3.119 \(\int \frac{c+d x}{(a-b x^4)^3} \, dx\)

Optimal. Leaf size=136 \[ \frac{x (7 c+6 d x)}{32 a^2 \left (a-b x^4\right )}+\frac{21 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} \sqrt [4]{b}}+\frac{21 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} \sqrt [4]{b}}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x (c+d x)}{8 a \left (a-b x^4\right )^2} \]

[Out]

(x*(c + d*x))/(8*a*(a - b*x^4)^2) + (x*(7*c + 6*d*x))/(32*a^2*(a - b*x^4)) + (21*c*ArcTan[(b^(1/4)*x)/a^(1/4)]
)/(64*a^(11/4)*b^(1/4)) + (21*c*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(64*a^(11/4)*b^(1/4)) + (3*d*ArcTanh[(Sqrt[b]*x^
2)/Sqrt[a]])/(16*a^(5/2)*Sqrt[b])

________________________________________________________________________________________

Rubi [A]  time = 0.110073, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{x (7 c+6 d x)}{32 a^2 \left (a-b x^4\right )}+\frac{21 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} \sqrt [4]{b}}+\frac{21 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} \sqrt [4]{b}}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}+\frac{x (c+d x)}{8 a \left (a-b x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(a - b*x^4)^3,x]

[Out]

(x*(c + d*x))/(8*a*(a - b*x^4)^2) + (x*(7*c + 6*d*x))/(32*a^2*(a - b*x^4)) + (21*c*ArcTan[(b^(1/4)*x)/a^(1/4)]
)/(64*a^(11/4)*b^(1/4)) + (21*c*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(64*a^(11/4)*b^(1/4)) + (3*d*ArcTanh[(Sqrt[b]*x^
2)/Sqrt[a]])/(16*a^(5/2)*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 )^3} \, dx &=\frac{x (c+d x)}{8 a \left (a-b x^4\right )^2}-\frac{\int \frac{-7 c-6 d x}{\left (a-b x^4\right )^2} \, dx}{8 a}\\ &=\frac{x (c+d x)}{8 a \left (a-b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a-b x^4\right )}+\frac{\int \frac{21 c+12 d x}{a-b x^4} \, dx}{32 a^2}\\ &=\frac{x (c+d x)}{8 a \left (a-b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a-b x^4\right )}+\frac{\int \left (\frac{21 c}{a-b x^4}+\frac{12 d x}{a-b x^4}\right ) \, dx}{32 a^2}\\ &=\frac{x (c+d x)}{8 a \left (a-b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a-b x^4\right )}+\frac{(21 c) \int \frac{1}{a-b x^4} \, dx}{32 a^2}+\frac{(3 d) \int \frac{x}{a-b x^4} \, dx}{8 a^2}\\ &=\frac{x (c+d x)}{8 a \left (a-b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a-b x^4\right )}+\frac{(21 c) \int \frac{1}{\sqrt{a}-\sqrt{b} x^2} \, dx}{64 a^{5/2}}+\frac{(21 c) \int \frac{1}{\sqrt{a}+\sqrt{b} x^2} \, dx}{64 a^{5/2}}+\frac{(3 d) \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )}{16 a^2}\\ &=\frac{x (c+d x)}{8 a \left (a-b x^4\right )^2}+\frac{x (7 c+6 d x)}{32 a^2 \left (a-b x^4\right )}+\frac{21 c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} \sqrt [4]{b}}+\frac{21 c \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} \sqrt [4]{b}}+\frac{3 d \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.146495, size = 193, normalized size = 1.42 \[ \frac{\frac{16 a^2 x (c+d x)}{\left (a-b x^4\right )^2}+\frac{4 a x (7 c+6 d x)}{a-b x^4}-\frac{3 \left (7 \sqrt [4]{a} \sqrt [4]{b} c+4 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{3 \left (7 \sqrt [4]{a} \sqrt [4]{b} c-4 \sqrt{a} d\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{\sqrt{b}}+\frac{42 \sqrt [4]{a} c \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{12 \sqrt{a} d \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt{b}}}{128 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a - b*x^4)^3,x]

[Out]

((16*a^2*x*(c + d*x))/(a - b*x^4)^2 + (4*a*x*(7*c + 6*d*x))/(a - b*x^4) + (42*a^(1/4)*c*ArcTan[(b^(1/4)*x)/a^(
1/4)])/b^(1/4) - (3*(7*a^(1/4)*b^(1/4)*c + 4*Sqrt[a]*d)*Log[a^(1/4) - b^(1/4)*x])/Sqrt[b] + (3*(7*a^(1/4)*b^(1
/4)*c - 4*Sqrt[a]*d)*Log[a^(1/4) + b^(1/4)*x])/Sqrt[b] + (12*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(1
28*a^3)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 180, normalized size = 1.3 \begin{align*}{\frac{cx}{8\,a \left ( b{x}^{4}-a \right ) ^{2}}}-{\frac{7\,cx}{32\,{a}^{2} \left ( b{x}^{4}-a \right ) }}+{\frac{21\,c}{128\,{a}^{3}}\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{21\,c}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{d{x}^{2}}{8\,a \left ( b{x}^{4}-a \right ) ^{2}}}-{\frac{3\,d{x}^{2}}{16\,{a}^{2} \left ( b{x}^{4}-a \right ) }}-{\frac{3\,d}{32\,{a}^{2}}\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)^3,x)

[Out]

1/8*c*x/a/(b*x^4-a)^2-7/32*c/a^2*x/(b*x^4-a)+21/128*c/a^3*(1/b*a)^(1/4)*ln((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4))
)+21/64*c/a^3*(1/b*a)^(1/4)*arctan(x/(1/b*a)^(1/4))+1/8*d*x^2/a/(b*x^4-a)^2-3/16*d/a^2*x^2/(b*x^4-a)-3/32*d/a^
2/(a*b)^(1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2)))

________________________________________________________________________________________

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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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)^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [A]  time = 2.1943, size = 194, normalized size = 1.43 \begin{align*} - \operatorname{RootSum}{\left (268435456 t^{4} a^{11} b^{2} - 4718592 t^{2} a^{6} b d^{2} - 2709504 t a^{3} b c^{2} d + 20736 a d^{4} - 194481 b c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 67108864 t^{3} a^{9} b d^{2} + 9633792 t^{2} a^{6} b c^{2} d + 589824 t a^{4} d^{4} - 2765952 t a^{3} b c^{4} + 423360 a c^{2} d^{3}}{193536 a c d^{4} + 453789 b c^{5}} \right )} \right )\right )} - \frac{- 11 a c x - 10 a d x^{2} + 7 b c x^{5} + 6 b d x^{6}}{32 a^{4} - 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(-b*x**4+a)**3,x)

[Out]

-RootSum(268435456*_t**4*a**11*b**2 - 4718592*_t**2*a**6*b*d**2 - 2709504*_t*a**3*b*c**2*d + 20736*a*d**4 - 19
4481*b*c**4, Lambda(_t, _t*log(x + (-67108864*_t**3*a**9*b*d**2 + 9633792*_t**2*a**6*b*c**2*d + 589824*_t*a**4
*d**4 - 2765952*_t*a**3*b*c**4 + 423360*a*c**2*d**3)/(193536*a*c*d**4 + 453789*b*c**5)))) - (-11*a*c*x - 10*a*
d*x**2 + 7*b*c*x**5 + 6*b*d*x**6)/(32*a**4 - 64*a**3*b*x**4 + 32*a**2*b**2*x**8)

________________________________________________________________________________________

Giac [B]  time = 1.08903, size = 367, normalized size = 2.7 \begin{align*} \frac{21 \, \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 )}{256 \, a^{3} b} - \frac{21 \, \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 )}{256 \, a^{3} b} + \frac{3 \, \sqrt{2}{\left (4 \, \sqrt{2} \sqrt{-a b} b d + 7 \, \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 )}{128 \, a^{3} b^{2}} + \frac{3 \, \sqrt{2}{\left (4 \, \sqrt{2} \sqrt{-a b} b d + 7 \, \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 )}{128 \, a^{3} b^{2}} - \frac{6 \, b d x^{6} + 7 \, b c x^{5} - 10 \, a d x^{2} - 11 \, a c x}{32 \,{\left (b x^{4} - a\right )}^{2} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(-b*x^4+a)^3,x, algorithm="giac")

[Out]

21/256*sqrt(2)*(-a*b^3)^(1/4)*c*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^3*b) - 21/256*sqrt(2)*(-a*b^
3)^(1/4)*c*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^3*b) + 3/128*sqrt(2)*(4*sqrt(2)*sqrt(-a*b)*b*d +
7*(-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^3*b^2) + 3/128*sqrt(2)*
(4*sqrt(2)*sqrt(-a*b)*b*d + 7*(-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^3*b^2) - 1/32*(6*b*d*x^6 + 7*b*c*x^5 - 10*a*d*x^2 - 11*a*c*x)/((b*x^4 - a)^2*a^2)