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

Optimal. Leaf size=83 \[ \frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.191917, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{\left (a+b x^2\right ) (b d-a e) \log \left (a+b x^2\right )}{2 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{e \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.5963, size = 76, normalized size = 0.92 \[ \frac{e \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{2 b^{2}} - \frac{\left (a + b x^{2}\right ) \left (a e - b d\right ) \log{\left (a + b x^{2} \right )}}{2 b^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0390962, size = 51, normalized size = 0.61 \[ \frac{\left (a+b x^2\right ) \left ((b d-a e) \log \left (a+b x^2\right )+b e x^2\right )}{2 b^2 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 55, normalized size = 0.7 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( -{x}^{2}be+\ln \left ( b{x}^{2}+a \right ) ae-\ln \left ( b{x}^{2}+a \right ) bd \right ) }{2\,{b}^{2}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.71101, size = 92, normalized size = 1.11 \[ \frac{1}{2} \, \sqrt{\frac{1}{b^{2}}} d \log \left (x^{2} + \frac{a}{b}\right ) - \frac{1}{2} \,{\left (\frac{a \sqrt{\frac{1}{b^{2}}} \log \left (x^{2} + \frac{a}{b}\right )}{b} - \frac{\sqrt{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}}{b^{2}}\right )} e \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(b^(-2))*d*log(x^2 + a/b) - 1/2*(a*sqrt(b^(-2))*log(x^2 + a/b)/b - sqrt(
b^2*x^4 + 2*a*b*x^2 + a^2)/b^2)*e

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Fricas [A]  time = 0.28958, size = 39, normalized size = 0.47 \[ \frac{b e x^{2} +{\left (b d - a e\right )} \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.51513, size = 27, normalized size = 0.33 \[ \frac{e x^{2}}{2 b} - \frac{\left (a e - b d\right ) \log{\left (a + b x^{2} \right )}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.263942, size = 57, normalized size = 0.69 \[ \frac{1}{2} \,{\left (\frac{x^{2} e}{b} + \frac{{\left (b d - a e\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{b^{2}}\right )}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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