Optimal. Leaf size=140 \[ \frac {\left (c d^2+a e^2\right )^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {4 c^2 d (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^5 (5+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711}
\begin {gather*} \frac {\left (a e^2+c d^2\right )^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac {4 c d \left (a e^2+c d^2\right ) (d+e x)^{m+2}}{e^5 (m+2)}+\frac {2 c \left (a e^2+3 c d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}-\frac {4 c^2 d (d+e x)^{m+4}}{e^5 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^5 (m+5)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 711
Rubi steps
\begin {align*} \int (d+e x)^m \left (a+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2 (d+e x)^m}{e^4}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^{1+m}}{e^4}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^{2+m}}{e^4}-\frac {4 c^2 d (d+e x)^{3+m}}{e^4}+\frac {c^2 (d+e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right )^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {4 c^2 d (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^5 (5+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 194, normalized size = 1.39 \begin {gather*} \frac {(d+e x)^{1+m} \left (a^2 e^4 \left (120+154 m+71 m^2+14 m^3+m^4\right )+2 a c e^2 \left (20+9 m+m^2\right ) \left (2 d^2-2 d e (1+m) x+e^2 \left (2+3 m+m^2\right ) x^2\right )+c^2 \left (24 d^4-24 d^3 e (1+m) x+12 d^2 e^2 \left (2+3 m+m^2\right ) x^2-4 d e^3 \left (6+11 m+6 m^2+m^3\right ) x^3+e^4 \left (24+50 m+35 m^2+10 m^3+m^4\right ) x^4\right )\right )}{e^5 (1+m) (2+m) (3+m) (4+m) (5+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(419\) vs.
\(2(140)=280\).
time = 0.44, size = 420, normalized size = 3.00
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{1+m} \left (c^{2} e^{4} m^{4} x^{4}+10 c^{2} e^{4} m^{3} x^{4}+2 a c \,e^{4} m^{4} x^{2}-4 c^{2} d \,e^{3} m^{3} x^{3}+35 c^{2} e^{4} m^{2} x^{4}+24 a c \,e^{4} m^{3} x^{2}-24 c^{2} d \,e^{3} m^{2} x^{3}+50 c^{2} e^{4} m \,x^{4}+a^{2} e^{4} m^{4}-4 a c d \,e^{3} m^{3} x +98 a c \,e^{4} m^{2} x^{2}+12 c^{2} d^{2} e^{2} m^{2} x^{2}-44 c^{2} d \,e^{3} m \,x^{3}+24 c^{2} e^{4} x^{4}+14 a^{2} e^{4} m^{3}-40 a c d \,e^{3} m^{2} x +156 a c \,e^{4} m \,x^{2}+36 c^{2} d^{2} e^{2} m \,x^{2}-24 c^{2} d \,x^{3} e^{3}+71 a^{2} e^{4} m^{2}+4 a c \,d^{2} e^{2} m^{2}-116 a c d \,e^{3} m x +80 a c \,e^{4} x^{2}-24 c^{2} d^{3} e m x +24 d^{2} e^{2} x^{2} c^{2}+154 a^{2} e^{4} m +36 a c \,d^{2} e^{2} m -80 a c d \,e^{3} x -24 c^{2} d^{3} e x +120 a^{2} e^{4}+80 a c \,d^{2} e^{2}+24 c^{2} d^{4}\right )}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\) | \(420\) |
norman | \(\frac {c^{2} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {d \left (a^{2} e^{4} m^{4}+14 a^{2} e^{4} m^{3}+71 a^{2} e^{4} m^{2}+4 a c \,d^{2} e^{2} m^{2}+154 a^{2} e^{4} m +36 a c \,d^{2} e^{2} m +120 a^{2} e^{4}+80 a c \,d^{2} e^{2}+24 c^{2} d^{4}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {\left (a^{2} e^{4} m^{4}+14 a^{2} e^{4} m^{3}-4 a c \,d^{2} e^{2} m^{3}+71 a^{2} e^{4} m^{2}-36 a c \,d^{2} e^{2} m^{2}+154 a^{2} e^{4} m -80 a c \,d^{2} e^{2} m -24 c^{2} d^{4} m +120 a^{2} e^{4}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {c^{2} d m \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+9 m +20\right )}+\frac {2 \left (a \,e^{2} m^{2}+9 a \,e^{2} m -2 c \,d^{2} m +20 e^{2} a \right ) c \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {2 \left (a \,e^{2} m^{2}+9 a \,e^{2} m +20 e^{2} a +6 c \,d^{2}\right ) c d m \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right )}\) | \(450\) |
risch | \(\frac {\left (c^{2} e^{5} m^{4} x^{5}+c^{2} d \,e^{4} m^{4} x^{4}+10 c^{2} e^{5} m^{3} x^{5}+2 a c \,e^{5} m^{4} x^{3}+6 c^{2} d \,e^{4} m^{3} x^{4}+35 c^{2} e^{5} m^{2} x^{5}+2 a c d \,e^{4} m^{4} x^{2}+24 a c \,e^{5} m^{3} x^{3}-4 c^{2} d^{2} e^{3} m^{3} x^{3}+11 c^{2} d \,e^{4} m^{2} x^{4}+50 c^{2} e^{5} m \,x^{5}+a^{2} e^{5} m^{4} x +20 a c d \,e^{4} m^{3} x^{2}+98 a c \,e^{5} m^{2} x^{3}-12 c^{2} d^{2} e^{3} m^{2} x^{3}+6 c^{2} d m \,x^{4} e^{4}+24 c^{2} x^{5} e^{5}+a^{2} d \,e^{4} m^{4}+14 a^{2} e^{5} m^{3} x -4 a c \,d^{2} e^{3} m^{3} x +58 a c d \,e^{4} m^{2} x^{2}+156 a c \,e^{5} m \,x^{3}+12 c^{2} d^{3} e^{2} m^{2} x^{2}-8 c^{2} d^{2} e^{3} m \,x^{3}+14 a^{2} d \,e^{4} m^{3}+71 a^{2} e^{5} m^{2} x -36 a c \,d^{2} e^{3} m^{2} x +40 a c d \,e^{4} m \,x^{2}+80 a c \,e^{5} x^{3}+12 c^{2} d^{3} e^{2} m \,x^{2}+71 a^{2} d \,e^{4} m^{2}+154 a^{2} e^{5} m x +4 a c \,d^{3} e^{2} m^{2}-80 a c \,d^{2} e^{3} m x -24 c^{2} d^{4} e m x +154 a^{2} d \,e^{4} m +120 a^{2} e^{5} x +36 a c \,d^{3} e^{2} m +120 a^{2} d \,e^{4}+80 a c \,d^{3} e^{2}+24 c^{2} d^{5}\right ) \left (e x +d \right )^{m}}{\left (4+m \right ) \left (5+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) e^{5}}\) | \(555\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.34, size = 235, normalized size = 1.68 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} a^{2} e^{\left (-1\right )}}{m + 1} + \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} x^{3} e^{3} + {\left (m^{2} + m\right )} d x^{2} e^{2} - 2 \, d^{2} m x e + 2 \, d^{3}\right )} a c e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} e^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d x^{4} e^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} x^{3} e^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} x^{2} e^{2} - 24 \, d^{4} m x e + 24 \, d^{5}\right )} c^{2} e^{\left (m \log \left (x e + d\right ) - 5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \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. 414 vs.
\(2 (137) = 274\).
time = 3.04, size = 414, normalized size = 2.96 \begin {gather*} -\frac {{\left (24 \, c^{2} d^{4} m x e - 24 \, c^{2} d^{5} - {\left ({\left (c^{2} m^{4} + 10 \, c^{2} m^{3} + 35 \, c^{2} m^{2} + 50 \, c^{2} m + 24 \, c^{2}\right )} x^{5} + 2 \, {\left (a c m^{4} + 12 \, a c m^{3} + 49 \, a c m^{2} + 78 \, a c m + 40 \, a c\right )} x^{3} + {\left (a^{2} m^{4} + 14 \, a^{2} m^{3} + 71 \, a^{2} m^{2} + 154 \, a^{2} m + 120 \, a^{2}\right )} x\right )} e^{5} - {\left (a^{2} d m^{4} + 14 \, a^{2} d m^{3} + 71 \, a^{2} d m^{2} + {\left (c^{2} d m^{4} + 6 \, c^{2} d m^{3} + 11 \, c^{2} d m^{2} + 6 \, c^{2} d m\right )} x^{4} + 154 \, a^{2} d m + 120 \, a^{2} d + 2 \, {\left (a c d m^{4} + 10 \, a c d m^{3} + 29 \, a c d m^{2} + 20 \, a c d m\right )} x^{2}\right )} e^{4} + 4 \, {\left ({\left (c^{2} d^{2} m^{3} + 3 \, c^{2} d^{2} m^{2} + 2 \, c^{2} d^{2} m\right )} x^{3} + {\left (a c d^{2} m^{3} + 9 \, a c d^{2} m^{2} + 20 \, a c d^{2} m\right )} x\right )} e^{3} - 4 \, {\left (a c d^{3} m^{2} + 9 \, a c d^{3} m + 20 \, a c d^{3} + 3 \, {\left (c^{2} d^{3} m^{2} + c^{2} d^{3} m\right )} x^{2}\right )} e^{2}\right )} {\left (x e + d\right )}^{m} e^{\left (-5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 5097 vs.
\(2 (128) = 256\).
time = 1.54, size = 5097, normalized size = 36.41 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 848 vs.
\(2 (137) = 274\).
time = 1.80, size = 848, normalized size = 6.06 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c^{2} m^{4} x^{5} e^{5} + {\left (x e + d\right )}^{m} c^{2} d m^{4} x^{4} e^{4} + 10 \, {\left (x e + d\right )}^{m} c^{2} m^{3} x^{5} e^{5} + 6 \, {\left (x e + d\right )}^{m} c^{2} d m^{3} x^{4} e^{4} - 4 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{3} x^{3} e^{3} + 2 \, {\left (x e + d\right )}^{m} a c m^{4} x^{3} e^{5} + 35 \, {\left (x e + d\right )}^{m} c^{2} m^{2} x^{5} e^{5} + 2 \, {\left (x e + d\right )}^{m} a c d m^{4} x^{2} e^{4} + 11 \, {\left (x e + d\right )}^{m} c^{2} d m^{2} x^{4} e^{4} - 12 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{2} x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m^{2} x^{2} e^{2} + 24 \, {\left (x e + d\right )}^{m} a c m^{3} x^{3} e^{5} + 50 \, {\left (x e + d\right )}^{m} c^{2} m x^{5} e^{5} + 20 \, {\left (x e + d\right )}^{m} a c d m^{3} x^{2} e^{4} + 6 \, {\left (x e + d\right )}^{m} c^{2} d m x^{4} e^{4} - 4 \, {\left (x e + d\right )}^{m} a c d^{2} m^{3} x e^{3} - 8 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m x^{2} e^{2} - 24 \, {\left (x e + d\right )}^{m} c^{2} d^{4} m x e + {\left (x e + d\right )}^{m} a^{2} m^{4} x e^{5} + 98 \, {\left (x e + d\right )}^{m} a c m^{2} x^{3} e^{5} + 24 \, {\left (x e + d\right )}^{m} c^{2} x^{5} e^{5} + {\left (x e + d\right )}^{m} a^{2} d m^{4} e^{4} + 58 \, {\left (x e + d\right )}^{m} a c d m^{2} x^{2} e^{4} - 36 \, {\left (x e + d\right )}^{m} a c d^{2} m^{2} x e^{3} + 4 \, {\left (x e + d\right )}^{m} a c d^{3} m^{2} e^{2} + 24 \, {\left (x e + d\right )}^{m} c^{2} d^{5} + 14 \, {\left (x e + d\right )}^{m} a^{2} m^{3} x e^{5} + 156 \, {\left (x e + d\right )}^{m} a c m x^{3} e^{5} + 14 \, {\left (x e + d\right )}^{m} a^{2} d m^{3} e^{4} + 40 \, {\left (x e + d\right )}^{m} a c d m x^{2} e^{4} - 80 \, {\left (x e + d\right )}^{m} a c d^{2} m x e^{3} + 36 \, {\left (x e + d\right )}^{m} a c d^{3} m e^{2} + 71 \, {\left (x e + d\right )}^{m} a^{2} m^{2} x e^{5} + 80 \, {\left (x e + d\right )}^{m} a c x^{3} e^{5} + 71 \, {\left (x e + d\right )}^{m} a^{2} d m^{2} e^{4} + 80 \, {\left (x e + d\right )}^{m} a c d^{3} e^{2} + 154 \, {\left (x e + d\right )}^{m} a^{2} m x e^{5} + 154 \, {\left (x e + d\right )}^{m} a^{2} d m e^{4} + 120 \, {\left (x e + d\right )}^{m} a^{2} x e^{5} + 120 \, {\left (x e + d\right )}^{m} a^{2} d e^{4}}{m^{5} e^{5} + 15 \, m^{4} e^{5} + 85 \, m^{3} e^{5} + 225 \, m^{2} e^{5} + 274 \, m e^{5} + 120 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.70, size = 496, normalized size = 3.54 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {c^2\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {d\,\left (a^2\,e^4\,m^4+14\,a^2\,e^4\,m^3+71\,a^2\,e^4\,m^2+154\,a^2\,e^4\,m+120\,a^2\,e^4+4\,a\,c\,d^2\,e^2\,m^2+36\,a\,c\,d^2\,e^2\,m+80\,a\,c\,d^2\,e^2+24\,c^2\,d^4\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x\,\left (a^2\,e^5\,m^4+14\,a^2\,e^5\,m^3+71\,a^2\,e^5\,m^2+154\,a^2\,e^5\,m+120\,a^2\,e^5-4\,a\,c\,d^2\,e^3\,m^3-36\,a\,c\,d^2\,e^3\,m^2-80\,a\,c\,d^2\,e^3\,m-24\,c^2\,d^4\,e\,m\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {2\,c\,x^3\,\left (m^2+3\,m+2\right )\,\left (-2\,c\,d^2\,m+a\,e^2\,m^2+9\,a\,e^2\,m+20\,a\,e^2\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c^2\,d\,m\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {2\,c\,d\,m\,x^2\,\left (m+1\right )\,\left (6\,c\,d^2+a\,e^2\,m^2+9\,a\,e^2\,m+20\,a\,e^2\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________