Optimal. Leaf size=119 \[ -\frac {6 c^2 d^2 (d+e x)^{11/2} \left (c d^2-a e^2\right )}{11 e^4}+\frac {2 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )^2}{3 e^4}-\frac {2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^3}{7 e^4}+\frac {2 c^3 d^3 (d+e x)^{13/2}}{13 e^4} \]
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Rubi [A] time = 0.06, 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 = {626, 43} \begin {gather*} -\frac {6 c^2 d^2 (d+e x)^{11/2} \left (c d^2-a e^2\right )}{11 e^4}+\frac {2 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )^2}{3 e^4}-\frac {2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^3}{7 e^4}+\frac {2 c^3 d^3 (d+e x)^{13/2}}{13 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
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}{\sqrt {d+e x}} \, dx &=\int (a e+c d x)^3 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3 (d+e x)^{5/2}}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{7/2}}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{9/2}}{e^3}+\frac {c^3 d^3 (d+e x)^{11/2}}{e^3}\right ) \, dx\\ &=-\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{7/2}}{7 e^4}+\frac {2 c d \left (c d^2-a e^2\right )^2 (d+e x)^{9/2}}{3 e^4}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{11/2}}{11 e^4}+\frac {2 c^3 d^3 (d+e x)^{13/2}}{13 e^4}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 98, normalized size = 0.82 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (-819 c^2 d^2 (d+e x)^2 \left (c d^2-a e^2\right )+1001 c d (d+e x) \left (c d^2-a e^2\right )^2-429 \left (c d^2-a e^2\right )^3+231 c^3 d^3 (d+e x)^3\right )}{3003 e^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 151, normalized size = 1.27 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (429 a^3 e^6-1287 a^2 c d^2 e^4+1001 a^2 c d e^4 (d+e x)+1287 a c^2 d^4 e^2-2002 a c^2 d^3 e^2 (d+e x)+819 a c^2 d^2 e^2 (d+e x)^2-429 c^3 d^6+1001 c^3 d^5 (d+e x)-819 c^3 d^4 (d+e x)^2+231 c^3 d^3 (d+e x)^3\right )}{3003 e^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 283, normalized size = 2.38 \begin {gather*} \frac {2 \, {\left (231 \, c^{3} d^{3} e^{6} x^{6} - 16 \, c^{3} d^{9} + 104 \, a c^{2} d^{7} e^{2} - 286 \, a^{2} c d^{5} e^{4} + 429 \, a^{3} d^{3} e^{6} + 63 \, {\left (9 \, c^{3} d^{4} e^{5} + 13 \, a c^{2} d^{2} e^{7}\right )} x^{5} + 7 \, {\left (53 \, c^{3} d^{5} e^{4} + 299 \, a c^{2} d^{3} e^{6} + 143 \, a^{2} c d e^{8}\right )} x^{4} + {\left (5 \, c^{3} d^{6} e^{3} + 1469 \, a c^{2} d^{4} e^{5} + 2717 \, a^{2} c d^{2} e^{7} + 429 \, a^{3} e^{9}\right )} x^{3} - 3 \, {\left (2 \, c^{3} d^{7} e^{2} - 13 \, a c^{2} d^{5} e^{4} - 715 \, a^{2} c d^{3} e^{6} - 429 \, a^{3} d e^{8}\right )} x^{2} + {\left (8 \, c^{3} d^{8} e - 52 \, a c^{2} d^{6} e^{3} + 143 \, a^{2} c d^{4} e^{5} + 1287 \, a^{3} d^{2} e^{7}\right )} x\right )} \sqrt {e x + d}}{3003 \, e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 929, normalized size = 7.81
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 131, normalized size = 1.10 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (231 c^{3} d^{3} e^{3} x^{3}+819 a \,c^{2} d^{2} e^{4} x^{2}-126 c^{3} d^{4} e^{2} x^{2}+1001 a^{2} c d \,e^{5} x -364 a \,c^{2} d^{3} e^{3} x +56 c^{3} d^{5} e x +429 a^{3} e^{6}-286 a^{2} c \,d^{2} e^{4}+104 a \,c^{2} d^{4} e^{2}-16 c^{3} d^{6}\right )}{3003 e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.23, size = 611, normalized size = 5.13 \begin {gather*} \frac {2 \, {\left (15015 \, \sqrt {e x + d} a^{3} d^{3} e^{3} + 3003 \, {\left (\frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c d}{e} + \frac {5 \, {\left (c d^{2} + a e^{2}\right )} {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )}}{e}\right )} a^{2} d^{2} e^{2} + \frac {5 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} c^{3} d^{3}}{e^{3}} + 143 \, {\left (\frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{2} d^{2}}{e^{2}} + \frac {18 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} {\left (c d^{2} + a e^{2}\right )} c d}{e^{2}} + \frac {21 \, {\left (c d^{2} + a e^{2}\right )}^{2} {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )}}{e^{2}}\right )} a d e + \frac {65 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} {\left (c d^{2} + a e^{2}\right )} c^{2} d^{2}}{e^{3}} + \frac {143 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} {\left (c d^{2} + a e^{2}\right )}^{2} c d}{e^{3}} + \frac {429 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} {\left (c d^{2} + a e^{2}\right )}^{3}}{e^{3}}\right )}}{15015 \, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 106, normalized size = 0.89 \begin {gather*} \frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {2\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{3\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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