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

Optimal. Leaf size=185 \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^2} (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) (b c-a d)}-\frac{\sqrt{c+d x^2}}{2 a c x^2 \left (a+b x^2\right )} \]

[Out]

-(b*(2*b*c - a*d)*Sqrt[c + d*x^2])/(2*a^2*c*(b*c - a*d)*(a + b*x^2)) - Sqrt[c +
d*x^2]/(2*a*c*x^2*(a + b*x^2)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]
)/(2*a^3*c^(3/2)) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/S
qrt[b*c - a*d]])/(2*a^3*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.676969, antiderivative size = 185, 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 \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 (b c-a d)^{3/2}}+\frac{(a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3 c^{3/2}}-\frac{b \sqrt{c+d x^2} (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) (b c-a d)}-\frac{\sqrt{c+d x^2}}{2 a c x^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(b*(2*b*c - a*d)*Sqrt[c + d*x^2])/(2*a^2*c*(b*c - a*d)*(a + b*x^2)) - Sqrt[c +
d*x^2]/(2*a*c*x^2*(a + b*x^2)) + ((4*b*c + a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]]
)/(2*a^3*c^(3/2)) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/S
qrt[b*c - a*d]])/(2*a^3*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 80.6159, size = 158, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{2}}}{2 a c x^{2} \left (a + b x^{2}\right )} - \frac{b \sqrt{c + d x^{2}} \left (a d - 2 b c\right )}{2 a^{2} c \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{b^{\frac{3}{2}} \left (5 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 a^{3} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{\left (a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 a^{3} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**2)/(2*a*c*x**2*(a + b*x**2)) - b*sqrt(c + d*x**2)*(a*d - 2*b*c)/(
2*a**2*c*(a + b*x**2)*(a*d - b*c)) + b**(3/2)*(5*a*d - 4*b*c)*atan(sqrt(b)*sqrt(
c + d*x**2)/sqrt(a*d - b*c))/(2*a**3*(a*d - b*c)**(3/2)) + (a*d + 4*b*c)*atanh(s
qrt(c + d*x**2)/sqrt(c))/(2*a**3*c**(3/2))

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Mathematica [C]  time = 2.11906, size = 387, normalized size = 2.09 \[ -\frac{\frac{b^{3/2} (4 b c-5 a d) \log \left (\frac{4 a^3 \left (-i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{3/2} \left (\sqrt{b} x+i \sqrt{a}\right ) (4 b c-5 a d)}\right )}{(b c-a d)^{3/2}}+\frac{b^{3/2} (4 b c-5 a d) \log \left (\frac{4 i a^3 \left (i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{3/2} \left (\sqrt{a}+i \sqrt{b} x\right ) (4 b c-5 a d)}\right )}{(b c-a d)^{3/2}}-2 a \sqrt{c+d x^2} \left (\frac{b^2}{\left (a+b x^2\right ) (a d-b c)}-\frac{1}{c x^2}\right )-\frac{2 (a d+4 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{c^{3/2}}+\frac{2 \log (x) (a d+4 b c)}{c^{3/2}}}{4 a^3} \]

Antiderivative was successfully verified.

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

[Out]

-(-2*a*Sqrt[c + d*x^2]*(-(1/(c*x^2)) + b^2/((-(b*c) + a*d)*(a + b*x^2))) + (2*(4
*b*c + a*d)*Log[x])/c^(3/2) - (2*(4*b*c + a*d)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])
/c^(3/2) + (b^(3/2)*(4*b*c - 5*a*d)*Log[(4*a^3*(Sqrt[b]*c*Sqrt[b*c - a*d] - I*Sq
rt[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)
*(4*b*c - 5*a*d)*(I*Sqrt[a] + Sqrt[b]*x))])/(b*c - a*d)^(3/2) + (b^(3/2)*(4*b*c
- 5*a*d)*Log[((4*I)*a^3*(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)*(4*b*c - 5*a*d)*(Sqrt[
a] + I*Sqrt[b]*x))])/(b*c - a*d)^(3/2))/(4*a^3)

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Maple [B]  time = 0.023, size = 899, normalized size = 4.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/a^2/c/x^2*(d*x^2+c)^(1/2)+1/2/a^2*d/c^(3/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/
2))/x)-b/a^3/(-(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)))-b/a^3/(-(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)))+2*b/a^3/c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^
2+c)^(1/2))/x)+1/4*b^2/a^2/(-a*b)^(1/2)/(a*d-b*c)/(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*
b/a^2*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*b^2/a^2
/(-a*b)^(1/2)/(a*d-b*c)/(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*b/a^2*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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.22973, 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^3),x, algorithm="fricas")

[Out]

[1/8*(((4*b^3*c^2 - 5*a*b^2*c*d)*x^4 + (4*a*b^2*c^2 - 5*a^2*b*c*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^
2*b*c - a^3*d + (2*a*b^2*c - a^2*b*d)*x^2)*sqrt(d*x^2 + c)*sqrt(c) + 2*((4*b^3*c
^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^4 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x^2)*l
og(-((d*x^2 + 2*c)*sqrt(c) + 2*sqrt(d*x^2 + c)*c)/x^2))/(((a^3*b^2*c^2 - a^4*b*c
*d)*x^4 + (a^4*b*c^2 - a^5*c*d)*x^2)*sqrt(c)), 1/8*(((4*b^3*c^2 - 5*a*b^2*c*d)*x
^4 + (4*a*b^2*c^2 - 5*a^2*b*c*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^2*b*c - a^3*d + (2*a*b^2*c - a^2*
b*d)*x^2)*sqrt(d*x^2 + c)*sqrt(-c) + 4*((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^
4 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x^2)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))
/(((a^3*b^2*c^2 - a^4*b*c*d)*x^4 + (a^4*b*c^2 - a^5*c*d)*x^2)*sqrt(-c)), 1/4*(((
4*b^3*c^2 - 5*a*b^2*c*d)*x^4 + (4*a*b^2*c^2 - 5*a^2*b*c*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^2*b*c - a^3*d + (2*a*b^2*c - a^2*b*d)*x^2)*sqrt(d*x^
2 + c)*sqrt(c) + ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^4 + (4*a*b^2*c^2 - 3*a
^2*b*c*d - a^3*d^2)*x^2)*log(-((d*x^2 + 2*c)*sqrt(c) + 2*sqrt(d*x^2 + c)*c)/x^2)
)/(((a^3*b^2*c^2 - a^4*b*c*d)*x^4 + (a^4*b*c^2 - a^5*c*d)*x^2)*sqrt(c)), 1/4*(((
4*b^3*c^2 - 5*a*b^2*c*d)*x^4 + (4*a*b^2*c^2 - 5*a^2*b*c*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)*s
qrt(-b/(b*c - a*d)))) - 2*(a^2*b*c - a^3*d + (2*a*b^2*c - a^2*b*d)*x^2)*sqrt(d*x
^2 + c)*sqrt(-c) + 2*((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^4 + (4*a*b^2*c^2 -
3*a^2*b*c*d - a^3*d^2)*x^2)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/(((a^3*b^2*c^2 -
a^4*b*c*d)*x^4 + (a^4*b*c^2 - a^5*c*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**3/(b*x**2+a)**2/(d*x**2+c)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.244442, size = 362, normalized size = 1.96 \[ \frac{1}{2} \, d^{3}{\left (\frac{{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt{-b^{2} c + a b d}} - \frac{2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c - 2 \, \sqrt{d x^{2} + c} b^{2} c^{2} -{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b d + 2 \, \sqrt{d x^{2} + c} a b c d - \sqrt{d x^{2} + c} a^{2} d^{2}}{{\left (a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}{\left ({\left (d x^{2} + c\right )}^{2} b - 2 \,{\left (d x^{2} + c\right )} b c + b c^{2} +{\left (d x^{2} + c\right )} a d - a c d\right )}} - \frac{{\left (4 \, b c + a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} c d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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