3.18 \(\int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx\)

Optimal. Leaf size=106 \[ \frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{2 \sqrt{2} \sqrt [4]{a}}-\frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{2 \sqrt{2} \sqrt [4]{a}} \]

[Out]

-(b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(2*Sqrt[2]*a^(
1/4)) + (b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(2*Sqrt
[2]*a^(1/4))

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Rubi [A]  time = 0.0875173, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{2 \sqrt{2} \sqrt [4]{a}}-\frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{2 \sqrt{2} \sqrt [4]{a}} \]

Antiderivative was successfully verified.

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

[Out]

-(b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(2*Sqrt[2]*a^(
1/4)) + (b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(2*Sqrt
[2]*a^(1/4))

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Rubi in Sympy [A]  time = 37.8359, size = 100, normalized size = 0.94 \[ - \frac{\sqrt{2} \sqrt [4]{b} \log{\left (- \sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{4 \sqrt [4]{a}} + \frac{\sqrt{2} \sqrt [4]{b} \log{\left (\sqrt{2} \sqrt [4]{a} b^{\frac{3}{4}} x + \sqrt{a} \sqrt{b} + b x^{2} \right )}}{4 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+a**(1/2)*b**(1/2))/(b*x**4+a),x)

[Out]

-sqrt(2)*b**(1/4)*log(-sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b) + b*x**2)/(
4*a**(1/4)) + sqrt(2)*b**(1/4)*log(sqrt(2)*a**(1/4)*b**(3/4)*x + sqrt(a)*sqrt(b)
+ b*x**2)/(4*a**(1/4))

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Mathematica [A]  time = 0.0422349, size = 91, normalized size = 0.86 \[ \frac{\sqrt [4]{b} \left (\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x-\sqrt{a}-\sqrt{b} x^2\right )\right )}{2 \sqrt{2} \sqrt [4]{a}} \]

Antiderivative was successfully verified.

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

[Out]

(b^(1/4)*(-Log[-Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x - Sqrt[b]*x^2] + Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2]))/(2*Sqrt[2]*a^(1/4))

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Maple [B]  time = 0.006, size = 254, normalized size = 2.4 \[{\frac{\sqrt{2}}{8}\sqrt{b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{a}}}}+{\frac{\sqrt{2}}{4}\sqrt{b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt{a}}}}+{\frac{\sqrt{2}}{4}\sqrt{b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt{a}}}}-{\frac{\sqrt{2}}{8}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{4}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{4}\arctan \left ({\sqrt{2}x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a^(1/2)*b^(1/2))/(b*x^4+a),x)

[Out]

1/8/a^(1/2)*b^(1/2)*(a/b)^(1/4)*2^(1/2)*ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2
))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+1/4/a^(1/2)*b^(1/2)*(a/b)^(1/4)*2^(1
/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+1/4/a^(1/2)*b^(1/2)*(a/b)^(1/4)*2^(1/2)*arct
an(2^(1/2)/(a/b)^(1/4)*x-1)-1/8/(a/b)^(1/4)*2^(1/2)*ln((x^2-(a/b)^(1/4)*x*2^(1/2
)+(a/b)^(1/2))/(x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))-1/4/(a/b)^(1/4)*2^(1/2)*
arctan(2^(1/2)/(a/b)^(1/4)*x+1)-1/4/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/
4)*x-1)

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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 - sqrt(a)*sqrt(b))/(b*x^4 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.329648, size = 1, normalized size = 0.01 \[ \left [\frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{\sqrt{b}}{\sqrt{a}}} \log \left (\frac{b x^{4} + 4 \, \sqrt{a} \sqrt{b} x^{2} + 4 \, \sqrt{\frac{1}{2}}{\left (\sqrt{a} \sqrt{b} x^{3} + a x\right )} \sqrt{\frac{\sqrt{b}}{\sqrt{a}}} + a}{b x^{4} + a}\right ), \sqrt{\frac{1}{2}} \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{b} x}{\sqrt{a} \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}}}\right ) + \sqrt{\frac{1}{2}} \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \arctan \left (-\frac{\sqrt{\frac{1}{2}}{\left (b x^{3} - \sqrt{a} \sqrt{b} x\right )}}{a \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*sqrt(1/2)*sqrt(sqrt(b)/sqrt(a))*log((b*x^4 + 4*sqrt(a)*sqrt(b)*x^2 + 4*sqrt
(1/2)*(sqrt(a)*sqrt(b)*x^3 + a*x)*sqrt(sqrt(b)/sqrt(a)) + a)/(b*x^4 + a)), sqrt(
1/2)*sqrt(-sqrt(b)/sqrt(a))*arctan(sqrt(1/2)*sqrt(b)*x/(sqrt(a)*sqrt(-sqrt(b)/sq
rt(a)))) + sqrt(1/2)*sqrt(-sqrt(b)/sqrt(a))*arctan(-sqrt(1/2)*(b*x^3 - sqrt(a)*s
qrt(b)*x)/(a*sqrt(-sqrt(b)/sqrt(a))))]

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Sympy [A]  time = 1.94154, size = 131, normalized size = 1.24 \[ - \frac{\sqrt{2} \sqrt{\frac{\sqrt{b}}{\sqrt{a}}} \log{\left (- \frac{\sqrt{2} \sqrt{a} x \sqrt{\frac{\sqrt{b}}{\sqrt{a}}}}{\sqrt{b}} + \frac{\sqrt{a}}{\sqrt{b}} + x^{2} \right )}}{4} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{b}}{\sqrt{a}}} \log{\left (\frac{\sqrt{2} \sqrt{a} x \sqrt{\frac{\sqrt{b}}{\sqrt{a}}}}{\sqrt{b}} + \frac{\sqrt{a}}{\sqrt{b}} + x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+a**(1/2)*b**(1/2))/(b*x**4+a),x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError