3.2.68 \(\int \frac {(c+d x^4)^3}{(a+b x^4)^2} \, dx\) [168]

Optimal. Leaf size=317 \[ \frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}-\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}-\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}} \]

[Out]

d^2*(-2*a*d+3*b*c)*x/b^3+1/5*d^3*x^5/b^2+1/4*(-a*d+b*c)^3*x/a/b^3/(b*x^4+a)+3/16*(-a*d+b*c)^2*(3*a*d+b*c)*arct
an(-1+b^(1/4)*x*2^(1/2)/a^(1/4))/a^(7/4)/b^(13/4)*2^(1/2)+3/16*(-a*d+b*c)^2*(3*a*d+b*c)*arctan(1+b^(1/4)*x*2^(
1/2)/a^(1/4))/a^(7/4)/b^(13/4)*2^(1/2)-3/32*(-a*d+b*c)^2*(3*a*d+b*c)*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2
*b^(1/2))/a^(7/4)/b^(13/4)*2^(1/2)+3/32*(-a*d+b*c)^2*(3*a*d+b*c)*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1
/2))/a^(7/4)/b^(13/4)*2^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {398, 393, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) (b c-a d)^2 (3 a d+b c)}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (3 a d+b c)}{8 \sqrt {2} a^{7/4} b^{13/4}}-\frac {3 (b c-a d)^2 (3 a d+b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (3 a d+b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {d^2 x (3 b c-2 a d)}{b^3}+\frac {x (b c-a d)^3}{4 a b^3 \left (a+b x^4\right )}+\frac {d^3 x^5}{5 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^4)^3/(a + b*x^4)^2,x]

[Out]

(d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^5)/(5*b^2) + ((b*c - a*d)^3*x)/(4*a*b^3*(a + b*x^4)) - (3*(b*c - a*d)^2*(
b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(13/4)) + (3*(b*c - a*d)^2*(b*c + 3
*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(13/4)) - (3*(b*c - a*d)^2*(b*c + 3*a*d)*L
og[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(13/4)) + (3*(b*c - a*d)^2*(b*c +
3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(13/4))

Rule 210

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

Rule 217

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 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 631

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 642

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 1176

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 1179

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]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx &=\int \left (\frac {d^2 (3 b c-2 a d)}{b^3}+\frac {d^3 x^4}{b^2}+\frac {(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^4}{b^3 \left (a+b x^4\right )^2}\right ) \, dx\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {\int \frac {(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^4}{\left (a+b x^4\right )^2} \, dx}{b^3}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {1}{a+b x^4} \, dx}{4 a b^3}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx}{8 a^{3/2} b^3}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx}{8 a^{3/2} b^3}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{7/2}}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} b^{7/2}}-\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \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}{16 \sqrt {2} a^{7/4} b^{13/4}}-\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \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}{16 \sqrt {2} a^{7/4} b^{13/4}}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}-\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}-\frac {\left (3 (b c-a d)^2 (b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^5}{5 b^2}+\frac {(b c-a d)^3 x}{4 a b^3 \left (a+b x^4\right )}-\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{13/4}}-\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}+\frac {3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{13/4}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 301, normalized size = 0.95 \begin {gather*} \frac {160 \sqrt [4]{b} d^2 (3 b c-2 a d) x+32 b^{5/4} d^3 x^5+\frac {40 \sqrt [4]{b} (b c-a d)^3 x}{a \left (a+b x^4\right )}-\frac {30 \sqrt {2} (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {30 \sqrt {2} (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac {15 \sqrt {2} (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}+\frac {15 \sqrt {2} (b c-a d)^2 (b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}}{160 b^{13/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^4)^3/(a + b*x^4)^2,x]

[Out]

(160*b^(1/4)*d^2*(3*b*c - 2*a*d)*x + 32*b^(5/4)*d^3*x^5 + (40*b^(1/4)*(b*c - a*d)^3*x)/(a*(a + b*x^4)) - (30*S
qrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (30*Sqrt[2]*(b*c - a*d)^
2*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) - (15*Sqrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*Log
[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (15*Sqrt[2]*(b*c - a*d)^2*(b*c + 3*a*d)*Log[Sqr
t[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4))/(160*b^(13/4))

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Maple [A]
time = 0.27, size = 220, normalized size = 0.69

method	result	size
risch	\(\frac {d^{3} x^{5}}{5 b^{2}}-\frac {2 d^{3} a x}{b^{3}}+\frac {3 d^{2} c x}{b^{2}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{4 a \,b^{3} \left (b \,x^{4}+a \right )}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (3 a^{3} d^{3}-5 a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{16 b^{4} a}\)	\(151\)
default	\(-\frac {d^{2} \left (-\frac {1}{5} b d \,x^{5}+2 a d x -3 b c x \right )}{b^{3}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{4 a \left (b \,x^{4}+a \right )}+\frac {3 \left (3 a^{3} d^{3}-5 a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}}{b^{3}}\)	\(220\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^4+c)^3/(b*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

-d^2/b^3*(-1/5*b*d*x^5+2*a*d*x-3*b*c*x)+1/b^3*(-1/4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/a*x/(b*x^4+a
)+3/32*(3*a^3*d^3-5*a^2*b*c*d^2+a*b^2*c^2*d+b^3*c^3)/a^2*(a/b)^(1/4)*2^(1/2)*(ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a
/b)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/b)^(
1/4)*x-1)))

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Maxima [A]
time = 0.51, size = 405, normalized size = 1.28 \begin {gather*} \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{4 \, {\left (a b^{4} x^{4} + a^{2} b^{3}\right )}} + \frac {b d^{3} x^{5} + 5 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x}{5 \, b^{3}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\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 )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\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 )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \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 {\sqrt {2} {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \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}}}\right )}}{32 \, a b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^3/(b*x^4+a)^2,x, algorithm="maxima")

[Out]

1/4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x/(a*b^4*x^4 + a^2*b^3) + 1/5*(b*d^3*x^5 + 5*(3*b*c*d^
2 - 2*a*d^3)*x)/b^3 + 3/32*(2*sqrt(2)*(b^3*c^3 + a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d^3)*arctan(1/2*sqrt(2)*(
2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(b^3
*c^3 + a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d^3)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqr
t(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(b^3*c^3 + a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d
^3)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b^3*c^3 + a*b^2*c^2*d
- 5*a^2*b*c*d^2 + 3*a^3*d^3)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a*b^3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1937 vs. \(2 (248) = 496\).
time = 2.66, size = 1937, normalized size = 6.11 \begin {gather*} \frac {16 \, a b^{2} d^{3} x^{9} + 48 \, {\left (5 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x^{5} + 60 \, {\left (a b^{4} x^{4} + a^{2} b^{3}\right )} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{4} b^{6} \sqrt {-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}} + {\left (b^{6} c^{6} + 2 \, a b^{5} c^{5} d - 9 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{3} b^{3} c^{3} d^{3} + 31 \, a^{4} b^{2} c^{2} d^{4} - 30 \, a^{5} b c d^{5} + 9 \, a^{6} d^{6}\right )} x^{2}} a^{5} b^{10} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {3}{4}} - {\left (a^{5} b^{13} c^{3} + a^{6} b^{12} c^{2} d - 5 \, a^{7} b^{11} c d^{2} + 3 \, a^{8} b^{10} d^{3}\right )} x \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {3}{4}}}{b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}\right ) + 15 \, {\left (a b^{4} x^{4} + a^{2} b^{3}\right )} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \log \left (3 \, a^{2} b^{3} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} + 3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} x\right ) - 15 \, {\left (a b^{4} x^{4} + a^{2} b^{3}\right )} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} \log \left (-3 \, a^{2} b^{3} \left (-\frac {b^{12} c^{12} + 4 \, a b^{11} c^{11} d - 14 \, a^{2} b^{10} c^{10} d^{2} - 44 \, a^{3} b^{9} c^{9} d^{3} + 127 \, a^{4} b^{8} c^{8} d^{4} + 136 \, a^{5} b^{7} c^{7} d^{5} - 644 \, a^{6} b^{6} c^{6} d^{6} + 328 \, a^{7} b^{5} c^{5} d^{7} + 1039 \, a^{8} b^{4} c^{4} d^{8} - 1932 \, a^{9} b^{3} c^{3} d^{9} + 1458 \, a^{10} b^{2} c^{2} d^{10} - 540 \, a^{11} b c d^{11} + 81 \, a^{12} d^{12}}{a^{7} b^{13}}\right )^{\frac {1}{4}} + 3 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d - 5 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} x\right ) + 20 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - 9 \, a^{3} d^{3}\right )} x}{80 \, {\left (a b^{4} x^{4} + a^{2} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^3/(b*x^4+a)^2,x, algorithm="fricas")

[Out]

1/80*(16*a*b^2*d^3*x^9 + 48*(5*a*b^2*c*d^2 - 3*a^2*b*d^3)*x^5 + 60*(a*b^4*x^4 + a^2*b^3)*(-(b^12*c^12 + 4*a*b^
11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^
6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a
^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(1/4)*arctan((sqrt(a^4*b^6*sqrt(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^
2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a
^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81
*a^12*d^12)/(a^7*b^13)) + (b^6*c^6 + 2*a*b^5*c^5*d - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d^
4 - 30*a^5*b*c*d^5 + 9*a^6*d^6)*x^2)*a^5*b^10*(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b
^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*
b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(3
/4) - (a^5*b^13*c^3 + a^6*b^12*c^2*d - 5*a^7*b^11*c*d^2 + 3*a^8*b^10*d^3)*x*(-(b^12*c^12 + 4*a*b^11*c^11*d - 1
4*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 3
28*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11
+ 81*a^12*d^12)/(a^7*b^13))^(3/4))/(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 +
127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 -
1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)) + 15*(a*b^4*x^4 + a^2*b^3)
*(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^
7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^1
0*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(1/4)*log(3*a^2*b^3*(-(b^12*c^12 + 4*a*b^11*c^1
1*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*
d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*
c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(1/4) + 3*(b^3*c^3 + a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d^3)*x) - 15*(a*b^
4*x^4 + a^2*b^3)*(-(b^12*c^12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*
d^4 + 136*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^
3*d^9 + 1458*a^10*b^2*c^2*d^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(1/4)*log(-3*a^2*b^3*(-(b^12*c^
12 + 4*a*b^11*c^11*d - 14*a^2*b^10*c^10*d^2 - 44*a^3*b^9*c^9*d^3 + 127*a^4*b^8*c^8*d^4 + 136*a^5*b^7*c^7*d^5 -
644*a^6*b^6*c^6*d^6 + 328*a^7*b^5*c^5*d^7 + 1039*a^8*b^4*c^4*d^8 - 1932*a^9*b^3*c^3*d^9 + 1458*a^10*b^2*c^2*d
^10 - 540*a^11*b*c*d^11 + 81*a^12*d^12)/(a^7*b^13))^(1/4) + 3*(b^3*c^3 + a*b^2*c^2*d - 5*a^2*b*c*d^2 + 3*a^3*d
^3)*x) + 20*(b^3*c^3 - 3*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 9*a^3*d^3)*x)/(a*b^4*x^4 + a^2*b^3)

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Sympy [A]
time = 82.86, size = 337, normalized size = 1.06 \begin {gather*} x \left (- \frac {2 a d^{3}}{b^{3}} + \frac {3 c d^{2}}{b^{2}}\right ) + \frac {x \left (- a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}\right )}{4 a^{2} b^{3} + 4 a b^{4} x^{4}} + \operatorname {RootSum} {\left (65536 t^{4} a^{7} b^{13} + 6561 a^{12} d^{12} - 43740 a^{11} b c d^{11} + 118098 a^{10} b^{2} c^{2} d^{10} - 156492 a^{9} b^{3} c^{3} d^{9} + 84159 a^{8} b^{4} c^{4} d^{8} + 26568 a^{7} b^{5} c^{5} d^{7} - 52164 a^{6} b^{6} c^{6} d^{6} + 11016 a^{5} b^{7} c^{7} d^{5} + 10287 a^{4} b^{8} c^{8} d^{4} - 3564 a^{3} b^{9} c^{9} d^{3} - 1134 a^{2} b^{10} c^{10} d^{2} + 324 a b^{11} c^{11} d + 81 b^{12} c^{12}, \left ( t \mapsto t \log {\left (\frac {16 t a^{2} b^{3}}{9 a^{3} d^{3} - 15 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + 3 b^{3} c^{3}} + x \right )} \right )\right )} + \frac {d^{3} x^{5}}{5 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**4+c)**3/(b*x**4+a)**2,x)

[Out]

x*(-2*a*d**3/b**3 + 3*c*d**2/b**2) + x*(-a**3*d**3 + 3*a**2*b*c*d**2 - 3*a*b**2*c**2*d + b**3*c**3)/(4*a**2*b*
*3 + 4*a*b**4*x**4) + RootSum(65536*_t**4*a**7*b**13 + 6561*a**12*d**12 - 43740*a**11*b*c*d**11 + 118098*a**10
*b**2*c**2*d**10 - 156492*a**9*b**3*c**3*d**9 + 84159*a**8*b**4*c**4*d**8 + 26568*a**7*b**5*c**5*d**7 - 52164*
a**6*b**6*c**6*d**6 + 11016*a**5*b**7*c**7*d**5 + 10287*a**4*b**8*c**8*d**4 - 3564*a**3*b**9*c**9*d**3 - 1134*
a**2*b**10*c**10*d**2 + 324*a*b**11*c**11*d + 81*b**12*c**12, Lambda(_t, _t*log(16*_t*a**2*b**3/(9*a**3*d**3 -
15*a**2*b*c*d**2 + 3*a*b**2*c**2*d + 3*b**3*c**3) + x))) + d**3*x**5/(5*b**2)

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Giac [A]
time = 0.64, size = 496, normalized size = 1.56 \begin {gather*} \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\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^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\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^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{4}} - \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{4}} + \frac {b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{4 \, {\left (b x^{4} + a\right )} a b^{3}} + \frac {b^{8} d^{3} x^{5} + 15 \, b^{8} c d^{2} x - 10 \, a b^{7} d^{3} x}{5 \, b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^4+c)^3/(b*x^4+a)^2,x, algorithm="giac")

[Out]

3/16*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 + (a*b^3)^(1/4)*a*b^2*c^2*d - 5*(a*b^3)^(1/4)*a^2*b*c*d^2 + 3*(a*b^3)^(1/4
)*a^3*d^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^4) + 3/16*sqrt(2)*((a*b^3)^(1/4)
*b^3*c^3 + (a*b^3)^(1/4)*a*b^2*c^2*d - 5*(a*b^3)^(1/4)*a^2*b*c*d^2 + 3*(a*b^3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(
2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^4) + 3/32*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 + (a*b^3)^(1/4)*a*
b^2*c^2*d - 5*(a*b^3)^(1/4)*a^2*b*c*d^2 + 3*(a*b^3)^(1/4)*a^3*d^3)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b)
)/(a^2*b^4) - 3/32*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 + (a*b^3)^(1/4)*a*b^2*c^2*d - 5*(a*b^3)^(1/4)*a^2*b*c*d^2 +
3*(a*b^3)^(1/4)*a^3*d^3)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^4) + 1/4*(b^3*c^3*x - 3*a*b^2*c^2
*d*x + 3*a^2*b*c*d^2*x - a^3*d^3*x)/((b*x^4 + a)*a*b^3) + 1/5*(b^8*d^3*x^5 + 15*b^8*c*d^2*x - 10*a*b^7*d^3*x)/
b^10

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Mupad [B]
time = 1.53, size = 1616, normalized size = 5.10 \begin {gather*} \frac {d^3\,x^5}{5\,b^2}-x\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )-\frac {x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{4\,a\,\left (b^4\,x^4+a\,b^3\right )}+\frac {\mathrm {atan}\left (\frac {\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}-\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}+\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}}{\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}-\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}-\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}+\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,3{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}}+\frac {3\,\mathrm {atan}\left (\frac {\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}+\frac {3\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}}{\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}-\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (\frac {9\,x\,\left (9\,a^6\,d^6-30\,a^5\,b\,c\,d^5+31\,a^4\,b^2\,c^2\,d^4-4\,a^3\,b^3\,c^3\,d^3-9\,a^2\,b^4\,c^4\,d^2+2\,a\,b^5\,c^5\,d+b^6\,c^6\right )}{4\,a^2\,b^3}+\frac {{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )\,\left (36\,a^3\,d^3-60\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+12\,b^3\,c^3\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}\right )\,3{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{13/4}}}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (3\,a\,d+b\,c\right )}{8\,{\left (-a\right )}^{7/4}\,b^{13/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^4)^3/(a + b*x^4)^2,x)

[Out]

(d^3*x^5)/(5*b^2) - x*((2*a*d^3)/b^3 - (3*c*d^2)/b^2) - (x*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)
)/(4*a*(a*b^3 + b^4*x^4)) + (atan((((a*d - b*c)^2*(3*a*d + b*c)*((9*x*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2
- 4*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(4*a^2*b^3) - (3*(a*d - b*c)^2*(3
*a*d + b*c)*(36*a^3*d^3 + 12*b^3*c^3 + 12*a*b^2*c^2*d - 60*a^2*b*c*d^2))/(16*(-a)^(7/4)*b^(13/4)))*3i)/(16*(-a
)^(7/4)*b^(13/4)) + ((a*d - b*c)^2*(3*a*d + b*c)*((9*x*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^
3*d^3 + 31*a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(4*a^2*b^3) + (3*(a*d - b*c)^2*(3*a*d + b*c)*(36
*a^3*d^3 + 12*b^3*c^3 + 12*a*b^2*c^2*d - 60*a^2*b*c*d^2))/(16*(-a)^(7/4)*b^(13/4)))*3i)/(16*(-a)^(7/4)*b^(13/4
)))/((3*(a*d - b*c)^2*(3*a*d + b*c)*((9*x*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31*a^
4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(4*a^2*b^3) - (3*(a*d - b*c)^2*(3*a*d + b*c)*(36*a^3*d^3 + 12
*b^3*c^3 + 12*a*b^2*c^2*d - 60*a^2*b*c*d^2))/(16*(-a)^(7/4)*b^(13/4))))/(16*(-a)^(7/4)*b^(13/4)) - (3*(a*d - b
*c)^2*(3*a*d + b*c)*((9*x*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d^4 +
2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(4*a^2*b^3) + (3*(a*d - b*c)^2*(3*a*d + b*c)*(36*a^3*d^3 + 12*b^3*c^3 + 12*a*
b^2*c^2*d - 60*a^2*b*c*d^2))/(16*(-a)^(7/4)*b^(13/4))))/(16*(-a)^(7/4)*b^(13/4))))*(a*d - b*c)^2*(3*a*d + b*c)
*3i)/(8*(-a)^(7/4)*b^(13/4)) + (3*atan(((3*(a*d - b*c)^2*(3*a*d + b*c)*((9*x*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*
c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(4*a^2*b^3) - ((a*d - b*c)
^2*(3*a*d + b*c)*(36*a^3*d^3 + 12*b^3*c^3 + 12*a*b^2*c^2*d - 60*a^2*b*c*d^2)*3i)/(16*(-a)^(7/4)*b^(13/4))))/(1
6*(-a)^(7/4)*b^(13/4)) + (3*(a*d - b*c)^2*(3*a*d + b*c)*((9*x*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4*a^3
*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(4*a^2*b^3) + ((a*d - b*c)^2*(3*a*d + b*c
)*(36*a^3*d^3 + 12*b^3*c^3 + 12*a*b^2*c^2*d - 60*a^2*b*c*d^2)*3i)/(16*(-a)^(7/4)*b^(13/4))))/(16*(-a)^(7/4)*b^
(13/4)))/(((a*d - b*c)^2*(3*a*d + b*c)*((9*x*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31
*a^4*b^2*c^2*d^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(4*a^2*b^3) - ((a*d - b*c)^2*(3*a*d + b*c)*(36*a^3*d^3 + 1
2*b^3*c^3 + 12*a*b^2*c^2*d - 60*a^2*b*c*d^2)*3i)/(16*(-a)^(7/4)*b^(13/4)))*3i)/(16*(-a)^(7/4)*b^(13/4)) - ((a*
d - b*c)^2*(3*a*d + b*c)*((9*x*(9*a^6*d^6 + b^6*c^6 - 9*a^2*b^4*c^4*d^2 - 4*a^3*b^3*c^3*d^3 + 31*a^4*b^2*c^2*d
^4 + 2*a*b^5*c^5*d - 30*a^5*b*c*d^5))/(4*a^2*b^3) + ((a*d - b*c)^2*(3*a*d + b*c)*(36*a^3*d^3 + 12*b^3*c^3 + 12
*a*b^2*c^2*d - 60*a^2*b*c*d^2)*3i)/(16*(-a)^(7/4)*b^(13/4)))*3i)/(16*(-a)^(7/4)*b^(13/4))))*(a*d - b*c)^2*(3*a
*d + b*c))/(8*(-a)^(7/4)*b^(13/4))

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