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

Optimal. Leaf size=75 \[ \frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a}} \]

[Out]

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

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Rubi [A]  time = 0.0367142, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1162, 617, 204} \[ \frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx &=\frac{1}{2} \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx+\frac{1}{2} \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx\\ &=\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a}}-\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a}}\\ &=-\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a}}\\ \end{align*}

Mathematica [A]  time = 0.0188539, size = 60, normalized size = 0.8 \[ \frac{\sqrt [4]{b} \left (\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )-\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right )}{\sqrt{2} \sqrt [4]{a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.047, size = 254, normalized size = 3.4 \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 = 1.50566, size = 458, normalized size = 6.11 \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 = 0.524981, size = 138, normalized size = 1.84 \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