3.86 \(\int \frac{d+e x^2}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=223 \[ -\frac{b d-a e}{4 a^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\log (x) \left (a+b x^2\right ) (3 b d-a e)}{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (3 b d-a e) \log \left (a+b x^2\right )}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 b d-a e}{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{2 a^3 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.429396, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{b d-a e}{4 a^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\log (x) \left (a+b x^2\right ) (3 b d-a e)}{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (3 b d-a e) \log \left (a+b x^2\right )}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 b d-a e}{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{2 a^3 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2)/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 59.0125, size = 223, normalized size = 1. \[ \frac{\left (a e - b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{4 a^{2} \left (a + b x^{2}\right )^{3}} - \frac{d \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{2 a^{3} x^{2} \left (a + b x^{2}\right )} + \frac{\left (a e - 2 b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{2 a^{3} \left (a + b x^{2}\right )^{2}} + \frac{\left (a e - 3 b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \log{\left (x^{2} \right )}}{2 a^{4} \left (a + b x^{2}\right )} - \frac{\left (a e - 3 b d\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \log{\left (a + b x^{2} \right )}}{2 a^{4} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d)/x**3/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

(a*e - b*d)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(4*a**2*(a + b*x**2)**3) - d*sqr
t(a**2 + 2*a*b*x**2 + b**2*x**4)/(2*a**3*x**2*(a + b*x**2)) + (a*e - 2*b*d)*sqrt
(a**2 + 2*a*b*x**2 + b**2*x**4)/(2*a**3*(a + b*x**2)**2) + (a*e - 3*b*d)*sqrt(a*
*2 + 2*a*b*x**2 + b**2*x**4)*log(x**2)/(2*a**4*(a + b*x**2)) - (a*e - 3*b*d)*sqr
t(a**2 + 2*a*b*x**2 + b**2*x**4)*log(a + b*x**2)/(2*a**4*(a + b*x**2))

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Mathematica [A]  time = 0.129722, size = 130, normalized size = 0.58 \[ \frac{a \left (a^2 \left (3 e x^2-2 d\right )+a b \left (2 e x^4-9 d x^2\right )-6 b^2 d x^4\right )+4 x^2 \log (x) \left (a+b x^2\right )^2 (a e-3 b d)+2 x^2 \left (a+b x^2\right )^2 (3 b d-a e) \log \left (a+b x^2\right )}{4 a^4 x^2 \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2)/(x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)),x]

[Out]

(a*(-6*b^2*d*x^4 + a^2*(-2*d + 3*e*x^2) + a*b*(-9*d*x^2 + 2*e*x^4)) + 4*(-3*b*d
+ a*e)*x^2*(a + b*x^2)^2*Log[x] + 2*(3*b*d - a*e)*x^2*(a + b*x^2)^2*Log[a + b*x^
2])/(4*a^4*x^2*(a + b*x^2)*Sqrt[(a + b*x^2)^2])

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Maple [A]  time = 0.033, size = 249, normalized size = 1.1 \[ -{\frac{ \left ( 2\,\ln \left ( b{x}^{2}+a \right ){x}^{6}a{b}^{2}e-6\,\ln \left ( b{x}^{2}+a \right ){x}^{6}{b}^{3}d-4\,\ln \left ( x \right ){x}^{6}a{b}^{2}e+12\,\ln \left ( x \right ){x}^{6}{b}^{3}d+4\,\ln \left ( b{x}^{2}+a \right ){x}^{4}{a}^{2}be-12\,\ln \left ( b{x}^{2}+a \right ){x}^{4}a{b}^{2}d-8\,\ln \left ( x \right ){x}^{4}{a}^{2}be+24\,\ln \left ( x \right ){x}^{4}a{b}^{2}d-2\,{x}^{4}{a}^{2}be+6\,{x}^{4}a{b}^{2}d+2\,\ln \left ( b{x}^{2}+a \right ){x}^{2}{a}^{3}e-6\,\ln \left ( b{x}^{2}+a \right ){x}^{2}{a}^{2}bd-4\,\ln \left ( x \right ){x}^{2}{a}^{3}e+12\,\ln \left ( x \right ){x}^{2}{a}^{2}bd-3\,{x}^{2}{a}^{3}e+9\,{x}^{2}{a}^{2}bd+2\,{a}^{3}d \right ) \left ( b{x}^{2}+a \right ) }{4\,{x}^{2}{a}^{4}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)/x^3/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

-1/4*(2*ln(b*x^2+a)*x^6*a*b^2*e-6*ln(b*x^2+a)*x^6*b^3*d-4*ln(x)*x^6*a*b^2*e+12*l
n(x)*x^6*b^3*d+4*ln(b*x^2+a)*x^4*a^2*b*e-12*ln(b*x^2+a)*x^4*a*b^2*d-8*ln(x)*x^4*
a^2*b*e+24*ln(x)*x^4*a*b^2*d-2*x^4*a^2*b*e+6*x^4*a*b^2*d+2*ln(b*x^2+a)*x^2*a^3*e
-6*ln(b*x^2+a)*x^2*a^2*b*d-4*ln(x)*x^2*a^3*e+12*ln(x)*x^2*a^2*b*d-3*x^2*a^3*e+9*
x^2*a^2*b*d+2*a^3*d)*(b*x^2+a)/x^2/a^4/((b*x^2+a)^2)^(3/2)

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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((e*x^2 + d)/((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274007, size = 277, normalized size = 1.24 \[ -\frac{2 \,{\left (3 \, a b^{2} d - a^{2} b e\right )} x^{4} + 2 \, a^{3} d + 3 \,{\left (3 \, a^{2} b d - a^{3} e\right )} x^{2} - 2 \,{\left ({\left (3 \, b^{3} d - a b^{2} e\right )} x^{6} + 2 \,{\left (3 \, a b^{2} d - a^{2} b e\right )} x^{4} +{\left (3 \, a^{2} b d - a^{3} e\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left ({\left (3 \, b^{3} d - a b^{2} e\right )} x^{6} + 2 \,{\left (3 \, a b^{2} d - a^{2} b e\right )} x^{4} +{\left (3 \, a^{2} b d - a^{3} e\right )} x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)/((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^3),x, algorithm="fricas")

[Out]

-1/4*(2*(3*a*b^2*d - a^2*b*e)*x^4 + 2*a^3*d + 3*(3*a^2*b*d - a^3*e)*x^2 - 2*((3*
b^3*d - a*b^2*e)*x^6 + 2*(3*a*b^2*d - a^2*b*e)*x^4 + (3*a^2*b*d - a^3*e)*x^2)*lo
g(b*x^2 + a) + 4*((3*b^3*d - a*b^2*e)*x^6 + 2*(3*a*b^2*d - a^2*b*e)*x^4 + (3*a^2
*b*d - a^3*e)*x^2)*log(x))/(a^4*b^2*x^6 + 2*a^5*b*x^4 + a^6*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**2+d)/x**3/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.604119, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)/((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^3),x, algorithm="giac")

[Out]

sage0*x