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

Optimal. Leaf size=130 \[ \frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^2}}{2 a \left (a+b x^2\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.393794, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 (b c-a d)^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2 \sqrt{c}}+\frac{b \sqrt{c+d x^2}}{2 a \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 51.5562, size = 110, normalized size = 0.85 \[ - \frac{b \sqrt{c + d x^{2}}}{2 a \left (a + b x^{2}\right ) \left (a d - b c\right )} - \frac{\sqrt{b} \left (\frac{3 a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a^{2} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a^{2} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*sqrt(c + d*x**2)/(2*a*(a + b*x**2)*(a*d - b*c)) - sqrt(b)*(3*a*d/2 - b*c)*ata
n(sqrt(b)*sqrt(c + d*x**2)/sqrt(a*d - b*c))/(a**2*(a*d - b*c)**(3/2)) - atanh(sq
rt(c + d*x**2)/sqrt(c))/(a**2*sqrt(c))

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Mathematica [C]  time = 1.19668, size = 360, normalized size = 2.77 \[ \frac{\frac{\sqrt{b} (2 b c-3 a d) \log \left (-\frac{4 i a^2 \left (i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{\sqrt{b} \left (\sqrt{a}+i \sqrt{b} x\right ) (2 b c-3 a d)}\right )}{(b c-a d)^{3/2}}+\frac{\sqrt{b} (2 b c-3 a d) \log \left (\frac{4 a^2 \left (i \sqrt{a} d x \sqrt{b c-a d}-\sqrt{b} c \sqrt{b c-a d}+a d \sqrt{c+d x^2}-b c \sqrt{c+d x^2}\right )}{\sqrt{b} \left (\sqrt{b} x+i \sqrt{a}\right ) (2 b c-3 a d)}\right )}{(b c-a d)^{3/2}}-\frac{2 a b \sqrt{c+d x^2}}{\left (a+b x^2\right ) (a d-b c)}-\frac{4 \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{\sqrt{c}}+\frac{4 \log (x)}{\sqrt{c}}}{4 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 838, normalized size = 6.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.728419, 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*sqrt(d*x^2 + c)*x),x, algorithm="fricas")

[Out]

[1/8*(4*sqrt(d*x^2 + c)*a*b*sqrt(c) + (2*a*b*c - 3*a^2*d + (2*b^2*c - 3*a*b*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*b*c - a^2*d + (b^2*c - a*b*d)*x^2)*log(-((d*x^2 + 2*c)*sqrt(c) - 2*s
qrt(d*x^2 + c)*c)/x^2))/((a^3*b*c - a^4*d + (a^2*b^2*c - a^3*b*d)*x^2)*sqrt(c)),
1/8*(4*sqrt(d*x^2 + c)*a*b*sqrt(-c) + (2*a*b*c - 3*a^2*d + (2*b^2*c - 3*a*b*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)) - 8*(a*b*c - a^2*d + (b^2*c - a*b*d)*x^2)*arctan(sqrt(-c)/sqrt(d*x^2 + c)
))/((a^3*b*c - a^4*d + (a^2*b^2*c - a^3*b*d)*x^2)*sqrt(-c)), 1/4*(2*sqrt(d*x^2 +
c)*a*b*sqrt(c) - (2*a*b*c - 3*a^2*d + (2*b^2*c - 3*a*b*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)*sq
rt(-b/(b*c - a*d)))) + 2*(a*b*c - a^2*d + (b^2*c - a*b*d)*x^2)*log(-((d*x^2 + 2*
c)*sqrt(c) - 2*sqrt(d*x^2 + c)*c)/x^2))/((a^3*b*c - a^4*d + (a^2*b^2*c - a^3*b*d
)*x^2)*sqrt(c)), 1/4*(2*sqrt(d*x^2 + c)*a*b*sqrt(-c) - (2*a*b*c - 3*a^2*d + (2*b
^2*c - 3*a*b*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)))) - 4*(a*b*c - a^2*d +
(b^2*c - a*b*d)*x^2)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/((a^3*b*c - a^4*d + (a^2*
b^2*c - a^3*b*d)*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)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.232729, size = 207, normalized size = 1.59 \[ -\frac{1}{2} \, d^{2}{\left (\frac{{\left (2 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{\sqrt{d x^{2} + c} b}{{\left (a b c d - a^{2} d^{2}\right )}{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} - \frac{2 \, \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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