Optimal. Leaf size=90 \[ \frac {\left (c d^2-a e^2\right )^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{m+4}}{e^3 (m+4)}+\frac {c^2 d^2 (d+e x)^{m+5}}{e^3 (m+5)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 43} \[ \frac {\left (c d^2-a e^2\right )^2 (d+e x)^{m+3}}{e^3 (m+3)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{m+4}}{e^3 (m+4)}+\frac {c^2 d^2 (d+e x)^{m+5}}{e^3 (m+5)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 626
Rubi steps
\begin {align*} \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx &=\int (a e+c d x)^2 (d+e x)^{2+m} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2 (d+e x)^{2+m}}{e^2}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{3+m}}{e^2}+\frac {c^2 d^2 (d+e x)^{4+m}}{e^2}\right ) \, dx\\ &=\frac {\left (c d^2-a e^2\right )^2 (d+e x)^{3+m}}{e^3 (3+m)}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{4+m}}{e^3 (4+m)}+\frac {c^2 d^2 (d+e x)^{5+m}}{e^3 (5+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 79, normalized size = 0.88 \[ \frac {(d+e x)^{m+3} \left (-\frac {2 c d (d+e x) \left (c d^2-a e^2\right )}{m+4}+\frac {\left (c d^2-a e^2\right )^2}{m+3}+\frac {c^2 d^2 (d+e x)^2}{m+5}\right )}{e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.20, size = 479, normalized size = 5.32 \[ \frac {{\left (a^{2} d^{3} e^{4} m^{2} + 2 \, c^{2} d^{7} - 10 \, a c d^{5} e^{2} + 20 \, a^{2} d^{3} e^{4} + {\left (c^{2} d^{2} e^{5} m^{2} + 7 \, c^{2} d^{2} e^{5} m + 12 \, c^{2} d^{2} e^{5}\right )} x^{5} + {\left (30 \, c^{2} d^{3} e^{4} + 30 \, a c d e^{6} + {\left (3 \, c^{2} d^{3} e^{4} + 2 \, a c d e^{6}\right )} m^{2} + {\left (19 \, c^{2} d^{3} e^{4} + 16 \, a c d e^{6}\right )} m\right )} x^{4} + {\left (20 \, c^{2} d^{4} e^{3} + 80 \, a c d^{2} e^{5} + 20 \, a^{2} e^{7} + {\left (3 \, c^{2} d^{4} e^{3} + 6 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} m^{2} + {\left (15 \, c^{2} d^{4} e^{3} + 46 \, a c d^{2} e^{5} + 9 \, a^{2} e^{7}\right )} m\right )} x^{3} + {\left (60 \, a c d^{3} e^{4} + 60 \, a^{2} d e^{6} + {\left (c^{2} d^{5} e^{2} + 6 \, a c d^{3} e^{4} + 3 \, a^{2} d e^{6}\right )} m^{2} + {\left (c^{2} d^{5} e^{2} + 42 \, a c d^{3} e^{4} + 27 \, a^{2} d e^{6}\right )} m\right )} x^{2} - {\left (2 \, a c d^{5} e^{2} - 9 \, a^{2} d^{3} e^{4}\right )} m + {\left (60 \, a^{2} d^{2} e^{5} + {\left (2 \, a c d^{4} e^{3} + 3 \, a^{2} d^{2} e^{5}\right )} m^{2} - {\left (2 \, c^{2} d^{6} e - 10 \, a c d^{4} e^{3} - 27 \, a^{2} d^{2} e^{5}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{3} m^{3} + 12 \, e^{3} m^{2} + 47 \, e^{3} m + 60 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.24, size = 804, normalized size = 8.93 \[ \frac {{\left (x e + d\right )}^{m} c^{2} d^{2} m^{2} x^{5} e^{5} + 3 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m^{2} x^{4} e^{4} + 3 \, {\left (x e + d\right )}^{m} c^{2} d^{4} m^{2} x^{3} e^{3} + {\left (x e + d\right )}^{m} c^{2} d^{5} m^{2} x^{2} e^{2} + 7 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m x^{5} e^{5} + 19 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m x^{4} e^{4} + 15 \, {\left (x e + d\right )}^{m} c^{2} d^{4} m x^{3} e^{3} + {\left (x e + d\right )}^{m} c^{2} d^{5} m x^{2} e^{2} - 2 \, {\left (x e + d\right )}^{m} c^{2} d^{6} m x e + 2 \, {\left (x e + d\right )}^{m} a c d m^{2} x^{4} e^{6} + 6 \, {\left (x e + d\right )}^{m} a c d^{2} m^{2} x^{3} e^{5} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{2} x^{5} e^{5} + 6 \, {\left (x e + d\right )}^{m} a c d^{3} m^{2} x^{2} e^{4} + 30 \, {\left (x e + d\right )}^{m} c^{2} d^{3} x^{4} e^{4} + 2 \, {\left (x e + d\right )}^{m} a c d^{4} m^{2} x e^{3} + 20 \, {\left (x e + d\right )}^{m} c^{2} d^{4} x^{3} e^{3} + 2 \, {\left (x e + d\right )}^{m} c^{2} d^{7} + 16 \, {\left (x e + d\right )}^{m} a c d m x^{4} e^{6} + 46 \, {\left (x e + d\right )}^{m} a c d^{2} m x^{3} e^{5} + 42 \, {\left (x e + d\right )}^{m} a c d^{3} m x^{2} e^{4} + 10 \, {\left (x e + d\right )}^{m} a c d^{4} m x e^{3} - 2 \, {\left (x e + d\right )}^{m} a c d^{5} m e^{2} + {\left (x e + d\right )}^{m} a^{2} m^{2} x^{3} e^{7} + 3 \, {\left (x e + d\right )}^{m} a^{2} d m^{2} x^{2} e^{6} + 30 \, {\left (x e + d\right )}^{m} a c d x^{4} e^{6} + 3 \, {\left (x e + d\right )}^{m} a^{2} d^{2} m^{2} x e^{5} + 80 \, {\left (x e + d\right )}^{m} a c d^{2} x^{3} e^{5} + {\left (x e + d\right )}^{m} a^{2} d^{3} m^{2} e^{4} + 60 \, {\left (x e + d\right )}^{m} a c d^{3} x^{2} e^{4} - 10 \, {\left (x e + d\right )}^{m} a c d^{5} e^{2} + 9 \, {\left (x e + d\right )}^{m} a^{2} m x^{3} e^{7} + 27 \, {\left (x e + d\right )}^{m} a^{2} d m x^{2} e^{6} + 27 \, {\left (x e + d\right )}^{m} a^{2} d^{2} m x e^{5} + 9 \, {\left (x e + d\right )}^{m} a^{2} d^{3} m e^{4} + 20 \, {\left (x e + d\right )}^{m} a^{2} x^{3} e^{7} + 60 \, {\left (x e + d\right )}^{m} a^{2} d x^{2} e^{6} + 60 \, {\left (x e + d\right )}^{m} a^{2} d^{2} x e^{5} + 20 \, {\left (x e + d\right )}^{m} a^{2} d^{3} e^{4}}{m^{3} e^{3} + 12 \, m^{2} e^{3} + 47 \, m e^{3} + 60 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 183, normalized size = 2.03 \[ \frac {\left (c^{2} d^{2} e^{2} m^{2} x^{2}+2 a c d \,e^{3} m^{2} x +7 c^{2} d^{2} e^{2} m \,x^{2}+a^{2} e^{4} m^{2}+16 a c d \,e^{3} m x -2 c^{2} d^{3} e m x +12 c^{2} d^{2} e^{2} x^{2}+9 a^{2} e^{4} m -2 a c \,d^{2} e^{2} m +30 a c d \,e^{3} x -6 c^{2} d^{3} e x +20 a^{2} e^{4}-10 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right ) \left (e x +d \right )^{m +3}}{\left (m^{3}+12 m^{2}+47 m +60\right ) e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.48, size = 691, normalized size = 7.68 \[ \frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a c d^{3}}{{\left (m^{2} + 3 \, m + 2\right )} e} + \frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a^{2} d e}{m^{2} + 3 \, m + 2} + \frac {{\left (e x + d\right )}^{m + 1} a^{2} d^{2} e}{m + 1} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c^{2} d^{4}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {4 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a c d^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a^{2} e}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c^{2} d^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} a c d}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{2} d^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.05, size = 486, normalized size = 5.40 \[ {\left (d+e\,x\right )}^m\,\left (\frac {x^3\,\left (a^2\,e^7\,m^2+9\,a^2\,e^7\,m+20\,a^2\,e^7+6\,a\,c\,d^2\,e^5\,m^2+46\,a\,c\,d^2\,e^5\,m+80\,a\,c\,d^2\,e^5+3\,c^2\,d^4\,e^3\,m^2+15\,c^2\,d^4\,e^3\,m+20\,c^2\,d^4\,e^3\right )}{e^3\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d^3\,\left (a^2\,e^4\,m^2+9\,a^2\,e^4\,m+20\,a^2\,e^4-2\,a\,c\,d^2\,e^2\,m-10\,a\,c\,d^2\,e^2+2\,c^2\,d^4\right )}{e^3\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d^2\,x\,\left (3\,a^2\,e^4\,m^2+27\,a^2\,e^4\,m+60\,a^2\,e^4+2\,a\,c\,d^2\,e^2\,m^2+10\,a\,c\,d^2\,e^2\,m-2\,c^2\,d^4\,m\right )}{e^2\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {d\,x^2\,\left (3\,a^2\,e^4\,m^2+27\,a^2\,e^4\,m+60\,a^2\,e^4+6\,a\,c\,d^2\,e^2\,m^2+42\,a\,c\,d^2\,e^2\,m+60\,a\,c\,d^2\,e^2+c^2\,d^4\,m^2+c^2\,d^4\,m\right )}{e\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {c^2\,d^2\,e^2\,x^5\,\left (m^2+7\,m+12\right )}{m^3+12\,m^2+47\,m+60}+\frac {c\,d\,e\,x^4\,\left (m+3\right )\,\left (10\,a\,e^2+10\,c\,d^2+2\,a\,e^2\,m+3\,c\,d^2\,m\right )}{m^3+12\,m^2+47\,m+60}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.70, size = 2494, normalized size = 27.71 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________