3.657 \(\int \frac{\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=145 \[ -\frac{a^2}{4 c x^4 \sqrt{c+d x^2}}-\frac{\left (8 b^2 c^2-3 a d (8 b c-5 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{7/2}}+\frac{8 b^2 c^2-3 a d (8 b c-5 a d)}{8 c^3 \sqrt{c+d x^2}}-\frac{a (8 b c-5 a d)}{8 c^2 x^2 \sqrt{c+d x^2}} \]

[Out]

(8*b^2*c^2 - 3*a*d*(8*b*c - 5*a*d))/(8*c^3*Sqrt[c + d*x^2]) - a^2/(4*c*x^4*Sqrt[
c + d*x^2]) - (a*(8*b*c - 5*a*d))/(8*c^2*x^2*Sqrt[c + d*x^2]) - ((8*b^2*c^2 - 3*
a*d*(8*b*c - 5*a*d))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c^(7/2))

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Rubi [A]  time = 0.43879, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2}{4 c x^4 \sqrt{c+d x^2}}+\frac{8 b^2-\frac{3 a d (8 b c-5 a d)}{c^2}}{8 c \sqrt{c+d x^2}}-\frac{\left (8 b^2 c^2-3 a d (8 b c-5 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{7/2}}-\frac{a (8 b c-5 a d)}{8 c^2 x^2 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/(x^5*(c + d*x^2)^(3/2)),x]

[Out]

(8*b^2 - (3*a*d*(8*b*c - 5*a*d))/c^2)/(8*c*Sqrt[c + d*x^2]) - a^2/(4*c*x^4*Sqrt[
c + d*x^2]) - (a*(8*b*c - 5*a*d))/(8*c^2*x^2*Sqrt[c + d*x^2]) - ((8*b^2*c^2 - 3*
a*d*(8*b*c - 5*a*d))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(8*c^(7/2))

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Rubi in Sympy [A]  time = 29.7308, size = 136, normalized size = 0.94 \[ - \frac{a^{2}}{4 c x^{4} \sqrt{c + d x^{2}}} + \frac{a \left (5 a d - 8 b c\right )}{8 c^{2} x^{2} \sqrt{c + d x^{2}}} + \frac{3 a d \left (5 a d - 8 b c\right ) + 8 b^{2} c^{2}}{8 c^{3} \sqrt{c + d x^{2}}} - \frac{\left (3 a d \left (5 a d - 8 b c\right ) + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{8 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x**5/(d*x**2+c)**(3/2),x)

[Out]

-a**2/(4*c*x**4*sqrt(c + d*x**2)) + a*(5*a*d - 8*b*c)/(8*c**2*x**2*sqrt(c + d*x*
*2)) + (3*a*d*(5*a*d - 8*b*c) + 8*b**2*c**2)/(8*c**3*sqrt(c + d*x**2)) - (3*a*d*
(5*a*d - 8*b*c) + 8*b**2*c**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(8*c**(7/2))

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Mathematica [A]  time = 0.319067, size = 143, normalized size = 0.99 \[ \frac{-\left (15 a^2 d^2-24 a b c d+8 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+\log (x) \left (15 a^2 d^2-24 a b c d+8 b^2 c^2\right )+\sqrt{c} \sqrt{c+d x^2} \left (-\frac{2 a^2 c}{x^4}+\frac{a (7 a d-8 b c)}{x^2}+\frac{8 (b c-a d)^2}{c+d x^2}\right )}{8 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/(x^5*(c + d*x^2)^(3/2)),x]

[Out]

(Sqrt[c]*Sqrt[c + d*x^2]*((-2*a^2*c)/x^4 + (a*(-8*b*c + 7*a*d))/x^2 + (8*(b*c -
a*d)^2)/(c + d*x^2)) + (8*b^2*c^2 - 24*a*b*c*d + 15*a^2*d^2)*Log[x] - (8*b^2*c^2
- 24*a*b*c*d + 15*a^2*d^2)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/(8*c^(7/2))

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Maple [A]  time = 0.018, size = 211, normalized size = 1.5 \[ -{\frac{{a}^{2}}{4\,c{x}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{5\,{a}^{2}d}{8\,{c}^{2}{x}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{15\,{a}^{2}{d}^{2}}{8\,{c}^{3}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{15\,{a}^{2}{d}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{7}{2}}}}+{\frac{{b}^{2}}{c}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{{b}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}-{\frac{ab}{c{x}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-3\,{\frac{abd}{{c}^{2}\sqrt{d{x}^{2}+c}}}+3\,{\frac{abd}{{c}^{5/2}}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^5/(d*x^2+c)^(3/2),x)

[Out]

-1/4*a^2/c/x^4/(d*x^2+c)^(1/2)+5/8*a^2*d/c^2/x^2/(d*x^2+c)^(1/2)+15/8*a^2*d^2/c^
3/(d*x^2+c)^(1/2)-15/8*a^2*d^2/c^(7/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+b^2
/c/(d*x^2+c)^(1/2)-b^2/c^(3/2)*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)-3*a*b*d/c^2/(d*x^2+c)^(1/2)+3*a*b*d/c^(5/2)*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/((d*x^2 + c)^(3/2)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242625, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left ({\left (8 \, b^{2} c^{2} - 24 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} -{\left (8 \, a b c^{2} - 5 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c} +{\left ({\left (8 \, b^{2} c^{2} d - 24 \, a b c d^{2} + 15 \, a^{2} d^{3}\right )} x^{6} +{\left (8 \, b^{2} c^{3} - 24 \, a b c^{2} d + 15 \, a^{2} c d^{2}\right )} x^{4}\right )} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right )}{16 \,{\left (c^{3} d x^{6} + c^{4} x^{4}\right )} \sqrt{c}}, \frac{{\left ({\left (8 \, b^{2} c^{2} - 24 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} -{\left (8 \, a b c^{2} - 5 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c} -{\left ({\left (8 \, b^{2} c^{2} d - 24 \, a b c d^{2} + 15 \, a^{2} d^{3}\right )} x^{6} +{\left (8 \, b^{2} c^{3} - 24 \, a b c^{2} d + 15 \, a^{2} c d^{2}\right )} x^{4}\right )} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right )}{8 \,{\left (c^{3} d x^{6} + c^{4} x^{4}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^5),x, algorithm="fricas")

[Out]

[1/16*(2*((8*b^2*c^2 - 24*a*b*c*d + 15*a^2*d^2)*x^4 - 2*a^2*c^2 - (8*a*b*c^2 - 5
*a^2*c*d)*x^2)*sqrt(d*x^2 + c)*sqrt(c) + ((8*b^2*c^2*d - 24*a*b*c*d^2 + 15*a^2*d
^3)*x^6 + (8*b^2*c^3 - 24*a*b*c^2*d + 15*a^2*c*d^2)*x^4)*log(-((d*x^2 + 2*c)*sqr
t(c) - 2*sqrt(d*x^2 + c)*c)/x^2))/((c^3*d*x^6 + c^4*x^4)*sqrt(c)), 1/8*(((8*b^2*
c^2 - 24*a*b*c*d + 15*a^2*d^2)*x^4 - 2*a^2*c^2 - (8*a*b*c^2 - 5*a^2*c*d)*x^2)*sq
rt(d*x^2 + c)*sqrt(-c) - ((8*b^2*c^2*d - 24*a*b*c*d^2 + 15*a^2*d^3)*x^6 + (8*b^2
*c^3 - 24*a*b*c^2*d + 15*a^2*c*d^2)*x^4)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/((c^3
*d*x^6 + c^4*x^4)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{2}}{x^{5} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x**5/(d*x**2+c)**(3/2),x)

[Out]

Integral((a + b*x**2)**2/(x**5*(c + d*x**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.243229, size = 220, normalized size = 1.52 \[ \frac{{\left (8 \, b^{2} c^{2} - 24 \, a b c d + 15 \, a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{8 \, \sqrt{-c} c^{3}} + \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{\sqrt{d x^{2} + c} c^{3}} - \frac{8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d - 8 \, \sqrt{d x^{2} + c} a b c^{2} d - 7 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{2} + 9 \, \sqrt{d x^{2} + c} a^{2} c d^{2}}{8 \, c^{3} d^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^5),x, algorithm="giac")

[Out]

1/8*(8*b^2*c^2 - 24*a*b*c*d + 15*a^2*d^2)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(sqrt
(-c)*c^3) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)/(sqrt(d*x^2 + c)*c^3) - 1/8*(8*(d*x^
2 + c)^(3/2)*a*b*c*d - 8*sqrt(d*x^2 + c)*a*b*c^2*d - 7*(d*x^2 + c)^(3/2)*a^2*d^2
+ 9*sqrt(d*x^2 + c)*a^2*c*d^2)/(c^3*d^2*x^4)