Optimal. Leaf size=228 \[ \frac {2 \left (B (c d-b e)^3+A c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (B (c d-b e)^2+A c e (2 c d-b e)\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}-\frac {2 A d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b}-\frac {2 (b B-A c) (c d-b e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b c^{9/2}} \]
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Rubi [A]
time = 0.38, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {838, 840, 1180,
214} \begin {gather*} \frac {2 \sqrt {d+e x} \left (A c e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )+B (c d-b e)^3\right )}{c^4}-\frac {2 (b B-A c) (c d-b e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b c^{9/2}}+\frac {2 (d+e x)^{3/2} \left (A c e (2 c d-b e)+B (c d-b e)^2\right )}{3 c^3}+\frac {2 (d+e x)^{5/2} (A c e-b B e+B c d)}{5 c^2}-\frac {2 A d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b}+\frac {2 B (d+e x)^{7/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 838
Rule 840
Rule 1180
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{7/2}}{b x+c x^2} \, dx &=\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {\int \frac {(d+e x)^{5/2} (A c d+(B c d-b B e+A c e) x)}{b x+c x^2} \, dx}{c}\\ &=\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {\int \frac {(d+e x)^{3/2} \left (A c^2 d^2+\left (B (c d-b e)^2+A c e (2 c d-b e)\right ) x\right )}{b x+c x^2} \, dx}{c^2}\\ &=\frac {2 \left (B (c d-b e)^2+A c e (2 c d-b e)\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {\int \frac {\sqrt {d+e x} \left (A c^3 d^3+\left (B (c d-b e)^3+A c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) x\right )}{b x+c x^2} \, dx}{c^3}\\ &=\frac {2 \left (B (c d-b e)^3+A c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (B (c d-b e)^2+A c e (2 c d-b e)\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {\int \frac {A c^4 d^4+\left (B (c d-b e)^4+A c e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{c^4}\\ &=\frac {2 \left (B (c d-b e)^3+A c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (B (c d-b e)^2+A c e (2 c d-b e)\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {2 \text {Subst}\left (\int \frac {A c^4 d^4 e-d \left (B (c d-b e)^4+A c e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right )+\left (B (c d-b e)^4+A c e \left (4 c^3 d^3-6 b c^2 d^2 e+4 b^2 c d e^2-b^3 e^3\right )\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{c^4}\\ &=\frac {2 \left (B (c d-b e)^3+A c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (B (c d-b e)^2+A c e (2 c d-b e)\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}+\frac {\left (2 A c d^4\right ) \text {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b}+\frac {\left (2 (b B-A c) (c d-b e)^4\right ) \text {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{b c^4}\\ &=\frac {2 \left (B (c d-b e)^3+A c e \left (3 c^2 d^2-3 b c d e+b^2 e^2\right )\right ) \sqrt {d+e x}}{c^4}+\frac {2 \left (B (c d-b e)^2+A c e (2 c d-b e)\right ) (d+e x)^{3/2}}{3 c^3}+\frac {2 (B c d-b B e+A c e) (d+e x)^{5/2}}{5 c^2}+\frac {2 B (d+e x)^{7/2}}{7 c}-\frac {2 A d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b}-\frac {2 (b B-A c) (c d-b e)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b c^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 239, normalized size = 1.05 \begin {gather*} \frac {2 \sqrt {d+e x} \left (7 A c e \left (15 b^2 e^2-5 b c e (10 d+e x)+c^2 \left (58 d^2+16 d e x+3 e^2 x^2\right )\right )+B \left (-105 b^3 e^3+35 b^2 c e^2 (10 d+e x)-7 b c^2 e \left (58 d^2+16 d e x+3 e^2 x^2\right )+c^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )\right )}{105 c^4}-\frac {2 (-b B+A c) (-c d+b e)^{7/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{b c^{9/2}}-\frac {2 A d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(440\) vs.
\(2(198)=396\).
time = 0.67, size = 441, normalized size = 1.93
method | result | size |
derivativedivides | \(\frac {\frac {2 B \left (e x +d \right )^{\frac {7}{2}} c^{3}}{7}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 B b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 B \,c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 A b \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 A \,c^{3} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {4 B b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} c \,e^{3} \sqrt {e x +d}-6 A b \,c^{2} d \,e^{2} \sqrt {e x +d}+6 A \,c^{3} d^{2} e \sqrt {e x +d}-2 B \,b^{3} e^{3} \sqrt {e x +d}+6 B \,b^{2} c d \,e^{2} \sqrt {e x +d}-6 B b \,c^{2} d^{2} e \sqrt {e x +d}+2 B \,c^{3} d^{3} \sqrt {e x +d}}{c^{4}}+\frac {2 \left (-A \,b^{4} c \,e^{4}+4 A \,b^{3} c^{2} d \,e^{3}-6 A \,b^{2} c^{3} d^{2} e^{2}+4 A \,c^{4} d^{3} e b -A \,d^{4} c^{5}+b^{5} B \,e^{4}-4 B \,b^{4} c d \,e^{3}+6 B \,b^{3} c^{2} d^{2} e^{2}-4 B \,b^{2} c^{3} d^{3} e +B b \,c^{4} d^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b \,c^{4} \sqrt {\left (b e -c d \right ) c}}-\frac {2 A \,d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b}\) | \(441\) |
default | \(\frac {\frac {2 B \left (e x +d \right )^{\frac {7}{2}} c^{3}}{7}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 B b \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 B \,c^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 A b \,c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 A \,c^{3} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,b^{2} c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {4 B b \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,c^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 A \,b^{2} c \,e^{3} \sqrt {e x +d}-6 A b \,c^{2} d \,e^{2} \sqrt {e x +d}+6 A \,c^{3} d^{2} e \sqrt {e x +d}-2 B \,b^{3} e^{3} \sqrt {e x +d}+6 B \,b^{2} c d \,e^{2} \sqrt {e x +d}-6 B b \,c^{2} d^{2} e \sqrt {e x +d}+2 B \,c^{3} d^{3} \sqrt {e x +d}}{c^{4}}+\frac {2 \left (-A \,b^{4} c \,e^{4}+4 A \,b^{3} c^{2} d \,e^{3}-6 A \,b^{2} c^{3} d^{2} e^{2}+4 A \,c^{4} d^{3} e b -A \,d^{4} c^{5}+b^{5} B \,e^{4}-4 B \,b^{4} c d \,e^{3}+6 B \,b^{3} c^{2} d^{2} e^{2}-4 B \,b^{2} c^{3} d^{3} e +B b \,c^{4} d^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b \,c^{4} \sqrt {\left (b e -c d \right ) c}}-\frac {2 A \,d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b}\) | \(441\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 33.61, size = 1454, normalized size = 6.38 \begin {gather*} \left [\frac {105 \, A c^{4} d^{\frac {7}{2}} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) - 105 \, {\left ({\left (B b c^{3} - A c^{4}\right )} d^{3} - 3 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2} e + 3 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d e^{2} - {\left (B b^{4} - A b^{3} c\right )} e^{3}\right )} \sqrt {\frac {c d - b e}{c}} \log \left (\frac {2 \, c d + 2 \, \sqrt {x e + d} c \sqrt {\frac {c d - b e}{c}} + {\left (c x - b\right )} e}{c x + b}\right ) + 2 \, {\left (176 \, B b c^{3} d^{3} + {\left (15 \, B b c^{3} x^{3} - 105 \, B b^{4} + 105 \, A b^{3} c - 21 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} x^{2} + 35 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} x\right )} e^{3} + 2 \, {\left (33 \, B b c^{3} d x^{2} - 56 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d x + 175 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d\right )} e^{2} + 2 \, {\left (61 \, B b c^{3} d^{2} x - 203 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{105 \, b c^{4}}, \frac {105 \, A c^{4} d^{\frac {7}{2}} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) - 210 \, {\left ({\left (B b c^{3} - A c^{4}\right )} d^{3} - 3 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2} e + 3 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d e^{2} - {\left (B b^{4} - A b^{3} c\right )} e^{3}\right )} \sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {x e + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + 2 \, {\left (176 \, B b c^{3} d^{3} + {\left (15 \, B b c^{3} x^{3} - 105 \, B b^{4} + 105 \, A b^{3} c - 21 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} x^{2} + 35 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} x\right )} e^{3} + 2 \, {\left (33 \, B b c^{3} d x^{2} - 56 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d x + 175 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d\right )} e^{2} + 2 \, {\left (61 \, B b c^{3} d^{2} x - 203 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{105 \, b c^{4}}, \frac {210 \, A c^{4} \sqrt {-d} d^{3} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) - 105 \, {\left ({\left (B b c^{3} - A c^{4}\right )} d^{3} - 3 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2} e + 3 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d e^{2} - {\left (B b^{4} - A b^{3} c\right )} e^{3}\right )} \sqrt {\frac {c d - b e}{c}} \log \left (\frac {2 \, c d + 2 \, \sqrt {x e + d} c \sqrt {\frac {c d - b e}{c}} + {\left (c x - b\right )} e}{c x + b}\right ) + 2 \, {\left (176 \, B b c^{3} d^{3} + {\left (15 \, B b c^{3} x^{3} - 105 \, B b^{4} + 105 \, A b^{3} c - 21 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} x^{2} + 35 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} x\right )} e^{3} + 2 \, {\left (33 \, B b c^{3} d x^{2} - 56 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d x + 175 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d\right )} e^{2} + 2 \, {\left (61 \, B b c^{3} d^{2} x - 203 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{105 \, b c^{4}}, \frac {2 \, {\left (105 \, A c^{4} \sqrt {-d} d^{3} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) - 105 \, {\left ({\left (B b c^{3} - A c^{4}\right )} d^{3} - 3 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2} e + 3 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d e^{2} - {\left (B b^{4} - A b^{3} c\right )} e^{3}\right )} \sqrt {-\frac {c d - b e}{c}} \arctan \left (-\frac {\sqrt {x e + d} c \sqrt {-\frac {c d - b e}{c}}}{c d - b e}\right ) + {\left (176 \, B b c^{3} d^{3} + {\left (15 \, B b c^{3} x^{3} - 105 \, B b^{4} + 105 \, A b^{3} c - 21 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} x^{2} + 35 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} x\right )} e^{3} + 2 \, {\left (33 \, B b c^{3} d x^{2} - 56 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d x + 175 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d\right )} e^{2} + 2 \, {\left (61 \, B b c^{3} d^{2} x - 203 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}\right )}}{105 \, b c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 86.22, size = 298, normalized size = 1.31 \begin {gather*} \frac {2 A d^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b \sqrt {- d}} + \frac {2 B \left (d + e x\right )^{\frac {7}{2}}}{7 c} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 A c e - 2 B b e + 2 B c d\right )}{5 c^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- 2 A b c e^{2} + 4 A c^{2} d e + 2 B b^{2} e^{2} - 4 B b c d e + 2 B c^{2} d^{2}\right )}{3 c^{3}} + \frac {\sqrt {d + e x} \left (2 A b^{2} c e^{3} - 6 A b c^{2} d e^{2} + 6 A c^{3} d^{2} e - 2 B b^{3} e^{3} + 6 B b^{2} c d e^{2} - 6 B b c^{2} d^{2} e + 2 B c^{3} d^{3}\right )}{c^{4}} + \frac {2 \left (- A c + B b\right ) \left (b e - c d\right )^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e - c d}{c}}} \right )}}{b c^{5} \sqrt {\frac {b e - c d}{c}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 476 vs.
\(2 (213) = 426\).
time = 1.13, size = 476, normalized size = 2.09 \begin {gather*} \frac {2 \, A d^{4} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b \sqrt {-d}} + \frac {2 \, {\left (B b c^{4} d^{4} - A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 4 \, A b c^{4} d^{3} e + 6 \, B b^{3} c^{2} d^{2} e^{2} - 6 \, A b^{2} c^{3} d^{2} e^{2} - 4 \, B b^{4} c d e^{3} + 4 \, A b^{3} c^{2} d e^{3} + B b^{5} e^{4} - A b^{4} c e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b c^{4}} + \frac {2 \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B c^{6} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{6} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{6} d^{2} + 105 \, \sqrt {x e + d} B c^{6} d^{3} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} B b c^{5} e + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{6} e - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} B b c^{5} d e + 70 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{6} d e - 315 \, \sqrt {x e + d} B b c^{5} d^{2} e + 315 \, \sqrt {x e + d} A c^{6} d^{2} e + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} c^{4} e^{2} - 35 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c^{5} e^{2} + 315 \, \sqrt {x e + d} B b^{2} c^{4} d e^{2} - 315 \, \sqrt {x e + d} A b c^{5} d e^{2} - 105 \, \sqrt {x e + d} B b^{3} c^{3} e^{3} + 105 \, \sqrt {x e + d} A b^{2} c^{4} e^{3}\right )}}{105 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.36, size = 2500, normalized size = 10.96 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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