Optimal. Leaf size=312 \[ \frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 c d}{21 \left (c d^2-a e^2\right )^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {128 c^3 d^3 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {1024 c^4 d^4 e \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {672, 628, 627}
\begin {gather*} \frac {1024 c^4 d^4 e \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {128 c^3 d^3 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {16 c^2 d^2}{21 (d+e x) \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 )^{3/2}}+\frac {8 c d}{21 (d+e x)^2 \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 )^{3/2}}+\frac {2}{9 (d+e x)^3 \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 )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 627
Rule 628
Rule 672
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(4 c d) \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{3 \left (c d^2-a e^2\right )}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 c d}{21 \left (c d^2-a e^2\right )^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {\left (40 c^2 d^2\right ) \int \frac {1}{(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}{21 \left (c d^2-a e^2\right )^2}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 c d}{21 \left (c d^2-a e^2\right )^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {\left (64 c^3 d^3\right ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{21 \left (c d^2-a e^2\right )^3}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 c d}{21 \left (c d^2-a e^2\right )^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {128 c^3 d^3 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {\left (512 c^4 d^4 e\right ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{63 \left (c d^2-a e^2\right )^5}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 c d}{21 \left (c d^2-a e^2\right )^2 (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {16 c^2 d^2}{21 \left (c d^2-a e^2\right )^3 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {128 c^3 d^3 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {1024 c^4 d^4 e \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^7 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.20, size = 216, normalized size = 0.69 \begin {gather*} -\frac {2 (a e+c d x)^7 \left (-7 e^6+\frac {54 c d e^5 (d+e x)}{a e+c d x}-\frac {189 c^2 d^2 e^4 (d+e x)^2}{(a e+c d x)^2}+\frac {420 c^3 d^3 e^3 (d+e x)^3}{(a e+c d x)^3}-\frac {945 c^4 d^4 e^2 (d+e x)^4}{(a e+c d x)^4}-\frac {378 c^5 d^5 e (d+e x)^5}{(a e+c d x)^5}+\frac {21 c^6 d^6 (d+e x)^6}{(a e+c d x)^6}\right )}{63 \left (c d^2-a e^2\right )^7 (d+e x)^2 ((a e+c d x) (d+e x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.74, size = 404, normalized size = 1.29
method | result | size |
default | \(\frac {-\frac {2}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3} \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 c d e \left (-\frac {2}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {10 c d e \left (-\frac {2}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {8 c d e \left (-\frac {2 \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (e^{2} a -c \,d^{2}\right )}\right )}{7 \left (e^{2} a -c \,d^{2}\right )}\right )}{3 \left (e^{2} a -c \,d^{2}\right )}}{e^{3}}\) | \(404\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (1024 c^{6} d^{6} e^{6} x^{6}+1536 a \,c^{5} d^{5} e^{7} x^{5}+4608 c^{6} d^{7} e^{5} x^{5}+384 a^{2} c^{4} d^{4} e^{8} x^{4}+6912 a \,c^{5} d^{6} e^{6} x^{4}+8064 c^{6} d^{8} e^{4} x^{4}-64 a^{3} c^{3} d^{3} e^{9} x^{3}+1728 a^{2} c^{4} d^{5} e^{7} x^{3}+12096 a \,c^{5} d^{7} e^{5} x^{3}+6720 c^{6} d^{9} e^{3} x^{3}+24 a^{4} c^{2} d^{2} e^{10} x^{2}-288 a^{3} c^{3} d^{4} e^{8} x^{2}+3024 a^{2} c^{4} d^{6} e^{6} x^{2}+10080 a \,c^{5} d^{8} e^{4} x^{2}+2520 c^{6} d^{10} e^{2} x^{2}-12 a^{5} c d \,e^{11} x +108 a^{4} c^{2} d^{3} e^{9} x -504 a^{3} c^{3} d^{5} e^{7} x +2520 a^{2} c^{4} d^{7} e^{5} x +3780 a \,c^{5} d^{9} e^{3} x +252 c^{6} d^{11} e x +7 a^{6} e^{12}-54 a^{5} c \,d^{2} e^{10}+189 a^{4} c^{2} d^{4} e^{8}-420 a^{3} c^{3} d^{6} e^{6}+945 a^{2} c^{4} d^{8} e^{4}+378 a \,c^{5} d^{10} e^{2}-21 c^{6} d^{12}\right )}{63 \left (e x +d \right )^{2} \left (a^{7} e^{14}-7 a^{6} c \,d^{2} e^{12}+21 a^{5} c^{2} d^{4} e^{10}-35 a^{4} c^{3} d^{6} e^{8}+35 a^{3} c^{4} d^{8} e^{6}-21 a^{2} c^{5} d^{10} e^{4}+7 a \,c^{6} d^{12} e^{2}-c^{7} d^{14}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\) | \(536\) |
trager | \(-\frac {2 \left (1024 c^{6} d^{6} e^{6} x^{6}+1536 a \,c^{5} d^{5} e^{7} x^{5}+4608 c^{6} d^{7} e^{5} x^{5}+384 a^{2} c^{4} d^{4} e^{8} x^{4}+6912 a \,c^{5} d^{6} e^{6} x^{4}+8064 c^{6} d^{8} e^{4} x^{4}-64 a^{3} c^{3} d^{3} e^{9} x^{3}+1728 a^{2} c^{4} d^{5} e^{7} x^{3}+12096 a \,c^{5} d^{7} e^{5} x^{3}+6720 c^{6} d^{9} e^{3} x^{3}+24 a^{4} c^{2} d^{2} e^{10} x^{2}-288 a^{3} c^{3} d^{4} e^{8} x^{2}+3024 a^{2} c^{4} d^{6} e^{6} x^{2}+10080 a \,c^{5} d^{8} e^{4} x^{2}+2520 c^{6} d^{10} e^{2} x^{2}-12 a^{5} c d \,e^{11} x +108 a^{4} c^{2} d^{3} e^{9} x -504 a^{3} c^{3} d^{5} e^{7} x +2520 a^{2} c^{4} d^{7} e^{5} x +3780 a \,c^{5} d^{9} e^{3} x +252 c^{6} d^{11} e x +7 a^{6} e^{12}-54 a^{5} c \,d^{2} e^{10}+189 a^{4} c^{2} d^{4} e^{8}-420 a^{3} c^{3} d^{6} e^{6}+945 a^{2} c^{4} d^{8} e^{4}+378 a \,c^{5} d^{10} e^{2}-21 c^{6} d^{12}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{63 \left (a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}\right ) \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{5}}\) | \(537\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1392 vs.
\(2 (292) = 584\).
time = 62.83, size = 1392, normalized size = 4.46 \begin {gather*} \frac {2 \, {\left (1024 \, c^{6} d^{6} e^{6} x^{6} - 21 \, c^{6} d^{12} + 378 \, a c^{5} d^{10} e^{2} + 945 \, a^{2} c^{4} d^{8} e^{4} - 420 \, a^{3} c^{3} d^{6} e^{6} + 189 \, a^{4} c^{2} d^{4} e^{8} - 54 \, a^{5} c d^{2} e^{10} + 7 \, a^{6} e^{12} + 1536 \, {\left (3 \, c^{6} d^{7} e^{5} + a c^{5} d^{5} e^{7}\right )} x^{5} + 384 \, {\left (21 \, c^{6} d^{8} e^{4} + 18 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} + 64 \, {\left (105 \, c^{6} d^{9} e^{3} + 189 \, a c^{5} d^{7} e^{5} + 27 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 24 \, {\left (105 \, c^{6} d^{10} e^{2} + 420 \, a c^{5} d^{8} e^{4} + 126 \, a^{2} c^{4} d^{6} e^{6} - 12 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 12 \, {\left (21 \, c^{6} d^{11} e + 315 \, a c^{5} d^{9} e^{3} + 210 \, a^{2} c^{4} d^{7} e^{5} - 42 \, a^{3} c^{3} d^{5} e^{7} + 9 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{63 \, {\left (a^{2} c^{7} d^{19} e^{2} - 7 \, a^{3} c^{6} d^{17} e^{4} + 21 \, a^{4} c^{5} d^{15} e^{6} - 35 \, a^{5} c^{4} d^{13} e^{8} + 35 \, a^{6} c^{3} d^{11} e^{10} - 21 \, a^{7} c^{2} d^{9} e^{12} + 7 \, a^{8} c d^{7} e^{14} - a^{9} d^{5} e^{16} + {\left (c^{9} d^{16} e^{5} - 7 \, a c^{8} d^{14} e^{7} + 21 \, a^{2} c^{7} d^{12} e^{9} - 35 \, a^{3} c^{6} d^{10} e^{11} + 35 \, a^{4} c^{5} d^{8} e^{13} - 21 \, a^{5} c^{4} d^{6} e^{15} + 7 \, a^{6} c^{3} d^{4} e^{17} - a^{7} c^{2} d^{2} e^{19}\right )} x^{7} + {\left (5 \, c^{9} d^{17} e^{4} - 33 \, a c^{8} d^{15} e^{6} + 91 \, a^{2} c^{7} d^{13} e^{8} - 133 \, a^{3} c^{6} d^{11} e^{10} + 105 \, a^{4} c^{5} d^{9} e^{12} - 35 \, a^{5} c^{4} d^{7} e^{14} - 7 \, a^{6} c^{3} d^{5} e^{16} + 9 \, a^{7} c^{2} d^{3} e^{18} - 2 \, a^{8} c d e^{20}\right )} x^{6} + {\left (10 \, c^{9} d^{18} e^{3} - 60 \, a c^{8} d^{16} e^{5} + 141 \, a^{2} c^{7} d^{14} e^{7} - 147 \, a^{3} c^{6} d^{12} e^{9} + 21 \, a^{4} c^{5} d^{10} e^{11} + 105 \, a^{5} c^{4} d^{8} e^{13} - 105 \, a^{6} c^{3} d^{6} e^{15} + 39 \, a^{7} c^{2} d^{4} e^{17} - 3 \, a^{8} c d^{2} e^{19} - a^{9} e^{21}\right )} x^{5} + 5 \, {\left (2 \, c^{9} d^{19} e^{2} - 10 \, a c^{8} d^{17} e^{4} + 15 \, a^{2} c^{7} d^{15} e^{6} + 7 \, a^{3} c^{6} d^{13} e^{8} - 49 \, a^{4} c^{5} d^{11} e^{10} + 63 \, a^{5} c^{4} d^{9} e^{12} - 35 \, a^{6} c^{3} d^{7} e^{14} + 5 \, a^{7} c^{2} d^{5} e^{16} + 3 \, a^{8} c d^{3} e^{18} - a^{9} d e^{20}\right )} x^{4} + 5 \, {\left (c^{9} d^{20} e - 3 \, a c^{8} d^{18} e^{3} - 5 \, a^{2} c^{7} d^{16} e^{5} + 35 \, a^{3} c^{6} d^{14} e^{7} - 63 \, a^{4} c^{5} d^{12} e^{9} + 49 \, a^{5} c^{4} d^{10} e^{11} - 7 \, a^{6} c^{3} d^{8} e^{13} - 15 \, a^{7} c^{2} d^{6} e^{15} + 10 \, a^{8} c d^{4} e^{17} - 2 \, a^{9} d^{2} e^{19}\right )} x^{3} + {\left (c^{9} d^{21} + 3 \, a c^{8} d^{19} e^{2} - 39 \, a^{2} c^{7} d^{17} e^{4} + 105 \, a^{3} c^{6} d^{15} e^{6} - 105 \, a^{4} c^{5} d^{13} e^{8} - 21 \, a^{5} c^{4} d^{11} e^{10} + 147 \, a^{6} c^{3} d^{9} e^{12} - 141 \, a^{7} c^{2} d^{7} e^{14} + 60 \, a^{8} c d^{5} e^{16} - 10 \, a^{9} d^{3} e^{18}\right )} x^{2} + {\left (2 \, a c^{8} d^{20} e - 9 \, a^{2} c^{7} d^{18} e^{3} + 7 \, a^{3} c^{6} d^{16} e^{5} + 35 \, a^{4} c^{5} d^{14} e^{7} - 105 \, a^{5} c^{4} d^{12} e^{9} + 133 \, a^{6} c^{3} d^{10} e^{11} - 91 \, a^{7} c^{2} d^{8} e^{13} + 33 \, a^{8} c d^{6} e^{15} - 5 \, a^{9} d^{4} e^{17}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 6.19, size = 2500, normalized size = 8.01 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________