Optimal. Leaf size=83 \[ \frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}{5 e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{7/2}}{7 e^3}+\frac {2 c^2 d^2 (d+e x)^{9/2}}{9 e^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {640, 45}
\begin {gather*} -\frac {4 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^3}+\frac {2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^3}+\frac {2 c^2 d^2 (d+e x)^{9/2}}{9 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{\sqrt {d+e x}} \, dx &=\int (a e+c d x)^2 (d+e x)^{3/2} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2 (d+e x)^{3/2}}{e^2}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{e^2}+\frac {c^2 d^2 (d+e x)^{7/2}}{e^2}\right ) \, dx\\ &=\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}{5 e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{7/2}}{7 e^3}+\frac {2 c^2 d^2 (d+e x)^{9/2}}{9 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 67, normalized size = 0.81 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (63 a^2 e^4+18 a c d e^2 (-2 d+5 e x)+c^2 d^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 68, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (e^{2} a -c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{3}}\) | \(68\) |
default | \(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (e^{2} a -c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{3}}\) | \(68\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (35 e^{2} x^{2} c^{2} d^{2}+90 a c d \,e^{3} x -20 c^{2} d^{3} e x +63 a^{2} e^{4}-36 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{315 e^{3}}\) | \(73\) |
trager | \(\frac {2 \left (35 c^{2} d^{2} e^{4} x^{4}+90 a c d \,e^{5} x^{3}+50 c^{2} d^{3} e^{3} x^{3}+63 a^{2} e^{6} x^{2}+144 a c \,d^{2} e^{4} x^{2}+3 c^{2} d^{4} e^{2} x^{2}+126 a^{2} d \,e^{5} x +18 a c \,d^{3} e^{3} x -4 c^{2} d^{5} e x +63 a^{2} d^{2} e^{4}-36 a \,d^{4} e^{2} c +8 d^{6} c^{2}\right ) \sqrt {e x +d}}{315 e^{3}}\) | \(151\) |
risch | \(\frac {2 \left (35 c^{2} d^{2} e^{4} x^{4}+90 a c d \,e^{5} x^{3}+50 c^{2} d^{3} e^{3} x^{3}+63 a^{2} e^{6} x^{2}+144 a c \,d^{2} e^{4} x^{2}+3 c^{2} d^{4} e^{2} x^{2}+126 a^{2} d \,e^{5} x +18 a c \,d^{3} e^{3} x -4 c^{2} d^{5} e x +63 a^{2} d^{2} e^{4}-36 a \,d^{4} e^{2} c +8 d^{6} c^{2}\right ) \sqrt {e x +d}}{315 e^{3}}\) | \(151\) |
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. 289 vs.
\(2 (69) = 138\).
time = 0.29, size = 289, normalized size = 3.48 \begin {gather*} \frac {2}{315} \, {\left ({\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^{2} d^{2} e^{\left (-2\right )} + 315 \, \sqrt {x e + d} a^{2} d^{2} e^{2} + 18 \, {\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 )} {\left (c d^{2} + a e^{2}\right )} c d e^{\left (-2\right )} + 42 \, {\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 )} + 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 )} a d e + 21 \, {\left (c d^{2} + a e^{2}\right )}^{2} {\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 )} e^{\left (-2\right )}\right )} e^{\left (-1\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. 142 vs.
\(2 (69) = 138\).
time = 2.59, size = 142, normalized size = 1.71 \begin {gather*} -\frac {2}{315} \, {\left (4 \, c^{2} d^{5} x e - 8 \, c^{2} d^{6} - 63 \, a^{2} x^{2} e^{6} - 18 \, {\left (5 \, a c d x^{3} + 7 \, a^{2} d x\right )} e^{5} - {\left (35 \, c^{2} d^{2} x^{4} + 144 \, a c d^{2} x^{2} + 63 \, a^{2} d^{2}\right )} e^{4} - 2 \, {\left (25 \, c^{2} d^{3} x^{3} + 9 \, a c d^{3} x\right )} e^{3} - 3 \, {\left (c^{2} d^{4} x^{2} - 12 \, a c d^{4}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-3\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. 631 vs.
\(2 (76) = 152\).
time = 43.42, size = 631, normalized size = 7.60 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{2} d^{3} e^{2}}{\sqrt {d + e x}} - 6 a^{2} d^{2} e^{2} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 6 a^{2} d e^{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 ) - 2 a^{2} e^{2} \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 ) - 4 a c d^{4} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 12 a c d^{3} \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 ) - 12 a c d^{2} \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 ) - 4 a c d \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right ) - \frac {2 c^{2} d^{5} \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^{2}} - \frac {6 c^{2} d^{4} \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^{2}} - \frac {6 c^{2} d^{3} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 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^{2} d^{2} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {c^{2} d^{\frac {7}{2}} x^{3}}{3} & \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. 388 vs.
\(2 (69) = 138\).
time = 2.02, size = 388, normalized size = 4.67 \begin {gather*} \frac {2}{315} \, {\left (21 \, {\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^{2} d^{4} e^{\left (-2\right )} + 18 \, {\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^{2} d^{3} e^{\left (-2\right )} + 210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a c d^{3} + {\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^{2} d^{2} e^{\left (-2\right )} + 315 \, \sqrt {x e + d} a^{2} d^{2} e^{2} + 84 \, {\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 c d^{2} + 210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} d e^{2} + 18 \, {\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 c d + 21 \, {\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^{2} e^{2}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 80, normalized size = 0.96 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (63\,a^2\,e^4+63\,c^2\,d^4+35\,c^2\,d^2\,{\left (d+e\,x\right )}^2-90\,c^2\,d^3\,\left (d+e\,x\right )-126\,a\,c\,d^2\,e^2+90\,a\,c\,d\,e^2\,\left (d+e\,x\right )\right )}{315\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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