Optimal. Leaf size=170 \[ \frac {f^2 \sqrt {\sqrt {b^2-4 a c}+b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} e \sqrt {b^2-4 a c}}-\frac {f^2 \sqrt {b-\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} e \sqrt {b^2-4 a c}} \]
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Rubi [A] time = 0.15, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1142, 1130, 205} \[ \frac {f^2 \sqrt {\sqrt {b^2-4 a c}+b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} e \sqrt {b^2-4 a c}}-\frac {f^2 \sqrt {b-\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} e \sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 1130
Rule 1142
Rubi steps
\begin {align*} \int \frac {(d f+e f x)^2}{a+b (d+e x)^2+c (d+e x)^4} \, dx &=\frac {f^2 \operatorname {Subst}\left (\int \frac {x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{e}\\ &=\frac {\left (\left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) f^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{2 e}+\frac {\left (\left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) f^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{2 e}\\ &=-\frac {\sqrt {b-\sqrt {b^2-4 a c}} f^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c} e}+\frac {\sqrt {b+\sqrt {b^2-4 a c}} f^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c} e}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 178, normalized size = 1.05 \[ \frac {f^2 \left (\left (\sqrt {b^2-4 a c}-b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )+\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {\sqrt {b^2-4 a c}+b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )\right )}{\sqrt {2} \sqrt {c} e \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 799, normalized size = 4.70 \[ \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b f^{4} + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{2}}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \log \left (e f^{6} x + d f^{6} + \sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{3} \sqrt {-\frac {b f^{4} + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{2}}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b f^{4} + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{2}}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \log \left (e f^{6} x + d f^{6} - \sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{3} \sqrt {-\frac {b f^{4} + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{2}}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b f^{4} - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{2}}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \log \left (e f^{6} x + d f^{6} + \sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{3} \sqrt {-\frac {b f^{4} - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{2}}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b f^{4} - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{2}}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \log \left (e f^{6} x + d f^{6} - \sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{3} \sqrt {-\frac {b f^{4} - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {f^{8}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} e^{2}}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.47, size = 1325, normalized size = 7.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 143, normalized size = 0.84 \[ \frac {f^{2} \left (\RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{2} e^{2}+2 \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right ) d e +d^{2}\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+x \right )}{2 e \left (2 c \,e^{3} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{3}+6 c d \,e^{2} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{2}+6 e c \,d^{2} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+2 c \,d^{3}+b e \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+b d \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e f x + d f\right )}^{2}}{{\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.79, size = 683, normalized size = 4.02 \[ -2\,\mathrm {atanh}\left (\frac {\sqrt {-\frac {b^3\,f^4+f^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c\,f^4}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}}\,\left (x\,\left (4\,a\,c^2\,e^{12}\,f^4-2\,b^2\,c\,e^{12}\,f^4\right )+\frac {\left (x\,\left (8\,b^3\,c^2\,e^{14}-32\,a\,b\,c^3\,e^{14}\right )+8\,b^3\,c^2\,d\,e^{13}-32\,a\,b\,c^3\,d\,e^{13}\right )\,\left (b^3\,f^4+f^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c\,f^4\right )}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}+4\,a\,c^2\,d\,e^{11}\,f^4-2\,b^2\,c\,d\,e^{11}\,f^4\right )}{a\,c\,e^{10}\,f^6}\right )\,\sqrt {-\frac {b^3\,f^4+f^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c\,f^4}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}}-2\,\mathrm {atanh}\left (\frac {\sqrt {\frac {f^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,f^4+4\,a\,b\,c\,f^4}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}}\,\left (x\,\left (4\,a\,c^2\,e^{12}\,f^4-2\,b^2\,c\,e^{12}\,f^4\right )-\frac {\left (x\,\left (8\,b^3\,c^2\,e^{14}-32\,a\,b\,c^3\,e^{14}\right )+8\,b^3\,c^2\,d\,e^{13}-32\,a\,b\,c^3\,d\,e^{13}\right )\,\left (f^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,f^4+4\,a\,b\,c\,f^4\right )}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}+4\,a\,c^2\,d\,e^{11}\,f^4-2\,b^2\,c\,d\,e^{11}\,f^4\right )}{a\,c\,e^{10}\,f^6}\right )\,\sqrt {\frac {f^4\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,f^4+4\,a\,b\,c\,f^4}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.59, size = 124, normalized size = 0.73 \[ \operatorname {RootSum} {\left (t^{4} \left (256 a^{2} c^{3} e^{4} - 128 a b^{2} c^{2} e^{4} + 16 b^{4} c e^{4}\right ) + t^{2} \left (- 16 a b c e^{2} f^{4} + 4 b^{3} e^{2} f^{4}\right ) + a f^{8}, \left (t \mapsto t \log {\left (x + \frac {64 t^{3} a c^{2} e^{3} - 16 t^{3} b^{2} c e^{3} - 2 t b e f^{4} + d f^{6}}{e f^{6}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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