Optimal. Leaf size=134 \[ -\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac {\sqrt {d} (4 c d-3 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {\sqrt {c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 \sqrt {c}} \]
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Rubi [A]
time = 0.13, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {752, 840, 1180,
214} \begin {gather*} \frac {\sqrt {d} (4 c d-3 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {\sqrt {c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 \sqrt {c}}-\frac {\sqrt {d+e x} (x (2 c d-b e)+b d)}{b^2 \left (b x+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 752
Rule 840
Rule 1180
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} d (4 c d-3 b e)+\frac {1}{2} e (2 c d-b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2}\\ &=-\frac {\sqrt {d+e x} (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} d e (4 c d-3 b e)-\frac {1}{2} d e (2 c d-b e)+\frac {1}{2} e (2 c d-b e) 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}\\ &=-\frac {\sqrt {d+e x} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}-\frac {(c d (4 c d-3 b e)) \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 {((c d-b e) (4 c d-b e)) \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 {\sqrt {d+e x} (b d+(2 c d-b e) x)}{b^2 \left (b x+c x^2\right )}+\frac {\sqrt {d} (4 c d-3 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}-\frac {\sqrt {c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 139, normalized size = 1.04 \begin {gather*} \frac {\frac {b \sqrt {d+e x} (-b d-2 c d x+b e x)}{x (b+c x)}+\frac {\left (4 c^2 d^2-5 b c d e+b^2 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{\sqrt {c} \sqrt {-c d+b e}}+\sqrt {d} (4 c d-3 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 = 0.53, size = 144, normalized size = 1.07
method | result | size |
derivativedivides | \(2 e^{3} \left (-\frac {d \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (3 b e -4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}+\frac {\left (b e -c d \right ) \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3}}\right )\) | \(144\) |
default | \(2 e^{3} \left (-\frac {d \left (\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (3 b e -4 c d \right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}\right )}{b^{3} e^{3}}+\frac {\left (b e -c d \right ) \left (\frac {b e \sqrt {e x +d}}{2 c \left (e x +d \right )+2 b e -2 c d}+\frac {\left (b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{2 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{3} e^{3}}\right )\) | \(144\) |
risch | \(-\frac {d \sqrt {e x +d}}{b^{2} x}+\frac {e^{2} \sqrt {e x +d}}{b \left (c e x +b e \right )}-\frac {e \sqrt {e x +d}\, c d}{b^{2} \left (c e x +b e \right )}+\frac {e^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b \sqrt {\left (b e -c d \right ) c}}-\frac {5 e \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) c d}{b^{2} \sqrt {\left (b e -c d \right ) c}}+\frac {4 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right ) d^{2} c^{2}}{b^{3} \sqrt {\left (b e -c d \right ) c}}-\frac {3 e \sqrt {d}\, \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}+\frac {4 d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{b^{3}}\) | \(237\) |
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 = 1.86, size = 806, normalized size = 6.01 \begin {gather*} \left [-\frac {{\left (4 \, c^{2} d x^{2} + 4 \, b c d x - {\left (b c x^{2} + b^{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^{2} d x^{2} + 4 \, b c d x - 3 \, {\left (b c x^{2} + b^{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 d x - b^{2} x e + b^{2} d\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c x^{2} + b^{4} x\right )}}, -\frac {2 \, {\left (4 \, c^{2} d x^{2} + 4 \, b c d x - {\left (b c x^{2} + b^{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^{2} d x^{2} + 4 \, b c d x - 3 \, {\left (b c x^{2} + b^{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 d x - b^{2} x e + b^{2} d\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c x^{2} + b^{4} x\right )}}, -\frac {2 \, {\left (4 \, c^{2} d x^{2} + 4 \, b c d x - 3 \, {\left (b c x^{2} + b^{2} x\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (4 \, c^{2} d x^{2} + 4 \, b c d x - {\left (b c x^{2} + b^{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 d x - b^{2} x e + b^{2} d\right )} \sqrt {x e + d}}{2 \, {\left (b^{3} c x^{2} + b^{4} x\right )}}, -\frac {{\left (4 \, c^{2} d x^{2} + 4 \, b c d x - {\left (b c x^{2} + b^{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^{2} d x^{2} + 4 \, b c d x - 3 \, {\left (b c x^{2} + b^{2} x\right )} e\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {x e + d} \sqrt {-d}}{d}\right ) + {\left (2 \, b c d x - b^{2} x e + b^{2} d\right )} \sqrt {x e + d}}{b^{3} c x^{2} + b^{4} 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. 1071 vs.
\(2 (119) = 238\).
time = 79.41, size = 1071, normalized size = 7.99 \begin {gather*} \frac {2 c^{2} d^{2} 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 {4 c d 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} - \frac {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 )}}{2} + \frac {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 )}}{2} + \frac {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 {c d 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 d 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^{2} 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^{2} 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^{2} 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^{2} e \sqrt {\frac {1}{d^{3}}} \log {\left (d^{2} \sqrt {\frac {1}{d^{3}}} + \sqrt {d + e x} \right )}}{2 b^{2}} - \frac {4 d 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 {4 d e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {- d}} \right )}}{b^{2} \sqrt {- d}} - \frac {d \sqrt {d + e x}}{b^{2} x} + \frac {4 c d^{2} \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^{2} \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 = 1.38, size = 211, normalized size = 1.57 \begin {gather*} \frac {{\left (4 \, c^{2} d^{2} - 5 \, b c d e + b^{2} e^{2}\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}} - \frac {{\left (4 \, c d^{2} - 3 \, b d e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c d e - 2 \, \sqrt {x e + d} c d^{2} e - {\left (x e + d\right )}^{\frac {3}{2}} b e^{2} + 2 \, \sqrt {x e + d} b d e^{2}}{{\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.42, size = 429, normalized size = 3.20 \begin {gather*} -\frac {\frac {2\,\left (b\,d\,e^2-c\,d^2\,e\right )\,\sqrt {d+e\,x}}{b^2}-\frac {e\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{b^2}}{\left (b\,e-2\,c\,d\right )\,\left (d+e\,x\right )+c\,{\left (d+e\,x\right )}^2+c\,d^2-b\,d\,e}-\frac {\sqrt {d}\,\mathrm {atanh}\left (\frac {6\,c\,\sqrt {d}\,e^7\,\sqrt {d+e\,x}}{6\,c\,d\,e^7-\frac {14\,c^2\,d^2\,e^6}{b}+\frac {8\,c^3\,d^3\,e^5}{b^2}}-\frac {14\,c^2\,d^{3/2}\,e^6\,\sqrt {d+e\,x}}{6\,b\,c\,d\,e^7-14\,c^2\,d^2\,e^6+\frac {8\,c^3\,d^3\,e^5}{b}}+\frac {8\,c^3\,d^{5/2}\,e^5\,\sqrt {d+e\,x}}{6\,b^2\,c\,d\,e^7-14\,b\,c^2\,d^2\,e^6+8\,c^3\,d^3\,e^5}\right )\,\left (3\,b\,e-4\,c\,d\right )}{b^3}-\frac {\mathrm {atanh}\left (\frac {2\,c\,d\,e^6\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{2\,b\,c\,d\,e^7-10\,c^2\,d^2\,e^6+\frac {8\,c^3\,d^3\,e^5}{b}}-\frac {8\,c^2\,d^2\,e^5\,\sqrt {c^2\,d-b\,c\,e}\,\sqrt {d+e\,x}}{2\,b^2\,c\,d\,e^7-10\,b\,c^2\,d^2\,e^6+8\,c^3\,d^3\,e^5}\right )\,\sqrt {-c\,\left (b\,e-c\,d\right )}\,\left (b\,e-4\,c\,d\right )}{b^3\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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