Optimal. Leaf size=127 \[ \frac {2 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{3 e^5}-\frac {8 c d \left (c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^5}-\frac {8 c^2 d (d+e x)^{9/2}}{9 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5} \]
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Rubi [A]
time = 0.03, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {711}
\begin {gather*} \frac {4 c (d+e x)^{7/2} \left (a e^2+3 c d^2\right )}{7 e^5}-\frac {8 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )}{5 e^5}+\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}{3 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5}-\frac {8 c^2 d (d+e x)^{9/2}}{9 e^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (a+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^4}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^{3/2}}{e^4}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^{5/2}}{e^4}-\frac {4 c^2 d (d+e x)^{7/2}}{e^4}+\frac {c^2 (d+e x)^{9/2}}{e^4}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{3 e^5}-\frac {8 c d \left (c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^5}-\frac {8 c^2 d (d+e x)^{9/2}}{9 e^5}+\frac {2 c^2 (d+e x)^{11/2}}{11 e^5}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 96, normalized size = 0.76 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (1155 a^2 e^4+66 a c e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 108, normalized size = 0.85
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 c^{2} e^{4} x^{4}-280 c^{2} d \,x^{3} e^{3}+990 a c \,e^{4} x^{2}+240 d^{2} e^{2} x^{2} c^{2}-792 a c d \,e^{3} x -192 c^{2} d^{3} e x +1155 a^{2} e^{4}+528 a c \,d^{2} e^{2}+128 c^{2} d^{4}\right )}{3465 e^{5}}\) | \(106\) |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {8 \left (e^{2} a +c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\) | \(108\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {8 \left (e^{2} a +c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\) | \(108\) |
trager | \(\frac {2 \left (315 c^{2} e^{5} x^{5}+35 c^{2} d \,e^{4} x^{4}+990 a c \,e^{5} x^{3}-40 c^{2} d^{2} e^{3} x^{3}+198 a c d \,e^{4} x^{2}+48 d^{3} e^{2} x^{2} c^{2}+1155 a^{2} e^{5} x -264 a c \,d^{2} e^{3} x -64 d^{4} e x \,c^{2}+1155 a^{2} d \,e^{4}+528 a c \,d^{3} e^{2}+128 c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\) | \(143\) |
risch | \(\frac {2 \left (315 c^{2} e^{5} x^{5}+35 c^{2} d \,e^{4} x^{4}+990 a c \,e^{5} x^{3}-40 c^{2} d^{2} e^{3} x^{3}+198 a c d \,e^{4} x^{2}+48 d^{3} e^{2} x^{2} c^{2}+1155 a^{2} e^{5} x -264 a c \,d^{2} e^{3} x -64 d^{4} e x \,c^{2}+1155 a^{2} d \,e^{4}+528 a c \,d^{3} e^{2}+128 c^{2} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 113, normalized size = 0.89 \begin {gather*} \frac {2}{3465} \, {\left (315 \, {\left (x e + d\right )}^{\frac {11}{2}} c^{2} - 1540 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{2} d + 990 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} {\left (x e + d\right )}^{\frac {7}{2}} - 2772 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 1155 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}}\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.16, size = 132, normalized size = 1.04 \begin {gather*} -\frac {2}{3465} \, {\left (64 \, c^{2} d^{4} x e - 128 \, c^{2} d^{5} - 15 \, {\left (21 \, c^{2} x^{5} + 66 \, a c x^{3} + 77 \, a^{2} x\right )} e^{5} - {\left (35 \, c^{2} d x^{4} + 198 \, a c d x^{2} + 1155 \, a^{2} d\right )} e^{4} + 8 \, {\left (5 \, c^{2} d^{2} x^{3} + 33 \, a c d^{2} x\right )} e^{3} - 48 \, {\left (c^{2} d^{3} x^{2} + 11 \, a c d^{3}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.37, size = 148, normalized size = 1.17 \begin {gather*} \frac {2 \left (- \frac {4 c^{2} d \left (d + e x\right )^{\frac {9}{2}}}{9 e^{4}} + \frac {c^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (2 a c e^{2} + 6 c^{2} d^{2}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 4 a c d e^{2} - 4 c^{2} d^{3}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}{3 e^{4}}\right )}{e} \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. 290 vs.
\(2 (104) = 208\).
time = 2.69, size = 290, normalized size = 2.28 \begin {gather*} \frac {2}{3465} \, {\left (462 \, {\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 e^{\left (-2\right )} + 11 \, {\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 e^{\left (-4\right )} + 198 \, {\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 e^{\left (-2\right )} + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} c^{2} e^{\left (-4\right )} + 3465 \, \sqrt {x e + d} a^{2} d + 1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{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 = 0.04, size = 114, normalized size = 0.90 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}-\frac {\left (8\,c^2\,d^3+8\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {\left (12\,c^2\,d^2+4\,a\,c\,e^2\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}-\frac {8\,c^2\,d\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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