Optimal. Leaf size=231 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 \left (c d^2-a e^2\right )^4 (d+e x)^7} \]
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Rubi [A]
time = 0.09, 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 )^{7/2}}{3003 (d+e x)^7 \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 )^{7/2}}{429 (d+e x)^8 \left (c d^2-a e^2\right )^3}+\frac {12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^9 \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 )^{7/2}}{13 (d+e x)^{10} \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 )^{5/2}}{(d+e x)^{10}} \, 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 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\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 )^{5/2}}{(d+e x)^9} \, dx}{13 \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 )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\frac {\left (24 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 )^{5/2}}{(d+e x)^8} \, dx}{143 \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 )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\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 )^{7/2}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\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 )^{5/2}}{(d+e x)^7} \, dx}{429 \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 )^{7/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^9}+\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 )^{7/2}}{429 \left (c d^2-a e^2\right )^3 (d+e x)^8}+\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 )^{7/2}}{3003 \left (c d^2-a e^2\right )^4 (d+e x)^7}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 132, normalized size = 0.57 \begin {gather*} \frac {2 (a e+c d x)^4 ((a e+c d x) (d+e x))^{5/2} \left (-231 e^3+\frac {819 c d e^2 (d+e x)}{a e+c d x}-\frac {1001 c^2 d^2 e (d+e x)^2}{(a e+c d x)^2}+\frac {429 c^3 d^3 (d+e x)^3}{(a e+c d x)^3}\right )}{3003 \left (c d^2-a e^2\right )^4 (d+e x)^9} \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 | \(-\frac {2 \left (c d x +a e \right ) \left (-16 c^{3} d^{3} e^{3} x^{3}+56 a \,c^{2} d^{2} e^{4} x^{2}-104 c^{3} d^{4} e^{2} x^{2}-126 a^{2} c d \,e^{5} x +364 a \,c^{2} d^{3} e^{3} x -286 c^{3} d^{5} e x +231 e^{6} a^{3}-819 e^{4} d^{2} a^{2} c +1001 d^{4} e^{2} c^{2} a -429 d^{6} c^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 \left (e x +d \right )^{9} \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right )}\) | \(217\) |
default | \(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{13 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{10}}-\frac {6 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{11 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{9}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{8}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{11 \left (e^{2} a -c \,d^{2}\right )}\right )}{13 \left (e^{2} a -c \,d^{2}\right )}}{e^{10}}\) | \(293\) |
trager | \(-\frac {2 \left (-16 c^{6} d^{6} e^{3} x^{6}+8 a \,c^{5} d^{5} e^{4} x^{5}-104 c^{6} d^{7} e^{2} x^{5}-6 a^{2} c^{4} e^{5} d^{4} x^{4}+52 a \,c^{5} d^{6} e^{3} x^{4}-286 c^{6} d^{8} e \,x^{4}+5 a^{3} c^{3} d^{3} e^{6} x^{3}-39 a^{2} c^{4} d^{5} e^{4} x^{3}+143 a \,c^{5} d^{7} e^{2} x^{3}-429 c^{6} d^{9} x^{3}+371 a^{4} c^{2} d^{2} e^{7} x^{2}-1469 a^{3} c^{3} d^{4} e^{5} x^{2}+2145 a^{2} c^{4} d^{6} e^{3} x^{2}-1287 a \,c^{5} d^{8} e \,x^{2}+567 a^{5} c d \,e^{8} x -2093 a^{4} c^{2} d^{3} e^{6} x +2717 a^{3} c^{3} d^{5} e^{4} x -1287 a^{2} c^{4} d^{7} e^{2} x +231 a^{6} e^{9}-819 a^{5} c \,d^{2} e^{7}+1001 a^{4} c^{2} d^{4} e^{5}-429 a^{3} c^{3} d^{6} e^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3003 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{7}}\) | \(407\) |
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. 834 vs.
\(2 (219) = 438\).
time = 165.13, size = 834, normalized size = 3.61 \begin {gather*} \frac {2 \, {\left (429 \, c^{6} d^{9} x^{3} - 567 \, a^{5} c d x e^{8} - 231 \, a^{6} e^{9} - 7 \, {\left (53 \, a^{4} c^{2} d^{2} x^{2} - 117 \, a^{5} c d^{2}\right )} e^{7} - {\left (5 \, a^{3} c^{3} d^{3} x^{3} - 2093 \, a^{4} c^{2} d^{3} x\right )} e^{6} + {\left (6 \, a^{2} c^{4} d^{4} x^{4} + 1469 \, a^{3} c^{3} d^{4} x^{2} - 1001 \, a^{4} c^{2} d^{4}\right )} e^{5} - {\left (8 \, a c^{5} d^{5} x^{5} - 39 \, a^{2} c^{4} d^{5} x^{3} + 2717 \, a^{3} c^{3} d^{5} x\right )} e^{4} + {\left (16 \, c^{6} d^{6} x^{6} - 52 \, a c^{5} d^{6} x^{4} - 2145 \, a^{2} c^{4} d^{6} x^{2} + 429 \, a^{3} c^{3} d^{6}\right )} e^{3} + 13 \, {\left (8 \, c^{6} d^{7} x^{5} - 11 \, a c^{5} d^{7} x^{3} + 99 \, a^{2} c^{4} d^{7} x\right )} e^{2} + 143 \, {\left (2 \, c^{6} d^{8} x^{4} + 9 \, a c^{5} d^{8} x^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{3003 \, {\left (7 \, c^{4} d^{14} x e + c^{4} d^{15} + a^{4} x^{7} e^{15} + 7 \, a^{4} d x^{6} e^{14} - {\left (4 \, a^{3} c d^{2} x^{7} - 21 \, a^{4} d^{2} x^{5}\right )} e^{13} - 7 \, {\left (4 \, a^{3} c d^{3} x^{6} - 5 \, a^{4} d^{3} x^{4}\right )} e^{12} + {\left (6 \, a^{2} c^{2} d^{4} x^{7} - 84 \, a^{3} c d^{4} x^{5} + 35 \, a^{4} d^{4} x^{3}\right )} e^{11} + 7 \, {\left (6 \, a^{2} c^{2} d^{5} x^{6} - 20 \, a^{3} c d^{5} x^{4} + 3 \, a^{4} d^{5} x^{2}\right )} e^{10} - {\left (4 \, a c^{3} d^{6} x^{7} - 126 \, a^{2} c^{2} d^{6} x^{5} + 140 \, a^{3} c d^{6} x^{3} - 7 \, a^{4} d^{6} x\right )} e^{9} - {\left (28 \, a c^{3} d^{7} x^{6} - 210 \, a^{2} c^{2} d^{7} x^{4} + 84 \, a^{3} c d^{7} x^{2} - a^{4} d^{7}\right )} e^{8} + {\left (c^{4} d^{8} x^{7} - 84 \, a c^{3} d^{8} x^{5} + 210 \, a^{2} c^{2} d^{8} x^{3} - 28 \, a^{3} c d^{8} x\right )} e^{7} + {\left (7 \, c^{4} d^{9} x^{6} - 140 \, a c^{3} d^{9} x^{4} + 126 \, a^{2} c^{2} d^{9} x^{2} - 4 \, a^{3} c d^{9}\right )} e^{6} + 7 \, {\left (3 \, c^{4} d^{10} x^{5} - 20 \, a c^{3} d^{10} x^{3} + 6 \, a^{2} c^{2} d^{10} x\right )} e^{5} + {\left (35 \, c^{4} d^{11} x^{4} - 84 \, a c^{3} d^{11} x^{2} + 6 \, a^{2} c^{2} d^{11}\right )} e^{4} + 7 \, {\left (5 \, c^{4} d^{12} x^{3} - 4 \, a c^{3} d^{12} x\right )} e^{3} + {\left (21 \, c^{4} d^{13} x^{2} - 4 \, a c^{3} d^{13}\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 = 9.57, size = 2500, normalized size = 10.82 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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