Optimal. Leaf size=233 \[ \frac {32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {12 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}} \]
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Rubi [A] time = 0.18, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {656, 648} \begin {gather*} \frac {12 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rubi steps
\begin {align*} \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}+\frac {\left (6 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx}{13 d}\\ &=\frac {12 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}+\frac {\left (24 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx}{143 d^2}\\ &=\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {12 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}+\frac {\left (16 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{429 d^3}\\ &=\frac {32 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 c^4 d^4 (d+e x)^{7/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {12 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 c^2 d^2 (d+e x)^{3/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 c d \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 142, normalized size = 0.61 \begin {gather*} \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (-16 a^3 e^6+8 a^2 c d e^4 (13 d+7 e x)-2 a c^2 d^2 e^2 \left (143 d^2+182 d e x+63 e^2 x^2\right )+c^3 d^3 \left (429 d^3+1001 d^2 e x+819 d e^2 x^2+231 e^3 x^3\right )\right )}{3003 c^4 d^4 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.03, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 354, normalized size = 1.52 \begin {gather*} \frac {2 \, {\left (231 \, c^{6} d^{6} e^{3} x^{6} + 429 \, a^{3} c^{3} d^{6} e^{3} - 286 \, a^{4} c^{2} d^{4} e^{5} + 104 \, a^{5} c d^{2} e^{7} - 16 \, a^{6} e^{9} + 63 \, {\left (13 \, c^{6} d^{7} e^{2} + 9 \, a c^{5} d^{5} e^{4}\right )} x^{5} + 7 \, {\left (143 \, c^{6} d^{8} e + 299 \, a c^{5} d^{6} e^{3} + 53 \, a^{2} c^{4} d^{4} e^{5}\right )} x^{4} + {\left (429 \, c^{6} d^{9} + 2717 \, a c^{5} d^{7} e^{2} + 1469 \, a^{2} c^{4} d^{5} e^{4} + 5 \, a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (429 \, a c^{5} d^{8} e + 715 \, a^{2} c^{4} d^{6} e^{3} + 13 \, a^{3} c^{3} d^{4} e^{5} - 2 \, a^{4} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (1287 \, a^{2} c^{4} d^{7} e^{2} + 143 \, a^{3} c^{3} d^{5} e^{4} - 52 \, a^{4} c^{2} d^{3} e^{6} + 8 \, a^{5} c d e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3003 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} \sqrt {e x + d}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 168, normalized size = 0.72 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-231 c^{3} d^{3} e^{3} x^{3}+126 a \,c^{2} d^{2} e^{4} x^{2}-819 c^{3} d^{4} e^{2} x^{2}-56 a^{2} c d \,e^{5} x +364 a \,c^{2} d^{3} e^{3} x -1001 c^{3} d^{5} e x +16 a^{3} e^{6}-104 a^{2} c \,d^{2} e^{4}+286 a \,c^{2} d^{4} e^{2}-429 c^{3} d^{6}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 \left (e x +d \right )^{\frac {5}{2}} c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.60, size = 335, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (231 \, c^{6} d^{6} e^{3} x^{6} + 429 \, a^{3} c^{3} d^{6} e^{3} - 286 \, a^{4} c^{2} d^{4} e^{5} + 104 \, a^{5} c d^{2} e^{7} - 16 \, a^{6} e^{9} + 63 \, {\left (13 \, c^{6} d^{7} e^{2} + 9 \, a c^{5} d^{5} e^{4}\right )} x^{5} + 7 \, {\left (143 \, c^{6} d^{8} e + 299 \, a c^{5} d^{6} e^{3} + 53 \, a^{2} c^{4} d^{4} e^{5}\right )} x^{4} + {\left (429 \, c^{6} d^{9} + 2717 \, a c^{5} d^{7} e^{2} + 1469 \, a^{2} c^{4} d^{5} e^{4} + 5 \, a^{3} c^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (429 \, a c^{5} d^{8} e + 715 \, a^{2} c^{4} d^{6} e^{3} + 13 \, a^{3} c^{3} d^{4} e^{5} - 2 \, a^{4} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (1287 \, a^{2} c^{4} d^{7} e^{2} + 143 \, a^{3} c^{3} d^{5} e^{4} - 52 \, a^{4} c^{2} d^{3} e^{6} + 8 \, a^{5} c d e^{8}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{3003 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.40, size = 383, normalized size = 1.64 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (x^4\,\sqrt {d+e\,x}\,\left (\frac {106\,a^2\,e^4}{429}+\frac {46\,a\,c\,d^2\,e^2}{33}+\frac {2\,c^2\,d^4}{3}\right )-\frac {\sqrt {d+e\,x}\,\left (32\,a^6\,e^9-208\,a^5\,c\,d^2\,e^7+572\,a^4\,c^2\,d^4\,e^5-858\,a^3\,c^3\,d^6\,e^3\right )}{3003\,c^4\,d^4\,e}+\frac {2\,c^2\,d^2\,e^2\,x^6\,\sqrt {d+e\,x}}{13}+\frac {x^3\,\sqrt {d+e\,x}\,\left (10\,a^3\,c^3\,d^3\,e^6+2938\,a^2\,c^4\,d^5\,e^4+5434\,a\,c^5\,d^7\,e^2+858\,c^6\,d^9\right )}{3003\,c^4\,d^4\,e}+\frac {6\,c\,d\,e\,x^5\,\left (13\,c\,d^2+9\,a\,e^2\right )\,\sqrt {d+e\,x}}{143}+\frac {2\,a\,x^2\,\sqrt {d+e\,x}\,\left (-2\,a^3\,e^6+13\,a^2\,c\,d^2\,e^4+715\,a\,c^2\,d^4\,e^2+429\,c^3\,d^6\right )}{1001\,c^2\,d^2}+\frac {2\,a^2\,e\,x\,\sqrt {d+e\,x}\,\left (8\,a^3\,e^6-52\,a^2\,c\,d^2\,e^4+143\,a\,c^2\,d^4\,e^2+1287\,c^3\,d^6\right )}{3003\,c^3\,d^3}\right )}{x+\frac {d}{e}} \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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