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

Optimal. Leaf size=110 \[ \frac{3 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x (c+d x)}{4 a \left (a-b x^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.0816042, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {1855, 1876, 212, 208, 205, 275} \[ \frac{3 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{3 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \sqrt{b}}+\frac{x (c+d x)}{4 a \left (a-b x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(a - b*x^4)^2,x]

[Out]

(x*(c + d*x))/(4*a*(a - b*x^4)) + (3*c*ArcTan[(b^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(1/4)) + (3*c*ArcTanh[(b^(1/4
)*x)/a^(1/4)])/(8*a^(7/4)*b^(1/4)) + (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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

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 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]

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]

Rubi steps

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

Mathematica [A]  time = 0.160481, size = 168, normalized size = 1.53 \[ \frac{\frac{4 a x (c+d x)}{a-b x^4}-\frac{\left (3 \sqrt [4]{a} \sqrt [4]{b} c+2 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{\left (3 \sqrt [4]{a} \sqrt [4]{b} c-2 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{6 \sqrt [4]{a} c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{2 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{16 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a - b*x^4)^2,x]

[Out]

((4*a*x*(c + d*x))/(a - b*x^4) + (6*a^(1/4)*c*ArcTan[(b^(1/4)*x)/a^(1/4)])/b^(1/4) - ((3*a^(1/4)*b^(1/4)*c + 2
*Sqrt[a]*d)*Log[a^(1/4) - b^(1/4)*x])/Sqrt[b] + ((3*a^(1/4)*b^(1/4)*c - 2*Sqrt[a]*d)*Log[a^(1/4) + b^(1/4)*x])
/Sqrt[b] + (2*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(16*a^2)

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Maple [A]  time = 0.004, size = 142, normalized size = 1.3 \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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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))
)

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

[Out]

Timed out

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Sympy [A]  time = 1.27555, size = 155, normalized size = 1.41 \begin{align*} \operatorname{RootSum}{\left (65536 t^{4} a^{7} b^{2} - 2048 t^{2} a^{4} b d^{2} + 1152 t a^{2} b c^{2} d + 16 a d^{4} - 81 b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{32768 t^{3} a^{6} b d^{2} + 4608 t^{2} a^{4} b c^{2} d - 512 t a^{3} d^{4} + 1296 t a^{2} b c^{4} + 360 a c^{2} d^{3}}{192 a c d^{4} + 243 b c^{5}} \right )} \right )\right )} - \frac{c x + d x^{2}}{- 4 a^{2} + 4 a b x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(65536*_t**4*a**7*b**2 - 2048*_t**2*a**4*b*d**2 + 1152*_t*a**2*b*c**2*d + 16*a*d**4 - 81*b*c**4, Lambda
(_t, _t*log(x + (32768*_t**3*a**6*b*d**2 + 4608*_t**2*a**4*b*c**2*d - 512*_t*a**3*d**4 + 1296*_t*a**2*b*c**4 +
360*a*c**2*d**3)/(192*a*c*d**4 + 243*b*c**5)))) - (c*x + d*x**2)/(-4*a**2 + 4*a*b*x**4)

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Giac [B]  time = 1.09117, size = 343, normalized size = 3.12 \begin{align*} \frac{3 \, \sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c \log \left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b} - \frac{3 \, \sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} c \log \left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{32 \, a^{2} b} - \frac{d x^{2} + c x}{4 \,{\left (b x^{4} - a\right )} a} - \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b d - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b c\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^{2}} - \frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-a b} b d - 3 \, \left (-a b^{3}\right )^{\frac{1}{4}} b c\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^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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