3.645 \(\int \frac{\left (a+b x^2\right )^2}{x^5 \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{5/2}}-\frac{a^2 \sqrt{c+d x^2}}{4 c x^4}-\frac{a \sqrt{c+d x^2} (8 b c-3 a d)}{8 c^2 x^2} \]

[Out]

-(a^2*Sqrt[c + d*x^2])/(4*c*x^4) - (a*(8*b*c - 3*a*d)*Sqrt[c + d*x^2])/(8*c^2*x^
2) - ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c
^(5/2))

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Rubi [A]  time = 0.294867, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{5/2}}-\frac{a^2 \sqrt{c+d x^2}}{4 c x^4}-\frac{a \sqrt{c+d x^2} (8 b c-3 a d)}{8 c^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/(x^5*Sqrt[c + d*x^2]),x]

[Out]

-(a^2*Sqrt[c + d*x^2])/(4*c*x^4) - (a*(8*b*c - 3*a*d)*Sqrt[c + d*x^2])/(8*c^2*x^
2) - ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c
^(5/2))

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Rubi in Sympy [A]  time = 24.5153, size = 95, normalized size = 0.9 \[ - \frac{a^{2} \sqrt{c + d x^{2}}}{4 c x^{4}} + \frac{a \sqrt{c + d x^{2}} \left (3 a d - 8 b c\right )}{8 c^{2} x^{2}} - \frac{\left (a d \left (3 a d - 8 b c\right ) + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{8 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x**5/(d*x**2+c)**(1/2),x)

[Out]

-a**2*sqrt(c + d*x**2)/(4*c*x**4) + a*sqrt(c + d*x**2)*(3*a*d - 8*b*c)/(8*c**2*x
**2) - (a*d*(3*a*d - 8*b*c) + 8*b**2*c**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(8*c*
*(5/2))

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Mathematica [A]  time = 0.150055, size = 126, normalized size = 1.19 \[ \frac{x^4 \log (x) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )+x^4 \left (-3 a^2 d^2+8 a b c d-8 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+a \sqrt{c} \sqrt{c+d x^2} \left (-2 a c+3 a d x^2-8 b c x^2\right )}{8 c^{5/2} x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/(x^5*Sqrt[c + d*x^2]),x]

[Out]

(a*Sqrt[c]*Sqrt[c + d*x^2]*(-2*a*c - 8*b*c*x^2 + 3*a*d*x^2) + (8*b^2*c^2 - 8*a*b
*c*d + 3*a^2*d^2)*x^4*Log[x] + (-8*b^2*c^2 + 8*a*b*c*d - 3*a^2*d^2)*x^4*Log[c +
Sqrt[c]*Sqrt[c + d*x^2]])/(8*c^(5/2)*x^4)

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Maple [A]  time = 0.016, size = 157, normalized size = 1.5 \[ -{\frac{{a}^{2}}{4\,c{x}^{4}}\sqrt{d{x}^{2}+c}}+{\frac{3\,{a}^{2}d}{8\,{c}^{2}{x}^{2}}\sqrt{d{x}^{2}+c}}-{\frac{3\,{a}^{2}{d}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}}-{{b}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}-{\frac{ab}{c{x}^{2}}\sqrt{d{x}^{2}+c}}+{abd\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^5/(d*x^2+c)^(1/2),x)

[Out]

-1/4*a^2*(d*x^2+c)^(1/2)/c/x^4+3/8*a^2*d/c^2/x^2*(d*x^2+c)^(1/2)-3/8*a^2*d^2/c^(
5/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-b^2/c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^2+
c)^(1/2))/x)-a*b/c/x^2*(d*x^2+c)^(1/2)+a*b*d/c^(3/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)
^(1/2))/x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261159, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \,{\left (2 \, a^{2} c +{\left (8 \, a b c - 3 \, a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c}}{16 \, c^{\frac{5}{2}} x^{4}}, -\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) +{\left (2 \, a^{2} c +{\left (8 \, a b c - 3 \, a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c}}{8 \, \sqrt{-c} c^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*x^5),x, algorithm="fricas")

[Out]

[1/16*((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x^4*log(-((d*x^2 + 2*c)*sqrt(c) - 2*s
qrt(d*x^2 + c)*c)/x^2) - 2*(2*a^2*c + (8*a*b*c - 3*a^2*d)*x^2)*sqrt(d*x^2 + c)*s
qrt(c))/(c^(5/2)*x^4), -1/8*((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x^4*arctan(sqrt
(-c)/sqrt(d*x^2 + c)) + (2*a^2*c + (8*a*b*c - 3*a^2*d)*x^2)*sqrt(d*x^2 + c)*sqrt
(-c))/(sqrt(-c)*c^2*x^4)]

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Sympy [A]  time = 74.4372, size = 178, normalized size = 1.68 \[ - \frac{a^{2}}{4 \sqrt{d} x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} \sqrt{d}}{8 c x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{3 a^{2} d^{\frac{3}{2}}}{8 c^{2} x \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a^{2} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{8 c^{\frac{5}{2}}} - \frac{a b \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{c x} + \frac{a b d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{c^{\frac{3}{2}}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x**5/(d*x**2+c)**(1/2),x)

[Out]

-a**2/(4*sqrt(d)*x**5*sqrt(c/(d*x**2) + 1)) + a**2*sqrt(d)/(8*c*x**3*sqrt(c/(d*x
**2) + 1)) + 3*a**2*d**(3/2)/(8*c**2*x*sqrt(c/(d*x**2) + 1)) - 3*a**2*d**2*asinh
(sqrt(c)/(sqrt(d)*x))/(8*c**(5/2)) - a*b*sqrt(d)*sqrt(c/(d*x**2) + 1)/(c*x) + a*
b*d*asinh(sqrt(c)/(sqrt(d)*x))/c**(3/2) - b**2*asinh(sqrt(c)/(sqrt(d)*x))/sqrt(c
)

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GIAC/XCAS [A]  time = 0.239426, size = 189, normalized size = 1.78 \[ \frac{\frac{{\left (8 \, b^{2} c^{2} d - 8 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} - \frac{8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d^{2} - 8 \, \sqrt{d x^{2} + c} a b c^{2} d^{2} - 3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{3} + 5 \, \sqrt{d x^{2} + c} a^{2} c d^{3}}{c^{2} d^{2} x^{4}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*x^5),x, algorithm="giac")

[Out]

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