Optimal. Leaf size=119 \[ -\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}{3 e^4}+\frac {6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}{5 e^4}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}{7 e^4}+\frac {2 c^3 d^3 (d+e x)^{9/2}}{9 e^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 119, 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 {6 c^2 d^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^4}+\frac {6 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^4}-\frac {2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}{3 e^4}+\frac {2 c^3 d^3 (d+e x)^{9/2}}{9 e^4} \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 )^3}{(d+e x)^{5/2}} \, dx &=\int (a e+c d x)^3 \sqrt {d+e x} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3 \sqrt {d+e x}}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{e^3}+\frac {c^3 d^3 (d+e x)^{7/2}}{e^3}\right ) \, dx\\ &=-\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}{3 e^4}+\frac {6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}{5 e^4}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}{7 e^4}+\frac {2 c^3 d^3 (d+e x)^{9/2}}{9 e^4}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 111, normalized size = 0.93 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (105 a^3 e^6-63 a^2 c d e^4 (2 d-3 e x)+9 a c^2 d^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )}{315 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.83, size = 97, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{4}}\) | \(97\) |
default | \(\frac {\frac {2 c^{3} d^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {6 \left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 \left (e^{2} a -c \,d^{2}\right )^{2} c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{4}}\) | \(97\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (35 c^{3} d^{3} e^{3} x^{3}+135 a \,c^{2} d^{2} e^{4} x^{2}-30 c^{3} d^{4} e^{2} x^{2}+189 a^{2} c d \,e^{5} x -108 a \,c^{2} d^{3} e^{3} x +24 c^{3} d^{5} e x +105 e^{6} a^{3}-126 e^{4} d^{2} a^{2} c +72 d^{4} e^{2} c^{2} a -16 d^{6} c^{3}\right )}{315 e^{4}}\) | \(131\) |
trager | \(\frac {2 \left (35 c^{3} d^{3} e^{4} x^{4}+135 a \,c^{2} d^{2} e^{5} x^{3}+5 c^{3} d^{4} e^{3} x^{3}+189 a^{2} c d \,e^{6} x^{2}+27 a \,c^{2} d^{3} e^{4} x^{2}-6 c^{3} d^{5} e^{2} x^{2}+105 a^{3} e^{7} x +63 a^{2} c \,d^{2} e^{5} x -36 a \,c^{2} d^{4} e^{3} x +8 c^{3} d^{6} e x +105 a^{3} d \,e^{6}-126 a^{2} c \,d^{3} e^{4}+72 a \,c^{2} d^{5} e^{2}-16 c^{3} d^{7}\right ) \sqrt {e x +d}}{315 e^{4}}\) | \(185\) |
risch | \(\frac {2 \left (35 c^{3} d^{3} e^{4} x^{4}+135 a \,c^{2} d^{2} e^{5} x^{3}+5 c^{3} d^{4} e^{3} x^{3}+189 a^{2} c d \,e^{6} x^{2}+27 a \,c^{2} d^{3} e^{4} x^{2}-6 c^{3} d^{5} e^{2} x^{2}+105 a^{3} e^{7} x +63 a^{2} c \,d^{2} e^{5} x -36 a \,c^{2} d^{4} e^{3} x +8 c^{3} d^{6} e x +105 a^{3} d \,e^{6}-126 a^{2} c \,d^{3} e^{4}+72 a \,c^{2} d^{5} e^{2}-16 c^{3} d^{7}\right ) \sqrt {e x +d}}{315 e^{4}}\) | \(185\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 134, normalized size = 1.13 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} d^{3} - 135 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 189 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (x e + d\right )}^{\frac {5}{2}} - 105 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left (x e + d\right )}^{\frac {3}{2}}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.53, size = 173, normalized size = 1.45 \begin {gather*} \frac {2}{315} \, {\left (8 \, c^{3} d^{6} x e - 16 \, c^{3} d^{7} + 105 \, a^{3} x e^{7} + 21 \, {\left (9 \, a^{2} c d x^{2} + 5 \, a^{3} d\right )} e^{6} + 9 \, {\left (15 \, a c^{2} d^{2} x^{3} + 7 \, a^{2} c d^{2} x\right )} e^{5} + {\left (35 \, c^{3} d^{3} x^{4} + 27 \, a c^{2} d^{3} x^{2} - 126 \, a^{2} c d^{3}\right )} e^{4} + {\left (5 \, c^{3} d^{4} x^{3} - 36 \, a c^{2} d^{4} x\right )} e^{3} - 6 \, {\left (c^{3} d^{5} x^{2} - 12 \, a c^{2} d^{5}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-4\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. 644 vs.
\(2 (110) = 220\).
time = 42.23, size = 644, normalized size = 5.41 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{3} d^{2} e^{3}}{\sqrt {d + e x}} - 4 a^{3} d e^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 2 a^{3} e^{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 ) - 6 a^{2} c d^{3} e \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 12 a^{2} c d^{2} 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 ) - 6 a^{2} c d e \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 ) - \frac {6 a c^{2} d^{4} \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 {12 a c^{2} d^{3} \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} - \frac {6 a c^{2} d^{2} \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} - \frac {2 c^{3} d^{5} \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^{3}} - \frac {4 c^{3} d^{4} \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^{3}} - \frac {2 c^{3} d^{3} \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^{3}}}{e} & \text {for}\: e \neq 0 \\\frac {c^{3} d^{\frac {7}{2}} x^{4}}{4} & \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. 356 vs.
\(2 (100) = 200\).
time = 1.01, size = 356, normalized size = 2.99 \begin {gather*} \frac {2}{315} \, {\left (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 )} c^{3} d^{4} e^{\left (-3\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 c^{2} d^{3} e^{\left (-1\right )} + {\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^{3} d^{3} e^{\left (-3\right )} + 315 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} c 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 )} a c^{2} 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^{2} c d e + 315 \, \sqrt {x e + d} a^{3} d e^{3} + 105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{3} e^{3}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 106, normalized size = 0.89 \begin {gather*} \frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {6\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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