Optimal. Leaf size=202 \[ -\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}+\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.24, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1261, 713,
1144, 214} \begin {gather*} \frac {\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}-\frac {\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 214
Rule 713
Rule 1144
Rule 1261
Rubi steps
\begin {align*} \int \frac {x \sqrt {d+e x^2}}{a+b x^2+c x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=e \text {Subst}\left (\int \frac {x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x^2}\right )\\ &=-\left (\frac {1}{2} \left (-e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x^2}\right )\right )+\frac {1}{2} \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x^2}\right )\\ &=-\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}+\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.57, size = 256, normalized size = 1.27 \begin {gather*} \frac {\frac {\left (-2 i c d+\left (i b+\sqrt {-b^2+4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (2 i c d+\left (-i b+\sqrt {-b^2+4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{\sqrt {2} \sqrt {c} \sqrt {-b^2+4 a c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.12, size = 177, normalized size = 0.88
method | result | size |
default | \(-\frac {e \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+\left (4 e b -4 c d \right ) \textit {\_Z}^{6}+\left (16 a \,e^{2}-8 d e b +6 c \,d^{2}\right ) \textit {\_Z}^{4}+\left (4 d^{2} e b -4 c \,d^{3}\right ) \textit {\_Z}^{2}+d^{4} c \right )}{\sum }\frac {\left (-\textit {\_R}^{6}-\textit {\_R}^{4} d +\textit {\_R}^{2} d^{2}+d^{3}\right ) \ln \left (\sqrt {e \,x^{2}+d}-\sqrt {e}\, x -\textit {\_R} \right )}{\textit {\_R}^{7} c +3 \textit {\_R}^{5} b e -3 \textit {\_R}^{5} c d +8 \textit {\_R}^{3} a \,e^{2}-4 \textit {\_R}^{3} b d e +3 \textit {\_R}^{3} c \,d^{2}+\textit {\_R} b \,d^{2} e -\textit {\_R} c \,d^{3}}\right )}{4}\) | \(177\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1037 vs.
\(2 (169) = 338\).
time = 6.19, size = 1037, normalized size = 5.13 \begin {gather*} -\frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {\frac {2 \, c d - b e + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {2 \, b d e + 2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d} {\left ({\left (b^{2} - 4 \, a c\right )} e + \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + {\left (b x^{2} - 2 \, a\right )} e^{2} + \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} e + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{x^{2}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {\frac {2 \, c d - b e + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {2 \, b d e - 2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d} {\left ({\left (b^{2} - 4 \, a c\right )} e + \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e + \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + {\left (b x^{2} - 2 \, a\right )} e^{2} + \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} e + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{x^{2}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {\frac {2 \, c d - b e - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {2 \, b d e + 2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d} {\left ({\left (b^{2} - 4 \, a c\right )} e - \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + {\left (b x^{2} - 2 \, a\right )} e^{2} - \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} e + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{x^{2}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {\frac {2 \, c d - b e - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac {2 \, b d e - 2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d} {\left ({\left (b^{2} - 4 \, a c\right )} e - \frac {{\left (b^{3} c - 4 \, a b c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right )} \sqrt {\frac {2 \, c d - b e - \frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + {\left (b x^{2} - 2 \, a\right )} e^{2} - \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} e + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d\right )} e}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \sqrt {d + e x^{2}}}{a + b x^{2} + c x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.96, size = 228, normalized size = 1.13 \begin {gather*} -\frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{2 \, \sqrt {b^{2} - 4 \, a c} {\left | c \right |}} + \frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{2} e + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{2 \, \sqrt {b^{2} - 4 \, a c} {\left | c \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.72, size = 717, normalized size = 3.55 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {e\,x^2+d}\,\left (-2\,b^2\,c\,e^4+4\,b\,c^2\,d\,e^3-4\,c^3\,d^2\,e^2+4\,a\,c^2\,e^4\right )+\frac {\sqrt {e\,x^2+d}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {-\frac {b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{2\,c^2\,d^2\,e^3-2\,b\,c\,d\,e^4+2\,a\,c\,e^5}\right )\,\sqrt {-\frac {b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}-2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {e\,x^2+d}\,\left (-2\,b^2\,c\,e^4+4\,b\,c^2\,d\,e^3-4\,c^3\,d^2\,e^2+4\,a\,c^2\,e^4\right )-\frac {\sqrt {e\,x^2+d}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{2\,c^2\,d^2\,e^3-2\,b\,c\,d\,e^4+2\,a\,c\,e^5}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________