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

Optimal. Leaf size=116 \[ \frac{\left (\sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a} e+\sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0954763, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {1876, 275, 208, 1167, 205} \[ \frac{\left (\sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a} e+\sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2)/(a - b*x^4),x]

[Out]

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

Rule 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 208

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

Rule 1167

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

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.050743, size = 187, normalized size = 1.61 \[ \frac{-\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (\sqrt [4]{a} \sqrt [4]{b} d+\sqrt{a} e+\sqrt{b} c\right )+2 \left (\sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt{b} c \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+\sqrt [4]{a} \sqrt [4]{b} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )-\sqrt [4]{a} \sqrt [4]{b} d \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+\sqrt{a} e \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{4 a^{3/4} b^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2)/(a - b*x^4),x]

[Out]

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

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Maple [B]  time = 0.002, size = 161, normalized size = 1.4 \begin{align*}{\frac{c}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{d}{4}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{e}{2\,b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{4\,b}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d*x+c)/(-b*x^4+a),x)

[Out]

1/4*c*(1/b*a)^(1/4)/a*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))+1/2*c*(1/b*a)^(1/4)/a*arctan(x/(1/b*a)^(1/4))-1/
4*d/(a*b)^(1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2)))-1/2*e/b/(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))+1/
4*e/b/(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))

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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((e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [B]  time = 5.49401, size = 471, normalized size = 4.06 \begin{align*} - \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{3} + t^{2} \left (- 64 a^{2} b^{2} c e - 32 a^{2} b^{2} d^{2}\right ) + t \left (- 16 a^{2} b d e^{2} - 16 a b^{2} c^{2} d\right ) - a^{2} e^{4} + 2 a b c^{2} e^{2} - 4 a b c d^{2} e + a b d^{4} - b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{4} b^{2} e^{3} - 64 t^{3} a^{3} b^{3} c^{2} e + 128 t^{3} a^{3} b^{3} c d^{2} + 48 t^{2} a^{3} b^{2} c d e^{2} - 32 t^{2} a^{3} b^{2} d^{3} e - 16 t^{2} a^{2} b^{3} c^{3} d + 12 t a^{3} b c e^{4} + 12 t a^{3} b d^{2} e^{3} + 16 t a^{2} b^{2} c^{3} e^{2} - 36 t a^{2} b^{2} c^{2} d^{2} e - 8 t a^{2} b^{2} c d^{4} + 4 t a b^{3} c^{5} + 3 a^{3} d e^{5} - 5 a^{2} b c d^{3} e^{2} + 2 a^{2} b d^{5} e + 5 a b^{2} c^{4} d e - 5 a b^{2} c^{3} d^{3}}{a^{3} e^{6} + a^{2} b c^{2} e^{4} - 8 a^{2} b c d^{2} e^{3} + 4 a^{2} b d^{4} e^{2} - a b^{2} c^{4} e^{2} + 8 a b^{2} c^{3} d^{2} e - 4 a b^{2} c^{2} d^{4} - b^{3} c^{6}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(256*_t**4*a**3*b**3 + _t**2*(-64*a**2*b**2*c*e - 32*a**2*b**2*d**2) + _t*(-16*a**2*b*d*e**2 - 16*a*b*
*2*c**2*d) - a**2*e**4 + 2*a*b*c**2*e**2 - 4*a*b*c*d**2*e + a*b*d**4 - b**2*c**4, Lambda(_t, _t*log(x + (-64*_
t**3*a**4*b**2*e**3 - 64*_t**3*a**3*b**3*c**2*e + 128*_t**3*a**3*b**3*c*d**2 + 48*_t**2*a**3*b**2*c*d*e**2 - 3
2*_t**2*a**3*b**2*d**3*e - 16*_t**2*a**2*b**3*c**3*d + 12*_t*a**3*b*c*e**4 + 12*_t*a**3*b*d**2*e**3 + 16*_t*a*
*2*b**2*c**3*e**2 - 36*_t*a**2*b**2*c**2*d**2*e - 8*_t*a**2*b**2*c*d**4 + 4*_t*a*b**3*c**5 + 3*a**3*d*e**5 - 5
*a**2*b*c*d**3*e**2 + 2*a**2*b*d**5*e + 5*a*b**2*c**4*d*e - 5*a*b**2*c**3*d**3)/(a**3*e**6 + a**2*b*c**2*e**4
- 8*a**2*b*c*d**2*e**3 + 4*a**2*b*d**4*e**2 - a*b**2*c**4*e**2 + 8*a*b**2*c**3*d**2*e - 4*a*b**2*c**2*d**4 - b
**3*c**6))))

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Giac [B]  time = 1.08251, size = 396, normalized size = 3.41 \begin{align*} -\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-a b} b^{2} d - \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\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 (\sqrt{2} \sqrt{-a b} b^{2} d - \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} e\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}} e\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}} e\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((e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="giac")

[Out]

-1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*b^2*d - (-a*b^3)^(1/4)*b^2*c - (-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sq
rt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a*b^3) - 1/4*sqrt(2)*(sqrt(2)*sqrt(-a*b)*b^2*d - (-a*b^3)^(1/4)*b^2*c - (-a
*b^3)^(3/4)*e)*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)*e)*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)*e)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a*b^3)