Optimal. Leaf size=129 \[ -\frac {1}{2 a d x^2}-\frac {c^{3/2} d \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \left (c d^2+a e^2\right )}-\frac {e \log (x)}{a d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2+a e^2\right )}+\frac {c e \log \left (a+c x^4\right )}{4 a \left (c d^2+a e^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1266, 908, 649,
211, 266} \begin {gather*} -\frac {c^{3/2} d \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )}+\frac {c e \log \left (a+c x^4\right )}{4 a \left (a e^2+c d^2\right )}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2+c d^2\right )}-\frac {e \log (x)}{a d^2}-\frac {1}{2 a d x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 266
Rule 649
Rule 908
Rule 1266
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a d x^2}-\frac {e}{a d^2 x}+\frac {e^4}{d^2 \left (c d^2+a e^2\right ) (d+e x)}-\frac {c^2 (d-e x)}{a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a d x^2}-\frac {e \log (x)}{a d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2+a e^2\right )}-\frac {c^2 \text {Subst}\left (\int \frac {d-e x}{a+c x^2} \, dx,x,x^2\right )}{2 a \left (c d^2+a e^2\right )}\\ &=-\frac {1}{2 a d x^2}-\frac {e \log (x)}{a d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2+a e^2\right )}-\frac {\left (c^2 d\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 a \left (c d^2+a e^2\right )}+\frac {\left (c^2 e\right ) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{2 a \left (c d^2+a e^2\right )}\\ &=-\frac {1}{2 a d x^2}-\frac {c^{3/2} d \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \left (c d^2+a e^2\right )}-\frac {e \log (x)}{a d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2+a e^2\right )}+\frac {c e \log \left (a+c x^4\right )}{4 a \left (c d^2+a e^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 169, normalized size = 1.31 \begin {gather*} \frac {2 c^{3/2} d^3 x^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 c^{3/2} d^3 x^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt {a} \left (-2 c d^3-2 a d e^2-4 e \left (c d^2+a e^2\right ) x^2 \log (x)+2 a e^3 x^2 \log \left (d+e x^2\right )+c d^2 e x^2 \log \left (a+c x^4\right )\right )}{4 a^{3/2} d^2 \left (c d^2+a e^2\right ) x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.18, size = 107, normalized size = 0.83
method | result | size |
default | \(\frac {e^{3} \ln \left (e \,x^{2}+d \right )}{2 d^{2} \left (a \,e^{2}+c \,d^{2}\right )}-\frac {c^{2} \left (-\frac {e \ln \left (c \,x^{4}+a \right )}{2 c}+\frac {d \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{\sqrt {a c}}\right )}{2 \left (a \,e^{2}+c \,d^{2}\right ) a}-\frac {1}{2 a d \,x^{2}}-\frac {e \ln \left (x \right )}{a \,d^{2}}\) | \(107\) |
risch | \(-\frac {1}{2 a d \,x^{2}}-\frac {e \ln \left (x \right )}{a \,d^{2}}+\frac {e^{3} \ln \left (-e \,x^{2}-d \right )}{2 d^{2} \left (a \,e^{2}+c \,d^{2}\right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (e^{2} a^{4}+a^{3} c \,d^{2}\right ) \textit {\_Z}^{2}-2 a^{2} c e \textit {\_Z} +c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-6 a^{5} d^{2} e^{4}-7 a^{4} c \,d^{4} e^{2}-5 a^{3} c^{2} d^{6}\right ) \textit {\_R}^{3}+\left (22 a^{3} c \,d^{2} e^{3}+3 a^{2} c^{2} d^{4} e \right ) \textit {\_R}^{2}+\left (-8 a^{2} c \,e^{4}-8 a \,c^{2} d^{2} e^{2}-4 c^{3} d^{4}\right ) \textit {\_R} +8 e^{3} c^{2}\right ) x^{2}+\left (-2 a^{5} d^{3} e^{3}+2 a^{4} c \,d^{5} e \right ) \textit {\_R}^{3}+\left (-8 a^{4} d \,e^{4}+8 a^{3} c \,d^{3} e^{2}-a^{2} c^{2} d^{5}\right ) \textit {\_R}^{2}-4 a \,c^{2} d^{3} e \textit {\_R} \right )\right )}{4}\) | \(285\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 119, normalized size = 0.92 \begin {gather*} -\frac {c^{2} d \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, {\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {a c}} + \frac {c e \log \left (c x^{4} + a\right )}{4 \, {\left (a c d^{2} + a^{2} e^{2}\right )}} + \frac {e^{3} \log \left (x^{2} e + d\right )}{2 \, {\left (c d^{4} + a d^{2} e^{2}\right )}} - \frac {e \log \left (x^{2}\right )}{2 \, a d^{2}} - \frac {1}{2 \, a d x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 22.84, size = 275, normalized size = 2.13 \begin {gather*} \left [\frac {c d^{3} x^{2} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{4} - 2 \, a x^{2} \sqrt {-\frac {c}{a}} - a}{c x^{4} + a}\right ) + c d^{2} x^{2} e \log \left (c x^{4} + a\right ) + 2 \, a x^{2} e^{3} \log \left (x^{2} e + d\right ) - 2 \, c d^{3} - 2 \, a d e^{2} - 4 \, {\left (c d^{2} x^{2} e + a x^{2} e^{3}\right )} \log \left (x\right )}{4 \, {\left (a c d^{4} x^{2} + a^{2} d^{2} x^{2} e^{2}\right )}}, \frac {2 \, c d^{3} x^{2} \sqrt {\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{a}}}{c x^{2}}\right ) + c d^{2} x^{2} e \log \left (c x^{4} + a\right ) + 2 \, a x^{2} e^{3} \log \left (x^{2} e + d\right ) - 2 \, c d^{3} - 2 \, a d e^{2} - 4 \, {\left (c d^{2} x^{2} e + a x^{2} e^{3}\right )} \log \left (x\right )}{4 \, {\left (a c d^{4} x^{2} + a^{2} d^{2} x^{2} e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.92, size = 132, normalized size = 1.02 \begin {gather*} -\frac {c^{2} d \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{2 \, {\left (a c d^{2} + a^{2} e^{2}\right )} \sqrt {a c}} + \frac {c e \log \left (c x^{4} + a\right )}{4 \, {\left (a c d^{2} + a^{2} e^{2}\right )}} + \frac {e^{4} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c d^{4} e + a d^{2} e^{3}\right )}} - \frac {e \log \left (x^{2}\right )}{2 \, a d^{2}} + \frac {x^{2} e - d}{2 \, a d^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.38, size = 820, normalized size = 6.36 \begin {gather*} \frac {\ln \left (a^6\,c^{12}\,d^{16}\,x^2+64\,a^{14}\,c^4\,e^{16}\,x^2+a^2\,c^7\,d^{16}\,{\left (-a^3\,c^3\right )}^{3/2}-64\,a^{13}\,c^2\,e^{16}\,\sqrt {-a^3\,c^3}+63\,a^3\,d^8\,e^8\,{\left (-a^3\,c^3\right )}^{5/2}+224\,a^9\,d^2\,e^{14}\,{\left (-a^3\,c^3\right )}^{3/2}-28\,c^3\,d^{14}\,e^2\,{\left (-a^3\,c^3\right )}^{5/2}+28\,a^7\,c^{11}\,d^{14}\,e^2\,x^2+114\,a^8\,c^{10}\,d^{12}\,e^4\,x^2+108\,a^9\,c^9\,d^{10}\,e^6\,x^2-63\,a^{10}\,c^8\,d^8\,e^8\,x^2-32\,a^{11}\,c^7\,d^6\,e^{10}\,x^2+212\,a^{12}\,c^6\,d^4\,e^{12}\,x^2+224\,a^{13}\,c^5\,d^2\,e^{14}\,x^2-114\,a\,c^2\,d^{12}\,e^4\,{\left (-a^3\,c^3\right )}^{5/2}-108\,a^2\,c\,d^{10}\,e^6\,{\left (-a^3\,c^3\right )}^{5/2}+212\,a^8\,c\,d^4\,e^{12}\,{\left (-a^3\,c^3\right )}^{3/2}-32\,a^7\,c^2\,d^6\,e^{10}\,{\left (-a^3\,c^3\right )}^{3/2}\right )\,\left (d\,\sqrt {-a^3\,c^3}+a^2\,c\,e\right )}{4\,a^4\,e^2+4\,c\,a^3\,d^2}-\frac {\ln \left (a^6\,c^{12}\,d^{16}\,x^2+64\,a^{14}\,c^4\,e^{16}\,x^2-a^2\,c^7\,d^{16}\,{\left (-a^3\,c^3\right )}^{3/2}+64\,a^{13}\,c^2\,e^{16}\,\sqrt {-a^3\,c^3}-63\,a^3\,d^8\,e^8\,{\left (-a^3\,c^3\right )}^{5/2}-224\,a^9\,d^2\,e^{14}\,{\left (-a^3\,c^3\right )}^{3/2}+28\,c^3\,d^{14}\,e^2\,{\left (-a^3\,c^3\right )}^{5/2}+28\,a^7\,c^{11}\,d^{14}\,e^2\,x^2+114\,a^8\,c^{10}\,d^{12}\,e^4\,x^2+108\,a^9\,c^9\,d^{10}\,e^6\,x^2-63\,a^{10}\,c^8\,d^8\,e^8\,x^2-32\,a^{11}\,c^7\,d^6\,e^{10}\,x^2+212\,a^{12}\,c^6\,d^4\,e^{12}\,x^2+224\,a^{13}\,c^5\,d^2\,e^{14}\,x^2+114\,a\,c^2\,d^{12}\,e^4\,{\left (-a^3\,c^3\right )}^{5/2}+108\,a^2\,c\,d^{10}\,e^6\,{\left (-a^3\,c^3\right )}^{5/2}-212\,a^8\,c\,d^4\,e^{12}\,{\left (-a^3\,c^3\right )}^{3/2}+32\,a^7\,c^2\,d^6\,e^{10}\,{\left (-a^3\,c^3\right )}^{3/2}\right )\,\left (d\,\sqrt {-a^3\,c^3}-a^2\,c\,e\right )}{4\,\left (a^4\,e^2+c\,a^3\,d^2\right )}+\frac {e^3\,\ln \left (e\,x^2+d\right )}{2\,c\,d^4+2\,a\,d^2\,e^2}-\frac {1}{2\,a\,d\,x^2}-\frac {e\,\ln \left (x\right )}{a\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________