\(\int \frac {(d+e x^2)^4}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx\) [214]

Optimal result

Integrand size = 39, antiderivative size = 121 \[ \int \frac {\left (d+e x^2\right )^4}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\frac {\left (7 c^2 d^2-5 b c d e+b^2 e^2\right ) x}{c^3}+\frac {e (4 c d-b e) x^3}{3 c^2}+\frac {e^2 x^5}{5 c}-\frac {(2 c d-b e)^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{7/2} \sqrt {e} \sqrt {c d-b e}} \]

[Out]

(b^2*e^2-5*b*c*d*e+7*c^2*d^2)*x/c^3+1/3*e*(-b*e+4*c*d)*x^3/c^2+1/5*e^2*x^5/c-(-b*e+2*c*d)^3*arctanh(x*c^(1/2)*
e^(1/2)/(-b*e+c*d)^(1/2))/c^(7/2)/e^(1/2)/(-b*e+c*d)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1163, 398, 214} \[ \int \frac {\left (d+e x^2\right )^4}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=-\frac {(2 c d-b e)^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{7/2} \sqrt {e} \sqrt {c d-b e}}+\frac {x \left (b^2 e^2-5 b c d e+7 c^2 d^2\right )}{c^3}+\frac {e x^3 (4 c d-b e)}{3 c^2}+\frac {e^2 x^5}{5 c} \]

[In]

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

[Out]

((7*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*x)/c^3 + (e*(4*c*d - b*e)*x^3)/(3*c^2) + (e^2*x^5)/(5*c) - ((2*c*d - b*e)^3
*ArcTanh[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(c^(7/2)*Sqrt[e]*Sqrt[c*d - b*e])

Rule 214

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

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 1163

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c/e)*x^2)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2
, 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (d+e x^2\right )^3}{\frac {-c d^2+b d e}{d}+c e x^2} \, dx \\ & = \int \left (\frac {7 c^2 d^2-5 b c d e+b^2 e^2}{c^3}+\frac {e (4 c d-b e) x^2}{c^2}+\frac {e^2 x^4}{c}+\frac {8 c^3 d^3-12 b c^2 d^2 e+6 b^2 c d e^2-b^3 e^3}{c^3 \left (-c d+b e+c e x^2\right )}\right ) \, dx \\ & = \frac {\left (7 c^2 d^2-5 b c d e+b^2 e^2\right ) x}{c^3}+\frac {e (4 c d-b e) x^3}{3 c^2}+\frac {e^2 x^5}{5 c}+\frac {(2 c d-b e)^3 \int \frac {1}{-c d+b e+c e x^2} \, dx}{c^3} \\ & = \frac {\left (7 c^2 d^2-5 b c d e+b^2 e^2\right ) x}{c^3}+\frac {e (4 c d-b e) x^3}{3 c^2}+\frac {e^2 x^5}{5 c}-\frac {(2 c d-b e)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{7/2} \sqrt {e} \sqrt {c d-b e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^4}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=-\frac {\left (-7 c^2 d^2+5 b c d e-b^2 e^2\right ) x}{c^3}-\frac {e (-4 c d+b e) x^3}{3 c^2}+\frac {e^2 x^5}{5 c}-\frac {(-2 c d+b e)^3 \arctan \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {-c d+b e}}\right )}{c^{7/2} \sqrt {e} \sqrt {-c d+b e}} \]

[In]

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

[Out]

-(((-7*c^2*d^2 + 5*b*c*d*e - b^2*e^2)*x)/c^3) - (e*(-4*c*d + b*e)*x^3)/(3*c^2) + (e^2*x^5)/(5*c) - ((-2*c*d +
b*e)^3*ArcTan[(Sqrt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(c^(7/2)*Sqrt[e]*Sqrt[-(c*d) + b*e])

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.11

method	result	size
default	\(\frac {\frac {1}{5} e^{2} x^{5} c^{2}-\frac {1}{3} b c \,e^{2} x^{3}+\frac {4}{3} c^{2} d e \,x^{3}+b^{2} e^{2} x -5 b c d e x +7 c^{2} d^{2} x}{c^{3}}+\frac {\left (-b^{3} e^{3}+6 b^{2} c d \,e^{2}-12 b \,c^{2} d^{2} e +8 c^{3} d^{3}\right ) \arctan \left (\frac {x c e}{\sqrt {\left (b e -c d \right ) e c}}\right )}{c^{3} \sqrt {\left (b e -c d \right ) e c}}\)	\(134\)
risch	\(\frac {e^{2} x^{5}}{5 c}-\frac {b \,e^{2} x^{3}}{3 c^{2}}+\frac {4 d e \,x^{3}}{3 c}+\frac {b^{2} e^{2} x}{c^{3}}-\frac {5 b d e x}{c^{2}}+\frac {7 d^{2} x}{c}-\frac {\ln \left (x c e -\sqrt {-\left (b e -c d \right ) e c}\right ) b^{3} e^{3}}{2 c^{3} \sqrt {-\left (b e -c d \right ) e c}}+\frac {3 \ln \left (x c e -\sqrt {-\left (b e -c d \right ) e c}\right ) b^{2} d \,e^{2}}{c^{2} \sqrt {-\left (b e -c d \right ) e c}}-\frac {6 \ln \left (x c e -\sqrt {-\left (b e -c d \right ) e c}\right ) b \,d^{2} e}{c \sqrt {-\left (b e -c d \right ) e c}}+\frac {4 \ln \left (x c e -\sqrt {-\left (b e -c d \right ) e c}\right ) d^{3}}{\sqrt {-\left (b e -c d \right ) e c}}+\frac {\ln \left (-x c e -\sqrt {-\left (b e -c d \right ) e c}\right ) b^{3} e^{3}}{2 c^{3} \sqrt {-\left (b e -c d \right ) e c}}-\frac {3 \ln \left (-x c e -\sqrt {-\left (b e -c d \right ) e c}\right ) b^{2} d \,e^{2}}{c^{2} \sqrt {-\left (b e -c d \right ) e c}}+\frac {6 \ln \left (-x c e -\sqrt {-\left (b e -c d \right ) e c}\right ) b \,d^{2} e}{c \sqrt {-\left (b e -c d \right ) e c}}-\frac {4 \ln \left (-x c e -\sqrt {-\left (b e -c d \right ) e c}\right ) d^{3}}{\sqrt {-\left (b e -c d \right ) e c}}\)	\(432\)

[In]

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

[Out]

1/c^3*(1/5*e^2*x^5*c^2-1/3*b*c*e^2*x^3+4/3*c^2*d*e*x^3+b^2*e^2*x-5*b*c*d*e*x+7*c^2*d^2*x)+(-b^3*e^3+6*b^2*c*d*
e^2-12*b*c^2*d^2*e+8*c^3*d^3)/c^3/((b*e-c*d)*e*c)^(1/2)*arctan(x*c*e/((b*e-c*d)*e*c)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (105) = 210\).

Time = 0.26 (sec) , antiderivative size = 446, normalized size of antiderivative = 3.69 \[ \int \frac {\left (d+e x^2\right )^4}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\left [\frac {6 \, {\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{5} + 10 \, {\left (4 \, c^{4} d^{2} e^{2} - 5 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{3} - 15 \, {\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \sqrt {c^{2} d e - b c e^{2}} \log \left (\frac {c e x^{2} + c d - b e + 2 \, \sqrt {c^{2} d e - b c e^{2}} x}{c e x^{2} - c d + b e}\right ) + 30 \, {\left (7 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x}{30 \, {\left (c^{5} d e - b c^{4} e^{2}\right )}}, \frac {3 \, {\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{5} + 5 \, {\left (4 \, c^{4} d^{2} e^{2} - 5 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{3} - 15 \, {\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \sqrt {-c^{2} d e + b c e^{2}} \arctan \left (-\frac {\sqrt {-c^{2} d e + b c e^{2}} x}{c d - b e}\right ) + 15 \, {\left (7 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x}{15 \, {\left (c^{5} d e - b c^{4} e^{2}\right )}}\right ] \]

[In]

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

[Out]

[1/30*(6*(c^4*d*e^3 - b*c^3*e^4)*x^5 + 10*(4*c^4*d^2*e^2 - 5*b*c^3*d*e^3 + b^2*c^2*e^4)*x^3 - 15*(8*c^3*d^3 -
12*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*sqrt(c^2*d*e - b*c*e^2)*log((c*e*x^2 + c*d - b*e + 2*sqrt(c^2*d*e -
b*c*e^2)*x)/(c*e*x^2 - c*d + b*e)) + 30*(7*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*x)/(c^5
*d*e - b*c^4*e^2), 1/15*(3*(c^4*d*e^3 - b*c^3*e^4)*x^5 + 5*(4*c^4*d^2*e^2 - 5*b*c^3*d*e^3 + b^2*c^2*e^4)*x^3 -
15*(8*c^3*d^3 - 12*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*sqrt(-c^2*d*e + b*c*e^2)*arctan(-sqrt(-c^2*d*e + b*
c*e^2)*x/(c*d - b*e)) + 15*(7*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*x)/(c^5*d*e - b*c^4*
e^2)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (110) = 220\).

Time = 0.45 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.85 \[ \int \frac {\left (d+e x^2\right )^4}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=x^{3} \left (- \frac {b e^{2}}{3 c^{2}} + \frac {4 d e}{3 c}\right ) + x \left (\frac {b^{2} e^{2}}{c^{3}} - \frac {5 b d e}{c^{2}} + \frac {7 d^{2}}{c}\right ) + \frac {\sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} \log {\left (x + \frac {- b c^{3} e \sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} + c^{4} d \sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3}}{b^{3} e^{3} - 6 b^{2} c d e^{2} + 12 b c^{2} d^{2} e - 8 c^{3} d^{3}} \right )}}{2} - \frac {\sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} \log {\left (x + \frac {b c^{3} e \sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} - c^{4} d \sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3}}{b^{3} e^{3} - 6 b^{2} c d e^{2} + 12 b c^{2} d^{2} e - 8 c^{3} d^{3}} \right )}}{2} + \frac {e^{2} x^{5}}{5 c} \]

[In]

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

[Out]

x**3*(-b*e**2/(3*c**2) + 4*d*e/(3*c)) + x*(b**2*e**2/c**3 - 5*b*d*e/c**2 + 7*d**2/c) + sqrt(-1/(c**7*e*(b*e -
c*d)))*(b*e - 2*c*d)**3*log(x + (-b*c**3*e*sqrt(-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3 + c**4*d*sqrt(-1/(c*
*7*e*(b*e - c*d)))*(b*e - 2*c*d)**3)/(b**3*e**3 - 6*b**2*c*d*e**2 + 12*b*c**2*d**2*e - 8*c**3*d**3))/2 - sqrt(
-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3*log(x + (b*c**3*e*sqrt(-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3 - c
**4*d*sqrt(-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3)/(b**3*e**3 - 6*b**2*c*d*e**2 + 12*b*c**2*d**2*e - 8*c**3
*d**3))/2 + e**2*x**5/(5*c)

Maxima [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^4}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.31 \[ \int \frac {\left (d+e x^2\right )^4}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\frac {{\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \arctan \left (\frac {c e x}{\sqrt {-c^{2} d e + b c e^{2}}}\right )}{\sqrt {-c^{2} d e + b c e^{2}} c^{3}} + \frac {3 \, c^{4} e^{7} x^{5} + 20 \, c^{4} d e^{6} x^{3} - 5 \, b c^{3} e^{7} x^{3} + 105 \, c^{4} d^{2} e^{5} x - 75 \, b c^{3} d e^{6} x + 15 \, b^{2} c^{2} e^{7} x}{15 \, c^{5} e^{5}} \]

[In]

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

[Out]

(8*c^3*d^3 - 12*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*arctan(c*e*x/sqrt(-c^2*d*e + b*c*e^2))/(sqrt(-c^2*d*e +
b*c*e^2)*c^3) + 1/15*(3*c^4*e^7*x^5 + 20*c^4*d*e^6*x^3 - 5*b*c^3*e^7*x^3 + 105*c^4*d^2*e^5*x - 75*b*c^3*d*e^6
*x + 15*b^2*c^2*e^7*x)/(c^5*e^5)

Mupad [B] (verification not implemented)

Time = 7.96 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.50 \[ \int \frac {\left (d+e x^2\right )^4}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=x\,\left (\frac {3\,d^2}{c}+\frac {\left (\frac {e\,\left (b\,e-c\,d\right )}{c^2}-\frac {3\,d\,e}{c}\right )\,\left (b\,e-c\,d\right )}{c\,e}\right )-x^3\,\left (\frac {e\,\left (b\,e-c\,d\right )}{3\,c^2}-\frac {d\,e}{c}\right )+\frac {e^2\,x^5}{5\,c}-\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,e\,x\,{\left (b\,e-2\,c\,d\right )}^3}{\sqrt {b\,e^2-c\,d\,e}\,\left (b^3\,e^3-6\,b^2\,c\,d\,e^2+12\,b\,c^2\,d^2\,e-8\,c^3\,d^3\right )}\right )\,{\left (b\,e-2\,c\,d\right )}^3}{c^{7/2}\,\sqrt {b\,e^2-c\,d\,e}} \]

[In]

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

[Out]

x*((3*d^2)/c + (((e*(b*e - c*d))/c^2 - (3*d*e)/c)*(b*e - c*d))/(c*e)) - x^3*((e*(b*e - c*d))/(3*c^2) - (d*e)/c
) + (e^2*x^5)/(5*c) - (atan((c^(1/2)*e*x*(b*e - 2*c*d)^3)/((b*e^2 - c*d*e)^(1/2)*(b^3*e^3 - 8*c^3*d^3 + 12*b*c
^2*d^2*e - 6*b^2*c*d*e^2)))*(b*e - 2*c*d)^3)/(c^(7/2)*(b*e^2 - c*d*e)^(1/2))