Optimal. Leaf size=170 \[ -\frac {b^2}{12 d e^5 (c+d x)^2}-\frac {b (a+b \text {ArcTan}(c+d x))}{6 d e^5 (c+d x)^3}+\frac {b (a+b \text {ArcTan}(c+d x))}{2 d e^5 (c+d x)}+\frac {(a+b \text {ArcTan}(c+d x))^2}{4 d e^5}-\frac {(a+b \text {ArcTan}(c+d x))^2}{4 d e^5 (c+d x)^4}-\frac {2 b^2 \log (c+d x)}{3 d e^5}+\frac {b^2 \log \left (1+(c+d x)^2\right )}{3 d e^5} \]
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Rubi [A]
time = 0.19, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5151, 12,
4946, 5038, 272, 46, 36, 29, 31, 5004} \begin {gather*} \frac {b (a+b \text {ArcTan}(c+d x))}{2 d e^5 (c+d x)}-\frac {b (a+b \text {ArcTan}(c+d x))}{6 d e^5 (c+d x)^3}-\frac {(a+b \text {ArcTan}(c+d x))^2}{4 d e^5 (c+d x)^4}+\frac {(a+b \text {ArcTan}(c+d x))^2}{4 d e^5}-\frac {b^2}{12 d e^5 (c+d x)^2}-\frac {2 b^2 \log (c+d x)}{3 d e^5}+\frac {b^2 \log \left ((c+d x)^2+1\right )}{3 d e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 46
Rule 272
Rule 4946
Rule 5004
Rule 5038
Rule 5151
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{(c e+d e x)^5} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b \tan ^{-1}(x)\right )^2}{e^5 x^5} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a+b \tan ^{-1}(x)\right )^2}{x^5} \, dx,x,c+d x\right )}{d e^5}\\ &=-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {b \text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{x^4 \left (1+x^2\right )} \, dx,x,c+d x\right )}{2 d e^5}\\ &=-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {b \text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{x^4} \, dx,x,c+d x\right )}{2 d e^5}-\frac {b \text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{x^2 \left (1+x^2\right )} \, dx,x,c+d x\right )}{2 d e^5}\\ &=-\frac {b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}-\frac {b \text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{2 d e^5}+\frac {b \text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{2 d e^5}+\frac {b^2 \text {Subst}\left (\int \frac {1}{x^3 \left (1+x^2\right )} \, dx,x,c+d x\right )}{6 d e^5}\\ &=-\frac {b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}+\frac {b \left (a+b \tan ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {b^2 \text {Subst}\left (\int \frac {1}{x^2 (1+x)} \, dx,x,(c+d x)^2\right )}{12 d e^5}-\frac {b^2 \text {Subst}\left (\int \frac {1}{x \left (1+x^2\right )} \, dx,x,c+d x\right )}{2 d e^5}\\ &=-\frac {b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}+\frac {b \left (a+b \tan ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}+\frac {b^2 \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {1}{x}+\frac {1}{1+x}\right ) \, dx,x,(c+d x)^2\right )}{12 d e^5}-\frac {b^2 \text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,(c+d x)^2\right )}{4 d e^5}\\ &=-\frac {b^2}{12 d e^5 (c+d x)^2}-\frac {b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}+\frac {b \left (a+b \tan ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}-\frac {b^2 \log (c+d x)}{6 d e^5}+\frac {b^2 \log \left (1+(c+d x)^2\right )}{12 d e^5}-\frac {b^2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,(c+d x)^2\right )}{4 d e^5}+\frac {b^2 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,(c+d x)^2\right )}{4 d e^5}\\ &=-\frac {b^2}{12 d e^5 (c+d x)^2}-\frac {b \left (a+b \tan ^{-1}(c+d x)\right )}{6 d e^5 (c+d x)^3}+\frac {b \left (a+b \tan ^{-1}(c+d x)\right )}{2 d e^5 (c+d x)}+\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5}-\frac {\left (a+b \tan ^{-1}(c+d x)\right )^2}{4 d e^5 (c+d x)^4}-\frac {2 b^2 \log (c+d x)}{3 d e^5}+\frac {b^2 \log \left (1+(c+d x)^2\right )}{3 d e^5}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 245, normalized size = 1.44 \begin {gather*} -\frac {3 a^2+2 a b (c+d x)+b^2 (c+d x)^2-6 a b (c+d x)^3-2 b \left (b \left (-c+3 c^3-d x+9 c^2 d x+9 c d^2 x^2+3 d^3 x^3\right )+3 a \left (-1+c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right )\right ) \text {ArcTan}(c+d x)-3 b^2 \left (-1+c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right ) \text {ArcTan}(c+d x)^2+8 b^2 (c+d x)^4 \log (c+d x)-4 b^2 (c+d x)^4 \log \left (1+c^2+2 c d x+d^2 x^2\right )}{12 d e^5 (c+d x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 210, normalized size = 1.24
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2}}{4 e^{5} \left (d x +c \right )^{4}}-\frac {b^{2} \arctan \left (d x +c \right )^{2}}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b^{2} \arctan \left (d x +c \right )^{2}}{4 e^{5}}-\frac {b^{2} \arctan \left (d x +c \right )}{6 e^{5} \left (d x +c \right )^{3}}+\frac {b^{2} \arctan \left (d x +c \right )}{2 e^{5} \left (d x +c \right )}+\frac {b^{2} \ln \left (1+\left (d x +c \right )^{2}\right )}{3 e^{5}}-\frac {b^{2}}{12 e^{5} \left (d x +c \right )^{2}}-\frac {2 b^{2} \ln \left (d x +c \right )}{3 e^{5}}-\frac {a b \arctan \left (d x +c \right )}{2 e^{5} \left (d x +c \right )^{4}}+\frac {a b \arctan \left (d x +c \right )}{2 e^{5}}-\frac {a b}{6 e^{5} \left (d x +c \right )^{3}}+\frac {a b}{2 e^{5} \left (d x +c \right )}}{d}\) | \(210\) |
default | \(\frac {-\frac {a^{2}}{4 e^{5} \left (d x +c \right )^{4}}-\frac {b^{2} \arctan \left (d x +c \right )^{2}}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b^{2} \arctan \left (d x +c \right )^{2}}{4 e^{5}}-\frac {b^{2} \arctan \left (d x +c \right )}{6 e^{5} \left (d x +c \right )^{3}}+\frac {b^{2} \arctan \left (d x +c \right )}{2 e^{5} \left (d x +c \right )}+\frac {b^{2} \ln \left (1+\left (d x +c \right )^{2}\right )}{3 e^{5}}-\frac {b^{2}}{12 e^{5} \left (d x +c \right )^{2}}-\frac {2 b^{2} \ln \left (d x +c \right )}{3 e^{5}}-\frac {a b \arctan \left (d x +c \right )}{2 e^{5} \left (d x +c \right )^{4}}+\frac {a b \arctan \left (d x +c \right )}{2 e^{5}}-\frac {a b}{6 e^{5} \left (d x +c \right )^{3}}+\frac {a b}{2 e^{5} \left (d x +c \right )}}{d}\) | \(210\) |
risch | \(\text {Expression too large to display}\) | \(1562\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 506 vs.
\(2 (149) = 298\).
time = 0.57, size = 506, normalized size = 2.98 \begin {gather*} \frac {1}{6} \, {\left (d {\left (\frac {3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} - 1}{d^{5} x^{3} e^{5} + 3 \, c d^{4} x^{2} e^{5} + 3 \, c^{2} d^{3} x e^{5} + c^{3} d^{2} e^{5}} + \frac {3 \, \arctan \left (\frac {d^{2} x + c d}{d}\right ) e^{\left (-5\right )}}{d^{2}}\right )} - \frac {3 \, \arctan \left (d x + c\right )}{d^{5} x^{4} e^{5} + 4 \, c d^{4} x^{3} e^{5} + 6 \, c^{2} d^{3} x^{2} e^{5} + 4 \, c^{3} d^{2} x e^{5} + c^{4} d e^{5}}\right )} a b + \frac {1}{12} \, {\left (2 \, d {\left (\frac {3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} - 1}{d^{5} x^{3} e^{5} + 3 \, c d^{4} x^{2} e^{5} + 3 \, c^{2} d^{3} x e^{5} + c^{3} d^{2} e^{5}} + \frac {3 \, \arctan \left (\frac {d^{2} x + c d}{d}\right ) e^{\left (-5\right )}}{d^{2}}\right )} \arctan \left (d x + c\right ) - \frac {{\left (3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \arctan \left (d x + c\right )^{2} - 4 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) + 8 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c\right ) + 1\right )} d^{2}}{d^{5} x^{2} e^{5} + 2 \, c d^{4} x e^{5} + c^{2} d^{3} e^{5}}\right )} b^{2} - \frac {b^{2} \arctan \left (d x + c\right )^{2}}{4 \, {\left (d^{5} x^{4} e^{5} + 4 \, c d^{4} x^{3} e^{5} + 6 \, c^{2} d^{3} x^{2} e^{5} + 4 \, c^{3} d^{2} x e^{5} + c^{4} d e^{5}\right )}} - \frac {a^{2}}{4 \, {\left (d^{5} x^{4} e^{5} + 4 \, c d^{4} x^{3} e^{5} + 6 \, c^{2} d^{3} x^{2} e^{5} + 4 \, c^{3} d^{2} x e^{5} + c^{4} d e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 435 vs.
\(2 (149) = 298\).
time = 2.71, size = 435, normalized size = 2.56 \begin {gather*} \frac {{\left (6 \, a b d^{3} x^{3} + 6 \, a b c^{3} + {\left (18 \, a b c - b^{2}\right )} d^{2} x^{2} - b^{2} c^{2} - 2 \, a b c + 2 \, {\left (9 \, a b c^{2} - b^{2} c - a b\right )} d x + 3 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - b^{2}\right )} \arctan \left (d x + c\right )^{2} - 3 \, a^{2} + 2 \, {\left (3 \, a b d^{4} x^{4} + 3 \, {\left (4 \, a b c + b^{2}\right )} d^{3} x^{3} + 3 \, a b c^{4} + 3 \, b^{2} c^{3} + 9 \, {\left (2 \, a b c^{2} + b^{2} c\right )} d^{2} x^{2} - b^{2} c + {\left (12 \, a b c^{3} + 9 \, b^{2} c^{2} - b^{2}\right )} d x - 3 \, a b\right )} \arctan \left (d x + c\right ) + 4 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) - 8 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (d x + c\right )\right )} e^{\left (-5\right )}}{12 \, {\left (d^{5} x^{4} + 4 \, c d^{4} x^{3} + 6 \, c^{2} d^{3} x^{2} + 4 \, c^{3} d^{2} x + c^{4} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.65, size = 438, normalized size = 2.58 \begin {gather*} {\mathrm {atan}\left (c+d\,x\right )}^2\,\left (\frac {b^2}{4\,d\,e^5}-\frac {b^2}{4\,d^3\,e^5\,\left (\frac {c^4}{d^2}+6\,c^2\,x^2+d^2\,x^4+\frac {4\,c^3\,x}{d}+4\,c\,d\,x^3\right )}\right )-\frac {x^2\,\left (\frac {b^2\,d}{2}-9\,a\,b\,c\,d\right )+x\,\left (b^2\,c-9\,a\,b\,c^2+a\,b\right )+\frac {3\,a^2-6\,a\,b\,c^3+2\,a\,b\,c+b^2\,c^2}{2\,d}-3\,a\,b\,d^2\,x^3}{6\,c^4\,e^5+24\,c^3\,d\,e^5\,x+36\,c^2\,d^2\,e^5\,x^2+24\,c\,d^3\,e^5\,x^3+6\,d^4\,e^5\,x^4}+\frac {\mathrm {atan}\left (c+d\,x\right )\,\left (\frac {b^2\,x^3}{2\,e^5}-\frac {a\,b}{2\,d^3\,e^5}+\frac {b^2\,c\,\left (\frac {c^2-1}{3\,d^2}+\frac {2\,c^2}{3\,d^2}\right )}{2\,d\,e^5}+\frac {b^2\,x\,\left (d\,\left (\frac {c^2-1}{3\,d^2}+\frac {2\,c^2}{3\,d^2}\right )+\frac {2\,c^2}{d}\right )}{2\,d\,e^5}+\frac {3\,b^2\,c\,x^2}{2\,d\,e^5}\right )}{\frac {c^4}{d^2}+6\,c^2\,x^2+d^2\,x^4+\frac {4\,c^3\,x}{d}+4\,c\,d\,x^3}-\frac {2\,b^2\,\ln \left (c+d\,x\right )}{3\,d\,e^5}-\frac {\ln \left (c+d\,x-\mathrm {i}\right )\,\left (-\frac {b^2}{3}+\frac {a\,b\,1{}\mathrm {i}}{4}\right )}{d\,e^5}+\frac {\ln \left (c+d\,x+1{}\mathrm {i}\right )\,\left (\frac {b^2}{3}+\frac {1{}\mathrm {i}\,a\,b}{4}\right )}{d\,e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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