Optimal. Leaf size=98 \[ \frac {2 c f \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{3/2}}-\frac {f \left (b+2 c (d+e x)^2\right )}{2 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1142, 1107, 614, 618, 206} \[ \frac {2 c f \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{e \left (b^2-4 a c\right )^{3/2}}-\frac {f \left (b+2 c (d+e x)^2\right )}{2 e \left (b^2-4 a c\right ) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \]
Antiderivative was successfully verified.
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Rule 206
Rule 614
Rule 618
Rule 1107
Rule 1142
Rubi steps
\begin {align*} \int \frac {d f+e f x}{\left (a+b (d+e x)^2+c (d+e x)^4\right )^2} \, dx &=\frac {f \operatorname {Subst}\left (\int \frac {x}{\left (a+b x^2+c x^4\right )^2} \, dx,x,d+e x\right )}{e}\\ &=\frac {f \operatorname {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx,x,(d+e x)^2\right )}{2 e}\\ &=-\frac {f \left (b+2 c (d+e x)^2\right )}{2 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}-\frac {(c f) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{\left (b^2-4 a c\right ) e}\\ &=-\frac {f \left (b+2 c (d+e x)^2\right )}{2 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {(2 c f) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{\left (b^2-4 a c\right ) e}\\ &=-\frac {f \left (b+2 c (d+e x)^2\right )}{2 \left (b^2-4 a c\right ) e \left (a+b (d+e x)^2+c (d+e x)^4\right )}+\frac {2 c f \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} e}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 99, normalized size = 1.01 \[ -\frac {f \left (\frac {4 c \tan ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+\frac {b+2 c (d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4}\right )}{2 e \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 1066, normalized size = 10.88 \[ \left [-\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} f x^{2} + 4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e f x + 2 \, {\left (c^{2} e^{4} f x^{4} + 4 \, c^{2} d e^{3} f x^{3} + {\left (6 \, c^{2} d^{2} + b c\right )} e^{2} f x^{2} + 2 \, {\left (2 \, c^{2} d^{3} + b c d\right )} e f x + {\left (c^{2} d^{4} + b c d^{2} + a c\right )} f\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \, {\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \, {\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c - {\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a}\right ) + {\left (b^{3} - 4 \, a b c + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2}\right )} f}{2 \, {\left ({\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} e^{5} x^{4} + 4 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d e^{4} x^{3} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2} + 6 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{2}\right )} e^{3} x^{2} + 2 \, {\left (2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{3} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d\right )} e^{2} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{2}\right )} e\right )}}, -\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} f x^{2} + 4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e f x - 4 \, {\left (c^{2} e^{4} f x^{4} + 4 \, c^{2} d e^{3} f x^{3} + {\left (6 \, c^{2} d^{2} + b c\right )} e^{2} f x^{2} + 2 \, {\left (2 \, c^{2} d^{3} + b c d\right )} e f x + {\left (c^{2} d^{4} + b c d^{2} + a c\right )} f\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (b^{3} - 4 \, a b c + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2}\right )} f}{2 \, {\left ({\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} e^{5} x^{4} + 4 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d e^{4} x^{3} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2} + 6 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{2}\right )} e^{3} x^{2} + 2 \, {\left (2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{3} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d\right )} e^{2} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{2}\right )} e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 211, normalized size = 2.15 \[ -\frac {2 \, c f \arctan \left (\frac {2 \, c d^{2} f + 2 \, {\left (f x^{2} e + 2 \, d f x\right )} c e + b f}{\sqrt {-b^{2} + 4 \, a c} f}\right ) e^{\left (-1\right )}}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c d^{2} f^{3} + 2 \, {\left (f x^{2} e + 2 \, d f x\right )} c f^{2} e + b f^{3}}{2 \, {\left (c d^{4} f^{2} + 2 \, {\left (f x^{2} e + 2 \, d f x\right )} c d^{2} f e + b d^{2} f^{2} + {\left (f x^{2} e + 2 \, d f x\right )}^{2} c e^{2} + {\left (f x^{2} e + 2 \, d f x\right )} b f e + a f^{2}\right )} {\left (b^{2} e - 4 \, a c e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 484, normalized size = 4.94 \[ \frac {c e f \,x^{2}}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+c \,d^{4}+2 b d e x +b \,d^{2}+a \right ) \left (4 a c -b^{2}\right )}+\frac {2 c d f x}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+c \,d^{4}+2 b d e x +b \,d^{2}+a \right ) \left (4 a c -b^{2}\right )}+\frac {c \,d^{2} f}{\left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+c \,d^{4}+2 b d e x +b \,d^{2}+a \right ) \left (4 a c -b^{2}\right ) e}+\frac {c f \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 ) e +d \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 )}{\left (4 a c -b^{2}\right ) 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 )}+\frac {b f}{2 \left (c \,e^{4} x^{4}+4 c d \,e^{3} x^{3}+6 c \,d^{2} e^{2} x^{2}+4 c \,d^{3} e x +b \,e^{2} x^{2}+c \,d^{4}+2 b d e x +b \,d^{2}+a \right ) \left (4 a c -b^{2}\right ) e} \]
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 \[ 2 \, c f \int -\frac {e x + d}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{4} x^{4} + 4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e^{3} x^{3} + {\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} + {\left (b^{3} - 4 \, a b c + 6 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2}\right )} e^{2} x^{2} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} d^{2} + 2 \, {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} + {\left (b^{3} - 4 \, a b c\right )} d\right )} e x}\,{d x} - \frac {2 \, c e^{2} f x^{2} + 4 \, c d e f x + {\left (2 \, c d^{2} + b\right )} f}{2 \, {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e^{5} x^{4} + 4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e^{4} x^{3} + {\left (b^{3} - 4 \, a b c + 6 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2}\right )} e^{3} x^{2} + 2 \, {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} + {\left (b^{3} - 4 \, a b c\right )} d\right )} e^{2} x + {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.91, size = 442, normalized size = 4.51 \[ \frac {\frac {f\,\left (2\,c\,d^2+b\right )}{2\,e\,\left (4\,a\,c-b^2\right )}+\frac {2\,c\,d\,f\,x}{4\,a\,c-b^2}+\frac {c\,e\,f\,x^2}{4\,a\,c-b^2}}{a+x^2\,\left (6\,c\,d^2\,e^2+b\,e^2\right )+b\,d^2+c\,d^4+x\,\left (4\,c\,e\,d^3+2\,b\,e\,d\right )+c\,e^4\,x^4+4\,c\,d\,e^3\,x^3}+\frac {2\,c\,f\,\mathrm {atan}\left (\frac {{\left (4\,a\,c-b^2\right )}^4\,\left (x\,\left (\frac {8\,c^4\,d\,e^7\,f^2}{a\,{\left (4\,a\,c-b^2\right )}^{7/2}}-\frac {8\,b\,c^2\,f^2\,\left (b^3\,c^2\,d\,e^9-4\,a\,b\,c^3\,d\,e^9\right )}{a\,e^2\,{\left (4\,a\,c-b^2\right )}^{11/2}}\right )+x^2\,\left (\frac {4\,c^4\,e^8\,f^2}{a\,{\left (4\,a\,c-b^2\right )}^{7/2}}-\frac {4\,b\,c^2\,f^2\,\left (b^3\,c^2\,e^{10}-4\,a\,b\,c^3\,e^{10}\right )}{a\,e^2\,{\left (4\,a\,c-b^2\right )}^{11/2}}\right )+\frac {4\,c^4\,d^2\,e^6\,f^2}{a\,{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {4\,b\,c^2\,f^2\,\left (8\,a^2\,c^3\,e^8-2\,a\,b^2\,c^2\,e^8+4\,a\,b\,c^3\,d^2\,e^8-b^3\,c^2\,d^2\,e^8\right )}{a\,e^2\,{\left (4\,a\,c-b^2\right )}^{11/2}}\right )}{8\,c^4\,e^6\,f^2}\right )}{e\,{\left (4\,a\,c-b^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.07, size = 525, normalized size = 5.36 \[ - \frac {c f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 16 a^{2} c^{3} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{2} c^{2} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - b^{4} c f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b c f + 2 c^{2} d^{2} f}{2 c^{2} e^{2} f} \right )}}{e} + \frac {c f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (\frac {2 d x}{e} + x^{2} + \frac {16 a^{2} c^{3} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{2} c^{2} f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b^{4} c f \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b c f + 2 c^{2} d^{2} f}{2 c^{2} e^{2} f} \right )}}{e} + \frac {b f + 2 c d^{2} f + 4 c d e f x + 2 c e^{2} f x^{2}}{8 a^{2} c e - 2 a b^{2} e + 8 a b c d^{2} e + 8 a c^{2} d^{4} e - 2 b^{3} d^{2} e - 2 b^{2} c d^{4} e + x^{4} \left (8 a c^{2} e^{5} - 2 b^{2} c e^{5}\right ) + x^{3} \left (32 a c^{2} d e^{4} - 8 b^{2} c d e^{4}\right ) + x^{2} \left (8 a b c e^{3} + 48 a c^{2} d^{2} e^{3} - 2 b^{3} e^{3} - 12 b^{2} c d^{2} e^{3}\right ) + x \left (16 a b c d e^{2} + 32 a c^{2} d^{3} e^{2} - 4 b^{3} d e^{2} - 8 b^{2} c d^{3} e^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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