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

Optimal. Leaf size=170 \[ \frac{b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 (b c-a d)^{5/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2 c^{3/2}}+\frac{b}{2 a \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c \sqrt{c+d x^2} (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.654345, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{b^{3/2} (2 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 (b c-a d)^{5/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2 c^{3/2}}+\frac{b}{2 a \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{d (2 a d+b c)}{2 a c \sqrt{c+d x^2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 82.1787, size = 144, normalized size = 0.85 \[ - \frac{b}{2 a \left (a + b x^{2}\right ) \sqrt{c + d x^{2}} \left (a d - b c\right )} + \frac{d \left (2 a d + b c\right )}{2 a c \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} + \frac{b^{\frac{3}{2}} \left (5 a d - 2 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 a^{2} \left (a d - b c\right )^{\frac{5}{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a^{2} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.8413, size = 406, normalized size = 2.39 \[ \frac{1}{4} \left (\frac{b^{3/2} (2 b c-5 a d) \log \left (-\frac{4 a^2 (b c-a d) \left (-i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{3/2} \left (\sqrt{b} x+i \sqrt{a}\right ) (2 b c-5 a d)}\right )}{a^2 (b c-a d)^{5/2}}+\frac{b^{3/2} (2 b c-5 a d) \log \left (-\frac{4 a^2 (b c-a d) \left (i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{3/2} \left (\sqrt{b} x-i \sqrt{a}\right ) (2 b c-5 a d)}\right )}{a^2 (b c-a d)^{5/2}}+\frac{2 \sqrt{c+d x^2} \left (\frac{b^2}{a^2+a b x^2}+\frac{2 d^2}{c^2+c d x^2}\right )}{(b c-a d)^2}-\frac{4 \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{a^2 c^{3/2}}+\frac{4 \log (x)}{a^2 c^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 1672, normalized size = 9.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.84598, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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),x, algorithm="fricas")

[Out]

[-1/8*((2*a*b^2*c^3 - 5*a^2*b*c^2*d + (2*b^3*c^2*d - 5*a*b^2*c*d^2)*x^4 + (2*b^3
*c^3 - 3*a*b^2*c^2*d - 5*a^2*b*c*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*(a*b^2*c^2 + 2*a^3*d^2 + (a*b^2*
c*d + 2*a^2*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(c) - 4*(a*b^2*c^3 - 2*a^2*b*c^2*d +
a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*
d - a^2*b*c*d^2 + a^3*d^3)*x^2)*log(-((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2 + c)*
c)/x^2))/((a^3*b^2*c^4 - 2*a^4*b*c^3*d + a^5*c^2*d^2 + (a^2*b^3*c^3*d - 2*a^3*b^
2*c^2*d^2 + a^4*b*c*d^3)*x^4 + (a^2*b^3*c^4 - a^3*b^2*c^3*d - a^4*b*c^2*d^2 + a^
5*c*d^3)*x^2)*sqrt(c)), -1/8*((2*a*b^2*c^3 - 5*a^2*b*c^2*d + (2*b^3*c^2*d - 5*a*
b^2*c*d^2)*x^4 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 5*a^2*b*c*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*(a*b^2*c
^2 + 2*a^3*d^2 + (a*b^2*c*d + 2*a^2*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(-c) + 8*(a*
b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^
4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*arctan(sqrt(-c)/sqrt(d*
x^2 + c)))/((a^3*b^2*c^4 - 2*a^4*b*c^3*d + a^5*c^2*d^2 + (a^2*b^3*c^3*d - 2*a^3*
b^2*c^2*d^2 + a^4*b*c*d^3)*x^4 + (a^2*b^3*c^4 - a^3*b^2*c^3*d - a^4*b*c^2*d^2 +
a^5*c*d^3)*x^2)*sqrt(-c)), -1/4*((2*a*b^2*c^3 - 5*a^2*b*c^2*d + (2*b^3*c^2*d - 5
*a*b^2*c*d^2)*x^4 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 5*a^2*b*c*d^2)*x^2)*sqrt(c)*sqr
t(-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*b^2*c^2 + 2*a^3*d^2 + (a*b^2*c*d + 2*a^2*b*d^2)
*x^2)*sqrt(d*x^2 + c)*sqrt(c) - 2*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*
c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 +
a^3*d^3)*x^2)*log(-((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2 + c)*c)/x^2))/((a^3*b^2
*c^4 - 2*a^4*b*c^3*d + a^5*c^2*d^2 + (a^2*b^3*c^3*d - 2*a^3*b^2*c^2*d^2 + a^4*b*
c*d^3)*x^4 + (a^2*b^3*c^4 - a^3*b^2*c^3*d - a^4*b*c^2*d^2 + a^5*c*d^3)*x^2)*sqrt
(c)), -1/4*((2*a*b^2*c^3 - 5*a^2*b*c^2*d + (2*b^3*c^2*d - 5*a*b^2*c*d^2)*x^4 + (
2*b^3*c^3 - 3*a*b^2*c^2*d - 5*a^2*b*c*d^2)*x^2)*sqrt(-c)*sqrt(-b/(b*c - a*d))*ar
ctan(-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*b^2*c^2 + 2*a^3*d^2 + (a*b^2*c*d + 2*a^2*b*d^2)*x^2)*sqrt(d*x^2 + c
)*sqrt(-c) + 4*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2 + (b^3*c^2*d - 2*a*b^2*c*d
^2 + a^2*b*d^3)*x^4 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*x^2)*arcta
n(sqrt(-c)/sqrt(d*x^2 + c)))/((a^3*b^2*c^4 - 2*a^4*b*c^3*d + a^5*c^2*d^2 + (a^2*
b^3*c^3*d - 2*a^3*b^2*c^2*d^2 + a^4*b*c*d^3)*x^4 + (a^2*b^3*c^4 - a^3*b^2*c^3*d
- a^4*b*c^2*d^2 + a^5*c*d^3)*x^2)*sqrt(-c))]

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.236842, size = 319, normalized size = 1.88 \[ -\frac{1}{2} \, d^{2}{\left (\frac{{\left (2 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b^{2} c^{2} d^{2} - 2 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-b^{2} c + a b d}} - \frac{{\left (d x^{2} + c\right )} b^{2} c + 2 \,{\left (d x^{2} + c\right )} a b d - 2 \, a b c d + 2 \, a^{2} d^{2}}{{\left (a b^{2} c^{3} d - 2 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3}\right )}{\left ({\left (d x^{2} + c\right )}^{\frac{3}{2}} b - \sqrt{d x^{2} + c} b c + \sqrt{d x^{2} + c} a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c d^{2}}\right )} \]

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),x, algorithm="giac")

[Out]

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