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

Optimal. Leaf size=156 \[ -\frac{b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 (b c-a d)^{3/2}}+\frac{(3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2 c^{5/2}}-\frac{d (b c-3 a d)}{2 a c^2 \sqrt{c+d x^2} (b c-a d)}-\frac{1}{2 a c x^2 \sqrt{c+d x^2}} \]

[Out]

-(d*(b*c - 3*a*d))/(2*a*c^2*(b*c - a*d)*Sqrt[c + d*x^2]) - 1/(2*a*c*x^2*Sqrt[c +
d*x^2]) + ((2*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2*c^(5/2)) -
(b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a^2*(b*c - a*d)^(3
/2))

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Rubi [A]  time = 0.646642, antiderivative size = 156, 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 \[ -\frac{b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 (b c-a d)^{3/2}}+\frac{(3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2 c^{5/2}}-\frac{d (b c-3 a d)}{2 a c^2 \sqrt{c+d x^2} (b c-a d)}-\frac{1}{2 a c x^2 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(b*c - 3*a*d))/(2*a*c^2*(b*c - a*d)*Sqrt[c + d*x^2]) - 1/(2*a*c*x^2*Sqrt[c +
d*x^2]) + ((2*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^2*c^(5/2)) -
(b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a^2*(b*c - a*d)^(3
/2))

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Rubi in Sympy [A]  time = 77.492, size = 136, normalized size = 0.87 \[ - \frac{1}{2 a c x^{2} \sqrt{c + d x^{2}}} - \frac{d \left (3 a d - b c\right )}{2 a c^{2} \sqrt{c + d x^{2}} \left (a d - b c\right )} - \frac{b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a^{2} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{\left (3 a d + 2 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 a^{2} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a*c*x**2*sqrt(c + d*x**2)) - d*(3*a*d - b*c)/(2*a*c**2*sqrt(c + d*x**2)*(a
*d - b*c)) - b**(5/2)*atan(sqrt(b)*sqrt(c + d*x**2)/sqrt(a*d - b*c))/(a**2*(a*d
- b*c)**(3/2)) + (3*a*d + 2*b*c)*atanh(sqrt(c + d*x**2)/sqrt(c))/(2*a**2*c**(5/2
))

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Mathematica [C]  time = 4.38667, size = 355, normalized size = 2.28 \[ \frac{1}{2} \left (-\frac{b^{5/2} \log \left (\frac{2 a^2 \left (-i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{5/2} \left (\sqrt{b} x+i \sqrt{a}\right )}\right )}{a^2 (b c-a d)^{3/2}}-\frac{b^{5/2} \log \left (\frac{2 a^2 \left (i \sqrt{a} d x \sqrt{b c-a d}+\sqrt{b} c \sqrt{b c-a d}-a d \sqrt{c+d x^2}+b c \sqrt{c+d x^2}\right )}{b^{5/2} \left (\sqrt{b} x-i \sqrt{a}\right )}\right )}{a^2 (b c-a d)^{3/2}}+\frac{(3 a d+2 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{a^2 c^{5/2}}-\frac{\log (x) (3 a d+2 b c)}{a^2 c^{5/2}}+\frac{\frac{2 d^2}{b c-a d}-\frac{\frac{c}{x^2}+d}{a}}{c^2 \sqrt{c+d x^2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 763, 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)/(d*x^2+c)^(3/2),x)

[Out]

-1/2/a/c/x^2/(d*x^2+c)^(1/2)-3/2/a*d/c^2/(d*x^2+c)^(1/2)+3/2/a*d/c^(5/2)*ln((2*c
+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-b/a^2/c/(d*x^2+c)^(1/2)+b/a^2/c^(3/2)*ln((2*c+2*c
^(1/2)*(d*x^2+c)^(1/2))/x)-1/2*b^2/a^2/(a*d-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)+1/2*b/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*b^2/a^2/(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)))-1/2*b^2/a^2/(a*d-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)-1/2*b/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*b^2/a^2/(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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.793519, 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^3),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.248932, size = 248, normalized size = 1.59 \[ \frac{1}{2} \,{\left (\frac{2 \, b^{3} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b^{2} c + a b d}} - \frac{{\left (d x^{2} + c\right )} b c - 3 \,{\left (d x^{2} + c\right )} a d + 2 \, a c d}{{\left (a b c^{3} d - a^{2} c^{2} d^{2}\right )}{\left ({\left (d x^{2} + c\right )}^{\frac{3}{2}} - \sqrt{d x^{2} + c} c\right )}} - \frac{{\left (2 \, b c + 3 \, a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{2} d^{2}}\right )} d^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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