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]

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

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Rubi [A]  time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.02, size = 60, normalized size = 0.80 \[ \frac {\sqrt [4]{b} \left (\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )-\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right )}{\sqrt {2} \sqrt [4]{a}} \]

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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fricas [A]  time = 0.64, size = 148, normalized size = 1.97 \[ \left [\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} \log \left (\frac {b x^{4} - 4 \, \sqrt {a} \sqrt {b} x^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (\sqrt {a} \sqrt {b} x^{3} - a x\right )} \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} + a}{b x^{4} + a}\right ), \sqrt {\frac {1}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}} \arctan \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\right ) + \sqrt {\frac {1}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}} \arctan \left (\frac {\sqrt {\frac {1}{2}} {\left (\sqrt {a} \sqrt {b} x^{3} + a x\right )} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}{a}\right )\right ] \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.00, size = 254, normalized size = 3.39 \[ \frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {b}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 \sqrt {a}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {b}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 \sqrt {a}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {b}\, \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{8 \sqrt {a}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}}} \]

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)*2^(1/2)*x+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*2^(1/2)*x+
(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)*arctan(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)*2^(1/2)*x+(a/b)^(1
/2))/(x^2+(a/b)^(1/4)*2^(1/2)*x+(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 [A]  time = 2.31, size = 100, normalized size = 1.33 \[ \frac {\sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {\sqrt {a} \sqrt {b}}} \]

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]

1/2*sqrt(2)*sqrt(b)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/sqrt(sqr
t(a)*sqrt(b)) + 1/2*sqrt(2)*sqrt(b)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sq
rt(b)))/sqrt(sqrt(a)*sqrt(b))

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mupad [B]  time = 4.79, size = 57, normalized size = 0.76 \[ \frac {\sqrt {2}\,b^{1/4}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/4}\,x}{2\,a^{1/4}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{3/4}\,x^3}{2\,a^{3/4}}+\frac {\sqrt {2}\,b^{1/4}\,x}{2\,a^{1/4}}\right )\right )}{4\,a^{1/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + a^(1/2)*b^(1/2))/(a + b*x^4),x)

[Out]

(2^(1/2)*b^(1/4)*(2*atan((2^(1/2)*b^(1/4)*x)/(2*a^(1/4))) + 2*atan((2^(1/2)*b^(3/4)*x^3)/(2*a^(3/4)) + (2^(1/2
)*b^(1/4)*x)/(2*a^(1/4)))))/(4*a^(1/4))

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sympy [A]  time = 0.39, size = 138, normalized size = 1.84 \[ - \frac {\sqrt {2} \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}} \log {\left (- \frac {\sqrt {2} \sqrt {a} x \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {b}} - \frac {\sqrt {a}}{\sqrt {b}} + x^{2} \right )}}{4} + \frac {\sqrt {2} \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}} \log {\left (\frac {\sqrt {2} \sqrt {a} x \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {b}} - \frac {\sqrt {a}}{\sqrt {b}} + x^{2} \right )}}{4} \]

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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