Optimal. Leaf size=178 \[ -\frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{16 \sqrt {2} d^{3/2} e}-\frac {c \sqrt {c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}+\frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {663, 673, 661, 208} \[ -\frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{16 \sqrt {2} d^{3/2} e}-\frac {c \sqrt {c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}+\frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}} \]
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)^{11/2}} \, dx &=-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}-\frac {1}{2} c \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{7/2}} \, dx\\ &=\frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}+\frac {1}{8} c^2 \int \frac {1}{(d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}} \, dx\\ &=\frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}+\frac {c^2 \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx}{32 d}\\ &=\frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}+\frac {\left (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 )}{16 d}\\ &=\frac {c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{5/2}}-\frac {c \sqrt {c d^2-c e^2 x^2}}{16 d e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}-\frac {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 )}{16 \sqrt {2} d^{3/2} e}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 134, normalized size = 0.75 \[ \frac {\left (c \left (d^2-e^2 x^2\right )\right )^{3/2} \left (-\frac {2 \sqrt {d} \left (7 d^2-22 d e x+3 e^2 x^2\right )}{(d-e x) (d+e x)^{9/2}}-\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{\left (d^2-e^2 x^2\right )^{3/2}}\right )}{96 d^{3/2} e} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 444, normalized size = 2.49 \[ \left [\frac {3 \, \sqrt {\frac {1}{2}} {\left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4}\right )} \sqrt {\frac {c}{d}} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 4 \, \sqrt {\frac {1}{2}} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {\frac {c}{d}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (3 \, c e^{2} x^{2} - 22 \, c d e x + 7 \, c d^{2}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{96 \, {\left (d e^{5} x^{4} + 4 \, d^{2} e^{4} x^{3} + 6 \, d^{3} e^{3} x^{2} + 4 \, d^{4} e^{2} x + d^{5} e\right )}}, -\frac {3 \, \sqrt {\frac {1}{2}} {\left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + 6 \, c d^{2} e^{2} x^{2} + 4 \, c d^{3} e x + c d^{4}\right )} \sqrt {-\frac {c}{d}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {-\frac {c}{d}}}{c e^{2} x^{2} - c d^{2}}\right ) + {\left (3 \, c e^{2} x^{2} - 22 \, c d e x + 7 \, c d^{2}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{48 \, {\left (d e^{5} x^{4} + 4 \, d^{2} e^{4} x^{3} + 6 \, d^{3} e^{3} x^{2} + 4 \, d^{4} e^{2} x + d^{5} e\right )}}\right ] \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 259, normalized size = 1.46 \[ -\frac {\sqrt {-\left (e^{2} x^{2}-d^{2}\right ) c}\, \left (3 \sqrt {2}\, c \,e^{3} x^{3} \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+9 \sqrt {2}\, c d \,e^{2} x^{2} \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+9 \sqrt {2}\, c \,d^{2} e x \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+3 \sqrt {2}\, c \,d^{3} \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+6 \sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\, e^{2} x^{2}-44 \sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\, d e x +14 \sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\, d^{2}\right ) c}{96 \left (e x +d \right )^{\frac {7}{2}} \sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\, d e} \]
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 \[ \int \frac {{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,d^2-c\,e^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]
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 \[ \int \frac {\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {11}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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