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]

1/4*x*(d*x+c)/a/(-b*x^4+a)+3/8*c*arctan(b^(1/4)*x/a^(1/4))/a^(7/4)/b^(1/4)+3/8*c*arctanh(b^(1/4)*x/a^(1/4))/a^
(7/4)/b^(1/4)+1/4*d*arctanh(x^2*b^(1/2)/a^(1/2))/a^(3/2)/b^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 110, normalized size of antiderivative = 1.00, 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 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 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 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 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 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

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 = 0.20, 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)

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________

giac [B]  time = 0.17, size = 254, normalized size = 2.31 \[ \frac {3 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{32 \, a^{2} b} - \frac {3 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{32 \, a^{2} b} - \frac {d x^{2} + c x}{4 \, {\left (b x^{4} - a\right )} a} - \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {-a b} b d - 3 \, \left (-a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{2}} - \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {-a b} b d - 3 \, \left (-a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{2}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 142, normalized size = 1.29 \[ -\frac {d \,x^{2}}{4 \left (b \,x^{4}-a \right ) a}-\frac {c x}{4 \left (b \,x^{4}-a \right ) a}-\frac {d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{8 \sqrt {a b}\, a}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 a^{2}} \]

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*(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+3/8*c/a^2*(a/b)^(1/4)*arctan(1
/(a/b)^(1/4)*x)-1/4*d*x^2/a/(b*x^4-a)-1/8*d/a/(a*b)^(1/2)*ln(((a*b)^(1/2)*x^2-a)/(-(a*b)^(1/2)*x^2-a))

________________________________________________________________________________________

maxima [A]  time = 3.04, size = 157, normalized size = 1.43 \[ -\frac {d x^{2} + c x}{4 \, {\left (a b x^{4} - a^{2}\right )}} + \frac {\frac {6 \, c \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {2 \, d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {3 \, c \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}}}{16 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(d*x^2 + c*x)/(a*b*x^4 - a^2) + 1/16*(6*c*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*s
qrt(b))) + 2*d*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqrt(b)) - 2*d*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*sqrt(b))
- 3*c*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt
(b))))/a

________________________________________________________________________________________

mupad [B]  time = 4.92, size = 283, normalized size = 2.57 \[ \left (\sum _{k=1}^4\ln \left (-\frac {b^2\,\left (3\,c\,d^2+2\,d^3\,x+{\mathrm {root}\left (65536\,a^7\,b^2\,z^4-2048\,a^4\,b\,d^2\,z^2+1152\,a^2\,b\,c^2\,d\,z-81\,b\,c^4+16\,a\,d^4,z,k\right )}^2\,a^3\,b\,c\,192-{\mathrm {root}\left (65536\,a^7\,b^2\,z^4-2048\,a^4\,b\,d^2\,z^2+1152\,a^2\,b\,c^2\,d\,z-81\,b\,c^4+16\,a\,d^4,z,k\right )}^2\,a^3\,b\,d\,x\,128+\mathrm {root}\left (65536\,a^7\,b^2\,z^4-2048\,a^4\,b\,d^2\,z^2+1152\,a^2\,b\,c^2\,d\,z-81\,b\,c^4+16\,a\,d^4,z,k\right )\,a\,b\,c^2\,x\,36\right )}{a^3\,16}\right )\,\mathrm {root}\left (65536\,a^7\,b^2\,z^4-2048\,a^4\,b\,d^2\,z^2+1152\,a^2\,b\,c^2\,d\,z-81\,b\,c^4+16\,a\,d^4,z,k\right )\right )+\frac {\frac {d\,x^2}{4\,a}+\frac {c\,x}{4\,a}}{a-b\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(a - b*x^4)^2,x)

[Out]

symsum(log(-(b^2*(3*c*d^2 + 2*d^3*x + 192*root(65536*a^7*b^2*z^4 - 2048*a^4*b*d^2*z^2 + 1152*a^2*b*c^2*d*z - 8
1*b*c^4 + 16*a*d^4, z, k)^2*a^3*b*c - 128*root(65536*a^7*b^2*z^4 - 2048*a^4*b*d^2*z^2 + 1152*a^2*b*c^2*d*z - 8
1*b*c^4 + 16*a*d^4, z, k)^2*a^3*b*d*x + 36*root(65536*a^7*b^2*z^4 - 2048*a^4*b*d^2*z^2 + 1152*a^2*b*c^2*d*z -
81*b*c^4 + 16*a*d^4, z, k)*a*b*c^2*x))/(16*a^3))*root(65536*a^7*b^2*z^4 - 2048*a^4*b*d^2*z^2 + 1152*a^2*b*c^2*
d*z - 81*b*c^4 + 16*a*d^4, z, k), k, 1, 4) + ((d*x^2)/(4*a) + (c*x)/(4*a))/(a - b*x^4)

________________________________________________________________________________________

sympy [A]  time = 1.80, size = 156, normalized size = 1.42 \[ \operatorname {RootSum} {\left (65536 t^{4} a^{7} b^{2} - 2048 t^{2} a^{4} b d^{2} + 1152 t a^{2} b c^{2} d + 16 a d^{4} - 81 b c^{4}, \left (t \mapsto t \log {\left (x + \frac {32768 t^{3} a^{6} b d^{2} + 4608 t^{2} a^{4} b c^{2} d - 512 t a^{3} d^{4} + 1296 t a^{2} b c^{4} + 360 a c^{2} d^{3}}{192 a c d^{4} + 243 b c^{5}} \right )} \right )\right )} + \frac {- c x - d x^{2}}{- 4 a^{2} + 4 a b x^{4}} \]

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)

________________________________________________________________________________________