Optimal. Leaf size=43 \[ \frac {2}{3} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{3/2}+\frac {2 c d (d+e x)^{5/2}}{5 e^2} \]
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Rubi [A]
time = 0.01, 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 = {640, 45}
\begin {gather*} \frac {2}{3} (d+e x)^{3/2} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{5/2}}{5 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{\sqrt {d+e x}} \, dx &=\int (a e+c d x) \sqrt {d+e x} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right ) \sqrt {d+e x}}{e}+\frac {c d (d+e x)^{3/2}}{e}\right ) \, dx\\ &=\frac {2}{3} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{3/2}+\frac {2 c d (d+e x)^{5/2}}{5 e^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 34, normalized size = 0.79 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (5 a e^2+c d (-2 d+3 e x)\right )}{15 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 39, normalized size = 0.91
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3 c d e x +5 e^{2} a -2 c \,d^{2}\right )}{15 e^{2}}\) | \(32\) |
derivativedivides | \(\frac {\frac {2 c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{2}}\) | \(39\) |
default | \(\frac {\frac {2 c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{2}}\) | \(39\) |
trager | \(\frac {2 \left (3 c d \,e^{2} x^{2}+5 a \,e^{3} x +c \,d^{2} e x +5 a d \,e^{2}-2 c \,d^{3}\right ) \sqrt {e x +d}}{15 e^{2}}\) | \(51\) |
risch | \(\frac {2 \left (3 c d \,e^{2} x^{2}+5 a \,e^{3} x +c \,d^{2} e x +5 a d \,e^{2}-2 c \,d^{3}\right ) \sqrt {e x +d}}{15 e^{2}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs.
\(2 (36) = 72\).
time = 0.28, size = 93, normalized size = 2.16 \begin {gather*} \frac {2}{15} \, {\left ({\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 e^{\left (-1\right )} + 15 \, \sqrt {x e + d} a d e + 5 \, {\left (c d^{2} + a e^{2}\right )} {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} e^{\left (-1\right )}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.42, size = 48, normalized size = 1.12 \begin {gather*} \frac {2}{15} \, {\left (c d^{2} x e - 2 \, c d^{3} + 5 \, a x e^{3} + {\left (3 \, c d x^{2} + 5 \, a d\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs.
\(2 (42) = 84\).
time = 11.41, size = 221, normalized size = 5.14 \begin {gather*} \begin {cases} \frac {- \frac {2 a d^{2} e}{\sqrt {d + e x}} - 4 a d e \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 2 a e \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right ) - \frac {2 c d^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {4 c d^{2} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {2 c d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e}}{e} & \text {for}\: e \neq 0 \\\frac {c d^{\frac {3}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs.
\(2 (36) = 72\).
time = 0.97, size = 112, normalized size = 2.60 \begin {gather*} \frac {2}{15} \, {\left (5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} c d^{2} e^{\left (-1\right )} + {\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 e^{\left (-1\right )} + 15 \, \sqrt {x e + d} a d e + 5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a e\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 34, normalized size = 0.79 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (5\,a\,e^2-5\,c\,d^2+3\,c\,d\,\left (d+e\,x\right )\right )}{15\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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