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

Optimal. Leaf size=146 \[ \frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (c+d x+e x^2\right )}{4 a \left (a-b x^4\right )} \]

[Out]

(x*(c + d*x + e*x^2))/(4*a*(a - b*x^4)) + ((3*Sqrt[b]*c - Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b
^(3/4)) + ((3*Sqrt[b]*c + Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(3/4)) + (d*ArcTanh[(Sqrt[b]*x
^2)/Sqrt[a]])/(4*a^(3/2)*Sqrt[b])

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Rubi [A]  time = 0.127515, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1855, 1876, 275, 208, 1167, 205} \[ \frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{b} c\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (c+d x+e x^2\right )}{4 a \left (a-b x^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c + d*x + e*x^2))/(4*a*(a - b*x^4)) + ((3*Sqrt[b]*c - Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b
^(3/4)) + ((3*Sqrt[b]*c + Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(3/4)) + (d*ArcTanh[(Sqrt[b]*x
^2)/Sqrt[a]])/(4*a^(3/2)*Sqrt[b])

Rule 1855

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(x*Pq*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Di
st[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b},
x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]

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}{\left (a-b x^4\right )^2} \, dx &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a-b x^4\right )}-\frac{\int \frac{-3 c-2 d x-e x^2}{a-b x^4} \, dx}{4 a}\\ &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a-b x^4\right )}-\frac{\int \left (-\frac{2 d x}{a-b x^4}+\frac{-3 c-e x^2}{a-b x^4}\right ) \, dx}{4 a}\\ &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a-b x^4\right )}-\frac{\int \frac{-3 c-e x^2}{a-b x^4} \, dx}{4 a}+\frac{d \int \frac{x}{a-b x^4} \, dx}{2 a}\\ &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a-b x^4\right )}+\frac{d \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )}{4 a}-\frac{\left (\frac{3 \sqrt{b} c}{\sqrt{a}}-e\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{8 a}+\frac{\left (3 \sqrt{b} c+\sqrt{a} e\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{8 a^{3/2}}\\ &=\frac{x \left (c+d x+e x^2\right )}{4 a \left (a-b x^4\right )}+\frac{\left (3 \sqrt{b} c-\sqrt{a} e\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{\left (3 \sqrt{b} c+\sqrt{a} e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{3/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.201098, size = 211, normalized size = 1.45 \[ \frac{-\frac{\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (a^{3/4} e+3 \sqrt [4]{a} \sqrt{b} c+2 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac{\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (a^{3/4} e+3 \sqrt [4]{a} \sqrt{b} c-2 \sqrt{a} \sqrt [4]{b} d\right )}{b^{3/4}}-\frac{2 \sqrt [4]{a} \left (\sqrt{a} e-3 \sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{4 a x (c+x (d+e x))}{a-b x^4}+\frac{2 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{16 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a*x*(c + x*(d + e*x)))/(a - b*x^4) - (2*a^(1/4)*(-3*Sqrt[b]*c + Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^
(3/4) - ((3*a^(1/4)*Sqrt[b]*c + 2*Sqrt[a]*b^(1/4)*d + a^(3/4)*e)*Log[a^(1/4) - b^(1/4)*x])/b^(3/4) + ((3*a^(1/
4)*Sqrt[b]*c - 2*Sqrt[a]*b^(1/4)*d + a^(3/4)*e)*Log[a^(1/4) + b^(1/4)*x])/b^(3/4) + (2*Sqrt[a]*d*Log[Sqrt[a] +
Sqrt[b]*x^2])/Sqrt[b])/(16*a^2)

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Maple [B]  time = 0.003, size = 228, normalized size = 1.6 \begin{align*} -{\frac{cx}{4\,a \left ( b{x}^{4}-a \right ) }}+{\frac{3\,c}{16\,{a}^{2}}\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{3\,c}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{d{x}^{2}}{4\,a \left ( b{x}^{4}-a \right ) }}-{\frac{d}{8\,a}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{e{x}^{3}}{4\,a \left ( b{x}^{4}-a \right ) }}-{\frac{e}{8\,ab}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{e}{16\,ab}\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)^2,x)

[Out]

-1/4*c*x/a/(b*x^4-a)+3/16*c/a^2*(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))+3/8*c/a^2*(1/b*a)^(1/4)*
arctan(x/(1/b*a)^(1/4))-1/4*d*x^2/a/(b*x^4-a)-1/8*d/a/(a*b)^(1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2))
)-1/4*e*x^3/a/(b*x^4-a)-1/8*e/a/b/(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))+1/16*e/a/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)^2,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)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [B]  time = 7.2781, size = 507, normalized size = 3.47 \begin{align*} \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{3} + t^{2} \left (- 3072 a^{4} b^{2} c e - 2048 a^{4} b^{2} d^{2}\right ) + t \left (128 a^{3} b d e^{2} + 1152 a^{2} b^{2} c^{2} d\right ) - a^{2} e^{4} + 18 a b c^{2} e^{2} - 48 a b c d^{2} e + 16 a b d^{4} - 81 b^{2} c^{4}, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{3} a^{7} b^{2} e^{3} + 36864 t^{3} a^{6} b^{3} c^{2} e - 98304 t^{3} a^{6} b^{3} c d^{2} + 4608 t^{2} a^{5} b^{2} c d e^{2} - 4096 t^{2} a^{5} b^{2} d^{3} e - 13824 t^{2} a^{4} b^{3} c^{3} d - 144 t a^{4} b c e^{4} - 192 t a^{4} b d^{2} e^{3} - 1728 t a^{3} b^{2} c^{3} e^{2} + 5184 t a^{3} b^{2} c^{2} d^{2} e + 1536 t a^{3} b^{2} c d^{4} - 3888 t a^{2} b^{3} c^{5} + 6 a^{3} d e^{5} - 120 a^{2} b c d^{3} e^{2} + 64 a^{2} b d^{5} e + 810 a b^{2} c^{4} d e - 1080 a b^{2} c^{3} d^{3}}{a^{3} e^{6} + 9 a^{2} b c^{2} e^{4} - 96 a^{2} b c d^{2} e^{3} + 64 a^{2} b d^{4} e^{2} - 81 a b^{2} c^{4} e^{2} + 864 a b^{2} c^{3} d^{2} e - 576 a b^{2} c^{2} d^{4} - 729 b^{3} c^{6}} \right )} \right )\right )} - \frac{c x + d x^{2} + e x^{3}}{- 4 a^{2} + 4 a b x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(65536*_t**4*a**7*b**3 + _t**2*(-3072*a**4*b**2*c*e - 2048*a**4*b**2*d**2) + _t*(128*a**3*b*d*e**2 + 11
52*a**2*b**2*c**2*d) - a**2*e**4 + 18*a*b*c**2*e**2 - 48*a*b*c*d**2*e + 16*a*b*d**4 - 81*b**2*c**4, Lambda(_t,
_t*log(x + (4096*_t**3*a**7*b**2*e**3 + 36864*_t**3*a**6*b**3*c**2*e - 98304*_t**3*a**6*b**3*c*d**2 + 4608*_t
**2*a**5*b**2*c*d*e**2 - 4096*_t**2*a**5*b**2*d**3*e - 13824*_t**2*a**4*b**3*c**3*d - 144*_t*a**4*b*c*e**4 - 1
92*_t*a**4*b*d**2*e**3 - 1728*_t*a**3*b**2*c**3*e**2 + 5184*_t*a**3*b**2*c**2*d**2*e + 1536*_t*a**3*b**2*c*d**
4 - 3888*_t*a**2*b**3*c**5 + 6*a**3*d*e**5 - 120*a**2*b*c*d**3*e**2 + 64*a**2*b*d**5*e + 810*a*b**2*c**4*d*e -
1080*a*b**2*c**3*d**3)/(a**3*e**6 + 9*a**2*b*c**2*e**4 - 96*a**2*b*c*d**2*e**3 + 64*a**2*b*d**4*e**2 - 81*a*b
**2*c**4*e**2 + 864*a*b**2*c**3*d**2*e - 576*a*b**2*c**2*d**4 - 729*b**3*c**6)))) - (c*x + d*x**2 + e*x**3)/(-
4*a**2 + 4*a*b*x**4)

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Giac [B]  time = 1.08587, size = 440, normalized size = 3.01 \begin{align*} -\frac{x^{3} e + d x^{2} + c x}{4 \,{\left (b x^{4} - a\right )} a} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b^{2} d + 3 \, \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 )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b^{2} d + 3 \, \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 )}{16 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (3 \, \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 )}{32 \, a^{2} b^{3}} - \frac{\sqrt{2}{\left (3 \, \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 )}{32 \, a^{2} 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)^2,x, algorithm="giac")

[Out]

-1/4*(x^3*e + d*x^2 + c*x)/((b*x^4 - a)*a) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(-a*b)*b^2*d + 3*(-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^2*b^3) + 1/16*sqrt(2)*(2
*sqrt(2)*sqrt(-a*b)*b^2*d + 3*(-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^2*b^3) + 1/32*sqrt(2)*(3*(-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^2*b^3) - 1/32*sqrt(2)*(3*(-a*b^3)^(1/4)*b^2*c - (-a*b^3)^(3/4)*e)*log(x^2 - sq
rt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^2*b^3)