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

Optimal. Leaf size=91 \[ -\frac{a^2}{c x \sqrt{c+d x^2}}-\frac{x \left (b^2 c^2-2 a d (b c-a d)\right )}{c^2 d \sqrt{c+d x^2}}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.184729, antiderivative size = 87, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{a^2}{c x \sqrt{c+d x^2}}-\frac{x \left (\frac{b^2}{d}-\frac{2 a (b c-a d)}{c^2}\right )}{\sqrt{c+d x^2}}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.8182, size = 78, normalized size = 0.86 \[ - \frac{a^{2}}{c x \sqrt{c + d x^{2}}} + \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{d^{\frac{3}{2}}} - \frac{x \left (2 a d \left (a d - b c\right ) + b^{2} c^{2}\right )}{c^{2} d \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.138154, size = 81, normalized size = 0.89 \[ \frac{b^2 \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{d^{3/2}}-\frac{\sqrt{c+d x^2} \left (a^2+\frac{x^2 (b c-a d)^2}{d \left (c+d x^2\right )}\right )}{c^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 99, normalized size = 1.1 \[ -{\frac{{b}^{2}x}{d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{{b}^{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}-{\frac{{a}^{2}}{cx}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-2\,{\frac{{a}^{2}dx}{{c}^{2}\sqrt{d{x}^{2}+c}}}+2\,{\frac{abx}{c\sqrt{d{x}^{2}+c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^2*x/d/(d*x^2+c)^(1/2)+b^2/d^(3/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-a^2/c/x/(d*x^
2+c)^(1/2)-2*a^2*d/c^2*x/(d*x^2+c)^(1/2)+2*a*b*x/c/(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/((d*x^2 + c)^(3/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236213, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (a^{2} c d +{\left (b^{2} c^{2} - 2 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{d} -{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{2 \,{\left (c^{2} d^{2} x^{3} + c^{3} d x\right )} \sqrt{d}}, -\frac{{\left (a^{2} c d +{\left (b^{2} c^{2} - 2 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-d} -{\left (b^{2} c^{2} d x^{3} + b^{2} c^{3} x\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{{\left (c^{2} d^{2} x^{3} + c^{3} d x\right )} \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.24361, size = 140, normalized size = 1.54 \[ -\frac{b^{2}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{2 \, d^{\frac{3}{2}}} + \frac{2 \, a^{2} \sqrt{d}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )} c} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{\sqrt{d x^{2} + c} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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