Optimal. Leaf size=43 \[ \frac {2}{7} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{7/2}+\frac {2 c d (d+e x)^{9/2}}{9 e^2} \]
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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 = {640, 45}
\begin {gather*} \frac {2}{7} (d+e x)^{7/2} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{9/2}}{9 e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \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)^{5/2} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^{5/2}}{e}+\frac {c d (d+e x)^{7/2}}{e}\right ) \, dx\\ &=\frac {2}{7} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{7/2}+\frac {2 c d (d+e x)^{9/2}}{9 e^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 34, normalized size = 0.79 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (9 a e^2+c d (-2 d+7 e x)\right )}{63 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 39, normalized size = 0.91
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (7 c d e x +9 e^{2} a -2 c \,d^{2}\right )}{63 e^{2}}\) | \(32\) |
derivativedivides | \(\frac {\frac {2 c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{2}}\) | \(39\) |
default | \(\frac {\frac {2 c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{2}}\) | \(39\) |
trager | \(\frac {2 \left (7 c d \,e^{4} x^{4}+9 a \,e^{5} x^{3}+19 c \,d^{2} e^{3} x^{3}+27 a d \,e^{4} x^{2}+15 c \,d^{3} e^{2} x^{2}+27 a \,e^{3} d^{2} x +c \,d^{4} e x +9 a \,e^{2} d^{3}-2 c \,d^{5}\right ) \sqrt {e x +d}}{63 e^{2}}\) | \(99\) |
risch | \(\frac {2 \left (7 c d \,e^{4} x^{4}+9 a \,e^{5} x^{3}+19 c \,d^{2} e^{3} x^{3}+27 a d \,e^{4} x^{2}+15 c \,d^{3} e^{2} x^{2}+27 a \,e^{3} d^{2} x +c \,d^{4} e x +9 a \,e^{2} d^{3}-2 c \,d^{5}\right ) \sqrt {e x +d}}{63 e^{2}}\) | \(99\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 38, normalized size = 0.88 \begin {gather*} \frac {2}{63} \, {\left (7 \, {\left (x e + d\right )}^{\frac {9}{2}} c d - 9 \, {\left (c d^{2} - a e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs.
\(2 (36) = 72\).
time = 3.66, size = 93, normalized size = 2.16 \begin {gather*} \frac {2}{63} \, {\left (c d^{4} x e - 2 \, c d^{5} + 9 \, a x^{3} e^{5} + {\left (7 \, c d x^{4} + 27 \, a d x^{2}\right )} e^{4} + {\left (19 \, c d^{2} x^{3} + 27 \, a d^{2} x\right )} e^{3} + 3 \, {\left (5 \, c d^{3} x^{2} + 3 \, a d^{3}\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 [A]
time = 5.38, size = 235, normalized size = 5.47 \begin {gather*} a d^{2} e \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + 4 a d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right ) + 2 a \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right ) + \frac {2 c d^{3} \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}} + \frac {4 c d^{2} \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}} + \frac {2 c d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{2}} \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. 336 vs.
\(2 (36) = 72\).
time = 1.46, size = 336, normalized size = 7.81 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} c d^{4} e^{\left (-1\right )} + 63 \, {\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^{3} e^{\left (-1\right )} + 315 \, \sqrt {x e + d} a d^{3} e + 315 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a d^{2} e + 27 \, {\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^{2} e^{\left (-1\right )} + 63 \, {\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 d e + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c d e^{\left (-1\right )} + 9 \, {\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 )} 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 = 0.61, size = 34, normalized size = 0.79 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (9\,a\,e^2-9\,c\,d^2+7\,c\,d\,\left (d+e\,x\right )\right )}{63\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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