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

Optimal. Leaf size=115 \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2 c^{3/2}}-\frac{\sqrt{c+d x^2}}{2 a c x^2} \]

[Out]

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

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Rubi [A]  time = 0.362275, antiderivative size = 115, 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 \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2 c^{3/2}}-\frac{\sqrt{c+d x^2}}{2 a c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 47.8614, size = 97, normalized size = 0.84 \[ - \frac{\sqrt{c + d x^{2}}}{2 a c x^{2}} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a^{2} \sqrt{a d - b c}} + \frac{\left (\frac{a d}{2} + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a^{2} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.668216, size = 292, normalized size = 2.54 \[ \frac{-\frac{\sqrt{c} \left (b^{3/2} c x^2 \log \left (\frac{2 a^2 \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{b^{3/2} \left (\sqrt{b} x+i \sqrt{a}\right ) \sqrt{b c-a d}}\right )+b^{3/2} c x^2 \log \left (\frac{2 a^2 \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{b^{3/2} \left (\sqrt{b} x-i \sqrt{a}\right ) \sqrt{b c-a d}}\right )+a \sqrt{c+d x^2} \sqrt{b c-a d}\right )}{x^2 \sqrt{b c-a d}}+(a d+2 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+\log (x) (-(a d+2 b c))}{2 a^2 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 385, normalized size = 3.4 \[ -{\frac{1}{2\,ac{x}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{d}{2\,a}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}+{\frac{b}{{a}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}-{\frac{b}{2\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{b}{2\,{a}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.394214, size = 1, normalized size = 0.01 \[ \left [\frac{b c^{\frac{3}{2}} x^{2} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \,{\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) +{\left (2 \, b c + a d\right )} x^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \, \sqrt{d x^{2} + c} a \sqrt{c}}{4 \, a^{2} c^{\frac{3}{2}} x^{2}}, \frac{b \sqrt{-c} c x^{2} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \,{\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \,{\left (2 \, b c + a d\right )} x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) - 2 \, \sqrt{d x^{2} + c} a \sqrt{-c}}{4 \, a^{2} \sqrt{-c} c x^{2}}, \frac{2 \, b c^{\frac{3}{2}} x^{2} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}\right ) +{\left (2 \, b c + a d\right )} x^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \, \sqrt{d x^{2} + c} a \sqrt{c}}{4 \, a^{2} c^{\frac{3}{2}} x^{2}}, \frac{b \sqrt{-c} c x^{2} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}\right ) +{\left (2 \, b c + a d\right )} x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) - \sqrt{d x^{2} + c} a \sqrt{-c}}{2 \, a^{2} \sqrt{-c} c x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.248291, size = 159, normalized size = 1.38 \[ \frac{1}{2} \, d^{2}{\left (\frac{2 \, b^{2} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c + a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c d^{2}} - \frac{\sqrt{d x^{2} + c}}{a c d^{2} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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