Optimal. Leaf size=192 \[ \frac {128 c^2 d^2 e \left (a e^2+c d^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {16 c d \left (a e^2+c d^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {2}{5 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {658, 614, 613} \begin {gather*} \frac {128 c^2 d^2 e \left (a e^2+c d^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {16 c d \left (a e^2+c d^2+2 c d e x\right )}{15 \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 {2}{5 (d+e x) \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 613
Rule 614
Rule 658
Rubi steps
\begin {align*} \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 &=\frac {2}{5 \left (c d^2-a e^2\right ) (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 {(8 c d) \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}{5 \left (c d^2-a e^2\right )}\\ &=\frac {2}{5 \left (c d^2-a e^2\right ) (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 {16 c d \left (c d^2+a e^2+2 c d e x\right )}{15 \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 )^{3/2}}-\frac {\left (64 c^2 d^2 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}{15 \left (c d^2-a e^2\right )^3}\\ &=\frac {2}{5 \left (c d^2-a e^2\right ) (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 {16 c d \left (c d^2+a e^2+2 c d e x\right )}{15 \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 )^{3/2}}+\frac {128 c^2 d^2 e \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 193, normalized size = 1.01 \begin {gather*} \frac {2 \left (3 a^4 e^8-4 a^3 c d e^6 (5 d+2 e x)+6 a^2 c^2 d^2 e^4 \left (15 d^2+20 d e x+8 e^2 x^2\right )+12 a c^3 d^3 e^2 \left (5 d^3+30 d^2 e x+40 d e^2 x^2+16 e^3 x^3\right )+c^4 d^4 \left (-5 d^4+40 d^3 e x+240 d^2 e^2 x^2+320 d e^3 x^3+128 e^4 x^4\right )\right )}{15 (d+e x) \left (c d^2-a e^2\right )^5 ((d+e x) (a e+c d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 180.03, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 24.47, size = 769, normalized size = 4.01 \begin {gather*} \frac {2 \, {\left (128 \, c^{4} d^{4} e^{4} x^{4} - 5 \, c^{4} d^{8} + 60 \, a c^{3} d^{6} e^{2} + 90 \, a^{2} c^{2} d^{4} e^{4} - 20 \, a^{3} c d^{2} e^{6} + 3 \, a^{4} e^{8} + 64 \, {\left (5 \, c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 48 \, {\left (5 \, c^{4} d^{6} e^{2} + 10 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (5 \, c^{4} d^{7} e + 45 \, a c^{3} d^{5} e^{3} + 15 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15 \, {\left (a^{2} c^{5} d^{13} e^{2} - 5 \, a^{3} c^{4} d^{11} e^{4} + 10 \, a^{4} c^{3} d^{9} e^{6} - 10 \, a^{5} c^{2} d^{7} e^{8} + 5 \, a^{6} c d^{5} e^{10} - a^{7} d^{3} e^{12} + {\left (c^{7} d^{12} e^{3} - 5 \, a c^{6} d^{10} e^{5} + 10 \, a^{2} c^{5} d^{8} e^{7} - 10 \, a^{3} c^{4} d^{6} e^{9} + 5 \, a^{4} c^{3} d^{4} e^{11} - a^{5} c^{2} d^{2} e^{13}\right )} x^{5} + {\left (3 \, c^{7} d^{13} e^{2} - 13 \, a c^{6} d^{11} e^{4} + 20 \, a^{2} c^{5} d^{9} e^{6} - 10 \, a^{3} c^{4} d^{7} e^{8} - 5 \, a^{4} c^{3} d^{5} e^{10} + 7 \, a^{5} c^{2} d^{3} e^{12} - 2 \, a^{6} c d e^{14}\right )} x^{4} + {\left (3 \, c^{7} d^{14} e - 9 \, a c^{6} d^{12} e^{3} + a^{2} c^{5} d^{10} e^{5} + 25 \, a^{3} c^{4} d^{8} e^{7} - 35 \, a^{4} c^{3} d^{6} e^{9} + 17 \, a^{5} c^{2} d^{4} e^{11} - a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} x^{3} + {\left (c^{7} d^{15} + a c^{6} d^{13} e^{2} - 17 \, a^{2} c^{5} d^{11} e^{4} + 35 \, a^{3} c^{4} d^{9} e^{6} - 25 \, a^{4} c^{3} d^{7} e^{8} - a^{5} c^{2} d^{5} e^{10} + 9 \, a^{6} c d^{3} e^{12} - 3 \, a^{7} d e^{14}\right )} x^{2} + {\left (2 \, a c^{6} d^{14} e - 7 \, a^{2} c^{5} d^{12} e^{3} + 5 \, a^{3} c^{4} d^{10} e^{5} + 10 \, a^{4} c^{3} d^{8} e^{7} - 20 \, a^{5} c^{2} d^{6} e^{9} + 13 \, a^{6} c d^{4} e^{11} - 3 \, a^{7} d^{2} e^{13}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 300, normalized size = 1.56 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (128 c^{4} d^{4} e^{4} x^{4}+192 a \,c^{3} d^{3} e^{5} x^{3}+320 c^{4} d^{5} e^{3} x^{3}+48 a^{2} c^{2} d^{2} e^{6} x^{2}+480 a \,c^{3} d^{4} e^{4} x^{2}+240 c^{4} d^{6} e^{2} x^{2}-8 a^{3} c d \,e^{7} x +120 a^{2} c^{2} d^{3} e^{5} x +360 a \,c^{3} d^{5} e^{3} x +40 c^{4} d^{7} e x +3 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}+90 a^{2} c^{2} d^{4} e^{4}+60 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right )}{15 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}} \end {gather*}
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]
________________________________________________________________________________________
mupad [B] time = 1.65, size = 253, normalized size = 1.32 \begin {gather*} -\frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (3\,a^4\,e^8-20\,a^3\,c\,d^2\,e^6-8\,a^3\,c\,d\,e^7\,x+90\,a^2\,c^2\,d^4\,e^4+120\,a^2\,c^2\,d^3\,e^5\,x+48\,a^2\,c^2\,d^2\,e^6\,x^2+60\,a\,c^3\,d^6\,e^2+360\,a\,c^3\,d^5\,e^3\,x+480\,a\,c^3\,d^4\,e^4\,x^2+192\,a\,c^3\,d^3\,e^5\,x^3-5\,c^4\,d^8+40\,c^4\,d^7\,e\,x+240\,c^4\,d^6\,e^2\,x^2+320\,c^4\,d^5\,e^3\,x^3+128\,c^4\,d^4\,e^4\,x^4\right )}{15\,{\left (a\,e+c\,d\,x\right )}^2\,{\left (a\,e^2-c\,d^2\right )}^5\,{\left (d+e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________