3.1 \(\int \frac{c+d x^2}{a+b x^4} \, dx\)

Optimal. Leaf size=247 \[ -\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left (\sqrt{a} d+\sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a} d+\sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{3/4}} \]

[Out]

-((Sqrt[b]*c + Sqrt[a]*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) + ((Sqrt[b]*c +
Sqrt[a]*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) - ((Sqrt[b]*c - Sqrt[a]*d)*Lo
g[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(3/4)) + ((Sqrt[b]*c - Sqrt[a]*d)*L
og[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(3/4))

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Rubi [A]  time = 0.150941, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left (\sqrt{a} d+\sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a} d+\sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} b^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/(a + b*x^4),x]

[Out]

-((Sqrt[b]*c + Sqrt[a]*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) + ((Sqrt[b]*c +
Sqrt[a]*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) - ((Sqrt[b]*c - Sqrt[a]*d)*Lo
g[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(3/4)) + ((Sqrt[b]*c - Sqrt[a]*d)*L
og[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(3/4))

Rule 1168

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

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])

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{c+d x^2}{a+b x^4} \, dx &=\frac{\left (\frac{\sqrt{b} c}{\sqrt{a}}-d\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{2 b}+\frac{\left (\frac{\sqrt{b} c}{\sqrt{a}}+d\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{2 b}\\ &=\frac{\left (\frac{\sqrt{b} c}{\sqrt{a}}+d\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b}+\frac{\left (\frac{\sqrt{b} c}{\sqrt{a}}+d\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{4 b}-\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \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}{4 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \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}{4 \sqrt{2} a^{3/4} b^{3/4}}\\ &=-\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left (\sqrt{b} c+\sqrt{a} d\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left (\sqrt{b} c+\sqrt{a} d\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4}}\\ &=-\frac{\left (\sqrt{b} c+\sqrt{a} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left (\sqrt{b} c+\sqrt{a} d\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} b^{3/4}}-\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}+\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{4 \sqrt{2} a^{3/4} b^{3/4}}\\ \end{align*}

Mathematica [A]  time = 0.074574, size = 183, normalized size = 0.74 \[ \frac{-\left (\sqrt{b} c-\sqrt{a} d\right ) \left (\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )-\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )\right )-2 \left (\sqrt{a} d+\sqrt{b} c\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt{a} d+\sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} b^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)/(a + b*x^4),x]

[Out]

(-2*(Sqrt[b]*c + Sqrt[a]*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(Sqrt[b]*c + Sqrt[a]*d)*ArcTan[1 + (Sq
rt[2]*b^(1/4)*x)/a^(1/4)] - (Sqrt[b]*c - Sqrt[a]*d)*(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]))/(4*Sqrt[2]*a^(3/4)*b^(3/4))

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Maple [A]  time = 0.044, size = 260, normalized size = 1.1 \begin{align*}{\frac{c\sqrt{2}}{8\,a}\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{c\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{c\sqrt{2}}{4\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{d\sqrt{2}}{8\,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 [4]{{\frac{a}{b}}}}}}+{\frac{d\sqrt{2}}{4\,b}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{d\sqrt{2}}{4\,b}\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((d*x^2+c)/(b*x^4+a),x)

[Out]

1/8*c*(1/b*a)^(1/4)/a*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*c*(1/b*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)+1/4*c*(1/b*a)^(1/4)/a*2^(1/2)*arct
an(2^(1/2)/(1/b*a)^(1/4)*x-1)+1/8*d/b/(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*d/b/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x+1)+1/4*
d/b/(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((d*x^2+c)/(b*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.78205, size = 1544, normalized size = 6.25 \begin{align*} -\frac{1}{4} \, \sqrt{-\frac{a b \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x +{\left (a^{3} b^{2} d \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt{-\frac{a b \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) + \frac{1}{4} \, \sqrt{-\frac{a b \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x -{\left (a^{3} b^{2} d \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt{-\frac{a b \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) + \frac{1}{4} \, \sqrt{\frac{a b \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x +{\left (a^{3} b^{2} d \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt{\frac{a b \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) - \frac{1}{4} \, \sqrt{\frac{a b \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x -{\left (a^{3} b^{2} d \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt{\frac{a b \sqrt{-\frac{b^{2} c^{4} - 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(b*x^4+a),x, algorithm="fricas")

[Out]

-1/4*sqrt(-(a*b*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + 2*c*d)/(a*b))*log(-(b^2*c^4 - a^2*d^4)*
x + (a^3*b^2*d*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + a*b^2*c^3 - a^2*b*c*d^2)*sqrt(-(a*b*sqrt
(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + 2*c*d)/(a*b))) + 1/4*sqrt(-(a*b*sqrt(-(b^2*c^4 - 2*a*b*c^2*
d^2 + a^2*d^4)/(a^3*b^3)) + 2*c*d)/(a*b))*log(-(b^2*c^4 - a^2*d^4)*x - (a^3*b^2*d*sqrt(-(b^2*c^4 - 2*a*b*c^2*d
^2 + a^2*d^4)/(a^3*b^3)) + a*b^2*c^3 - a^2*b*c*d^2)*sqrt(-(a*b*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*
b^3)) + 2*c*d)/(a*b))) + 1/4*sqrt((a*b*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) - 2*c*d)/(a*b))*lo
g(-(b^2*c^4 - a^2*d^4)*x + (a^3*b^2*d*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) - a*b^2*c^3 + a^2*b
*c*d^2)*sqrt((a*b*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) - 2*c*d)/(a*b))) - 1/4*sqrt((a*b*sqrt(-
(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) - 2*c*d)/(a*b))*log(-(b^2*c^4 - a^2*d^4)*x - (a^3*b^2*d*sqrt(-(
b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) - a*b^2*c^3 + a^2*b*c*d^2)*sqrt((a*b*sqrt(-(b^2*c^4 - 2*a*b*c^2*
d^2 + a^2*d^4)/(a^3*b^3)) - 2*c*d)/(a*b)))

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Sympy [A]  time = 0.59846, size = 109, normalized size = 0.44 \begin{align*} \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{3} + 64 t^{2} a^{2} b^{2} c d + a^{2} d^{4} + 2 a b c^{2} d^{2} + b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{64 t^{3} a^{3} b^{2} d + 12 t a^{2} b c d^{2} - 4 t a b^{2} c^{3}}{a^{2} d^{4} - b^{2} c^{4}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)/(b*x**4+a),x)

[Out]

RootSum(256*_t**4*a**3*b**3 + 64*_t**2*a**2*b**2*c*d + a**2*d**4 + 2*a*b*c**2*d**2 + b**2*c**4, Lambda(_t, _t*
log(x + (64*_t**3*a**3*b**2*d + 12*_t*a**2*b*c*d**2 - 4*_t*a*b**2*c**3)/(a**2*d**4 - b**2*c**4))))

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Giac [A]  time = 1.18441, size = 325, normalized size = 1.32 \begin{align*} \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac{3}{4}} d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{3}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac{3}{4}} d\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{3}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac{3}{4}} d\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{8 \, a b^{3}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac{3}{4}} d\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{8 \, a b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(b*x^4+a),x, algorithm="giac")

[Out]

1/4*sqrt(2)*((a*b^3)^(1/4)*b^2*c + (a*b^3)^(3/4)*d)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4)
)/(a*b^3) + 1/4*sqrt(2)*((a*b^3)^(1/4)*b^2*c + (a*b^3)^(3/4)*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))
/(a/b)^(1/4))/(a*b^3) + 1/8*sqrt(2)*((a*b^3)^(1/4)*b^2*c - (a*b^3)^(3/4)*d)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) +
sqrt(a/b))/(a*b^3) - 1/8*sqrt(2)*((a*b^3)^(1/4)*b^2*c - (a*b^3)^(3/4)*d)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqr
t(a/b))/(a*b^3)