Optimal. Leaf size=139 \[ \frac {3 c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}-\frac {3 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{4 \sqrt {2} \sqrt {d} e} \]
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Rubi [A]
time = 0.05, antiderivative size = 139, 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 = {677, 675, 214}
\begin {gather*} -\frac {3 c^{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 )}{4 \sqrt {2} \sqrt {d} e}+\frac {3 c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 675
Rule 677
Rubi steps
\begin {align*} \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx &=-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}-\frac {1}{4} (3 c) \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx\\ &=\frac {3 c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}+\frac {1}{8} \left (3 c^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx\\ &=\frac {3 c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}+\frac {1}{4} \left (3 c^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 {3 c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}-\frac {3 c^{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 )}{4 \sqrt {2} \sqrt {d} e}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 109, normalized size = 0.78 \begin {gather*} \frac {c \sqrt {c \left (d^2-e^2 x^2\right )} \left (\frac {2 (d+5 e x)}{(d+e x)^{5/2}}-\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {d+e x}}{\sqrt {d^2-e^2 x^2}}\right )}{\sqrt {d} \sqrt {d^2-e^2 x^2}}\right )}{8 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 176, normalized size = 1.27
method | result | size |
default | \(-\frac {\sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, c \left (3 \arctanh \left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {2}\, c \,e^{2} x^{2}+6 \arctanh \left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {2}\, c d e x +3 \arctanh \left (\frac {\sqrt {c \left (-e x +d \right )}\, \sqrt {2}}{2 \sqrt {c d}}\right ) \sqrt {2}\, c \,d^{2}-10 e x \sqrt {c d}\, \sqrt {c \left (-e x +d \right )}-2 \sqrt {c \left (-e x +d \right )}\, \sqrt {c d}\, d \right )}{8 \left (e x +d \right )^{\frac {5}{2}} \sqrt {c \left (-e x +d \right )}\, e \sqrt {c d}}\) | \(176\) |
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 = 2.80, size = 362, normalized size = 2.60 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{2}} {\left (c x^{3} e^{3} + 3 \, c d x^{2} e^{2} + 3 \, c d^{2} x e + c d^{3}\right )} \sqrt {\frac {c}{d}} \log \left (-\frac {c x^{2} e^{2} - 2 \, c d x e - 3 \, c d^{2} + 4 \, \sqrt {\frac {1}{2}} \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d} d \sqrt {\frac {c}{d}}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, \sqrt {-c x^{2} e^{2} + c d^{2}} {\left (5 \, c x e + c d\right )} \sqrt {x e + d}}{8 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} e\right )}}, -\frac {3 \, \sqrt {\frac {1}{2}} {\left (c x^{3} e^{3} + 3 \, c d x^{2} e^{2} + 3 \, c d^{2} x e + c d^{3}\right )} \sqrt {-\frac {c}{d}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d} d \sqrt {-\frac {c}{d}}}{c x^{2} e^{2} - c d^{2}}\right ) - \sqrt {-c x^{2} e^{2} + c d^{2}} {\left (5 \, c x e + c d\right )} \sqrt {x e + d}}{4 \, {\left (x^{3} e^{4} + 3 \, d x^{2} e^{3} + 3 \, d^{2} x e^{2} + d^{3} 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 {9}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.54, size = 104, normalized size = 0.75 \begin {gather*} \frac {1}{8} \, {\left (\frac {3 \, \sqrt {2} c^{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 {2 \, {\left (6 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{3} d - 5 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{2}\right )}}{{\left (x e + d\right )}^{2} c^{2}}\right )} e^{\left (-1\right )} \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 )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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