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

Optimal. Leaf size=75 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{(b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 \sqrt{c+d x^2}}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.206644, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{c^{3/2}}+\frac{(b c-a d)^2}{c d^2 \sqrt{c+d x^2}}+\frac{b^2 \sqrt{c+d x^2}}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.416, size = 88, normalized size = 1.17 \[ \frac{a^{2} \sqrt{c + d x^{2}}}{c^{2}} - \frac{a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{c^{\frac{3}{2}}} - \sqrt{c + d x^{2}} \left (\frac{a^{2}}{c^{2}} - \frac{b^{2}}{d^{2}}\right ) + \frac{\left (a d - b c\right )^{2}}{c d^{2} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.190104, size = 88, normalized size = 1.17 \[ -\frac{a^2 \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{c^{3/2}}+\frac{a^2 \log (x)}{c^{3/2}}+\sqrt{c+d x^2} \left (\frac{(b c-a d)^2}{c d^2 \left (c+d x^2\right )}+\frac{b^2}{d^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 102, normalized size = 1.4 \[{\frac{{a}^{2}}{c}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{{a}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}+{\frac{{b}^{2}{x}^{2}}{d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+2\,{\frac{{b}^{2}c}{{d}^{2}\sqrt{d{x}^{2}+c}}}-2\,{\frac{ab}{d\sqrt{d{x}^{2}+c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237702, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (b^{2} c d x^{2} + 2 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c} +{\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right )}{2 \,{\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \sqrt{c}}, \frac{{\left (b^{2} c d x^{2} + 2 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c} -{\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right )}{{\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.236193, size = 111, normalized size = 1.48 \[ \frac{a^{2} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} + \frac{\sqrt{d x^{2} + c} b^{2}}{d^{2}} + \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{\sqrt{d x^{2} + c} c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(sqrt(-c)*c) + sqrt(d*x^2 + c)*b^2/d^2 + (b
^2*c^2 - 2*a*b*c*d + a^2*d^2)/(sqrt(d*x^2 + c)*c*d^2)