3.172 \(\int \frac{c+d x+e x^2+f x^3+g x^4}{(a-b x^4)^2} \, dx\)

Optimal. Leaf size=172 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]

[Out]

(x*(b*c + a*g + b*d*x + b*e*x^2 + b*f*x^3))/(4*a*b*(a - b*x^4)) + ((3*b*c - Sqrt[a]*Sqrt[b]*e - a*g)*ArcTan[(b
^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(5/4)) + ((3*b*c + Sqrt[a]*Sqrt[b]*e - a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(8*
a^(7/4)*b^(5/4)) + (d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*Sqrt[b])

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Rubi [A]  time = 0.164898, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {1858, 1876, 275, 208, 1167, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^2,x]

[Out]

(x*(b*c + a*g + b*d*x + b*e*x^2 + b*f*x^3))/(4*a*b*(a - b*x^4)) + ((3*b*c - Sqrt[a]*Sqrt[b]*e - a*g)*ArcTan[(b
^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(5/4)) + ((3*b*c + Sqrt[a]*Sqrt[b]*e - a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(8*
a^(7/4)*b^(5/4)) + (d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*Sqrt[b])

Rule 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -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+f x^3+g x^4}{\left (a-b x^4\right )^2} \, dx &=\frac{x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac{\int \frac{3 b c-a g+2 b d x+b e x^2}{a-b x^4} \, dx}{4 a b}\\ &=\frac{x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac{\int \left (\frac{2 b d x}{a-b x^4}+\frac{3 b c-a g+b e x^2}{a-b x^4}\right ) \, dx}{4 a b}\\ &=\frac{x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac{\int \frac{3 b c-a g+b e x^2}{a-b x^4} \, dx}{4 a b}+\frac{d \int \frac{x}{a-b x^4} \, dx}{2 a}\\ &=\frac{x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{4 a b \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 (3 b c-\sqrt{a} \sqrt{b} e-a g\right ) \int \frac{1}{-\sqrt{a} \sqrt{b}-b x^2} \, dx}{8 a^{3/2} \sqrt{b}}+\frac{\left (3 b c+\sqrt{a} \sqrt{b} e-a g\right ) \int \frac{1}{\sqrt{a} \sqrt{b}-b x^2} \, dx}{8 a^{3/2} \sqrt{b}}\\ &=\frac{x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac{\left (3 b c-\sqrt{a} \sqrt{b} e-a g\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{5/4}}+\frac{\left (3 b c+\sqrt{a} \sqrt{b} e-a g\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{5/4}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.314004, size = 221, normalized size = 1.28 \[ \frac{\frac{4 a^{3/4} \sqrt [4]{b} (a (f+g x)+b x (c+x (d+e x)))}{a-b x^4}-\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (2 \sqrt [4]{a} b^{3/4} d+\sqrt{a} \sqrt{b} e-a g+3 b c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-2 \sqrt [4]{a} b^{3/4} d+\sqrt{a} \sqrt{b} e-a g+3 b c\right )+2 \sqrt [4]{a} b^{3/4} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )-2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt{a} \sqrt{b} e+a g-3 b c\right )}{16 a^{7/4} b^{5/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^2,x]

[Out]

((4*a^(3/4)*b^(1/4)*(a*(f + g*x) + b*x*(c + x*(d + e*x))))/(a - b*x^4) - 2*(-3*b*c + Sqrt[a]*Sqrt[b]*e + a*g)*
ArcTan[(b^(1/4)*x)/a^(1/4)] - (3*b*c + 2*a^(1/4)*b^(3/4)*d + Sqrt[a]*Sqrt[b]*e - a*g)*Log[a^(1/4) - b^(1/4)*x]
+ (3*b*c - 2*a^(1/4)*b^(3/4)*d + Sqrt[a]*Sqrt[b]*e - a*g)*Log[a^(1/4) + b^(1/4)*x] + 2*a^(1/4)*b^(3/4)*d*Log[
Sqrt[a] + Sqrt[b]*x^2])/(16*a^(7/4)*b^(5/4))

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Maple [B]  time = 0.008, size = 289, normalized size = 1.7 \begin{align*}{\frac{1}{b{x}^{4}-a} \left ( -{\frac{e{x}^{3}}{4\,a}}-{\frac{d{x}^{2}}{4\,a}}-{\frac{ \left ( ag+bc \right ) x}{4\,ab}}-{\frac{f}{4\,b}} \right ) }-{\frac{g}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{3\,c}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{g}{16\,ab}\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}{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{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}{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((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^2,x)

[Out]

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

[Out]

Timed out

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Sympy [B]  time = 169.346, size = 1406, normalized size = 8.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**2,x)

[Out]

RootSum(65536*_t**4*a**7*b**5 + _t**2*(1024*a**5*b**3*e*g - 3072*a**4*b**4*c*e - 2048*a**4*b**4*d**2) + _t*(12
8*a**4*b**2*d*g**2 - 768*a**3*b**3*c*d*g + 128*a**3*b**3*d*e**2 + 1152*a**2*b**4*c**2*d) - a**4*g**4 + 12*a**3
*b*c*g**3 + 2*a**3*b*e**2*g**2 - 54*a**2*b**2*c**2*g**2 - 12*a**2*b**2*c*e**2*g + 16*a**2*b**2*d**2*e*g - a**2
*b**2*e**4 + 108*a*b**3*c**3*g + 18*a*b**3*c**2*e**2 - 48*a*b**3*c*d**2*e + 16*a*b**3*d**4 - 81*b**4*c**4, Lam
bda(_t, _t*log(x + (-4096*_t**3*a**8*b**4*e*g**2 + 24576*_t**3*a**7*b**5*c*e*g - 32768*_t**3*a**7*b**5*d**2*g
- 4096*_t**3*a**7*b**5*e**3 - 36864*_t**3*a**6*b**6*c**2*e + 98304*_t**3*a**6*b**6*c*d**2 - 512*_t**2*a**7*b**
3*d*g**3 + 4608*_t**2*a**6*b**4*c*d*g**2 + 1536*_t**2*a**6*b**4*d*e**2*g - 13824*_t**2*a**5*b**5*c**2*d*g - 46
08*_t**2*a**5*b**5*c*d*e**2 + 4096*_t**2*a**5*b**5*d**3*e + 13824*_t**2*a**4*b**6*c**3*d - 16*_t*a**7*b*g**5 +
240*_t*a**6*b**2*c*g**4 - 64*_t*a**6*b**2*e**2*g**3 - 1440*_t*a**5*b**3*c**2*g**3 + 576*_t*a**5*b**3*c*e**2*g
**2 - 576*_t*a**5*b**3*d**2*e*g**2 - 48*_t*a**5*b**3*e**4*g + 4320*_t*a**4*b**4*c**3*g**2 - 1728*_t*a**4*b**4*
c**2*e**2*g + 3456*_t*a**4*b**4*c*d**2*e*g + 144*_t*a**4*b**4*c*e**4 + 512*_t*a**4*b**4*d**4*g + 192*_t*a**4*b
**4*d**2*e**3 - 6480*_t*a**3*b**5*c**4*g + 1728*_t*a**3*b**5*c**3*e**2 - 5184*_t*a**3*b**5*c**2*d**2*e - 1536*
_t*a**3*b**5*c*d**4 + 3888*_t*a**2*b**6*c**5 - 10*a**5*b*d*e*g**4 + 120*a**4*b**2*c*d*e*g**3 - 40*a**4*b**2*d*
*3*g**3 - 540*a**3*b**3*c**2*d*e*g**2 + 360*a**3*b**3*c*d**3*g**2 - 40*a**3*b**3*d**3*e**2*g - 6*a**3*b**3*d*e
**5 + 1080*a**2*b**4*c**3*d*e*g - 1080*a**2*b**4*c**2*d**3*g + 120*a**2*b**4*c*d**3*e**2 - 64*a**2*b**4*d**5*e
- 810*a*b**5*c**4*d*e + 1080*a*b**5*c**3*d**3)/(a**6*g**6 - 18*a**5*b*c*g**5 + a**5*b*e**2*g**4 + 135*a**4*b*
*2*c**2*g**4 - 12*a**4*b**2*c*e**2*g**3 + 32*a**4*b**2*d**2*e*g**3 - a**4*b**2*e**4*g**2 - 540*a**3*b**3*c**3*
g**3 + 54*a**3*b**3*c**2*e**2*g**2 - 288*a**3*b**3*c*d**2*e*g**2 + 6*a**3*b**3*c*e**4*g + 64*a**3*b**3*d**4*g*
*2 - 32*a**3*b**3*d**2*e**3*g - a**3*b**3*e**6 + 1215*a**2*b**4*c**4*g**2 - 108*a**2*b**4*c**3*e**2*g + 864*a*
*2*b**4*c**2*d**2*e*g - 9*a**2*b**4*c**2*e**4 - 384*a**2*b**4*c*d**4*g + 96*a**2*b**4*c*d**2*e**3 - 64*a**2*b*
*4*d**4*e**2 - 1458*a*b**5*c**5*g + 81*a*b**5*c**4*e**2 - 864*a*b**5*c**3*d**2*e + 576*a*b**5*c**2*d**4 + 729*
b**6*c**6)))) - (a*f + b*d*x**2 + b*e*x**3 + x*(a*g + b*c))/(-4*a**2*b + 4*a*b**2*x**4)

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Giac [B]  time = 1.08497, size = 528, normalized size = 3.07 \begin{align*} -\frac{b x^{3} e + b d x^{2} + b c x + a g x + a f}{4 \,{\left (b x^{4} - a\right )} a b} + \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{1}{4}} a b g + \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{1}{4}} a b g + \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{1}{4}} a b g - \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{1}{4}} a b g - \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((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^2,x, algorithm="giac")

[Out]

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