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

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

[Out]

1/8*x*(d*x+c)/a/(-b*x^4+a)^2+1/32*x*(6*d*x+7*c)/a^2/(-b*x^4+a)+21/64*c*arctan(b^(1/4)*x/a^(1/4))/a^(11/4)/b^(1
/4)+21/64*c*arctanh(b^(1/4)*x/a^(1/4))/a^(11/4)/b^(1/4)+3/16*d*arctanh(x^2*b^(1/2)/a^(1/2))/a^(5/2)/b^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 136, normalized size of antiderivative = 1.00, 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 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 )^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*}

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Mathematica [A]  time = 0.21, 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)

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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)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.19, size = 272, normalized size = 2.00 \[ \frac {21 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, a^{3} b} - \frac {21 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, a^{3} b} + \frac {3 \, \sqrt {2} {\left (4 \, \sqrt {2} \sqrt {-a b} b d + 7 \, \left (-a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{2}} + \frac {3 \, \sqrt {2} {\left (4 \, \sqrt {2} \sqrt {-a b} b d + 7 \, \left (-a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{2}} - \frac {6 \, b d x^{6} + 7 \, b c x^{5} - 10 \, a d x^{2} - 11 \, a c x}{32 \, {\left (b x^{4} - a\right )}^{2} a^{2}} \]

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)

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maple [A]  time = 0.05, size = 180, normalized size = 1.32 \[ \frac {d \,x^{2}}{8 \left (b \,x^{4}-a \right )^{2} a}+\frac {c x}{8 \left (b \,x^{4}-a \right )^{2} a}-\frac {3 d \,x^{2}}{16 \left (b \,x^{4}-a \right ) a^{2}}-\frac {7 c x}{32 \left (b \,x^{4}-a \right ) a^{2}}-\frac {3 d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{32 \sqrt {a b}\, a^{2}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 a^{3}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 a^{3}} \]

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*(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+21/6
4*c/a^3*(a/b)^(1/4)*arctan(1/(a/b)^(1/4)*x)+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*b)^(1/2)*x^2-a)/(-(a*b)^(1/2)*x^2-a))

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maxima [A]  time = 3.01, size = 186, normalized size = 1.37 \[ -\frac {6 \, b d x^{6} + 7 \, b c x^{5} - 10 \, a d x^{2} - 11 \, a c x}{32 \, {\left (a^{2} b^{2} x^{8} - 2 \, a^{3} b x^{4} + a^{4}\right )}} + \frac {3 \, {\left (\frac {14 \, c \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {4 \, d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {4 \, d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {7 \, c \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}}\right )}}{128 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(6*b*d*x^6 + 7*b*c*x^5 - 10*a*d*x^2 - 11*a*c*x)/(a^2*b^2*x^8 - 2*a^3*b*x^4 + a^4) + 3/128*(14*c*arctan(s
qrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 4*d*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqrt
(b)) - 4*d*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*sqrt(b)) - 7*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^2

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mupad [B]  time = 4.98, size = 315, normalized size = 2.32 \[ \frac {\frac {5\,d\,x^2}{16\,a}+\frac {11\,c\,x}{32\,a}-\frac {7\,b\,c\,x^5}{32\,a^2}-\frac {3\,b\,d\,x^6}{16\,a^2}}{a^2-2\,a\,b\,x^4+b^2\,x^8}+\left (\sum _{k=1}^4\ln \left (-\frac {b^2\,\left (63\,c\,d^2+36\,d^3\,x+{\mathrm {root}\left (268435456\,a^{11}\,b^2\,z^4-4718592\,a^6\,b\,d^2\,z^2+2709504\,a^3\,b\,c^2\,d\,z-194481\,b\,c^4+20736\,a\,d^4,z,k\right )}^2\,a^5\,b\,c\,7168+\mathrm {root}\left (268435456\,a^{11}\,b^2\,z^4-4718592\,a^6\,b\,d^2\,z^2+2709504\,a^3\,b\,c^2\,d\,z-194481\,b\,c^4+20736\,a\,d^4,z,k\right )\,a^2\,b\,c^2\,x\,1176-{\mathrm {root}\left (268435456\,a^{11}\,b^2\,z^4-4718592\,a^6\,b\,d^2\,z^2+2709504\,a^3\,b\,c^2\,d\,z-194481\,b\,c^4+20736\,a\,d^4,z,k\right )}^2\,a^5\,b\,d\,x\,4096\right )\,3}{a^6\,2048}\right )\,\mathrm {root}\left (268435456\,a^{11}\,b^2\,z^4-4718592\,a^6\,b\,d^2\,z^2+2709504\,a^3\,b\,c^2\,d\,z-194481\,b\,c^4+20736\,a\,d^4,z,k\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(a - b*x^4)^3,x)

[Out]

((5*d*x^2)/(16*a) + (11*c*x)/(32*a) - (7*b*c*x^5)/(32*a^2) - (3*b*d*x^6)/(16*a^2))/(a^2 + b^2*x^8 - 2*a*b*x^4)
+ symsum(log(-(3*b^2*(63*c*d^2 + 36*d^3*x + 7168*root(268435456*a^11*b^2*z^4 - 4718592*a^6*b*d^2*z^2 + 270950
4*a^3*b*c^2*d*z - 194481*b*c^4 + 20736*a*d^4, z, k)^2*a^5*b*c + 1176*root(268435456*a^11*b^2*z^4 - 4718592*a^6
*b*d^2*z^2 + 2709504*a^3*b*c^2*d*z - 194481*b*c^4 + 20736*a*d^4, z, k)*a^2*b*c^2*x - 4096*root(268435456*a^11*
b^2*z^4 - 4718592*a^6*b*d^2*z^2 + 2709504*a^3*b*c^2*d*z - 194481*b*c^4 + 20736*a*d^4, z, k)^2*a^5*b*d*x))/(204
8*a^6))*root(268435456*a^11*b^2*z^4 - 4718592*a^6*b*d^2*z^2 + 2709504*a^3*b*c^2*d*z - 194481*b*c^4 + 20736*a*d
^4, z, k), k, 1, 4)

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sympy [A]  time = 1.97, size = 194, normalized size = 1.43 \[ - \operatorname {RootSum} {\left (268435456 t^{4} a^{11} b^{2} - 4718592 t^{2} a^{6} b d^{2} - 2709504 t a^{3} b c^{2} d + 20736 a d^{4} - 194481 b c^{4}, \left (t \mapsto t \log {\left (x + \frac {- 67108864 t^{3} a^{9} b d^{2} + 9633792 t^{2} a^{6} b c^{2} d + 589824 t a^{4} d^{4} - 2765952 t a^{3} b c^{4} + 423360 a c^{2} d^{3}}{193536 a c d^{4} + 453789 b c^{5}} \right )} \right )\right )} - \frac {- 11 a c x - 10 a d x^{2} + 7 b c x^{5} + 6 b d x^{6}}{32 a^{4} - 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \]

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)

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