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

Optimal. Leaf size=291 \[ -\frac {77 c \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {77 c \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {77 c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {77 c \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac {x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac {x (c+d x)}{12 a \left (a+b x^4\right )^3} \]

[Out]

1/12*x*(d*x+c)/a/(b*x^4+a)^3+1/96*x*(10*d*x+11*c)/a^2/(b*x^4+a)^2+1/384*x*(60*d*x+77*c)/a^3/(b*x^4+a)+77/512*c
*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))/a^(15/4)/b^(1/4)*2^(1/2)+77/512*c*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))/a^
(15/4)/b^(1/4)*2^(1/2)-77/1024*c*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))/a^(15/4)/b^(1/4)*2^(1/2)+7
7/1024*c*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))/a^(15/4)/b^(1/4)*2^(1/2)+5/32*d*arctan(x^2*b^(1/2)/
a^(1/2))/a^(7/2)/b^(1/2)

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Rubi [A]  time = 0.27, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1855, 1876, 211, 1165, 628, 1162, 617, 204, 275, 205} \[ \frac {x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac {x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}-\frac {77 c \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {77 c \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {77 c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {77 c \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {x (c+d x)}{12 a \left (a+b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(a + b*x^4)^4,x]

[Out]

(x*(c + d*x))/(12*a*(a + b*x^4)^3) + (x*(11*c + 10*d*x))/(96*a^2*(a + b*x^4)^2) + (x*(77*c + 60*d*x))/(384*a^3
*(a + b*x^4)) + (5*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*Sqrt[b]) - (77*c*ArcTan[1 - (Sqrt[2]*b^(1/4)*x
)/a^(1/4)])/(256*Sqrt[2]*a^(15/4)*b^(1/4)) + (77*c*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(256*Sqrt[2]*a^(15
/4)*b^(1/4)) - (77*c*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(1/4)) +
(77*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(512*Sqrt[2]*a^(15/4)*b^(1/4))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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 211

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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 )^4} \, dx &=\frac {x (c+d x)}{12 a \left (a+b x^4\right )^3}-\frac {\int \frac {-11 c-10 d x}{\left (a+b x^4\right )^3} \, dx}{12 a}\\ &=\frac {x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac {\int \frac {77 c+60 d x}{\left (a+b x^4\right )^2} \, dx}{96 a^2}\\ &=\frac {x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}-\frac {\int \frac {-231 c-120 d x}{a+b x^4} \, dx}{384 a^3}\\ &=\frac {x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}-\frac {\int \left (-\frac {231 c}{a+b x^4}-\frac {120 d x}{a+b x^4}\right ) \, dx}{384 a^3}\\ &=\frac {x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac {(77 c) \int \frac {1}{a+b x^4} \, dx}{128 a^3}+\frac {(5 d) \int \frac {x}{a+b x^4} \, dx}{16 a^3}\\ &=\frac {x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac {(77 c) \int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx}{256 a^{7/2}}+\frac {(77 c) \int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx}{256 a^{7/2}}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{32 a^3}\\ &=\frac {x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {(77 c) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^{7/2} \sqrt {b}}+\frac {(77 c) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^{7/2} \sqrt {b}}-\frac {(77 c) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{512 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(77 c) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{512 \sqrt {2} a^{15/4} \sqrt [4]{b}}\\ &=\frac {x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}-\frac {77 c \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {77 c \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {(77 c) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {(77 c) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b}}\\ &=\frac {x (c+d x)}{12 a \left (a+b x^4\right )^3}+\frac {x (11 c+10 d x)}{96 a^2 \left (a+b x^4\right )^2}+\frac {x (77 c+60 d x)}{384 a^3 \left (a+b x^4\right )}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}-\frac {77 c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {77 c \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {77 c \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} \sqrt [4]{b}}+\frac {77 c \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} \sqrt [4]{b}}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 274, normalized size = 0.94 \[ \frac {\frac {256 a^{11/4} x (c+d x)}{\left (a+b x^4\right )^3}+\frac {32 a^{7/4} x (11 c+10 d x)}{\left (a+b x^4\right )^2}+\frac {8 a^{3/4} x (77 c+60 d x)}{a+b x^4}-\frac {6 \left (80 \sqrt [4]{a} d+77 \sqrt {2} \sqrt [4]{b} c\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {b}}+\frac {6 \left (77 \sqrt {2} \sqrt [4]{b} c-80 \sqrt [4]{a} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {b}}-\frac {231 \sqrt {2} c \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt [4]{b}}+\frac {231 \sqrt {2} c \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt [4]{b}}}{3072 a^{15/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a + b*x^4)^4,x]

[Out]

((256*a^(11/4)*x*(c + d*x))/(a + b*x^4)^3 + (32*a^(7/4)*x*(11*c + 10*d*x))/(a + b*x^4)^2 + (8*a^(3/4)*x*(77*c
+ 60*d*x))/(a + b*x^4) - (6*(77*Sqrt[2]*b^(1/4)*c + 80*a^(1/4)*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqr
t[b] + (6*(77*Sqrt[2]*b^(1/4)*c - 80*a^(1/4)*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] - (231*Sqrt[2
]*c*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4) + (231*Sqrt[2]*c*Log[Sqrt[a] + Sqrt[2]*a^(
1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4))/(3072*a^(15/4))

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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)^4,x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.18, size = 280, normalized size = 0.96 \[ \frac {77 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b} - \frac {77 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b} + \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b d + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b d + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{2}} + \frac {60 \, b^{2} d x^{10} + 77 \, b^{2} c x^{9} + 160 \, a b d x^{6} + 198 \, a b c x^{5} + 132 \, a^{2} d x^{2} + 153 \, a^{2} c x}{384 \, {\left (b x^{4} + a\right )}^{3} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

77/1024*sqrt(2)*(a*b^3)^(1/4)*c*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b) - 77/1024*sqrt(2)*(a*b^3)
^(1/4)*c*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b) + 1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b*d + 77*(
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^4*b^2) + 1/512*sqrt(2)*(40*sq
rt(2)*sqrt(a*b)*b*d + 77*(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^4*b
^2) + 1/384*(60*b^2*d*x^10 + 77*b^2*c*x^9 + 160*a*b*d*x^6 + 198*a*b*c*x^5 + 132*a^2*d*x^2 + 153*a^2*c*x)/((b*x
^4 + a)^3*a^3)

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maple [A]  time = 0.07, size = 225, normalized size = 0.77 \[ \frac {5 d \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{32 \sqrt {a b}\, a^{3}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{512 a^{4}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{512 a^{4}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{1024 a^{4}}+\frac {\frac {5 b^{2} d \,x^{10}}{32 a^{3}}+\frac {77 b^{2} c \,x^{9}}{384 a^{3}}+\frac {5 b d \,x^{6}}{12 a^{2}}+\frac {33 b c \,x^{5}}{64 a^{2}}+\frac {11 d \,x^{2}}{32 a}+\frac {51 c x}{128 a}}{\left (b \,x^{4}+a \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)/(b*x^4+a)^4,x)

[Out]

(5/32/a^3*b^2*d*x^10+77/384/a^3*b^2*c*x^9+5/12/a^2*b*d*x^6+33/64/a^2*b*c*x^5+11/32/a*d*x^2+51/128/a*c*x)/(b*x^
4+a)^3+77/1024/a^4*c*(a/b)^(1/4)*2^(1/2)*ln((x^2+(a/b)^(1/4)*2^(1/2)*x+(a/b)^(1/2))/(x^2-(a/b)^(1/4)*2^(1/2)*x
+(a/b)^(1/2)))+77/512/a^4*c*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+77/512/a^4*c*(a/b)^(1/4)*2^(1/
2)*arctan(2^(1/2)/(a/b)^(1/4)*x-1)+5/32/a^3*d/(a*b)^(1/2)*arctan((1/a*b)^(1/2)*x^2)

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maxima [A]  time = 3.20, size = 304, normalized size = 1.04 \[ \frac {60 \, b^{2} d x^{10} + 77 \, b^{2} c x^{9} + 160 \, a b d x^{6} + 198 \, a b c x^{5} + 132 \, a^{2} d x^{2} + 153 \, a^{2} c x}{384 \, {\left (a^{3} b^{3} x^{12} + 3 \, a^{4} b^{2} x^{8} + 3 \, a^{5} b x^{4} + a^{6}\right )}} + \frac {\frac {77 \, \sqrt {2} c \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {77 \, \sqrt {2} c \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} c - 80 \, \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {1}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} c + 80 \, \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {1}{4}}}}{1024 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(60*b^2*d*x^10 + 77*b^2*c*x^9 + 160*a*b*d*x^6 + 198*a*b*c*x^5 + 132*a^2*d*x^2 + 153*a^2*c*x)/(a^3*b^3*x^
12 + 3*a^4*b^2*x^8 + 3*a^5*b*x^4 + a^6) + 1/1024*(77*sqrt(2)*c*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + s
qrt(a))/(a^(3/4)*b^(1/4)) - 77*sqrt(2)*c*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/
4)) + 2*(77*sqrt(2)*a^(1/4)*b^(1/4)*c - 80*sqrt(a)*d)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4
))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(1/4)) + 2*(77*sqrt(2)*a^(1/4)*b^(1/4)*c + 80*sqrt(
a)*d)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*
sqrt(b))*b^(1/4)))/a^3

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mupad [B]  time = 0.31, size = 350, normalized size = 1.20 \[ \left (\sum _{k=1}^4\ln \left (\frac {b^2\,\left (1925\,c\,d^2+1000\,d^3\,x-{\mathrm {root}\left (68719476736\,a^{15}\,b^2\,z^4+838860800\,a^8\,b\,d^2\,z^2-485703680\,a^4\,b\,c^2\,d\,z+35153041\,b\,c^4+2560000\,a\,d^4,z,k\right )}^2\,a^7\,b\,c\,315392-\mathrm {root}\left (68719476736\,a^{15}\,b^2\,z^4+838860800\,a^8\,b\,d^2\,z^2-485703680\,a^4\,b\,c^2\,d\,z+35153041\,b\,c^4+2560000\,a\,d^4,z,k\right )\,a^3\,b\,c^2\,x\,47432+{\mathrm {root}\left (68719476736\,a^{15}\,b^2\,z^4+838860800\,a^8\,b\,d^2\,z^2-485703680\,a^4\,b\,c^2\,d\,z+35153041\,b\,c^4+2560000\,a\,d^4,z,k\right )}^2\,a^7\,b\,d\,x\,163840\right )}{a^9\,32768}\right )\,\mathrm {root}\left (68719476736\,a^{15}\,b^2\,z^4+838860800\,a^8\,b\,d^2\,z^2-485703680\,a^4\,b\,c^2\,d\,z+35153041\,b\,c^4+2560000\,a\,d^4,z,k\right )\right )+\frac {\frac {11\,d\,x^2}{32\,a}+\frac {51\,c\,x}{128\,a}+\frac {77\,b^2\,c\,x^9}{384\,a^3}+\frac {5\,b^2\,d\,x^{10}}{32\,a^3}+\frac {33\,b\,c\,x^5}{64\,a^2}+\frac {5\,b\,d\,x^6}{12\,a^2}}{a^3+3\,a^2\,b\,x^4+3\,a\,b^2\,x^8+b^3\,x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(a + b*x^4)^4,x)

[Out]

symsum(log((b^2*(1925*c*d^2 + 1000*d^3*x - 315392*root(68719476736*a^15*b^2*z^4 + 838860800*a^8*b*d^2*z^2 - 48
5703680*a^4*b*c^2*d*z + 35153041*b*c^4 + 2560000*a*d^4, z, k)^2*a^7*b*c - 47432*root(68719476736*a^15*b^2*z^4
+ 838860800*a^8*b*d^2*z^2 - 485703680*a^4*b*c^2*d*z + 35153041*b*c^4 + 2560000*a*d^4, z, k)*a^3*b*c^2*x + 1638
40*root(68719476736*a^15*b^2*z^4 + 838860800*a^8*b*d^2*z^2 - 485703680*a^4*b*c^2*d*z + 35153041*b*c^4 + 256000
0*a*d^4, z, k)^2*a^7*b*d*x))/(32768*a^9))*root(68719476736*a^15*b^2*z^4 + 838860800*a^8*b*d^2*z^2 - 485703680*
a^4*b*c^2*d*z + 35153041*b*c^4 + 2560000*a*d^4, z, k), k, 1, 4) + ((11*d*x^2)/(32*a) + (51*c*x)/(128*a) + (77*
b^2*c*x^9)/(384*a^3) + (5*b^2*d*x^10)/(32*a^3) + (33*b*c*x^5)/(64*a^2) + (5*b*d*x^6)/(12*a^2))/(a^3 + b^3*x^12
+ 3*a^2*b*x^4 + 3*a*b^2*x^8)

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sympy [A]  time = 1.81, size = 231, normalized size = 0.79 \[ \operatorname {RootSum} {\left (68719476736 t^{4} a^{15} b^{2} + 838860800 t^{2} a^{8} b d^{2} - 485703680 t a^{4} b c^{2} d + 2560000 a d^{4} + 35153041 b c^{4}, \left (t \mapsto t \log {\left (x + \frac {- 429496729600 t^{3} a^{12} b d^{2} - 62170071040 t^{2} a^{8} b c^{2} d - 2621440000 t a^{5} d^{4} - 17998356992 t a^{4} b c^{4} + 1897280000 a c^{2} d^{3}}{788480000 a c d^{4} - 2706784157 b c^{5}} \right )} \right )\right )} + \frac {153 a^{2} c x + 132 a^{2} d x^{2} + 198 a b c x^{5} + 160 a b d x^{6} + 77 b^{2} c x^{9} + 60 b^{2} d x^{10}}{384 a^{6} + 1152 a^{5} b x^{4} + 1152 a^{4} b^{2} x^{8} + 384 a^{3} b^{3} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(68719476736*_t**4*a**15*b**2 + 838860800*_t**2*a**8*b*d**2 - 485703680*_t*a**4*b*c**2*d + 2560000*a*d*
*4 + 35153041*b*c**4, Lambda(_t, _t*log(x + (-429496729600*_t**3*a**12*b*d**2 - 62170071040*_t**2*a**8*b*c**2*
d - 2621440000*_t*a**5*d**4 - 17998356992*_t*a**4*b*c**4 + 1897280000*a*c**2*d**3)/(788480000*a*c*d**4 - 27067
84157*b*c**5)))) + (153*a**2*c*x + 132*a**2*d*x**2 + 198*a*b*c*x**5 + 160*a*b*d*x**6 + 77*b**2*c*x**9 + 60*b**
2*d*x**10)/(384*a**6 + 1152*a**5*b*x**4 + 1152*a**4*b**2*x**8 + 384*a**3*b**3*x**12)

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