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 = 0.03, 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 = {657, 649} \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 649
Rule 657
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 = 0.05, 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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IntegrateAlgebraic [A] time = 5.92, size = 57, normalized size = 0.77 \begin {gather*} \frac {2 (e x-3 d) \sqrt {2 c d (d+e x)-c (d+e x)^2}}{c^2 e (e x-d) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 55, normalized size = 0.74 \begin {gather*} \frac {2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} {\left (e x - 3 \, d\right )}}{c^{2} e^{3} x^{2} - c^{2} d^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 44, normalized size = 0.59 \begin {gather*} \frac {2 \left (-e x +d \right ) \left (-e x +3 d \right ) \left (e x +d \right )^{\frac {3}{2}}}{\left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.52, size = 23, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (e x - 3 \, d\right )}}{\sqrt {-e x + d} c^{\frac {3}{2}} e} \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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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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