Optimal. Leaf size=43 \[ \frac {2}{5} (d+e x)^{5/2} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{7/2}}{7 e^2} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 43} \begin {gather*} \frac {2}{5} (d+e x)^{5/2} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{7/2}}{7 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx &=\int (a e+c d x) (d+e x)^{3/2} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^{3/2}}{e}+\frac {c d (d+e x)^{5/2}}{e}\right ) \, dx\\ &=\frac {2}{5} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{5/2}+\frac {2 c d (d+e x)^{7/2}}{7 e^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 0.79 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (7 a e^2+c d (5 e x-2 d)\right )}{35 e^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 38, normalized size = 0.88 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (7 a e^2-7 c d^2+5 c d (d+e x)\right )}{35 e^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 74, normalized size = 1.72 \begin {gather*} \frac {2 \, {\left (5 \, c d e^{3} x^{3} - 2 \, c d^{4} + 7 \, a d^{2} e^{2} + {\left (8 \, c d^{2} e^{2} + 7 \, a e^{4}\right )} x^{2} + {\left (c d^{3} e + 14 \, a d e^{3}\right )} x\right )} \sqrt {e x + d}}{35 \, e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 212, normalized size = 4.93 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} c d^{3} e^{\left (-1\right )} + 14 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} c d^{2} e^{\left (-1\right )} + 105 \, \sqrt {x e + d} a d^{2} e + 70 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a d e + 3 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c d e^{\left (-1\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a e\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 0.74 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (5 c d e x +7 a \,e^{2}-2 c \,d^{2}\right )}{35 e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 38, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} c d - 7 \, {\left (c d^{2} - a e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{35 \, e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 34, normalized size = 0.79 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (7\,a\,e^2-7\,c\,d^2+5\,c\,d\,\left (d+e\,x\right )\right )}{35\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.64, size = 41, normalized size = 0.95 \begin {gather*} \frac {2 \left (\frac {c d \left (d + e x\right )^{\frac {7}{2}}}{7 e} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a e^{2} - c d^{2}\right )}{5 e}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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