Optimal. Leaf size=63 \[ -\frac {a+b \tan ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac {b}{2 d e^3 (c+d x)}-\frac {b \tan ^{-1}(c+d x)}{2 d e^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5043, 12, 4852, 325, 203} \[ -\frac {a+b \tan ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac {b}{2 d e^3 (c+d x)}-\frac {b \tan ^{-1}(c+d x)}{2 d e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 203
Rule 325
Rule 4852
Rule 5043
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c+d x)}{(c e+d e x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {a+b \tan ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+x^2\right )} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac {b}{2 d e^3 (c+d x)}-\frac {a+b \tan ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac {b \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac {b}{2 d e^3 (c+d x)}-\frac {b \tan ^{-1}(c+d x)}{2 d e^3}-\frac {a+b \tan ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 51, normalized size = 0.81 \[ -\frac {a+b (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-(c+d x)^2\right )+b \tan ^{-1}(c+d x)}{2 d e^3 (c+d x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 70, normalized size = 1.11 \[ -\frac {b d x + b c + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + b\right )} \arctan \left (d x + c\right ) + a}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 71, normalized size = 1.13 \[ -\frac {a}{2 d \,e^{3} \left (d x +c \right )^{2}}-\frac {b \arctan \left (d x +c \right )}{2 d \,e^{3} \left (d x +c \right )^{2}}-\frac {b}{2 d \,e^{3} \left (d x +c \right )}-\frac {b \arctan \left (d x +c \right )}{2 d \,e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.42, size = 120, normalized size = 1.90 \[ -\frac {1}{2} \, {\left (d {\left (\frac {1}{d^{3} e^{3} x + c d^{2} e^{3}} + \frac {\arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{2} e^{3}}\right )} + \frac {\arctan \left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} b - \frac {a}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.75, size = 103, normalized size = 1.63 \[ -\frac {\frac {a+b\,c}{d}+b\,x}{2\,c^2\,e^3+4\,c\,d\,e^3\,x+2\,d^2\,e^3\,x^2}-\frac {b\,\mathrm {atan}\left (\frac {b\,c+b\,d\,x}{b}\right )}{2\,d\,e^3}-\frac {b\,\mathrm {atan}\left (c+d\,x\right )}{2\,d^3\,e^3\,\left (x^2+\frac {c^2}{d^2}+\frac {2\,c\,x}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 12.39, size = 314, normalized size = 4.98 \[ \begin {cases} - \frac {a}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b c^{2} \operatorname {atan}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {2 b c d x \operatorname {atan}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b c}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b d^{2} x^{2} \operatorname {atan}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b d x}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b \operatorname {atan}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} & \text {for}\: d \neq 0 \\\frac {x \left (a + b \operatorname {atan}{\relax (c )}\right )}{c^{3} e^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________