3.739 \(\int \frac{\sqrt{c+d x^2}}{x^4 \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=147 \[ \frac{b (5 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} (15 b c-2 a d)}{6 a^3 c x}-\frac{5 \sqrt{c+d x^2}}{6 a^2 x^3}+\frac{\sqrt{c+d x^2}}{2 a x^3 \left (a+b x^2\right )} \]

[Out]

(-5*Sqrt[c + d*x^2])/(6*a^2*x^3) + ((15*b*c - 2*a*d)*Sqrt[c + d*x^2])/(6*a^3*c*x
) + Sqrt[c + d*x^2]/(2*a*x^3*(a + b*x^2)) + (b*(5*b*c - 4*a*d)*ArcTan[(Sqrt[b*c
- a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*Sqrt[b*c - a*d])

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Rubi [A]  time = 0.59739, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{b (5 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} (15 b c-2 a d)}{6 a^3 c x}-\frac{5 \sqrt{c+d x^2}}{6 a^2 x^3}+\frac{\sqrt{c+d x^2}}{2 a x^3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x^2]/(x^4*(a + b*x^2)^2),x]

[Out]

(-5*Sqrt[c + d*x^2])/(6*a^2*x^3) + ((15*b*c - 2*a*d)*Sqrt[c + d*x^2])/(6*a^3*c*x
) + Sqrt[c + d*x^2]/(2*a*x^3*(a + b*x^2)) + (b*(5*b*c - 4*a*d)*ArcTan[(Sqrt[b*c
- a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(7/2)*Sqrt[b*c - a*d])

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Rubi in Sympy [A]  time = 77.589, size = 129, normalized size = 0.88 \[ \frac{\sqrt{c + d x^{2}}}{2 a x^{3} \left (a + b x^{2}\right )} - \frac{5 \sqrt{c + d x^{2}}}{6 a^{2} x^{3}} - \frac{\sqrt{c + d x^{2}} \left (2 a d - 15 b c\right )}{6 a^{3} c x} - \frac{b \left (4 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{7}{2}} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**(1/2)/x**4/(b*x**2+a)**2,x)

[Out]

sqrt(c + d*x**2)/(2*a*x**3*(a + b*x**2)) - 5*sqrt(c + d*x**2)/(6*a**2*x**3) - sq
rt(c + d*x**2)*(2*a*d - 15*b*c)/(6*a**3*c*x) - b*(4*a*d - 5*b*c)*atanh(x*sqrt(a*
d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/(2*a**(7/2)*sqrt(a*d - b*c))

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Mathematica [A]  time = 0.24808, size = 120, normalized size = 0.82 \[ \frac{b (5 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{7/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^2} \left (3 b x^2 \left (\frac{b x^2}{a+b x^2}+4\right )-\frac{2 a \left (c+d x^2\right )}{c}\right )}{6 a^3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c + d*x^2]/(x^4*(a + b*x^2)^2),x]

[Out]

(Sqrt[c + d*x^2]*((-2*a*(c + d*x^2))/c + 3*b*x^2*(4 + (b*x^2)/(a + b*x^2))))/(6*
a^3*x^3) + (b*(5*b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2
])])/(2*a^(7/2)*Sqrt[b*c - a*d])

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Maple [B]  time = 0.026, size = 2667, normalized size = 18.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^(1/2)/x^4/(b*x^2+a)^2,x)

[Out]

-1/4*b/a^2*d^(3/2)/(a*d-b*c)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(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^2/a^3/(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)^(3/2)-1/4*b/a^2*d^(3/2)/(a*d-b
*c)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(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))+5/4*b^2/a^3/(-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)+5/4*b/a^3*d^(1/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(
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))-5/4*b^2/a^3/(-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)+5/4*b/a^3*d^(1/2)*ln((-d*(-a*b)^(1/2)/b
+(x+1/b*(-a*b)^(1/2))*d)/d^(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))-1/3/a^2/c/x^3*(d*x^2+c)^(3/2)-2*b/a^3*d^(
1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))+1/4*b^2/a^3*d/(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)*x+1/4*b^2/a^3*d^
(1/2)/(a*d-b*c)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(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))*c+1/4
*b/a^3*d*(-a*b)^(1/2)/(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)
))*c-1/4*b/a^3*d*(-a*b)^(1/2)/(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)))*c-2*b/a^3*d/c*x*(d*x^2+c)^(1/2)-1/4/a^2*d^2*(-a*b)^(1/2)/(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)))+1/4*b^2/a^3*d/(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)*x-
1/4*b^2/a^3/(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)^(3/2)+2*b/a^3/c/x*(d*x^2+c)^(3/2)+1/4*
b^2/a^3*d^(1/2)/(a*d-b*c)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(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))*c+5/4*b/a^2/(-a*b)^(1/2)/(-(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))
)*d-5/4*b^2/a^3/(-a*b)^(1/2)/(-(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)))*
c+1/4*b/a^3*d*(-a*b)^(1/2)/(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)-1/4*b/a^3*d*(-a*b)^(1/2)/(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
)-5/4*b/a^2/(-a*b)^(1/2)/(-(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)))*d+5/
4*b^2/a^3/(-a*b)^(1/2)/(-(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)))*c+1/4/
a^2*d^2*(-a*b)^(1/2)/(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. \[ \int \frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )}^{2} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/((b*x^2 + a)^2*x^4),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^2 + c)/((b*x^2 + a)^2*x^4), x)

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Fricas [A]  time = 0.385319, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (15 \, b^{2} c - 2 \, a b d\right )} x^{4} - 2 \, a^{2} c + 2 \,{\left (5 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} - 3 \,{\left ({\left (5 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{5} +{\left (5 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{3}\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} - 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{24 \,{\left (a^{3} b c x^{5} + a^{4} c x^{3}\right )} \sqrt{-a b c + a^{2} d}}, \frac{2 \,{\left ({\left (15 \, b^{2} c - 2 \, a b d\right )} x^{4} - 2 \, a^{2} c + 2 \,{\left (5 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} + 3 \,{\left ({\left (5 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{5} +{\left (5 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{3}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{12 \,{\left (a^{3} b c x^{5} + a^{4} c x^{3}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/((b*x^2 + a)^2*x^4),x, algorithm="fricas")

[Out]

[1/24*(4*((15*b^2*c - 2*a*b*d)*x^4 - 2*a^2*c + 2*(5*a*b*c - a^2*d)*x^2)*sqrt(-a*
b*c + a^2*d)*sqrt(d*x^2 + c) - 3*((5*b^3*c^2 - 4*a*b^2*c*d)*x^5 + (5*a*b^2*c^2 -
4*a^2*b*c*d)*x^3)*log((((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*
a*b*c^2 - 4*a^2*c*d)*x^2)*sqrt(-a*b*c + a^2*d) - 4*((a*b^2*c^2 - 3*a^2*b*c*d + 2
*a^3*d^2)*x^3 - (a^2*b*c^2 - a^3*c*d)*x)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 +
a^2)))/((a^3*b*c*x^5 + a^4*c*x^3)*sqrt(-a*b*c + a^2*d)), 1/12*(2*((15*b^2*c - 2
*a*b*d)*x^4 - 2*a^2*c + 2*(5*a*b*c - a^2*d)*x^2)*sqrt(a*b*c - a^2*d)*sqrt(d*x^2
+ c) + 3*((5*b^3*c^2 - 4*a*b^2*c*d)*x^5 + (5*a*b^2*c^2 - 4*a^2*b*c*d)*x^3)*arcta
n(1/2*((b*c - 2*a*d)*x^2 - a*c)/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x)))/((a^3*
b*c*x^5 + a^4*c*x^3)*sqrt(a*b*c - a^2*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{2}}}{x^{4} \left (a + b x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**(1/2)/x**4/(b*x**2+a)**2,x)

[Out]

Integral(sqrt(c + d*x**2)/(x**4*(a + b*x**2)**2), x)

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GIAC/XCAS [A]  time = 7.38974, size = 487, normalized size = 3.31 \[ -\frac{{\left (5 \, b^{2} c \sqrt{d} - 4 \, a b d^{\frac{3}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt{a b c d - a^{2} d^{2}} a^{3}} - \frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c \sqrt{d} - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b d^{\frac{3}{2}} - b^{2} c^{2} \sqrt{d}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} a^{3}} - \frac{2 \,{\left (6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b c \sqrt{d} - 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a d^{\frac{3}{2}} - 12 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c^{2} \sqrt{d} + 6 \, b c^{3} \sqrt{d} - a c^{2} d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/((b*x^2 + a)^2*x^4),x, algorithm="giac")

[Out]

-1/2*(5*b^2*c*sqrt(d) - 4*a*b*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))
^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*a^3) - ((s
qrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c*sqrt(d) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*
a*b*d^(3/2) - b^2*c^2*sqrt(d))/(((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*
x - sqrt(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)*a^3)
- 2/3*(6*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b*c*sqrt(d) - 3*(sqrt(d)*x - sqrt(d*x^
2 + c))^4*a*d^(3/2) - 12*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c^2*sqrt(d) + 6*b*c^3
*sqrt(d) - a*c^2*d^(3/2))/(((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^3*a^3)