(a + b*x**2)*sqrt(c + d*x**2)) - b*(a*d - 2*b*c)/(2*a**2*c*(a + b
*x**2)*sqrt(c + d*x**2)*(a*d - b*c)) - d*(3*a**2*d**2 - 2*a*b*c*d + 2*b**2*c**2)
/(2*a**2*c**2*sqrt(c + d*x**2)*(a*d - b*c)**2) - b**(5/2)*(7*a*d - 4*b*c)*atan(s
qrt(b)*sqrt(c + d*x**2)/sqrt(a*d - b*c))/(2*a**3*(a*d - b*c)**(5/2)) + (3*a*d +
4*b*c)*atanh(sqrt(c + d*x**2)/sqrt(c))/(2*a**3*c**(5/2))

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Mathematica [C]  time = 3.81412, size = 451, normalized size = 1.87 \[ \frac{1}{4} \left (-\frac{b^{5/2} (4 b c-7 a d) \log \left (\frac{4 a^3 (b c-a d) \left (-i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{5/2} \left (\sqrt{b} x+i \sqrt{a}\right ) (4 b c-7 a d)}\right )}{a^3 (b c-a d)^{5/2}}-\frac{b^{5/2} (4 b c-7 a d) \log \left (\frac{4 a^3 (b c-a d) \left (i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{5/2} \left (\sqrt{b} x-i \sqrt{a}\right ) (4 b c-7 a d)}\right )}{a^3 (b c-a d)^{5/2}}+\frac{2 (3 a d+4 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{a^3 c^{5/2}}-\frac{2 \log (x) (3 a d+4 b c)}{a^3 c^{5/2}}+4 \sqrt{c+d x^2} \left (\frac{-\frac{b^3}{2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{1}{2 c^2 x^2}}{a^2}-\frac{d^3}{c^2 \left (c+d x^2\right ) (b c-a d)^2}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[c + d*x^2]*(-(d^3/(c^2*(b*c - a*d)^2*(c + d*x^2))) + (-1/(2*c^2*x^2) - b
^3/(2*(b*c - a*d)^2*(a + b*x^2)))/a^2) - (2*(4*b*c + 3*a*d)*Log[x])/(a^3*c^(5/2)
) + (2*(4*b*c + 3*a*d)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/(a^3*c^(5/2)) - (b^(5/2
)*(4*b*c - 7*a*d)*Log[(4*a^3*(b*c - a*d)*(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^(5/2)*(4*b*
c - 7*a*d)*(I*Sqrt[a] + Sqrt[b]*x))])/(a^3*(b*c - a*d)^(5/2)) - (b^(5/2)*(4*b*c
- 7*a*d)*Log[(4*a^3*(b*c - a*d)*(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^(5/2)*(4*b*c - 7*a*d
)*((-I)*Sqrt[a] + Sqrt[b]*x))])/(a^3*(b*c - a*d)^(5/2)))/4

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Maple [B]  time = 0.025, size = 1778, normalized size = 7.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/a^2/c/x^2/(d*x^2+c)^(1/2)-3/2/a^2*d/c^2/(d*x^2+c)^(1/2)+3/2/a^2*d/c^(5/2)*l
n((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-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)^(1/2)+b/a^3*(-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)^(1/2)*x*d+b^2/a^3/(a*d-b*c)/(-(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^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)^(1/2)-b/a^3*(-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)^(1/2)*x*d+b^2/a^3/
(a*d-b*c)/(-(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)))-2*b/a^3/c/(d*x^2+c)
^(1/2)+2*b/a^3/c^(3/2)*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)^(1/2)-3/4*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)^(1/2)-3/4*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)^(1/2)*x+3/4*b^2/a^2*d/(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)))+1/2*b^2/a^2/(-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)^(1/2)*
x*d-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)^(1/2)-3/4*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)^(1/2)+3/4*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)^(1/2)*x+3/4*b^2/a^2*d/(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)))-1/2*b^2/a^2/(-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)^(1/2)*x*d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 5.38081, 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)^(3/2)*x^3),x, algorithm="fricas")

[Out]

[-1/8*(((4*b^4*c^3*d - 7*a*b^3*c^2*d^2)*x^6 + (4*b^4*c^4 - 3*a*b^3*c^3*d - 7*a^2
*b^2*c^2*d^2)*x^4 + (4*a*b^3*c^4 - 7*a^2*b^2*c^3*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*(a^2*b^2*c^3 - 2*a
^3*b*c^2*d + a^4*c*d^2 + (2*a*b^3*c^2*d - 2*a^2*b^2*c*d^2 + 3*a^3*b*d^3)*x^4 + (
2*a*b^3*c^3 - a^2*b^2*c^2*d - a^3*b*c*d^2 + 3*a^4*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt
(c) - 2*((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^6 + (
4*b^4*c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^4 + (4*
a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x^2)*log(-((d*x^2 +
2*c)*sqrt(c) + 2*sqrt(d*x^2 + c)*c)/x^2))/(((a^3*b^3*c^4*d - 2*a^4*b^2*c^3*d^2
+ a^5*b*c^2*d^3)*x^6 + (a^3*b^3*c^5 - a^4*b^2*c^4*d - a^5*b*c^3*d^2 + a^6*c^2*d^
3)*x^4 + (a^4*b^2*c^5 - 2*a^5*b*c^4*d + a^6*c^3*d^2)*x^2)*sqrt(c)), -1/8*(((4*b^
4*c^3*d - 7*a*b^3*c^2*d^2)*x^6 + (4*b^4*c^4 - 3*a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2)
*x^4 + (4*a*b^3*c^4 - 7*a^2*b^2*c^3*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*(a^2*b^2*c^3 - 2*a^3*b*c^2*d +
a^4*c*d^2 + (2*a*b^3*c^2*d - 2*a^2*b^2*c*d^2 + 3*a^3*b*d^3)*x^4 + (2*a*b^3*c^3
- a^2*b^2*c^2*d - a^3*b*c*d^2 + 3*a^4*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt(-c) - 4*((4
*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^6 + (4*b^4*c^4 -
a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^4 + (4*a*b^3*c^4 -
5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x^2)*arctan(sqrt(-c)/sqrt(d*x^
2 + c)))/(((a^3*b^3*c^4*d - 2*a^4*b^2*c^3*d^2 + a^5*b*c^2*d^3)*x^6 + (a^3*b^3*c^
5 - a^4*b^2*c^4*d - a^5*b*c^3*d^2 + a^6*c^2*d^3)*x^4 + (a^4*b^2*c^5 - 2*a^5*b*c^
4*d + a^6*c^3*d^2)*x^2)*sqrt(-c)), 1/4*(((4*b^4*c^3*d - 7*a*b^3*c^2*d^2)*x^6 + (
4*b^4*c^4 - 3*a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2)*x^4 + (4*a*b^3*c^4 - 7*a^2*b^2*c^
3*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*(a^2*b^2*c^3 - 2*a^3*b*c^2*d
+ a^4*c*d^2 + (2*a*b^3*c^2*d - 2*a^2*b^2*c*d^2 + 3*a^3*b*d^3)*x^4 + (2*a*b^3*c^3
- a^2*b^2*c^2*d - a^3*b*c*d^2 + 3*a^4*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt(c) + ((4*b
^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^6 + (4*b^4*c^4 - a
*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^4 + (4*a*b^3*c^4 - 5
*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x^2)*log(-((d*x^2 + 2*c)*sqrt(c)
+ 2*sqrt(d*x^2 + c)*c)/x^2))/(((a^3*b^3*c^4*d - 2*a^4*b^2*c^3*d^2 + a^5*b*c^2*d
^3)*x^6 + (a^3*b^3*c^5 - a^4*b^2*c^4*d - a^5*b*c^3*d^2 + a^6*c^2*d^3)*x^4 + (a^4
*b^2*c^5 - 2*a^5*b*c^4*d + a^6*c^3*d^2)*x^2)*sqrt(c)), 1/4*(((4*b^4*c^3*d - 7*a*
b^3*c^2*d^2)*x^6 + (4*b^4*c^4 - 3*a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2)*x^4 + (4*a*b^
3*c^4 - 7*a^2*b^2*c^3*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*(a^2*b^2
*c^3 - 2*a^3*b*c^2*d + a^4*c*d^2 + (2*a*b^3*c^2*d - 2*a^2*b^2*c*d^2 + 3*a^3*b*d^
3)*x^4 + (2*a*b^3*c^3 - a^2*b^2*c^2*d - a^3*b*c*d^2 + 3*a^4*d^3)*x^2)*sqrt(d*x^2
+ c)*sqrt(-c) + 2*((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d
^4)*x^6 + (4*b^4*c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4
)*x^4 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x^2)*arc
tan(sqrt(-c)/sqrt(d*x^2 + c)))/(((a^3*b^3*c^4*d - 2*a^4*b^2*c^3*d^2 + a^5*b*c^2*
d^3)*x^6 + (a^3*b^3*c^5 - a^4*b^2*c^4*d - a^5*b*c^3*d^2 + a^6*c^2*d^3)*x^4 + (a^
4*b^2*c^5 - 2*a^5*b*c^4*d + a^6*c^3*d^2)*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.238508, size = 510, normalized size = 2.12 \[ \frac{1}{2} \, d^{3}{\left (\frac{{\left (4 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b^{2} c^{2} d^{3} - 2 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x^{2} + c\right )}^{2} b^{3} c^{2} - 2 \,{\left (d x^{2} + c\right )} b^{3} c^{3} - 2 \,{\left (d x^{2} + c\right )}^{2} a b^{2} c d + 3 \,{\left (d x^{2} + c\right )} a b^{2} c^{2} d + 3 \,{\left (d x^{2} + c\right )}^{2} a^{2} b d^{2} - 7 \,{\left (d x^{2} + c\right )} a^{2} b c d^{2} + 2 \, a^{2} b c^{2} d^{2} + 3 \,{\left (d x^{2} + c\right )} a^{3} d^{3} - 2 \, a^{3} c d^{3}}{{\left (a^{2} b^{2} c^{4} d^{2} - 2 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\right )}{\left ({\left (d x^{2} + c\right )}^{\frac{5}{2}} b - 2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b c + \sqrt{d x^{2} + c} b c^{2} +{\left (d x^{2} + c\right )}^{\frac{3}{2}} a d - \sqrt{d x^{2} + c} a c d\right )}} - \frac{{\left (4 \, b c + 3 \, a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c^{2} d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d^3*((4*b^4*c - 7*a*b^3*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/((
a^3*b^2*c^2*d^3 - 2*a^4*b*c*d^4 + a^5*d^5)*sqrt(-b^2*c + a*b*d)) - (2*(d*x^2 + c
)^2*b^3*c^2 - 2*(d*x^2 + c)*b^3*c^3 - 2*(d*x^2 + c)^2*a*b^2*c*d + 3*(d*x^2 + c)*
a*b^2*c^2*d + 3*(d*x^2 + c)^2*a^2*b*d^2 - 7*(d*x^2 + c)*a^2*b*c*d^2 + 2*a^2*b*c^
2*d^2 + 3*(d*x^2 + c)*a^3*d^3 - 2*a^3*c*d^3)/((a^2*b^2*c^4*d^2 - 2*a^3*b*c^3*d^3
+ a^4*c^2*d^4)*((d*x^2 + c)^(5/2)*b - 2*(d*x^2 + c)^(3/2)*b*c + sqrt(d*x^2 + c)
*b*c^2 + (d*x^2 + c)^(3/2)*a*d - sqrt(d*x^2 + c)*a*c*d)) - (4*b*c + 3*a*d)*arcta
n(sqrt(d*x^2 + c)/sqrt(-c))/(a^3*sqrt(-c)*c^2*d^3))