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

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 = 0.0472239, 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, Rules used = {1165, 628} \[ \frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{2 \sqrt{2} \sqrt [4]{a}}-\frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{2 \sqrt{2} \sqrt [4]{a}} \]

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))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx &=-\frac{\sqrt [4]{b} \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{2 \sqrt{2} \sqrt [4]{a}}-\frac{\sqrt [4]{b} \int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{2 \sqrt{2} \sqrt [4]{a}}\\ &=-\frac{\sqrt [4]{b} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{2 \sqrt{2} \sqrt [4]{a}}+\frac{\sqrt [4]{b} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{2 \sqrt{2} \sqrt [4]{a}}\\ \end{align*}

Mathematica [A]  time = 0.0223496, 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 = 0.048, size = 254, normalized size = 2.4 \begin{align*}{\frac{\sqrt{2}}{8}\sqrt{b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \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 ({x\sqrt{2}{\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 ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt{a}}}}-{\frac{\sqrt{2}}{8}\ln \left ({ \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 ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{4}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48736, size = 462, normalized size = 4.36 \begin{align*} \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 (\sqrt{\frac{1}{2}} x \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 (\sqrt{a} \sqrt{b} x^{3} - a x\right )} \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}}}{a}\right )\right ] \end{align*}

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, 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)*x*sqrt(-sqrt(b
)/sqrt(a))) + sqrt(1/2)*sqrt(-sqrt(b)/sqrt(a))*arctan(sqrt(1/2)*(sqrt(a)*sqrt(b)*x^3 - a*x)*sqrt(-sqrt(b)/sqrt
(a))/a)]

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Sympy [A]  time = 0.534222, size = 131, normalized size = 1.24 \begin{align*} - \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} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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, algorithm="giac")

[Out]

Exception raised: TypeError