Optimal. Leaf size=136 \[ \frac {4 c d \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {4 \sqrt {2} c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e} \]
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Rubi [A]
time = 0.05, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {679, 675, 214}
\begin {gather*} -\frac {4 \sqrt {2} c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e}+\frac {4 c d \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 675
Rule 679
Rubi steps
\begin {align*} \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx &=\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+(2 c d) \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{3/2}} \, dx\\ &=\frac {4 c d \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\left (4 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx\\ &=\frac {4 c d \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\left (8 c^2 d^2 e\right ) \text {Subst}\left (\int \frac {1}{-2 c d e^2+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=\frac {4 c d \sqrt {c d^2-c e^2 x^2}}{e \sqrt {d+e x}}+\frac {2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {4 \sqrt {2} c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{e}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 110, normalized size = 0.81 \begin {gather*} \frac {2 c \sqrt {c \left (d^2-e^2 x^2\right )} \left (\frac {7 d-e x}{\sqrt {d+e x}}-\frac {6 \sqrt {2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{\sqrt {d^2-e^2 x^2}}\right )}{3 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.51, size = 112, normalized size = 0.82
method | result | size |
default | \(-\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, c \left (6 \arctanh \left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {2}\, c \,d^{2}+e x \sqrt {c d}\, \sqrt {c \left (-e x +d \right )}-7 \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}\, d \right )}{3 \sqrt {e x +d}\, \sqrt {c \left (-e x +d \right )}\, e \sqrt {c d}}\) | \(112\) |
risch | \(\frac {2 \left (-e x +7 d \right ) \left (-e x +d \right ) \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2}}{3 e \sqrt {-c \left (e x -d \right )}\, \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}}-\frac {4 d^{2} \sqrt {2}\, \arctanh \left (\frac {\sqrt {-c e x +c d}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}\, c^{2}}{e \sqrt {c d}\, \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}}\) | \(177\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.70, size = 272, normalized size = 2.00 \begin {gather*} \left [\frac {2 \, {\left (3 \, \sqrt {2} {\left (c d x e + c d^{2}\right )} \sqrt {c d} \log \left (-\frac {c x^{2} e^{2} - 2 \, c d x e - 3 \, c d^{2} + 2 \, \sqrt {2} \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {c d} \sqrt {x e + d}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) - \sqrt {-c x^{2} e^{2} + c d^{2}} {\left (c x e - 7 \, c d\right )} \sqrt {x e + d}\right )}}{3 \, {\left (x e^{2} + d e\right )}}, -\frac {2 \, {\left (6 \, \sqrt {2} {\left (c d x e + c d^{2}\right )} \sqrt {-c d} \arctan \left (\frac {\sqrt {2} \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {-c d} \sqrt {x e + d}}{c x^{2} e^{2} - c d^{2}}\right ) + \sqrt {-c x^{2} e^{2} + c d^{2}} {\left (c x e - 7 \, c d\right )} \sqrt {x e + d}\right )}}{3 \, {\left (x e^{2} + d e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 147, normalized size = 1.08 \begin {gather*} \frac {2}{3} \, {\left (\frac {6 \, \sqrt {2} c d^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d}} + \frac {6 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{3} d + {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{2}}{c^{3}}\right )} c e^{\left (-1\right )} - \frac {4 \, \sqrt {2} {\left (3 \, c^{2} d^{2} \arctan \left (\frac {\sqrt {c d}}{\sqrt {-c d}}\right ) + 4 \, \sqrt {c d} \sqrt {-c d} c d\right )} e^{\left (-1\right )}}{3 \, \sqrt {-c d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,d^2-c\,e^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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