Optimal. Leaf size=231 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^5} \]
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Rubi [A]
time = 0.08, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664}
\begin {gather*} \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^5 \left (c d^2-a e^2\right )^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^6 \left (c d^2-a e^2\right )^3}+\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^7 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^8 \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 672
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 )^{3/2}}{(d+e x)^8} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {(6 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^7} \, dx}{11 \left (c d^2-a e^2\right )}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {\left (8 c^2 d^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^6} \, dx}{33 \left (c d^2-a e^2\right )^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\frac {\left (16 c^3 d^3\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx}{231 \left (c d^2-a e^2\right )^3}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^8}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 \left (c d^2-a e^2\right )^2 (d+e x)^7}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 \left (c d^2-a e^2\right )^3 (d+e x)^6}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^5}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 132, normalized size = 0.57 \begin {gather*} \frac {2 (a e+c d x)^4 ((a e+c d x) (d+e x))^{3/2} \left (-105 e^3+\frac {385 c d e^2 (d+e x)}{a e+c d x}-\frac {495 c^2 d^2 e (d+e x)^2}{(a e+c d x)^2}+\frac {231 c^3 d^3 (d+e x)^3}{(a e+c d x)^3}\right )}{1155 \left (c d^2-a e^2\right )^4 (d+e x)^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 293, normalized size = 1.27
method | result | size |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+40 a \,c^{2} d^{2} e^{4} x^{2}-88 c^{3} d^{4} e^{2} x^{2}-70 a^{2} c d \,e^{5} x +220 a \,c^{2} d^{3} e^{3} x -198 c^{3} d^{5} e x +105 e^{6} a^{3}-385 e^{4} d^{2} a^{2} c +495 d^{4} e^{2} c^{2} a -231 d^{6} c^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{1155 \left (e x +d \right )^{7} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}\) | \(217\) |
default | \(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{11 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {6 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{7}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{9 \left (e^{2} a -c \,d^{2}\right )}\right )}{11 \left (e^{2} a -c \,d^{2}\right )}}{e^{8}}\) | \(293\) |
trager | \(-\frac {2 \left (-16 c^{5} d^{5} e^{3} x^{5}+8 a \,c^{4} d^{4} e^{4} x^{4}-88 c^{5} d^{6} e^{2} x^{4}-6 a^{2} c^{3} d^{3} e^{5} x^{3}+44 a \,c^{4} d^{5} e^{3} x^{3}-198 c^{5} d^{7} e \,x^{3}+5 a^{3} c^{2} d^{2} e^{6} x^{2}-33 a^{2} c^{3} d^{4} e^{4} x^{2}+99 a \,c^{4} d^{6} e^{2} x^{2}-231 c^{5} d^{8} x^{2}+140 a^{4} c d \,e^{7} x -550 a^{3} c^{2} d^{3} e^{5} x +792 a^{2} c^{3} d^{5} e^{3} x -462 a \,c^{4} d^{7} e x +105 a^{5} e^{8}-385 a^{4} c \,d^{2} e^{6}+495 a^{3} c^{2} d^{4} e^{4}-231 a^{2} c^{3} d^{6} e^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{1155 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{6}}\) | \(339\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 710 vs.
\(2 (219) = 438\).
time = 79.95, size = 710, normalized size = 3.07 \begin {gather*} \frac {2 \, {\left (231 \, c^{5} d^{8} x^{2} - 140 \, a^{4} c d x e^{7} - 105 \, a^{5} e^{8} - 5 \, {\left (a^{3} c^{2} d^{2} x^{2} - 77 \, a^{4} c d^{2}\right )} e^{6} + 2 \, {\left (3 \, a^{2} c^{3} d^{3} x^{3} + 275 \, a^{3} c^{2} d^{3} x\right )} e^{5} - {\left (8 \, a c^{4} d^{4} x^{4} - 33 \, a^{2} c^{3} d^{4} x^{2} + 495 \, a^{3} c^{2} d^{4}\right )} e^{4} + 4 \, {\left (4 \, c^{5} d^{5} x^{5} - 11 \, a c^{4} d^{5} x^{3} - 198 \, a^{2} c^{3} d^{5} x\right )} e^{3} + 11 \, {\left (8 \, c^{5} d^{6} x^{4} - 9 \, a c^{4} d^{6} x^{2} + 21 \, a^{2} c^{3} d^{6}\right )} e^{2} + 66 \, {\left (3 \, c^{5} d^{7} x^{3} + 7 \, a c^{4} d^{7} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{1155 \, {\left (6 \, c^{4} d^{13} x e + c^{4} d^{14} + a^{4} x^{6} e^{14} + 6 \, a^{4} d x^{5} e^{13} - {\left (4 \, a^{3} c d^{2} x^{6} - 15 \, a^{4} d^{2} x^{4}\right )} e^{12} - 4 \, {\left (6 \, a^{3} c d^{3} x^{5} - 5 \, a^{4} d^{3} x^{3}\right )} e^{11} + 3 \, {\left (2 \, a^{2} c^{2} d^{4} x^{6} - 20 \, a^{3} c d^{4} x^{4} + 5 \, a^{4} d^{4} x^{2}\right )} e^{10} + 2 \, {\left (18 \, a^{2} c^{2} d^{5} x^{5} - 40 \, a^{3} c d^{5} x^{3} + 3 \, a^{4} d^{5} x\right )} e^{9} - {\left (4 \, a c^{3} d^{6} x^{6} - 90 \, a^{2} c^{2} d^{6} x^{4} + 60 \, a^{3} c d^{6} x^{2} - a^{4} d^{6}\right )} e^{8} - 24 \, {\left (a c^{3} d^{7} x^{5} - 5 \, a^{2} c^{2} d^{7} x^{3} + a^{3} c d^{7} x\right )} e^{7} + {\left (c^{4} d^{8} x^{6} - 60 \, a c^{3} d^{8} x^{4} + 90 \, a^{2} c^{2} d^{8} x^{2} - 4 \, a^{3} c d^{8}\right )} e^{6} + 2 \, {\left (3 \, c^{4} d^{9} x^{5} - 40 \, a c^{3} d^{9} x^{3} + 18 \, a^{2} c^{2} d^{9} x\right )} e^{5} + 3 \, {\left (5 \, c^{4} d^{10} x^{4} - 20 \, a c^{3} d^{10} x^{2} + 2 \, a^{2} c^{2} d^{10}\right )} e^{4} + 4 \, {\left (5 \, c^{4} d^{11} x^{3} - 6 \, a c^{3} d^{11} x\right )} e^{3} + {\left (15 \, c^{4} d^{12} x^{2} - 4 \, a c^{3} d^{12}\right )} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.15, size = 2657, normalized size = 11.50 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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