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

Optimal. Leaf size=225 \[ \frac{b^{5/2} (2 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 (b c-a d)^{7/2}}+\frac{d \left (-2 a^2 d^2+6 a b c d+b^2 c^2\right )}{2 a c^2 \sqrt{c+d x^2} (b c-a d)^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2 c^{5/2}}+\frac{b}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 1.02758, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{b^{5/2} (2 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 (b c-a d)^{7/2}}+\frac{d \left (-2 a^2 d^2+6 a b c d+b^2 c^2\right )}{2 a c^2 \sqrt{c+d x^2} (b c-a d)^3}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2 c^{5/2}}+\frac{b}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c \left (c+d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 124.364, size = 197, normalized size = 0.88 \[ - \frac{b}{2 a \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{d \left (2 a d + 3 b c\right )}{6 a c \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{d \left (2 a^{2} d^{2} - 6 a b c d - b^{2} c^{2}\right )}{2 a c^{2} \sqrt{c + d x^{2}} \left (a d - b c\right )^{3}} - \frac{b^{\frac{5}{2}} \left (7 a d - 2 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 a^{2} \left (a d - b c\right )^{\frac{7}{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a^{2} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 2.41167, size = 461, normalized size = 2.05 \[ \frac{b^{5/2} (2 b c-7 a d) \log \left (-\frac{4 a^2 (b c-a d)^2 \left (i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{5/2} \left (\sqrt{b} x-i \sqrt{a}\right ) (2 b c-7 a d)}\right )}{4 a^2 (b c-a d)^{7/2}}+\frac{b^{5/2} (2 b c-7 a d) \log \left (\frac{4 a^2 (b c-a d)^2 \left (i \sqrt{a} d x \sqrt{b c-a d}-\sqrt{b} c \sqrt{b c-a d}+a d \sqrt{c+d x^2}-b c \sqrt{c+d x^2}\right )}{b^{5/2} \left (\sqrt{b} x+i \sqrt{a}\right ) (2 b c-7 a d)}\right )}{4 a^2 (b c-a d)^{7/2}}-\frac{\log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{a^2 c^{5/2}}+\frac{\log (x)}{a^2 c^{5/2}}+\sqrt{c+d x^2} \left (-\frac{b^3}{2 a \left (a+b x^2\right ) (a d-b c)^3}+\frac{d^2 (3 b c-a d)}{c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{d^2}{3 c \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 2837, normalized size = 12.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 7.74842, 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),x, algorithm="fricas")

[Out]

[1/24*(3*(2*a*b^3*c^5 - 7*a^2*b^2*c^4*d + (2*b^4*c^3*d^2 - 7*a*b^3*c^2*d^3)*x^6
+ (4*b^4*c^4*d - 12*a*b^3*c^3*d^2 - 7*a^2*b^2*c^2*d^3)*x^4 + (2*b^4*c^5 - 3*a*b^
3*c^4*d - 14*a^2*b^2*c^3*d^2)*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*b^3*c^4 + 20*a^3*b*c^2*d^2 - 8*a^4*
c*d^3 + 3*(a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 - 2*a^3*b*d^4)*x^4 + 2*(3*a*b^3*c^3*d
+ 10*a^2*b^2*c^2*d^2 + 5*a^3*b*c*d^3 - 3*a^4*d^4)*x^2)*sqrt(d*x^2 + c)*sqrt(c)
+ 12*(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)*log(-((d*x^2 + 2*c)*s
qrt(c) - 2*sqrt(d*x^2 + c)*c)/x^2))/((a^3*b^3*c^7 - 3*a^4*b^2*c^6*d + 3*a^5*b*c^
5*d^2 - a^6*c^4*d^3 + (a^2*b^4*c^5*d^2 - 3*a^3*b^3*c^4*d^3 + 3*a^4*b^2*c^3*d^4 -
a^5*b*c^2*d^5)*x^6 + (2*a^2*b^4*c^6*d - 5*a^3*b^3*c^5*d^2 + 3*a^4*b^2*c^4*d^3 +
a^5*b*c^3*d^4 - a^6*c^2*d^5)*x^4 + (a^2*b^4*c^7 - a^3*b^3*c^6*d - 3*a^4*b^2*c^5
*d^2 + 5*a^5*b*c^4*d^3 - 2*a^6*c^3*d^4)*x^2)*sqrt(c)), 1/24*(3*(2*a*b^3*c^5 - 7*
a^2*b^2*c^4*d + (2*b^4*c^3*d^2 - 7*a*b^3*c^2*d^3)*x^6 + (4*b^4*c^4*d - 12*a*b^3*
c^3*d^2 - 7*a^2*b^2*c^2*d^3)*x^4 + (2*b^4*c^5 - 3*a*b^3*c^4*d - 14*a^2*b^2*c^3*d
^2)*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*b^3*c^4 + 20*a^3*b*c^2*d^2 - 8*a^4*c*d^3 + 3*(a*b^3*c^2*d^2
+ 6*a^2*b^2*c*d^3 - 2*a^3*b*d^4)*x^4 + 2*(3*a*b^3*c^3*d + 10*a^2*b^2*c^2*d^2 + 5
*a^3*b*c*d^3 - 3*a^4*d^4)*x^2)*sqrt(d*x^2 + c)*sqrt(-c) - 24*(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)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/((a^3*b^3*c
^7 - 3*a^4*b^2*c^6*d + 3*a^5*b*c^5*d^2 - a^6*c^4*d^3 + (a^2*b^4*c^5*d^2 - 3*a^3*
b^3*c^4*d^3 + 3*a^4*b^2*c^3*d^4 - a^5*b*c^2*d^5)*x^6 + (2*a^2*b^4*c^6*d - 5*a^3*
b^3*c^5*d^2 + 3*a^4*b^2*c^4*d^3 + a^5*b*c^3*d^4 - a^6*c^2*d^5)*x^4 + (a^2*b^4*c^
7 - a^3*b^3*c^6*d - 3*a^4*b^2*c^5*d^2 + 5*a^5*b*c^4*d^3 - 2*a^6*c^3*d^4)*x^2)*sq
rt(-c)), -1/12*(3*(2*a*b^3*c^5 - 7*a^2*b^2*c^4*d + (2*b^4*c^3*d^2 - 7*a*b^3*c^2*
d^3)*x^6 + (4*b^4*c^4*d - 12*a*b^3*c^3*d^2 - 7*a^2*b^2*c^2*d^3)*x^4 + (2*b^4*c^5
- 3*a*b^3*c^4*d - 14*a^2*b^2*c^3*d^2)*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*b^3*c^4 + 20*a^3*b*c^2*d^2 - 8*a^4*c*d^3 + 3*(a*b^3*c^2*d^2 + 6*a^2*b^
2*c*d^3 - 2*a^3*b*d^4)*x^4 + 2*(3*a*b^3*c^3*d + 10*a^2*b^2*c^2*d^2 + 5*a^3*b*c*d
^3 - 3*a^4*d^4)*x^2)*sqrt(d*x^2 + c)*sqrt(c) - 6*(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)*log(-((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2 + c)*c)/x^2))/
((a^3*b^3*c^7 - 3*a^4*b^2*c^6*d + 3*a^5*b*c^5*d^2 - a^6*c^4*d^3 + (a^2*b^4*c^5*d
^2 - 3*a^3*b^3*c^4*d^3 + 3*a^4*b^2*c^3*d^4 - a^5*b*c^2*d^5)*x^6 + (2*a^2*b^4*c^6
*d - 5*a^3*b^3*c^5*d^2 + 3*a^4*b^2*c^4*d^3 + a^5*b*c^3*d^4 - a^6*c^2*d^5)*x^4 +
(a^2*b^4*c^7 - a^3*b^3*c^6*d - 3*a^4*b^2*c^5*d^2 + 5*a^5*b*c^4*d^3 - 2*a^6*c^3*d
^4)*x^2)*sqrt(c)), -1/12*(3*(2*a*b^3*c^5 - 7*a^2*b^2*c^4*d + (2*b^4*c^3*d^2 - 7*
a*b^3*c^2*d^3)*x^6 + (4*b^4*c^4*d - 12*a*b^3*c^3*d^2 - 7*a^2*b^2*c^2*d^3)*x^4 +
(2*b^4*c^5 - 3*a*b^3*c^4*d - 14*a^2*b^2*c^3*d^2)*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*b^3*c^4 + 20*a^3*b*c^2*d^2 - 8*a^4*c*d^3 + 3*(a*b^3*c^2*d^2
+ 6*a^2*b^2*c*d^3 - 2*a^3*b*d^4)*x^4 + 2*(3*a*b^3*c^3*d + 10*a^2*b^2*c^2*d^2 +
5*a^3*b*c*d^3 - 3*a^4*d^4)*x^2)*sqrt(d*x^2 + c)*sqrt(-c) + 12*(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)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/((a^3*b^3*
c^7 - 3*a^4*b^2*c^6*d + 3*a^5*b*c^5*d^2 - a^6*c^4*d^3 + (a^2*b^4*c^5*d^2 - 3*a^3
*b^3*c^4*d^3 + 3*a^4*b^2*c^3*d^4 - a^5*b*c^2*d^5)*x^6 + (2*a^2*b^4*c^6*d - 5*a^3
*b^3*c^5*d^2 + 3*a^4*b^2*c^4*d^3 + a^5*b*c^3*d^4 - a^6*c^2*d^5)*x^4 + (a^2*b^4*c
^7 - a^3*b^3*c^6*d - 3*a^4*b^2*c^5*d^2 + 5*a^5*b*c^4*d^3 - 2*a^6*c^3*d^4)*x^2)*s
qrt(-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/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.239995, size = 410, normalized size = 1.82 \[ \frac{1}{6} \,{\left (\frac{3 \, \sqrt{d x^{2} + c} b^{3}}{{\left (a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2} + 3 \, a^{3} b c d^{3} - a^{4} d^{4}\right )}{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} - \frac{3 \,{\left (2 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b^{3} c^{3} d^{2} - 3 \, a^{3} b^{2} c^{2} d^{3} + 3 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \,{\left (9 \,{\left (d x^{2} + c\right )} b c + b c^{2} - 3 \,{\left (d x^{2} + c\right )} a d - a c d\right )}}{{\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}}} + \frac{6 \, \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{2} d^{2}}\right )} d^{2} \]

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),x, algorithm="giac")

[Out]

1/6*(3*sqrt(d*x^2 + c)*b^3/((a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3 - a
^4*d^4)*((d*x^2 + c)*b - b*c + a*d)) - 3*(2*b^4*c - 7*a*b^3*d)*arctan(sqrt(d*x^2
+ c)*b/sqrt(-b^2*c + a*b*d))/((a^2*b^3*c^3*d^2 - 3*a^3*b^2*c^2*d^3 + 3*a^4*b*c*
d^4 - a^5*d^5)*sqrt(-b^2*c + a*b*d)) + 2*(9*(d*x^2 + c)*b*c + b*c^2 - 3*(d*x^2 +
c)*a*d - a*c*d)/((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*(d*x
^2 + c)^(3/2)) + 6*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^2*sqrt(-c)*c^2*d^2))*d^2