3.80 \(\int \frac{d+e x^2}{x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=137 \[ -\frac{\log (x) \left (a+b x^2\right ) (b d-a e)}{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{2 a x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.262672, antiderivative size = 137, 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 \[ -\frac{\log (x) \left (a+b x^2\right ) (b d-a e)}{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \left (a+b x^2\right )}{2 a x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d)/x**3/((b*x**2+a)**2)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0560914, size = 70, normalized size = 0.51 \[ \frac{\left (a+b x^2\right ) \left (2 x^2 \log (x) (a e-b d)+x^2 (b d-a e) \log \left (a+b x^2\right )-a d\right )}{2 a^2 x^2 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 78, normalized size = 0.6 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( \ln \left ( b{x}^{2}+a \right ){x}^{2}ae-\ln \left ( b{x}^{2}+a \right ){x}^{2}bd-2\,\ln \left ( x \right ){x}^{2}ae+2\,\ln \left ( x \right ){x}^{2}bd+ad \right ) }{2\,{x}^{2}{a}^{2}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)/x^3/((b*x^2+a)^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266636, size = 65, normalized size = 0.47 \[ \frac{{\left (b d - a e\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \,{\left (b d - a e\right )} x^{2} \log \left (x\right ) - a d}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*((b*d - a*e)*x^2*log(b*x^2 + a) - 2*(b*d - a*e)*x^2*log(x) - a*d)/(a^2*x^2)

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Sympy [A]  time = 2.9685, size = 41, normalized size = 0.3 \[ - \frac{d}{2 a x^{2}} + \frac{\left (a e - b d\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a e - b d\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**2+d)/x**3/((b*x**2+a)**2)**(1/2),x)

[Out]

-d/(2*a*x**2) + (a*e - b*d)*log(x)/a**2 - (a*e - b*d)*log(a/b + x**2)/(2*a**2)

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GIAC/XCAS [A]  time = 0.265249, size = 177, normalized size = 1.29 \[ -\frac{{\left (b d{\rm sign}\left (b x^{2} + a\right ) - a e{\rm sign}\left (b x^{2} + a\right )\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{2} d{\rm sign}\left (b x^{2} + a\right ) - a b e{\rm sign}\left (b x^{2} + a\right )\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} + \frac{b d x^{2}{\rm sign}\left (b x^{2} + a\right ) - a x^{2} e{\rm sign}\left (b x^{2} + a\right ) - a d{\rm sign}\left (b x^{2} + a\right )}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b*d*sign(b*x^2 + a) - a*e*sign(b*x^2 + a))*ln(x^2)/a^2 + 1/2*(b^2*d*sign(b
*x^2 + a) - a*b*e*sign(b*x^2 + a))*ln(abs(b*x^2 + a))/(a^2*b) + 1/2*(b*d*x^2*sig
n(b*x^2 + a) - a*x^2*e*sign(b*x^2 + a) - a*d*sign(b*x^2 + a))/(a^2*x^2)