Optimal. Leaf size=195 \[ -\frac {1}{a e (d+e x)}-\frac {\sqrt {c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b+\sqrt {b^2-4 a c}} e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.21, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1156, 1137,
1180, 211} \begin {gather*} -\frac {\sqrt {c} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a e \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} a e \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {1}{a e (d+e x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 1137
Rule 1156
Rule 1180
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{e}\\ &=-\frac {1}{a e (d+e x)}+\frac {\text {Subst}\left (\int \frac {-b-c x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{a e}\\ &=-\frac {1}{a e (d+e x)}-\frac {\left (c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right )\right ) \text {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 a e}-\frac {\left (c \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right )\right ) \text {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 a e}\\ &=-\frac {1}{a e (d+e x)}-\frac {\sqrt {c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b+\sqrt {b^2-4 a c}} e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.24, size = 206, normalized size = 1.06 \begin {gather*} -\frac {\frac {2}{d+e x}+\frac {\sqrt {2} \sqrt {c} \left (b+\sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-b+\sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{2 a e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.19, size = 168, normalized size = 0.86
method | result | size |
default | \(-\frac {1}{a e \left (e x +d \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (-\textit {\_R}^{2} c \,e^{2}-2 \textit {\_R} c d e -c \,d^{2}-b \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{2 a e}\) | \(168\) |
risch | \(-\frac {1}{a e \left (e x +d \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (16 a^{5} c^{2} e^{4}-8 e^{4} b^{2} c \,a^{4}+b^{4} e^{4} a^{3}\right ) \textit {\_Z}^{4}+\left (12 a^{2} b \,c^{2} e^{2}-7 a \,b^{3} c \,e^{2}+b^{5} e^{2}\right ) \textit {\_Z}^{2}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 a^{5} c^{2} e^{5}-22 a^{4} b^{2} c \,e^{5}+3 a^{3} b^{4} e^{5}\right ) \textit {\_R}^{4}+\left (25 a^{2} b \,c^{2} e^{3}-14 a \,b^{3} c \,e^{3}+2 b^{5} e^{3}\right ) \textit {\_R}^{2}+2 c^{3} e \right ) x +\left (40 a^{5} c^{2} d \,e^{4}-22 a^{4} b^{2} c d \,e^{4}+3 a^{3} b^{4} d \,e^{4}\right ) \textit {\_R}^{4}+\left (4 a^{4} c^{2} e^{3}-5 a^{3} b^{2} c \,e^{3}+a^{2} b^{4} e^{3}\right ) \textit {\_R}^{3}+\left (25 a^{2} b \,c^{2} d \,e^{2}-14 a \,b^{3} c d \,e^{2}+2 b^{5} d \,e^{2}\right ) \textit {\_R}^{2}+2 c^{3} d \right )\right )}{2}\) | \(310\) |
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. 1275 vs.
\(2 (158) = 316\).
time = 0.44, size = 1275, normalized size = 6.54 \begin {gather*} \frac {\sqrt {\frac {1}{2}} {\left (a x e^{2} + a d e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} e^{\left (-2\right )}}{a^{3} b^{2} - 4 \, a^{4} c}} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x e - 2 \, {\left (b^{2} c^{2} - a c^{3}\right )} d + \sqrt {\frac {1}{2}} {\left ({\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}} e - {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2}\right )} e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} e^{\left (-2\right )}}{a^{3} b^{2} - 4 \, a^{4} c}}\right ) - \sqrt {\frac {1}{2}} {\left (a x e^{2} + a d e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} e^{\left (-2\right )}}{a^{3} b^{2} - 4 \, a^{4} c}} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x e - 2 \, {\left (b^{2} c^{2} - a c^{3}\right )} d - \sqrt {\frac {1}{2}} {\left ({\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}} e - {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2}\right )} e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} e^{\left (-2\right )}}{a^{3} b^{2} - 4 \, a^{4} c}}\right ) - \sqrt {\frac {1}{2}} {\left (a x e^{2} + a d e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} e^{\left (-2\right )}}{a^{3} b^{2} - 4 \, a^{4} c}} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x e - 2 \, {\left (b^{2} c^{2} - a c^{3}\right )} d + \sqrt {\frac {1}{2}} {\left ({\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}} e + {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2}\right )} e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} e^{\left (-2\right )}}{a^{3} b^{2} - 4 \, a^{4} c}}\right ) + \sqrt {\frac {1}{2}} {\left (a x e^{2} + a d e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} e^{\left (-2\right )}}{a^{3} b^{2} - 4 \, a^{4} c}} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x e - 2 \, {\left (b^{2} c^{2} - a c^{3}\right )} d - \sqrt {\frac {1}{2}} {\left ({\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}} e + {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2}\right )} e\right )} \sqrt {-\frac {{\left (b^{3} - 3 \, a b c - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} e^{\left (-2\right )}}{a^{3} b^{2} - 4 \, a^{4} c}}\right ) - 2}{2 \, {\left (a x e^{2} + a d e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 3.01, size = 211, normalized size = 1.08 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{5} c^{2} e^{4} - 128 a^{4} b^{2} c e^{4} + 16 a^{3} b^{4} e^{4}\right ) + t^{2} \cdot \left (48 a^{2} b c^{2} e^{2} - 28 a b^{3} c e^{2} + 4 b^{5} e^{2}\right ) + c^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{5} c^{2} e^{3} + 48 t^{3} a^{4} b^{2} c e^{3} - 8 t^{3} a^{3} b^{4} e^{3} - 10 t a^{2} b c^{2} e + 10 t a b^{3} c e - 2 t b^{5} e + a c^{3} d - b^{2} c^{2} d}{a c^{3} e - b^{2} c^{2} e} \right )} \right )\right )} - \frac {1}{a d e + a e^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 2.39, size = 2500, normalized size = 12.82 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________