Optimal. Leaf size=23 \[ \frac {\left (a+b (c+d x)^4\right )^3}{12 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {372, 261} \[ \frac {\left (a+b (c+d x)^4\right )^3}{12 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 372
Rubi steps
\begin {align*} \int (c+d x)^3 \left (a+b (c+d x)^4\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int x^3 \left (a+b x^4\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {\left (a+b (c+d x)^4\right )^3}{12 b d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.04, size = 172, normalized size = 7.48 \[ \frac {1}{12} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \left (3 a^2+3 a b \left (2 c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right )+b^2 \left (3 c^8+12 c^7 d x+34 c^6 d^2 x^2+60 c^5 d^3 x^3+71 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.89, size = 289, normalized size = 12.57 \[ \frac {1}{12} x^{12} d^{11} b^{2} + x^{11} d^{10} c b^{2} + \frac {11}{2} x^{10} d^{9} c^{2} b^{2} + \frac {55}{3} x^{9} d^{8} c^{3} b^{2} + \frac {165}{4} x^{8} d^{7} c^{4} b^{2} + 66 x^{7} d^{6} c^{5} b^{2} + 77 x^{6} d^{5} c^{6} b^{2} + 66 x^{5} d^{4} c^{7} b^{2} + \frac {165}{4} x^{4} d^{3} c^{8} b^{2} + \frac {1}{4} x^{8} d^{7} b a + \frac {55}{3} x^{3} d^{2} c^{9} b^{2} + 2 x^{7} d^{6} c b a + \frac {11}{2} x^{2} d c^{10} b^{2} + 7 x^{6} d^{5} c^{2} b a + x c^{11} b^{2} + 14 x^{5} d^{4} c^{3} b a + \frac {35}{2} x^{4} d^{3} c^{4} b a + 14 x^{3} d^{2} c^{5} b a + 7 x^{2} d c^{6} b a + 2 x c^{7} b a + \frac {1}{4} x^{4} d^{3} a^{2} + x^{3} d^{2} c a^{2} + \frac {3}{2} x^{2} d c^{2} a^{2} + x c^{3} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 21, normalized size = 0.91 \[ \frac {{\left ({\left (d x + c\right )}^{4} b + a\right )}^{3}}{12 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.00, size = 622, normalized size = 27.04 \[ \frac {b^{2} d^{11} x^{12}}{12}+b^{2} c \,d^{10} x^{11}+\frac {11 b^{2} c^{2} d^{9} x^{10}}{2}+\frac {55 b^{2} c^{3} d^{8} x^{9}}{3}+\frac {\left (260 b^{2} c^{4} d^{7}+\left (68 b^{2} c^{4} d^{4}+2 \left (b \,c^{4}+a \right ) b \,d^{4}\right ) d^{3}\right ) x^{8}}{8}+\frac {\left (196 b^{2} c^{5} d^{6}+3 \left (68 b^{2} c^{4} d^{4}+2 \left (b \,c^{4}+a \right ) b \,d^{4}\right ) c \,d^{2}+\left (48 b^{2} c^{5} d^{3}+8 \left (b \,c^{4}+a \right ) b c \,d^{3}\right ) d^{3}\right ) x^{7}}{7}+\frac {\left (56 b^{2} c^{6} d^{5}+3 \left (68 b^{2} c^{4} d^{4}+2 \left (b \,c^{4}+a \right ) b \,d^{4}\right ) c^{2} d +3 \left (48 b^{2} c^{5} d^{3}+8 \left (b \,c^{4}+a \right ) b c \,d^{3}\right ) c \,d^{2}+\left (16 b^{2} c^{6} d^{2}+12 \left (b \,c^{4}+a \right ) b \,c^{2} d^{2}\right ) d^{3}\right ) x^{6}}{6}+\frac {\left (8 \left (b \,c^{4}+a \right ) b \,c^{3} d^{4}+\left (68 b^{2} c^{4} d^{4}+2 \left (b \,c^{4}+a \right ) b \,d^{4}\right ) c^{3}+3 \left (48 b^{2} c^{5} d^{3}+8 \left (b \,c^{4}+a \right ) b c \,d^{3}\right ) c^{2} d +3 \left (16 b^{2} c^{6} d^{2}+12 \left (b \,c^{4}+a \right ) b \,c^{2} d^{2}\right ) c \,d^{2}\right ) x^{5}}{5}+\left (b \,c^{4}+a \right )^{2} c^{3} x +\frac {\left (24 \left (b \,c^{4}+a \right ) b \,c^{4} d^{3}+\left (48 b^{2} c^{5} d^{3}+8 \left (b \,c^{4}+a \right ) b c \,d^{3}\right ) c^{3}+3 \left (16 b^{2} c^{6} d^{2}+12 \left (b \,c^{4}+a \right ) b \,c^{2} d^{2}\right ) c^{2} d +\left (b \,c^{4}+a \right )^{2} d^{3}\right ) x^{4}}{4}+\frac {\left (24 \left (b \,c^{4}+a \right ) b \,c^{5} d^{2}+\left (16 b^{2} c^{6} d^{2}+12 \left (b \,c^{4}+a \right ) b \,c^{2} d^{2}\right ) c^{3}+3 \left (b \,c^{4}+a \right )^{2} c \,d^{2}\right ) x^{3}}{3}+\frac {\left (8 \left (b \,c^{4}+a \right ) b \,c^{6} d +3 \left (b \,c^{4}+a \right )^{2} c^{2} d \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.55, size = 21, normalized size = 0.91 \[ \frac {{\left ({\left (d x + c\right )}^{4} b + a\right )}^{3}}{12 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.15, size = 229, normalized size = 9.96 \[ c^3\,x\,{\left (b\,c^4+a\right )}^2+\frac {b^2\,d^{11}\,x^{12}}{12}+\frac {d^3\,x^4\,\left (a^2+70\,a\,b\,c^4+165\,b^2\,c^8\right )}{4}+b^2\,c\,d^{10}\,x^{11}+\frac {55\,b^2\,c^3\,d^8\,x^9}{3}+\frac {11\,b^2\,c^2\,d^9\,x^{10}}{2}+\frac {b\,d^7\,x^8\,\left (165\,b\,c^4+a\right )}{4}+\frac {c^2\,d\,x^2\,\left (3\,a^2+14\,a\,b\,c^4+11\,b^2\,c^8\right )}{2}+\frac {c\,d^2\,x^3\,\left (3\,a^2+42\,a\,b\,c^4+55\,b^2\,c^8\right )}{3}+2\,b\,c\,d^6\,x^7\,\left (33\,b\,c^4+a\right )+7\,b\,c^2\,d^5\,x^6\,\left (11\,b\,c^4+a\right )+2\,b\,c^3\,d^4\,x^5\,\left (33\,b\,c^4+7\,a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.14, size = 299, normalized size = 13.00 \[ \frac {55 b^{2} c^{3} d^{8} x^{9}}{3} + \frac {11 b^{2} c^{2} d^{9} x^{10}}{2} + b^{2} c d^{10} x^{11} + \frac {b^{2} d^{11} x^{12}}{12} + x^{8} \left (\frac {a b d^{7}}{4} + \frac {165 b^{2} c^{4} d^{7}}{4}\right ) + x^{7} \left (2 a b c d^{6} + 66 b^{2} c^{5} d^{6}\right ) + x^{6} \left (7 a b c^{2} d^{5} + 77 b^{2} c^{6} d^{5}\right ) + x^{5} \left (14 a b c^{3} d^{4} + 66 b^{2} c^{7} d^{4}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4} + \frac {35 a b c^{4} d^{3}}{2} + \frac {165 b^{2} c^{8} d^{3}}{4}\right ) + x^{3} \left (a^{2} c d^{2} + 14 a b c^{5} d^{2} + \frac {55 b^{2} c^{9} d^{2}}{3}\right ) + x^{2} \left (\frac {3 a^{2} c^{2} d}{2} + 7 a b c^{6} d + \frac {11 b^{2} c^{10} d}{2}\right ) + x \left (a^{2} c^{3} + 2 a b c^{7} + b^{2} c^{11}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________