Optimal. Leaf size=123 \[ \frac {e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac {2 d e^3 x^2}{c}+\frac {e^4 x^3}{3 c}+\frac {\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{5/2}}+\frac {2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {716, 649, 211,
266} \begin {gather*} \frac {\text {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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 266
Rule 649
Rule 716
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{a+c x^2} \, dx &=\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]
time = 0.06, size = 111, normalized size = 0.90 \begin {gather*} \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 {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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.50, size = 117, normalized size = 0.95
method | result | size |
default | \(-\frac {e^{2} \left (-\frac {1}{3} c \,e^{2} x^{3}-2 c d e \,x^{2}+a \,e^{2} x -6 c \,d^{2} x \right )}{c^{2}}+\frac {\frac {\left (-4 a c d \,e^{3}+4 c^{2} d^{3} e \right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{c^{2}}\) | \(117\) |
risch | \(\frac {e^{4} x^{3}}{3 c}+\frac {2 d \,e^{3} x^{2}}{c}-\frac {e^{4} a x}{c^{2}}+\frac {6 e^{2} d^{2} x}{c}-\frac {2 a \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}+\frac {2 \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d^{3} e}{c}+\frac {\ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}}{2 c^{3} a}-\frac {2 a \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}+\frac {2 \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d^{3} e}{c}-\frac {\ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}}{2 c^{3} a}\) | \(554\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.54, size = 108, normalized size = 0.88 \begin {gather*} \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 x^{3} e^{4} + 6 \, c d x^{2} e^{3} + 3 \, {\left (6 \, c d^{2} e^{2} - a e^{4}\right )} x}{3 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.05, size = 267, normalized size = 2.17 \begin {gather*} \left [\frac {12 \, a c^{2} d x^{2} e^{3} + 36 \, a c^{2} d^{2} x e^{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 ) + 2 \, {\left (a c^{2} x^{3} - 3 \, a^{2} c x\right )} e^{4} + 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 {6 \, a c^{2} d x^{2} e^{3} + 18 \, a c^{2} d^{2} x e^{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 ) + {\left (a c^{2} x^{3} - 3 \, a^{2} c x\right )} e^{4} + 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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 401 vs.
\(2 (117) = 234\).
time = 0.52, size = 401, normalized size = 3.26 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.25, size = 113, normalized size = 0.92 \begin {gather*} \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} x^{3} e^{4} + 6 \, c^{2} d x^{2} e^{3} + 18 \, c^{2} d^{2} x e^{2} - 3 \, a c x e^{4}}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.11, size = 127, normalized size = 1.03 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________