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

Optimal. Leaf size=107 \[ \frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a (b c-a d)^{3/2}}-\frac{d}{c \sqrt{c+d x^2} (b c-a d)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.346962, antiderivative size = 107, 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 \[ \frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a (b c-a d)^{3/2}}-\frac{d}{c \sqrt{c+d x^2} (b c-a d)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.0304, size = 87, normalized size = 0.81 \[ \frac{d}{c \sqrt{c + d x^{2}} \left (a d - b c\right )} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a \left (a d - b c\right )^{\frac{3}{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 2.32741, size = 316, normalized size = 2.95 \[ \frac{\log (x)}{a c^{3/2}}+\frac{1}{2} \left (\frac{b^{3/2} \log \left (-\frac{2 a \left (-i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{3/2} \left (\sqrt{b} x+i \sqrt{a}\right )}\right )}{a (b c-a d)^{3/2}}+\frac{b^{3/2} \log \left (-\frac{2 a \left (i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{3/2} \left (\sqrt{b} x-i \sqrt{a}\right )}\right )}{a (b c-a d)^{3/2}}+\frac{2 d}{c \sqrt{c+d x^2} (a d-b c)}-\frac{2 \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{a c^{3/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.018, size = 681, normalized size = 6.4 \[{\frac{1}{ac}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{1}{a}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}+{\frac{b}{2\,a \left ( ad-bc \right ) }{\frac{1}{\sqrt{ \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{dx}{2\,a \left ( ad-bc \right ) c}\sqrt{-ab}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{b}{2\,a \left ( ad-bc \right ) }\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{b}{2\,a \left ( ad-bc \right ) }{\frac{1}{\sqrt{ \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}+{\frac{dx}{2\,a \left ( ad-bc \right ) c}\sqrt{-ab}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}}}}}-{\frac{b}{2\,a \left ( ad-bc \right ) }\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.526308, 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)*(d*x^2 + c)^(3/2)*x),x, algorithm="fricas")

[Out]

[-1/4*(4*sqrt(d*x^2 + c)*a*sqrt(c)*d + (b*c*d*x^2 + b*c^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)) - 2*(b*c^2 - a*c*d +
(b*c*d - a*d^2)*x^2)*log(-((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2 + c)*c)/x^2))/(
(a*b*c^3 - a^2*c^2*d + (a*b*c^2*d - a^2*c*d^2)*x^2)*sqrt(c)), -1/4*(4*sqrt(d*x^2
+ c)*a*sqrt(-c)*d + (b*c*d*x^2 + b*c^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*(b*c^2 - a*c*d + (b*c*d - a*d^2)*
x^2)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/((a*b*c^3 - a^2*c^2*d + (a*b*c^2*d - a^2*
c*d^2)*x^2)*sqrt(-c)), -1/2*(2*sqrt(d*x^2 + c)*a*sqrt(c)*d + (b*c*d*x^2 + b*c^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)))) - (b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)*lo
g(-((d*x^2 + 2*c)*sqrt(c) - 2*sqrt(d*x^2 + c)*c)/x^2))/((a*b*c^3 - a^2*c^2*d + (
a*b*c^2*d - a^2*c*d^2)*x^2)*sqrt(c)), -1/2*(2*sqrt(d*x^2 + c)*a*sqrt(-c)*d + (b*
c*d*x^2 + b*c^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*(b*c^2 - a*c*d + (b*c
*d - a*d^2)*x^2)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/((a*b*c^3 - a^2*c^2*d + (a*b*
c^2*d - a^2*c*d^2)*x^2)*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232371, size = 158, normalized size = 1.48 \[ -{\left (\frac{b^{2} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a b c d - a^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{1}{{\left (b c^{2} - a c d\right )} \sqrt{d x^{2} + c}} - \frac{\arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a \sqrt{-c} c d}\right )} d \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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