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

Optimal. Leaf size=84 \[ -\frac{a^2 \sqrt{c+d x^2}}{3 c x^3}-\frac{2 a \sqrt{c+d x^2} (3 b c-a d)}{3 c^2 x}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{\sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.184589, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{a^2 \sqrt{c+d x^2}}{3 c x^3}-\frac{2 a \sqrt{c+d x^2} (3 b c-a d)}{3 c^2 x}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{\sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.214, size = 75, normalized size = 0.89 \[ - \frac{a^{2} \sqrt{c + d x^{2}}}{3 c x^{3}} + \frac{2 a \sqrt{c + d x^{2}} \left (a d - 3 b c\right )}{3 c^{2} x} + \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{\sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.117537, size = 72, normalized size = 0.86 \[ \frac{b^2 \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}-\frac{a \sqrt{c+d x^2} \left (a \left (c-2 d x^2\right )+6 b c x^2\right )}{3 c^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(a*Sqrt[c + d*x^2]*(6*b*c*x^2 + a*(c - 2*d*x^2)))/(3*c^2*x^3) + (b^2*Log[d*x +
Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d]

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Maple [A]  time = 0.015, size = 85, normalized size = 1. \[{{b}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}}-{\frac{{a}^{2}}{3\,c{x}^{3}}\sqrt{d{x}^{2}+c}}+{\frac{2\,{a}^{2}d}{3\,{c}^{2}x}\sqrt{d{x}^{2}+c}}-2\,{\frac{\sqrt{d{x}^{2}+c}ab}{cx}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312928, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} c^{2} x^{3} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) - 2 \,{\left (a^{2} c + 2 \,{\left (3 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{d}}{6 \, c^{2} \sqrt{d} x^{3}}, \frac{3 \, b^{2} c^{2} x^{3} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) -{\left (a^{2} c + 2 \,{\left (3 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{3 \, c^{2} \sqrt{-d} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*b^2*c^2*x^3*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)) - 2*(a^2
*c + 2*(3*a*b*c - a^2*d)*x^2)*sqrt(d*x^2 + c)*sqrt(d))/(c^2*sqrt(d)*x^3), 1/3*(3
*b^2*c^2*x^3*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - (a^2*c + 2*(3*a*b*c - a^2*d)*x
^2)*sqrt(d*x^2 + c)*sqrt(-d))/(c^2*sqrt(-d)*x^3)]

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Sympy [A]  time = 5.53914, size = 158, normalized size = 1.88 \[ - \frac{a^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 c x^{2}} + \frac{2 a^{2} d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3 c^{2}} - \frac{2 a b \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{c} + b^{2} \left (\begin{cases} \frac{\sqrt{- \frac{c}{d}} \operatorname{asin}{\left (x \sqrt{- \frac{d}{c}} \right )}}{\sqrt{c}} & \text{for}\: c > 0 \wedge d < 0 \\\frac{\sqrt{\frac{c}{d}} \operatorname{asinh}{\left (x \sqrt{\frac{d}{c}} \right )}}{\sqrt{c}} & \text{for}\: c > 0 \wedge d > 0 \\\frac{\sqrt{- \frac{c}{d}} \operatorname{acosh}{\left (x \sqrt{- \frac{d}{c}} \right )}}{\sqrt{- c}} & \text{for}\: d > 0 \wedge c < 0 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(3*c*x**2) + 2*a**2*d**(3/2)*sqrt(c/(d*x**2)
+ 1)/(3*c**2) - 2*a*b*sqrt(d)*sqrt(c/(d*x**2) + 1)/c + b**2*Piecewise((sqrt(-c/d
)*asin(x*sqrt(-d/c))/sqrt(c), (c > 0) & (d < 0)), (sqrt(c/d)*asinh(x*sqrt(d/c))/
sqrt(c), (c > 0) & (d > 0)), (sqrt(-c/d)*acosh(x*sqrt(-d/c))/sqrt(-c), (d > 0) &
(c < 0)))

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GIAC/XCAS [A]  time = 0.252527, size = 211, normalized size = 2.51 \[ -\frac{b^{2}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{2 \, \sqrt{d}} + \frac{4 \,{\left (3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b \sqrt{d} - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c \sqrt{d} + 3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} d^{\frac{3}{2}} + 3 \, a b c^{2} \sqrt{d} - a^{2} c d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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