Optimal. Leaf size=74 \[ \frac {8 d \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {c d^2-c e^2 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663}
\begin {gather*} \frac {8 d \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {c d^2-c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 663
Rule 671
Rubi steps
\begin {align*} \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^{3/2}}{c e \sqrt {c d^2-c e^2 x^2}}+(4 d) \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {8 d \sqrt {d+e x}}{c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {c d^2-c e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 43, normalized size = 0.58 \begin {gather*} \frac {2 (3 d-e x) \sqrt {d+e x}}{c e \sqrt {c \left (d^2-e^2 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 48, normalized size = 0.65
method | result | size |
gosper | \(\frac {2 \left (-e x +d \right ) \left (-e x +3 d \right ) \left (e x +d \right )^{\frac {3}{2}}}{e \left (-x^{2} c \,e^{2}+c \,d^{2}\right )^{\frac {3}{2}}}\) | \(44\) |
default | \(\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (-e x +3 d \right )}{\sqrt {e x +d}\, c^{2} \left (-e x +d \right ) e}\) | \(48\) |
risch | \(\frac {2 \left (-e x +d \right ) \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}}{e \sqrt {-c \left (e x -d \right )}\, \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, c}+\frac {4 d \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}}{e \sqrt {c \left (-e x +d \right )}\, \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, c}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 24, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (x e - 3 \, d\right )} e^{\left (-1\right )}}{\sqrt {-x e + d} c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.81, size = 56, normalized size = 0.76 \begin {gather*} \frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2}} \sqrt {x e + d} {\left (x e - 3 \, d\right )}}{c^{2} x^{2} e^{3} - c^{2} d^{2} e} \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 (d + e x\right )^{\frac {5}{2}}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 66, normalized size = 0.89 \begin {gather*} -\frac {4 \, \sqrt {2} d e^{\left (-1\right )}}{\sqrt {c d} c} + \frac {2 \, {\left (\frac {2 \, d e^{\left (-1\right )}}{\sqrt {-{\left (x e + d\right )} c + 2 \, c d}} + \frac {\sqrt {-{\left (x e + d\right )} c + 2 \, c d} e^{\left (-1\right )}}{c}\right )}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 66, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {6\,d\,\sqrt {d+e\,x}}{c^2\,e^3}-\frac {2\,x\,\sqrt {d+e\,x}}{c^2\,e^2}\right )}{x^2-\frac {d^2}{e^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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