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

Optimal. Leaf size=151 \[ -\frac{b d \left (3 a c^2-d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2} \left (a c^2-d^2\right )^2}-\frac{a c-d \sqrt{a+b x^2}}{2 a x^2 \left (a c^2-d^2\right )}+\frac{b c^3 \log \left (c \sqrt{a+b x^2}+d\right )}{\left (a c^2-d^2\right )^2}-\frac{b c^3 \log (x)}{\left (a c^2-d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.637981, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{b d \left (3 a c^2-d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{3/2} \left (a c^2-d^2\right )^2}-\frac{a c-d \sqrt{a+b x^2}}{2 a x^2 \left (a c^2-d^2\right )}+\frac{b c^3 \log \left (c \sqrt{a+b x^2}+d\right )}{\left (a c^2-d^2\right )^2}-\frac{b c^3 \log (x)}{\left (a c^2-d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.6553, size = 133, normalized size = 0.88 \[ - \frac{b c^{3} \log{\left (- b x^{2} \right )}}{2 \left (- a c^{2} + d^{2}\right )^{2}} + \frac{b c^{3} \log{\left (c \sqrt{a + b x^{2}} + d \right )}}{\left (- a c^{2} + d^{2}\right )^{2}} + \frac{a c - d \sqrt{a + b x^{2}}}{2 a x^{2} \left (- a c^{2} + d^{2}\right )} + \frac{b d \left (- 3 a c^{2} + d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \left (- a c^{2} + d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.94616, size = 430, normalized size = 2.85 \[ \frac{a^{3/2} b c^3 x^2 \log \left (-\frac{2 \left (d^2-a c^2\right )^2 \left (-i \sqrt{b} x \sqrt{a c^2-d^2}+d \sqrt{a+b x^2}+a c\right )}{b^{3/2} c^3 d^2 \left (\sqrt{b} c x+i \sqrt{a c^2-d^2}\right )}\right )+a^{3/2} b c^3 x^2 \log \left (-\frac{2 \left (d^2-a c^2\right )^2 \left (i \sqrt{b} x \sqrt{a c^2-d^2}+d \sqrt{a+b x^2}+a c\right )}{b^{3/2} c^3 d^2 \left (\sqrt{b} c x-i \sqrt{a c^2-d^2}\right )}\right )+a^{3/2} c^2 d \sqrt{a+b x^2}-b x^2 \log (x) \left (2 a^{3/2} c^3-3 a c^2 d+d^3\right )+a^{3/2} b c^3 x^2 \log \left (a c^2+b c^2 x^2-d^2\right )-a^{5/2} c^3+a^{3/2} c d^2-3 a b c^2 d x^2 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )-\sqrt{a} d^3 \sqrt{a+b x^2}+b d^3 x^2 \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{2 a^{3/2} x^2 \left (d^2-a c^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.067, size = 2459, normalized size = 16.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c/(a*c^2-d^2)/x^2-2*b*c^3*ln(x)/(a*c^2-d^2)^2+1/a*c*b/(a*c^2-d^2)^2*ln(x)*d
^2+1/2*a*c^5*b/(a*c^2-d^2)^2/d^2*ln(b*c^2*x^2+a*c^2-d^2)+b*c/a/(a*c^2-d^2)*ln(x)
-1/2*b*c^3/(a*c^2-d^2)/d^2*ln(b*c^2*x^2+a*c^2-d^2)+1/2*d/a^2/(a*c^2-d^2)/x^2*(b*
x^2+a)^(3/2)+1/2*d/a^(3/2)/(a*c^2-d^2)*b*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-1
/2*d/a^2/(a*c^2-d^2)*b*(b*x^2+a)^(1/2)-2*d*b/a^(1/2)/(a*c^2-d^2)^2*ln((2*a+2*a^(
1/2)*(b*x^2+a)^(1/2))/x)*c^2+2*d*b/a/(a*c^2-d^2)^2*(b*x^2+a)^(1/2)*c^2+b/a^(3/2)
/(a*c^2-d^2)^2*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)*d^3-b/a^2/(a*c^2-d^2)^2*(b*
x^2+a)^(1/2)*d^3-1/2*d*b^2*c^2/a^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))
/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*((x-1/b*(-a*b)^(1/2))^2*b+2*(-a*b
)^(1/2)*(x-1/b*(-a*b)^(1/2)))^(1/2)-1/2*d*b^(3/2)*c^2/a^2/((-a*b)^(1/2)*c^2+(-c^
2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*(-a*b)^(1/
2)*ln(((x-1/b*(-a*b)^(1/2))*b+(-a*b)^(1/2))/b^(1/2)+((x-1/b*(-a*b)^(1/2))^2*b+2*
(-a*b)^(1/2)*(x-1/b*(-a*b)^(1/2)))^(1/2))-1/2*d*b^2*c^2/a^2/((-a*b)^(1/2)*c^2+(-
c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*((x+1/b*
(-a*b)^(1/2))^2*b-2*(-a*b)^(1/2)*(x+1/b*(-a*b)^(1/2)))^(1/2)+1/2*d*b^(3/2)*c^2/a
^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^
2-d^2))^(1/2))*(-a*b)^(1/2)*ln(((x+1/b*(-a*b)^(1/2))*b-(-a*b)^(1/2))/b^(1/2)+((x
+1/b*(-a*b)^(1/2))^2*b-2*(-a*b)^(1/2)*(x+1/b*(-a*b)^(1/2)))^(1/2))+1/2*d*b^2*c^6
/(a*c^2-d^2)^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(
-c^2*b*(a*c^2-d^2))^(1/2))*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(
a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)+1/2*d*
b^(3/2)*c^4/(a*c^2-d^2)^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^
(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*(-c^2*b*(a*c^2-d^2))^(1/2)*ln(((-c^2*b*(a*
c^2-d^2))^(1/2)/c^2+(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2)+((x-(-c^2*b*
(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2
-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))-1/2*b^2*c^4/(a*c^2-d^2)^2/((-a*b)^(1/2)*c^2+
(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*d^3/(d
^2/c^2)^(1/2)*ln((2*d^2/c^2+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d
^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*
(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/
2))/(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))+1/2*d*b^2*c^6/(a*c^2-d^2)^2/((-a*b)^(1
/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2)
)*((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+
(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)-1/2*d*b^(3/2)*c^4/(a*c^2-d^2)^2
/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-
d^2))^(1/2))*(-c^2*b*(a*c^2-d^2))^(1/2)*ln((-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x+(
-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2)+((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b
)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/
c^2)^(1/2))-1/2*b^2*c^4/(a*c^2-d^2)^2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/
2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))*d^3/(d^2/c^2)^(1/2)*ln((2*d^2/
c^2-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(d^2
/c^2)^(1/2)*((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/
2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))/(x+(-c^2*b*(a*c^2-d^
2))^(1/2)/c^2/b))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.9211, size = 1, normalized size = 0.01 \[ \left [\frac{a^{\frac{3}{2}} b c^{3} x^{2} \log \left (-\frac{b c^{2} x^{2} + a c^{2} + 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - a^{\frac{3}{2}} b c^{3} x^{2} \log \left (-\frac{b c^{2} x^{2} + a c^{2} - 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) -{\left (3 \, a b c^{2} d - b d^{3}\right )} x^{2} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{2} + a} \sqrt{a} + 2 \,{\left (a b c^{3} x^{2} \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) - 2 \, a b c^{3} x^{2} \log \left (x\right ) - a^{2} c^{3} + a c d^{2}\right )} \sqrt{a}}{4 \,{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{a} x^{2}}, \frac{\sqrt{-a} a b c^{3} x^{2} \log \left (-\frac{b c^{2} x^{2} + a c^{2} + 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - \sqrt{-a} a b c^{3} x^{2} \log \left (-\frac{b c^{2} x^{2} + a c^{2} - 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - 2 \,{\left (3 \, a b c^{2} d - b d^{3}\right )} x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) + 2 \,{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{2} + a} \sqrt{-a} + 2 \,{\left (a b c^{3} x^{2} \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) - 2 \, a b c^{3} x^{2} \log \left (x\right ) - a^{2} c^{3} + a c d^{2}\right )} \sqrt{-a}}{4 \,{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{-a} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.278815, size = 275, normalized size = 1.82 \[ \frac{1}{2} \,{\left (\frac{2 \, c^{4}{\rm ln}\left ({\left | \sqrt{b x^{2} + a} c + d \right |}\right )}{a^{2} c^{5} - 2 \, a c^{3} d^{2} + c d^{4}} - \frac{c^{3}{\rm ln}\left (b x^{2}\right )}{a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}} + \frac{{\left (3 \, a c^{2} d - d^{3}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{-a}} - \frac{a^{2} c^{3} - a c d^{2} -{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{2} + a}}{{\left (a c^{2} - d^{2}\right )}^{2} a b x^{2}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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