3.7.13 \(\int \frac {(d+e x)^4}{a+b (d+e x)^2+c (d+e x)^4} \, dx\) [613]

Optimal. Leaf size=193 \[ \frac {x}{c}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} e} \]

[Out]

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

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Rubi [A]
time = 0.29, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1156, 1136, 1180, 211} \begin {gather*} -\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} e \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} c^{3/2} e \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {x}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/c - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/
(Sqrt[2]*c^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[
c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)

Rule 211

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

Rule 1136

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*
x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rule 1180

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^4}{a+b (d+e x)^2+c (d+e x)^4} \, dx &=\frac {\text {Subst}\left (\int \frac {x^4}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{e}\\ &=\frac {x}{c}-\frac {\text {Subst}\left (\int \frac {a+b x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{c e}\\ &=\frac {x}{c}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{2 c e}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{2 c e}\\ &=\frac {x}{c}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b+\sqrt {b^2-4 a c}} e}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 219, normalized size = 1.13 \begin {gather*} \frac {2 \sqrt {c} (d+e x)-\frac {\sqrt {2} \left (-b^2+2 a c+b \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \left (b^2-2 a c+b \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{2 c^{3/2} e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*(d + e*x) - (Sqrt[2]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b
- Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*(b^2 - 2*a*c + b*Sqrt[b^2 -
4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2
- 4*a*c]]))/(2*c^(3/2)*e)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.15, size = 158, normalized size = 0.82

method	result	size
default	\(\frac {x}{c}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (-\textit {\_R}^{2} b \,e^{2}-2 \textit {\_R} b d e -d^{2} b -a \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{2 c e}\)	\(158\)
risch	\(\frac {x}{c}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (-\textit {\_R}^{2} b \,e^{2}-2 \textit {\_R} b d e -d^{2} b -a \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{2 c e}\)	\(158\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c+1/2/c/e*sum((-_R^2*b*e^2-2*_R*b*d*e-b*d^2-a)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)
*ln(x-_R),_R=RootOf(e^4*c*_Z^4+4*d*e^3*c*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+d^4*c+d^2*b+a))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/c - integrate((b*x^2*e^2 + 2*b*d*x*e + b*d^2 + a)/(c*x^4*e^4 + 4*c*d*x^3*e^3 + c*d^4 + b*d^2 + (6*c*d^2*e^2
+ b*e^2)*x^2 + 2*(2*c*d^3*e + b*d*e)*x + a), x)/c

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1167 vs. \(2 (157) = 314\).
time = 0.38, size = 1167, normalized size = 6.05 \begin {gather*} \frac {\sqrt {\frac {1}{2}} c \sqrt {-\frac {{\left (b^{3} - 3 \, a b c + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} x e - 2 \, {\left (a b^{2} - a^{2} c\right )} d + \sqrt {\frac {1}{2}} {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}} e - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}}\right ) - \sqrt {\frac {1}{2}} c \sqrt {-\frac {{\left (b^{3} - 3 \, a b c + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} x e - 2 \, {\left (a b^{2} - a^{2} c\right )} d - \sqrt {\frac {1}{2}} {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}} e - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}}\right ) - \sqrt {\frac {1}{2}} c \sqrt {-\frac {{\left (b^{3} - 3 \, a b c - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} x e - 2 \, {\left (a b^{2} - a^{2} c\right )} d + \sqrt {\frac {1}{2}} {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}} e + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}}\right ) + \sqrt {\frac {1}{2}} c \sqrt {-\frac {{\left (b^{3} - 3 \, a b c - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-2 \, {\left (a b^{2} - a^{2} c\right )} x e - 2 \, {\left (a b^{2} - a^{2} c\right )} d - \sqrt {\frac {1}{2}} {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}} e + {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} e^{\left (-2\right )}}{b^{2} c^{3} - 4 \, a c^{4}}}\right ) + 2 \, x}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(1/2)*c*sqrt(-(b^3 - 3*a*b*c + (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^
7)))*e^(-2)/(b^2*c^3 - 4*a*c^4))*log(-2*(a*b^2 - a^2*c)*x*e - 2*(a*b^2 - a^2*c)*d + sqrt(1/2)*((b^3*c^3 - 4*a*
b*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7))*e - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*sqrt(-(b^3 -
3*a*b*c + (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7)))*e^(-2)/(b^2*c^3 - 4*a*c^
4))) - sqrt(1/2)*c*sqrt(-(b^3 - 3*a*b*c + (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*
c^7)))*e^(-2)/(b^2*c^3 - 4*a*c^4))*log(-2*(a*b^2 - a^2*c)*x*e - 2*(a*b^2 - a^2*c)*d - sqrt(1/2)*((b^3*c^3 - 4*
a*b*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7))*e - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*sqrt(-(b^3
- 3*a*b*c + (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7)))*e^(-2)/(b^2*c^3 - 4*a*
c^4))) - sqrt(1/2)*c*sqrt(-(b^3 - 3*a*b*c - (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*
a*c^7)))*e^(-2)/(b^2*c^3 - 4*a*c^4))*log(-2*(a*b^2 - a^2*c)*x*e - 2*(a*b^2 - a^2*c)*d + sqrt(1/2)*((b^3*c^3 -
4*a*b*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7))*e + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*sqrt(-(b
^3 - 3*a*b*c - (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7)))*e^(-2)/(b^2*c^3 - 4*
a*c^4))) + sqrt(1/2)*c*sqrt(-(b^3 - 3*a*b*c - (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 -
4*a*c^7)))*e^(-2)/(b^2*c^3 - 4*a*c^4))*log(-2*(a*b^2 - a^2*c)*x*e - 2*(a*b^2 - a^2*c)*d - sqrt(1/2)*((b^3*c^3
- 4*a*b*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7))*e + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e)*sqrt(-
(b^3 - 3*a*b*c - (b^2*c^3 - 4*a*c^4)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(b^2*c^6 - 4*a*c^7)))*e^(-2)/(b^2*c^3 -
4*a*c^4))) + 2*x)/c

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Sympy [A]
time = 1.73, size = 178, normalized size = 0.92 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} c^{5} e^{4} - 128 a b^{2} c^{4} e^{4} + 16 b^{4} c^{3} e^{4}\right ) + t^{2} \cdot \left (48 a^{2} b c^{2} e^{2} - 28 a b^{3} c e^{2} + 4 b^{5} e^{2}\right ) + a^{3}, \left ( t \mapsto t \log {\left (x + \frac {32 t^{3} a b c^{4} e^{3} - 8 t^{3} b^{3} c^{3} e^{3} - 4 t a^{2} c^{2} e + 8 t a b^{2} c e - 2 t b^{4} e + a^{2} c d - a b^{2} d}{a^{2} c e - a b^{2} e} \right )} \right )\right )} + \frac {x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**4*(256*a**2*c**5*e**4 - 128*a*b**2*c**4*e**4 + 16*b**4*c**3*e**4) + _t**2*(48*a**2*b*c**2*e**2 - 2
8*a*b**3*c*e**2 + 4*b**5*e**2) + a**3, Lambda(_t, _t*log(x + (32*_t**3*a*b*c**4*e**3 - 8*_t**3*b**3*c**3*e**3
- 4*_t*a**2*c**2*e + 8*_t*a*b**2*c*e - 2*_t*b**4*e + a**2*c*d - a*b**2*d)/(a**2*c*e - a*b**2*e)))) + x/c

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1194 vs. \(2 (157) = 314\).
time = 4.14, size = 1194, normalized size = 6.19 \begin {gather*} \frac {{\left (\frac {{\left ({\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} b e^{6} - 2 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )} b d e^{5} + b d^{2} e^{4} + a e^{4}\right )} \log \left (d e^{\left (-1\right )} + x + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}{2 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{3} c e^{4} - 6 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} c d e^{3} - 2 \, c d^{3} e - b d e + {\left (6 \, c d^{2} e^{2} + b e^{2}\right )} {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}} + \frac {{\left ({\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} b e^{6} - 2 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )} b d e^{5} + b d^{2} e^{4} + a e^{4}\right )} \log \left (d e^{\left (-1\right )} + x - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}{2 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{3} c e^{4} - 6 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} c d e^{3} - 2 \, c d^{3} e - b d e + {\left (6 \, c d^{2} e^{2} + b e^{2}\right )} {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} + \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}} + \frac {{\left ({\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} b e^{6} - 2 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )} b d e^{5} + b d^{2} e^{4} + a e^{4}\right )} \log \left (d e^{\left (-1\right )} + x + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}{2 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{3} c e^{4} - 6 \, {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} c d e^{3} - 2 \, c d^{3} e - b d e + {\left (6 \, c d^{2} e^{2} + b e^{2}\right )} {\left (d e^{\left (-1\right )} + \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}} + \frac {{\left ({\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} b e^{6} - 2 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )} b d e^{5} + b d^{2} e^{4} + a e^{4}\right )} \log \left (d e^{\left (-1\right )} + x - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}{2 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{3} c e^{4} - 6 \, {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}^{2} c d e^{3} - 2 \, c d^{3} e - b d e + {\left (6 \, c d^{2} e^{2} + b e^{2}\right )} {\left (d e^{\left (-1\right )} - \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b e^{2} - \sqrt {b^{2} - 4 \, a c} e^{2}\right )} e^{\left (-4\right )}}{c}}\right )}}\right )} e^{\left (-4\right )}}{2 \, c} + \frac {x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(((d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b*e^6 - 2*(d*e^(-1) + sqrt(1/2)
*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*d*e^5 + b*d^2*e^4 + a*e^4)*log(d*e^(-1) + x + sqrt(1/2)*sq
rt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*
e^(-4)/c))^3*c*e^4 - 6*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*c*d*e^3 - 2*c*
d^3*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c)))
+ ((d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b*e^6 - 2*(d*e^(-1) - sqrt(1/2)*sq
rt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*d*e^5 + b*d^2*e^4 + a*e^4)*log(d*e^(-1) + x - sqrt(1/2)*sqrt(
-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(
-4)/c))^3*c*e^4 - 6*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*c*d*e^3 - 2*c*d^3
*e - b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 + sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))) + (
(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b*e^6 - 2*(d*e^(-1) + sqrt(1/2)*sqrt(
-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*d*e^5 + b*d^2*e^4 + a*e^4)*log(d*e^(-1) + x + sqrt(1/2)*sqrt(-(b
*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)
/c))^3*c*e^4 - 6*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*c*d*e^3 - 2*c*d^3*e
- b*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) + sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))) + ((d*
e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*b*e^6 - 2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b
*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))*b*d*e^5 + b*d^2*e^4 + a*e^4)*log(d*e^(-1) + x - sqrt(1/2)*sqrt(-(b*e^
2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))/(2*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c)
)^3*c*e^4 - 6*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))^2*c*d*e^3 - 2*c*d^3*e - b
*d*e + (6*c*d^2*e^2 + b*e^2)*(d*e^(-1) - sqrt(1/2)*sqrt(-(b*e^2 - sqrt(b^2 - 4*a*c)*e^2)*e^(-4)/c))))*e^(-4)/c
+ x/c

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Mupad [B]
time = 2.32, size = 2500, normalized size = 12.95 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4/(a + b*(d + e*x)^2 + c*(d + e*x)^4),x)

[Out]

atan(((-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*
a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2)*(((16*a^2*c^3*e^12 - 4*a*b^2*c^2*e^12)/c + ((8*b^3*c^3*d*
e^13 - 32*a*b*c^4*d*e^13)/c + (2*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2
) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^
2)))^(1/2))*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(
8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2) + (2*b^4*d*e^11 + 4*a^2*c^2*d*e^11 - 8*a*b^2*c*d*e^
11)/c + (2*x*(b^4*e^12 + 2*a^2*c^2*e^12 - 4*a*b^2*c*e^12))/c)*1i + (-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*
a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1
/2)*((2*b^4*d*e^11 + 4*a^2*c^2*d*e^11 - 8*a*b^2*c*d*e^11)/c - ((16*a^2*c^3*e^12 - 4*a*b^2*c^2*e^12)/c - ((8*b^
3*c^3*d*e^13 - 32*a*b*c^4*d*e^13)/c + (2*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5 + b^2*(-(4*a*c - b^2)
^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^
2*c^4*e^2)))^(1/2))*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^
(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2) + (2*x*(b^4*e^12 + 2*a^2*c^2*e^12 - 4*a*b^2
*c*e^12))/c)*1i)/((-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1
/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2)*(((16*a^2*c^3*e^12 - 4*a*b^2*c^2*e^12)/c + ((
8*b^3*c^3*d*e^13 - 32*a*b*c^4*d*e^13)/c + (2*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5 + b^2*(-(4*a*c -
b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*
a*b^2*c^4*e^2)))^(1/2))*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)
^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2) + (2*b^4*d*e^11 + 4*a^2*c^2*d*e^11 - 8*
a*b^2*c*d*e^11)/c + (2*x*(b^4*e^12 + 2*a^2*c^2*e^12 - 4*a*b^2*c*e^12))/c) - (-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1
/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*
e^2)))^(1/2)*((2*b^4*d*e^11 + 4*a^2*c^2*d*e^11 - 8*a*b^2*c*d*e^11)/c - ((16*a^2*c^3*e^12 - 4*a*b^2*c^2*e^12)/c
- ((8*b^3*c^3*d*e^13 - 32*a*b*c^4*d*e^13)/c + (2*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5 + b^2*(-(4*a
*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2
- 8*a*b^2*c^4*e^2)))^(1/2))*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c -
b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2) + (2*x*(b^4*e^12 + 2*a^2*c^2*e^12
- 4*a*b^2*c*e^12))/c) + (2*a^2*b*e^10)/c))*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c -
a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2)*2i + atan(((-(b^5 -
b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b
^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2)*(((16*a^2*c^3*e^12 - 4*a*b^2*c^2*e^12)/c + ((8*b^3*c^3*d*e^13 - 32*a*b*c
^4*d*e^13)/c + (2*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^
2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2))*(-(
b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e
^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2) + (2*b^4*d*e^11 + 4*a^2*c^2*d*e^11 - 8*a*b^2*c*d*e^11)/c + (2*x*(b
^4*e^12 + 2*a^2*c^2*e^12 - 4*a*b^2*c*e^12))/c)*1i + (-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a
*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2)*((2*b^4*d*e
^11 + 4*a^2*c^2*d*e^11 - 8*a*b^2*c*d*e^11)/c - ((16*a^2*c^3*e^12 - 4*a*b^2*c^2*e^12)/c - ((8*b^3*c^3*d*e^13 -
32*a*b*c^4*d*e^13)/c + (2*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*
a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1
/2))*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a
^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2) + (2*x*(b^4*e^12 + 2*a^2*c^2*e^12 - 4*a*b^2*c*e^12))/c)*1i
)/((-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2
*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2)))^(1/2)*(((16*a^2*c^3*e^12 - 4*a*b^2*c^2*e^12)/c + ((8*b^3*c^3*d*e^1
3 - 32*a*b*c^4*d*e^13)/c + (2*x*(4*b^3*c^3*e^14 - 16*a*b*c^4*e^14))/c)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) +
12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^5*e^2 + b^4*c^3*e^2 - 8*a*b^2*c^4*e^2))
)^(1/2))*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) ...

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