Optimal. Leaf size=217 \[ -\frac {3 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{256 \sqrt {2} d^{5/2} e}-\frac {3 c \sqrt {c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}+\frac {c \sqrt {c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {663, 673, 661, 208} \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 )}{256 \sqrt {2} d^{5/2} e}-\frac {3 c \sqrt {c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}+\frac {c \sqrt {c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rule 663
Rule 673
Rubi steps
\begin {align*} \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx &=-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}-\frac {1}{8} (3 c) \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{9/2}} \, dx\\ &=\frac {c \sqrt {c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac {1}{16} c^2 \int \frac {1}{(d+e x)^{5/2} \sqrt {c d^2-c e^2 x^2}} \, dx\\ &=\frac {c \sqrt {c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac {\left (3 c^2\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}} \, dx}{128 d}\\ &=\frac {c \sqrt {c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}-\frac {3 c \sqrt {c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx}{512 d^2}\\ &=\frac {c \sqrt {c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}-\frac {3 c \sqrt {c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac {\left (3 c^2 e\right ) \operatorname {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 )}{256 d^2}\\ &=\frac {c \sqrt {c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}-\frac {3 c \sqrt {c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}-\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 )}{256 \sqrt {2} d^{5/2} e}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 145, normalized size = 0.67 \begin {gather*} \frac {\left (c \left (d^2-e^2 x^2\right )\right )^{3/2} \left (-\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{\left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 \sqrt {d} \left (39 d^3-79 d^2 e x+13 d e^2 x^2+3 e^3 x^3\right )}{(d-e x) (d+e x)^{11/2}}\right )}{512 d^{5/2} e} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.10, size = 161, normalized size = 0.74 \begin {gather*} \frac {3 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {2 c d (d+e x)-c (d+e x)^2}}{\sqrt {c} (e x-d) \sqrt {d+e x}}\right )}{256 \sqrt {2} d^{5/2} e}-\frac {c \left (128 d^3-96 d^2 (d+e x)+4 d (d+e x)^2+3 (d+e x)^3\right ) \sqrt {2 c d (d+e x)-c (d+e x)^2}}{256 d^2 e (d+e x)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 518, normalized size = 2.39 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{2}} {\left (c e^{5} x^{5} + 5 \, c d e^{4} x^{4} + 10 \, c d^{2} e^{3} x^{3} + 10 \, c d^{3} e^{2} x^{2} + 5 \, c d^{4} e x + c d^{5}\right )} \sqrt {\frac {c}{d}} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 4 \, \sqrt {\frac {1}{2}} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {\frac {c}{d}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (3 \, c e^{3} x^{3} + 13 \, c d e^{2} x^{2} - 79 \, c d^{2} e x + 39 \, c d^{3}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{512 \, {\left (d^{2} e^{6} x^{5} + 5 \, d^{3} e^{5} x^{4} + 10 \, d^{4} e^{4} x^{3} + 10 \, d^{5} e^{3} x^{2} + 5 \, d^{6} e^{2} x + d^{7} e\right )}}, -\frac {3 \, \sqrt {\frac {1}{2}} {\left (c e^{5} x^{5} + 5 \, c d e^{4} x^{4} + 10 \, c d^{2} e^{3} x^{3} + 10 \, c d^{3} e^{2} x^{2} + 5 \, c d^{4} e x + c d^{5}\right )} \sqrt {-\frac {c}{d}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {-\frac {c}{d}}}{c e^{2} x^{2} - c d^{2}}\right ) + {\left (3 \, c e^{3} x^{3} + 13 \, c d e^{2} x^{2} - 79 \, c d^{2} e x + 39 \, c d^{3}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{256 \, {\left (d^{2} e^{6} x^{5} + 5 \, d^{3} e^{5} x^{4} + 10 \, d^{4} e^{4} x^{3} + 10 \, d^{5} e^{3} x^{2} + 5 \, d^{6} e^{2} x + d^{7} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 325, normalized size = 1.50 \begin {gather*} -\frac {\sqrt {-\left (e^{2} x^{2}-d^{2}\right ) c}\, \left (3 \sqrt {2}\, c \,e^{4} x^{4} \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+12 \sqrt {2}\, c d \,e^{3} x^{3} \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+18 \sqrt {2}\, c \,d^{2} e^{2} x^{2} \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+12 \sqrt {2}\, c \,d^{3} e x \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+3 \sqrt {2}\, c \,d^{4} \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+6 \sqrt {c d}\, \sqrt {-\left (e x -d \right ) c}\, e^{3} x^{3}+26 \sqrt {c d}\, \sqrt {-\left (e x -d \right ) c}\, d \,e^{2} x^{2}-158 \sqrt {c d}\, \sqrt {-\left (e x -d \right ) c}\, d^{2} e x +78 \sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\, d^{3}\right ) c}{512 \left (e x +d \right )^{\frac {9}{2}} \sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\, d^{2} e} \end {gather*}
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*} \int \frac {{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {13}{2}}}\,{d x} \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 = -0.00 \begin {gather*} \int \frac {{\left (c\,d^2-c\,e^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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