Optimal. Leaf size=200 \[ \frac {e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{3/2}}{b^2 c}-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac {d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{5/2} (4 c d+3 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{5/2}} \]
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Rubi [A]
time = 0.26, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {752, 838, 840,
1180, 214} \begin {gather*} -\frac {(c d-b e)^{5/2} (3 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{5/2}}+\frac {d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}+\frac {e \sqrt {d+e x} \left (3 b^2 e^2-2 b c d e+2 c^2 d^2\right )}{b^2 c^2}-\frac {(d+e x)^{5/2} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )}+\frac {e (d+e x)^{3/2} (2 c d-b e)}{b^2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 752
Rule 838
Rule 840
Rule 1180
Rubi steps
\begin {align*} \int \frac {(d+e x)^{7/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} d (4 c d-7 b e)-\frac {3}{2} e (2 c d-b e) x\right )}{b x+c x^2} \, dx}{b^2}\\ &=\frac {e (2 c d-b e) (d+e x)^{3/2}}{b^2 c}-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} c d^2 (4 c d-7 b e)-\frac {1}{2} e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right ) x\right )}{b x+c x^2} \, dx}{b^2 c}\\ &=\frac {e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{3/2}}{b^2 c}-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} c^2 d^3 (4 c d-7 b e)+\frac {1}{2} e (2 c d-b e) \left (c^2 d^2-b c d e-3 b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c^2}\\ &=\frac {e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{3/2}}{b^2 c}-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} c^2 d^3 e (4 c d-7 b e)-\frac {1}{2} d e (2 c d-b e) \left (c^2 d^2-b c d e-3 b^2 e^2\right )+\frac {1}{2} e (2 c d-b e) \left (c^2 d^2-b c d e-3 b^2 e^2\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 )}{b^2 c^2}\\ &=\frac {e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{3/2}}{b^2 c}-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\left (c d^3 (4 c d-7 b e)\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^3}+\frac {\left ((c d-b e)^3 (4 c d+3 b e)\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^3 c^2}\\ &=\frac {e \left (2 c^2 d^2-2 b c d e+3 b^2 e^2\right ) \sqrt {d+e x}}{b^2 c^2}+\frac {e (2 c d-b e) (d+e x)^{3/2}}{b^2 c}-\frac {(d+e x)^{5/2} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac {d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {(c d-b e)^{5/2} (4 c d+3 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.63, size = 167, normalized size = 0.84 \begin {gather*} \frac {-\frac {b \sqrt {d+e x} \left (2 c^3 d^3 x-3 b^3 e^3 x+b c^2 d^2 (d-3 e x)+b^2 c e^2 x (3 d-2 e x)\right )}{c^2 x (b+c x)}-\frac {(-c d+b e)^{5/2} (4 c d+3 b e) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{5/2}}+d^{5/2} (4 c d-7 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 236, normalized size = 1.18
method | result | size |
derivativedivides | \(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {d^{3} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (7 b e -4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}-\frac {\frac {\left (-\frac {1}{2} b^{4} e^{4}+\frac {3}{2} b^{3} c d \,e^{3}-\frac {3}{2} b^{2} c^{2} d^{2} e^{2}+\frac {1}{2} b \,c^{3} d^{3} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (3 b^{4} e^{4}-5 b^{3} c d \,e^{3}-3 b^{2} c^{2} d^{2} e^{2}+9 b \,c^{3} d^{3} e -4 c^{4} d^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{c^{2} b^{3} e^{3}}\right )\) | \(236\) |
default | \(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {d^{3} \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (7 b e -4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}-\frac {\frac {\left (-\frac {1}{2} b^{4} e^{4}+\frac {3}{2} b^{3} c d \,e^{3}-\frac {3}{2} b^{2} c^{2} d^{2} e^{2}+\frac {1}{2} b \,c^{3} d^{3} e \right ) \sqrt {e x +d}}{c \left (e x +d \right )+b e -c d}+\frac {\left (3 b^{4} e^{4}-5 b^{3} c d \,e^{3}-3 b^{2} c^{2} d^{2} e^{2}+9 b \,c^{3} d^{3} e -4 c^{4} d^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}}{c^{2} b^{3} e^{3}}\right )\) | \(236\) |
risch | \(-\frac {d^{3} \sqrt {e x +d}}{b^{2} x}+\frac {e^{4} b \sqrt {e x +d}}{c^{2} \left (c e x +b e \right )}-\frac {3 e^{3} \sqrt {e x +d}\, d}{c \left (c e x +b e \right )}+\frac {3 e^{2} \sqrt {e x +d}\, d^{2}}{b \left (c e x +b e \right )}-\frac {e c \sqrt {e x +d}\, d^{3}}{b^{2} \left (c e x +b e \right )}-\frac {3 e^{4} b \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{c^{2} \sqrt {\left (b e -c d \right ) c}}+\frac {5 e^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d}{c \sqrt {\left (b e -c d \right ) c}}+\frac {3 e^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{2}}{b \sqrt {\left (b e -c d \right ) c}}-\frac {9 e c \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{3}}{b^{2} \sqrt {\left (b e -c d \right ) c}}+\frac {4 c^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{4}}{b^{3} \sqrt {\left (b e -c d \right ) c}}+\frac {2 e^{3} \sqrt {e x +d}}{c^{2}}-\frac {7 e \,d^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}+\frac {4 d^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{b^{3}}\) | \(403\) |
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 = 3.15, size = 1308, normalized size = 6.54 \begin {gather*} \left [\frac {{\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + 3 \, {\left (b^{3} c x^{2} + b^{4} x\right )} e^{3} - 2 \, {\left (b^{2} c^{2} d x^{2} + b^{3} c d x\right )} e^{2} - 5 \, {\left (b c^{3} d^{2} x^{2} + b^{2} c^{2} d^{2} x\right )} e\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 ) - {\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x - 7 \, {\left (b c^{3} d^{2} x^{2} + b^{2} c^{2} d^{2} x\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) - 2 \, {\left (2 \, b c^{3} d^{3} x - 3 \, b^{2} c^{2} d^{2} x e + b^{2} c^{2} d^{3} + 3 \, b^{3} c d x e^{2} - {\left (2 \, b^{3} c x^{2} + 3 \, b^{4} x\right )} e^{3}\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c^{3} x^{2} + b^{4} c^{2} x\right )}}, -\frac {2 \, {\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + 3 \, {\left (b^{3} c x^{2} + b^{4} x\right )} e^{3} - 2 \, {\left (b^{2} c^{2} d x^{2} + b^{3} c d x\right )} e^{2} - 5 \, {\left (b c^{3} d^{2} x^{2} + b^{2} c^{2} d^{2} x\right )} e\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 (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x - 7 \, {\left (b c^{3} d^{2} x^{2} + b^{2} c^{2} d^{2} x\right )} e\right )} \sqrt {d} \log \left (\frac {x e - 2 \, \sqrt {x e + d} \sqrt {d} + 2 \, d}{x}\right ) + 2 \, {\left (2 \, b c^{3} d^{3} x - 3 \, b^{2} c^{2} d^{2} x e + b^{2} c^{2} d^{3} + 3 \, b^{3} c d x e^{2} - {\left (2 \, b^{3} c x^{2} + 3 \, b^{4} x\right )} e^{3}\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c^{3} x^{2} + b^{4} c^{2} x\right )}}, -\frac {2 \, {\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x - 7 \, {\left (b c^{3} d^{2} x^{2} + b^{2} c^{2} d^{2} x\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) - {\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + 3 \, {\left (b^{3} c x^{2} + b^{4} x\right )} e^{3} - 2 \, {\left (b^{2} c^{2} d x^{2} + b^{3} c d x\right )} e^{2} - 5 \, {\left (b c^{3} d^{2} x^{2} + b^{2} c^{2} d^{2} x\right )} e\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 (2 \, b c^{3} d^{3} x - 3 \, b^{2} c^{2} d^{2} x e + b^{2} c^{2} d^{3} + 3 \, b^{3} c d x e^{2} - {\left (2 \, b^{3} c x^{2} + 3 \, b^{4} x\right )} e^{3}\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c^{3} x^{2} + b^{4} c^{2} x\right )}}, -\frac {{\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + 3 \, {\left (b^{3} c x^{2} + b^{4} x\right )} e^{3} - 2 \, {\left (b^{2} c^{2} d x^{2} + b^{3} c d x\right )} e^{2} - 5 \, {\left (b c^{3} d^{2} x^{2} + b^{2} c^{2} d^{2} x\right )} e\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 (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x - 7 \, {\left (b c^{3} d^{2} x^{2} + b^{2} c^{2} d^{2} x\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (2 \, b c^{3} d^{3} x - 3 \, b^{2} c^{2} d^{2} x e + b^{2} c^{2} d^{3} + 3 \, b^{3} c d x e^{2} - {\left (2 \, b^{3} c x^{2} + 3 \, b^{4} x\right )} e^{3}\right )} \sqrt {x e + d}}{b^{3} c^{3} x^{2} + b^{4} c^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1760 vs.
\(2 (187) = 374\).
time = 212.09, size = 1760, normalized size = 8.80 \begin {gather*} \frac {2 b^{2} e^{5} \sqrt {d + e x}}{2 b^{2} c^{2} e^{2} - 2 b c^{3} d e + 2 b c^{3} e^{2} x - 2 c^{4} d e x} - \frac {b^{2} e^{5} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (- b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 c^{2}} + \frac {b^{2} e^{5} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 c^{2}} - \frac {8 b d e^{4} \sqrt {d + e x}}{2 b^{2} c e^{2} - 2 b c^{2} d e + 2 b c^{2} e^{2} x - 2 c^{3} d e x} + \frac {2 b d e^{4} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (- b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{c} - \frac {2 b d e^{4} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{c} - \frac {4 b e^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e}{c} - d}} \right )}}{c^{3} \sqrt {\frac {b e}{c} - d}} + \frac {2 c^{2} d^{4} e \sqrt {d + e x}}{2 b^{4} e^{2} - 2 b^{3} c d e + 2 b^{3} c e^{2} x - 2 b^{2} c^{2} d e x} - \frac {8 c d^{3} e^{2} \sqrt {d + e x}}{2 b^{3} e^{2} - 2 b^{2} c d e + 2 b^{2} c e^{2} x - 2 b c^{2} d e x} - 3 d^{2} e^{3} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (- b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )} + 3 d^{2} e^{3} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )} + \frac {12 d^{2} e^{3} \sqrt {d + e x}}{2 b^{2} e^{2} - 2 b c d e + 2 b c e^{2} x - 2 c^{2} d e x} + \frac {8 d e^{3} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e}{c} - d}} \right )}}{c^{2} \sqrt {\frac {b e}{c} - d}} + \frac {2 e^{3} \sqrt {d + e x}}{c^{2}} + \frac {2 c d^{3} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (- b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{b} - \frac {2 c d^{3} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{b} - \frac {c^{2} d^{4} e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (- b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} + \frac {c^{2} d^{4} e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} \log {\left (b^{2} e^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} - 2 b c d e \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + c^{2} d^{2} \sqrt {- \frac {1}{c \left (b e - c d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} - \frac {d^{4} e \sqrt {\frac {1}{d^{3}}} \log {\left (- d^{2} \sqrt {\frac {1}{d^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} + \frac {d^{4} e \sqrt {\frac {1}{d^{3}}} \log {\left (d^{2} \sqrt {\frac {1}{d^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} - \frac {8 d^{3} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e}{c} - d}} \right )}}{b^{2} \sqrt {\frac {b e}{c} - d}} + \frac {8 d^{3} e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b^{2} \sqrt {- d}} - \frac {d^{3} \sqrt {d + e x}}{b^{2} x} + \frac {4 c d^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {b e}{c} - d}} \right )}}{b^{3} \sqrt {\frac {b e}{c} - d}} - \frac {4 c d^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b^{3} \sqrt {- d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.54, size = 344, normalized size = 1.72 \begin {gather*} \frac {2 \, \sqrt {x e + d} e^{3}}{c^{2}} - \frac {{\left (4 \, c d^{4} - 7 \, b d^{3} e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} + \frac {{\left (4 \, c^{4} d^{4} - 9 \, b c^{3} d^{3} e + 3 \, b^{2} c^{2} d^{2} e^{2} + 5 \, b^{3} c d e^{3} - 3 \, b^{4} 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^{3} c^{2}} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{3} e - 2 \, \sqrt {x e + d} c^{3} d^{4} e - 3 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{2} d^{2} e^{2} + 4 \, \sqrt {x e + d} b c^{2} d^{3} e^{2} + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c d e^{3} - 3 \, \sqrt {x e + d} b^{2} c d^{2} e^{3} - {\left (x e + d\right )}^{\frac {3}{2}} b^{3} e^{4} + \sqrt {x e + d} b^{3} d e^{4}}{{\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )} b^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.78, size = 2913, normalized size = 14.56 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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