Optimal. Leaf size=150 \[ -\frac {\sqrt {c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac {3 \sqrt {c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{16 \sqrt {2} \sqrt {c} d^{5/2} e} \]
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Rubi [A]
time = 0.05, antiderivative size = 150, 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 = {687, 675, 214}
\begin {gather*} -\frac {3 \sqrt {c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}-\frac {\sqrt {c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{16 \sqrt {2} \sqrt {c} d^{5/2} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 675
Rule 687
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{5/2} \sqrt {c d^2-c e^2 x^2}} \, dx &=-\frac {\sqrt {c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}+\frac {3 \int \frac {1}{(d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}} \, dx}{8 d}\\ &=-\frac {\sqrt {c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac {3 \sqrt {c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}+\frac {3 \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx}{32 d^2}\\ &=-\frac {\sqrt {c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac {3 \sqrt {c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}+\frac {(3 e) \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 )}{16 d^2}\\ &=-\frac {\sqrt {c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac {3 \sqrt {c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{16 \sqrt {2} \sqrt {c} d^{5/2} e}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 135, normalized size = 0.90 \begin {gather*} \frac {2 \sqrt {d} \left (-7 d^2+4 d e x+3 e^2 x^2\right )-3 \sqrt {2} (d+e x)^{3/2} \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{32 d^{5/2} e (d+e x)^{3/2} \sqrt {c \left (d^2-e^2 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 181, normalized size = 1.21
method | result | size |
default | \(-\frac {\sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (3 \arctanh \left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {2}\, c \,e^{2} x^{2}+6 \arctanh \left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {2}\, c d e x +3 \arctanh \left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {2}\, c \,d^{2}+6 e x \sqrt {c d}\, \sqrt {c \left (-e x +d \right )}+14 \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}\, d \right )}{32 \left (e x +d \right )^{\frac {5}{2}} c \sqrt {c \left (-e x +d \right )}\, e \,d^{2} \sqrt {c d}}\) | \(181\) |
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 = 3.08, size = 363, normalized size = 2.42 \begin {gather*} \left [\frac {3 \, \sqrt {2} {\left (x^{3} e^{3} + 3 \, d x^{2} e^{2} + 3 \, d^{2} x e + d^{3}\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 ) - 4 \, \sqrt {-c x^{2} e^{2} + c d^{2}} {\left (3 \, d x e + 7 \, d^{2}\right )} \sqrt {x e + d}}{64 \, {\left (c d^{3} x^{3} e^{4} + 3 \, c d^{4} x^{2} e^{3} + 3 \, c d^{5} x e^{2} + c d^{6} e\right )}}, -\frac {3 \, \sqrt {2} {\left (x^{3} e^{3} + 3 \, d x^{2} e^{2} + 3 \, d^{2} x e + d^{3}\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 ) + 2 \, \sqrt {-c x^{2} e^{2} + c d^{2}} {\left (3 \, d x e + 7 \, d^{2}\right )} \sqrt {x e + d}}{32 \, {\left (c d^{3} x^{3} e^{4} + 3 \, c d^{4} x^{2} e^{3} + 3 \, c d^{5} x e^{2} + c d^{6} 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 {1}{\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )} \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 = 3.48, size = 109, normalized size = 0.73 \begin {gather*} \frac {{\left (\frac {3 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{2 \, \sqrt {-c d}}\right )}{\sqrt {-c d} d^{2}} - \frac {2 \, {\left (10 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{2} d - 3 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c\right )}}{{\left (x e + d\right )}^{2} c^{2} d^{2}}\right )} e^{\left (-1\right )}}{32 \, c} \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 {1}{\sqrt {c\,d^2-c\,e^2\,x^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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