Optimal. Leaf size=75 \[ \frac {2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}{5 e^3}-\frac {2 (2 c d-b e) (d+e x)^{7/2}}{7 e^3}+\frac {2 c (d+e x)^{9/2}}{9 e^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712}
\begin {gather*} \frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}{5 e^3}-\frac {2 (d+e x)^{7/2} (2 c d-b e)}{7 e^3}+\frac {2 c (d+e x)^{9/2}}{9 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int (d+e x)^{3/2} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}{e^2}+\frac {(-2 c d+b e) (d+e x)^{5/2}}{e^2}+\frac {c (d+e x)^{7/2}}{e^2}\right ) \, dx\\ &=\frac {2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}{5 e^3}-\frac {2 (2 c d-b e) (d+e x)^{7/2}}{7 e^3}+\frac {2 c (d+e x)^{9/2}}{9 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 55, normalized size = 0.73 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (9 e (-2 b d+7 a e+5 b e x)+c \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 = 0.48, size = 59, normalized size = 0.79
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (35 x^{2} c \,e^{2}+45 b \,e^{2} x -20 c d e x +63 e^{2} a -18 b d e +8 c \,d^{2}\right )}{315 e^{3}}\) | \(53\) |
derivativedivides | \(\frac {\frac {2 c \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{3}}\) | \(59\) |
default | \(\frac {\frac {2 c \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{3}}\) | \(59\) |
trager | \(\frac {2 \left (35 c \,e^{4} x^{4}+45 b \,e^{4} x^{3}+50 d \,e^{3} c \,x^{3}+63 a \,e^{4} x^{2}+72 b d \,e^{3} x^{2}+3 c \,d^{2} e^{2} x^{2}+126 a d \,e^{3} x +9 b \,d^{2} e^{2} x -4 c \,d^{3} e x +63 a \,d^{2} e^{2}-18 b \,d^{3} e +8 c \,d^{4}\right ) \sqrt {e x +d}}{315 e^{3}}\) | \(121\) |
risch | \(\frac {2 \left (35 c \,e^{4} x^{4}+45 b \,e^{4} x^{3}+50 d \,e^{3} c \,x^{3}+63 a \,e^{4} x^{2}+72 b d \,e^{3} x^{2}+3 c \,d^{2} e^{2} x^{2}+126 a d \,e^{3} x +9 b \,d^{2} e^{2} x -4 c \,d^{3} e x +63 a \,d^{2} e^{2}-18 b \,d^{3} e +8 c \,d^{4}\right ) \sqrt {e x +d}}{315 e^{3}}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 62, normalized size = 0.83 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} c - 45 \, {\left (2 \, c d - b e\right )} {\left (x e + d\right )}^{\frac {7}{2}} + 63 \, {\left (c d^{2} - b d e + a e^{2}\right )} {\left (x e + d\right )}^{\frac {5}{2}}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.58, size = 109, normalized size = 1.45 \begin {gather*} \frac {2}{315} \, {\left (8 \, c d^{4} + {\left (35 \, c x^{4} + 45 \, b x^{3} + 63 \, a x^{2}\right )} e^{4} + 2 \, {\left (25 \, c d x^{3} + 36 \, b d x^{2} + 63 \, a d x\right )} e^{3} + 3 \, {\left (c d^{2} x^{2} + 3 \, b d^{2} x + 21 \, a d^{2}\right )} e^{2} - 2 \, {\left (2 \, c d^{3} x + 9 \, b d^{3}\right )} e\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 [A]
time = 5.72, size = 230, normalized size = 3.07 \begin {gather*} a d \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 ) + \frac {2 a \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {2 b d \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 {2 b \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^{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^{3}} + \frac {2 c \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^{3}} \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. 365 vs.
\(2 (64) = 128\).
time = 1.41, size = 365, normalized size = 4.87 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} b d^{2} e^{\left (-1\right )} + 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 d^{2} e^{\left (-2\right )} + 42 \, {\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 )} b d e^{\left (-1\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 d e^{\left (-2\right )} + 315 \, \sqrt {x e + d} a d^{2} + 210 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a d + 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 )} b 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 e^{\left (-2\right )} + 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\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 58, normalized size = 0.77 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (35\,c\,{\left (d+e\,x\right )}^2+63\,a\,e^2+63\,c\,d^2+45\,b\,e\,\left (d+e\,x\right )-90\,c\,d\,\left (d+e\,x\right )-63\,b\,d\,e\right )}{315\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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