\(\int \frac {(d+e x)^4}{a+c x^2} \, dx\) [498]

Optimal result

Integrand size = 17, antiderivative size = 123 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac {2 d e^3 x^2}{c}+\frac {e^4 x^3}{3 c}+\frac {\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{5/2}}+\frac {2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2} \]

[Out]

e^2*(-a*e^2+6*c*d^2)*x/c^2+2*d*e^3*x^2/c+1/3*e^4*x^3/c+2*d*e*(-a*e^2+c*d^2)*ln(c*x^2+a)/c^2+(a^2*e^4-6*a*c*d^2
*e^2+c^2*d^4)*arctan(x*c^(1/2)/a^(1/2))/c^(5/2)/a^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {716, 649, 211, 266} \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )}{\sqrt {a} c^{5/2}}+\frac {2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2}+\frac {e^2 x \left (6 c d^2-a e^2\right )}{c^2}+\frac {2 d e^3 x^2}{c}+\frac {e^4 x^3}{3 c} \]

[In]

Int[(d + e*x)^4/(a + c*x^2),x]

[Out]

(e^2*(6*c*d^2 - a*e^2)*x)/c^2 + (2*d*e^3*x^2)/c + (e^4*x^3)/(3*c) + ((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*ArcTa
n[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(5/2)) + (2*d*e*(c*d^2 - a*e^2)*Log[a + c*x^2])/c^2

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 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 716

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^2 \left (6 c d^2-a e^2\right )}{c^2}+\frac {4 d e^3 x}{c}+\frac {e^4 x^2}{c}+\frac {c^2 d^4-6 a c d^2 e^2+a^2 e^4+4 c d e \left (c d^2-a e^2\right ) x}{c^2 \left (a+c x^2\right )}\right ) \, dx \\ & = \frac {e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac {2 d e^3 x^2}{c}+\frac {e^4 x^3}{3 c}+\frac {\int \frac {c^2 d^4-6 a c d^2 e^2+a^2 e^4+4 c d e \left (c d^2-a e^2\right ) x}{a+c x^2} \, dx}{c^2} \\ & = \frac {e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac {2 d e^3 x^2}{c}+\frac {e^4 x^3}{3 c}+\frac {\left (4 d e \left (c d^2-a e^2\right )\right ) \int \frac {x}{a+c x^2} \, dx}{c}+\frac {\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \int \frac {1}{a+c x^2} \, dx}{c^2} \\ & = \frac {e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac {2 d e^3 x^2}{c}+\frac {e^4 x^3}{3 c}+\frac {\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{5/2}}+\frac {2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{5/2}}+\frac {e \left (-3 a e^3 x+c e x \left (18 d^2+6 d e x+e^2 x^2\right )+6 \left (c d^3-a d e^2\right ) \log \left (a+c x^2\right )\right )}{3 c^2} \]

[In]

Integrate[(d + e*x)^4/(a + c*x^2),x]

[Out]

((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(5/2)) + (e*(-3*a*e^3*x + c*e*x*(
18*d^2 + 6*d*e*x + e^2*x^2) + 6*(c*d^3 - a*d*e^2)*Log[a + c*x^2]))/(3*c^2)

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95

method	result	size
default	\(-\frac {e^{2} \left (-\frac {1}{3} c \,e^{2} x^{3}-2 c d e \,x^{2}+a \,e^{2} x -6 c \,d^{2} x \right )}{c^{2}}+\frac {\frac {\left (-4 d \,e^{3} a c +4 d^{3} e \,c^{2}\right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{c^{2}}\)	\(117\)
risch	\(\frac {e^{4} x^{3}}{3 c}+\frac {2 d \,e^{3} x^{2}}{c}-\frac {e^{4} a x}{c^{2}}+\frac {6 e^{2} d^{2} x}{c}-\frac {2 a \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}+\frac {2 \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d^{3} e}{c}+\frac {\ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}}{2 c^{3} a}-\frac {2 a \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}+\frac {2 \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d^{3} e}{c}-\frac {\ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}}{2 c^{3} a}\)	\(554\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.30 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\left [\frac {2 \, a c^{2} e^{4} x^{3} + 12 \, a c^{2} d e^{3} x^{2} - 3 \, {\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 6 \, {\left (6 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x + 12 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} \log \left (c x^{2} + a\right )}{6 \, a c^{3}}, \frac {a c^{2} e^{4} x^{3} + 6 \, a c^{2} d e^{3} x^{2} + 3 \, {\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + 3 \, {\left (6 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x + 6 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} \log \left (c x^{2} + a\right )}{3 \, a c^{3}}\right ] \]

[In]

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

[Out]

[1/6*(2*a*c^2*e^4*x^3 + 12*a*c^2*d*e^3*x^2 - 3*(c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*sqrt(-a*c)*log((c*x^2 - 2*s
qrt(-a*c)*x - a)/(c*x^2 + a)) + 6*(6*a*c^2*d^2*e^2 - a^2*c*e^4)*x + 12*(a*c^2*d^3*e - a^2*c*d*e^3)*log(c*x^2 +
a))/(a*c^3), 1/3*(a*c^2*e^4*x^3 + 6*a*c^2*d*e^3*x^2 + 3*(c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*sqrt(a*c)*arctan(
sqrt(a*c)*x/a) + 3*(6*a*c^2*d^2*e^2 - a^2*c*e^4)*x + 6*(a*c^2*d^3*e - a^2*c*d*e^3)*log(c*x^2 + a))/(a*c^3)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (117) = 234\).

Time = 0.53 (sec) , antiderivative size = 401, normalized size of antiderivative = 3.26 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=x \left (- \frac {a e^{4}}{c^{2}} + \frac {6 d^{2} e^{2}}{c}\right ) + \left (- \frac {2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} - \frac {\sqrt {- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) \log {\left (x + \frac {4 a^{2} d e^{3} + 2 a c^{2} \left (- \frac {2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} - \frac {\sqrt {- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) - 4 a c d^{3} e}{a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}} \right )} + \left (- \frac {2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} + \frac {\sqrt {- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) \log {\left (x + \frac {4 a^{2} d e^{3} + 2 a c^{2} \left (- \frac {2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} + \frac {\sqrt {- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) - 4 a c d^{3} e}{a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}} \right )} + \frac {2 d e^{3} x^{2}}{c} + \frac {e^{4} x^{3}}{3 c} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {2 \, {\left (c d^{3} e - a d e^{3}\right )} \log \left (c x^{2} + a\right )}{c^{2}} + \frac {{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c^{2}} + \frac {c e^{4} x^{3} + 6 \, c d e^{3} x^{2} + 3 \, {\left (6 \, c d^{2} e^{2} - a e^{4}\right )} x}{3 \, c^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {2 \, {\left (c d^{3} e - a d e^{3}\right )} \log \left (c x^{2} + a\right )}{c^{2}} + \frac {{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c^{2}} + \frac {c^{2} e^{4} x^{3} + 6 \, c^{2} d e^{3} x^{2} + 18 \, c^{2} d^{2} e^{2} x - 3 \, a c e^{4} x}{3 \, c^{3}} \]

[In]

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

[Out]

2*(c*d^3*e - a*d*e^3)*log(c*x^2 + a)/c^2 + (c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c
)*c^2) + 1/3*(c^2*e^4*x^3 + 6*c^2*d*e^3*x^2 + 18*c^2*d^2*e^2*x - 3*a*c*e^4*x)/c^3

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {e^4\,x^3}{3\,c}-x\,\left (\frac {a\,e^4}{c^2}-\frac {6\,d^2\,e^2}{c}\right )+\frac {2\,d\,e^3\,x^2}{c}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (a^2\,e^4-6\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{\sqrt {a}\,c^{5/2}}-\frac {\ln \left (c\,x^2+a\right )\,\left (16\,a^2\,c^3\,d\,e^3-16\,a\,c^4\,d^3\,e\right )}{8\,a\,c^5} \]

[In]

int((d + e*x)^4/(a + c*x^2),x)

[Out]

(e^4*x^3)/(3*c) - x*((a*e^4)/c^2 - (6*d^2*e^2)/c) + (2*d*e^3*x^2)/c + (atan((c^(1/2)*x)/a^(1/2))*(a^2*e^4 + c^
2*d^4 - 6*a*c*d^2*e^2))/(a^(1/2)*c^(5/2)) - (log(a + c*x^2)*(16*a^2*c^3*d*e^3 - 16*a*c^4*d^3*e))/(8*a*c^5)