Optimal. Leaf size=138 \[ \frac {a^3 (d+e x)^4}{4 e}+\frac {a^2 b (d+e x)^6}{2 e}+\frac {3 a \left (b^2+a c\right ) (d+e x)^8}{8 e}+\frac {b \left (b^2+6 a c\right ) (d+e x)^{10}}{10 e}+\frac {c \left (b^2+a c\right ) (d+e x)^{12}}{4 e}+\frac {3 b c^2 (d+e x)^{14}}{14 e}+\frac {c^3 (d+e x)^{16}}{16 e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.25, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1156, 1128,
645} \begin {gather*} \frac {a^3 (d+e x)^4}{4 e}+\frac {a^2 b (d+e x)^6}{2 e}+\frac {c \left (a c+b^2\right ) (d+e x)^{12}}{4 e}+\frac {b \left (6 a c+b^2\right ) (d+e x)^{10}}{10 e}+\frac {3 a \left (a c+b^2\right ) (d+e x)^8}{8 e}+\frac {3 b c^2 (d+e x)^{14}}{14 e}+\frac {c^3 (d+e x)^{16}}{16 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 645
Rule 1128
Rule 1156
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^3 \, dx &=\frac {\text {Subst}\left (\int x^3 \left (a+b x^2+c x^4\right )^3 \, dx,x,d+e x\right )}{e}\\ &=\frac {\text {Subst}\left (\int x \left (a+b x+c x^2\right )^3 \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac {\text {Subst}\left (\int \left (a^3 x+3 a^2 b x^2+3 a \left (b^2+a c\right ) x^3+b \left (b^2+6 a c\right ) x^4+3 c \left (b^2+a c\right ) x^5+3 b c^2 x^6+c^3 x^7\right ) \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac {a^3 (d+e x)^4}{4 e}+\frac {a^2 b (d+e x)^6}{2 e}+\frac {3 a \left (b^2+a c\right ) (d+e x)^8}{8 e}+\frac {b \left (b^2+6 a c\right ) (d+e x)^{10}}{10 e}+\frac {c \left (b^2+a c\right ) (d+e x)^{12}}{4 e}+\frac {3 b c^2 (d+e x)^{14}}{14 e}+\frac {c^3 (d+e x)^{16}}{16 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(797\) vs. \(2(138)=276\).
time = 0.19, size = 797, normalized size = 5.78 \begin {gather*} d^3 \left (a+b d^2+c d^4\right )^3 x+\frac {3}{2} d^2 \left (a+b d^2+c d^4\right )^2 \left (a+3 b d^2+5 c d^4\right ) e x^2+d \left (a^3+10 a^2 b d^2+21 a b^2 d^4+21 a^2 c d^4+12 b^3 d^6+72 a b c d^6+55 b^2 c d^8+55 a c^2 d^8+78 b c^2 d^{10}+35 c^3 d^{12}\right ) e^2 x^3+\frac {1}{4} \left (a^3+30 a^2 b d^2+105 a b^2 d^4+105 a^2 c d^4+84 b^3 d^6+504 a b c d^6+495 b^2 c d^8+495 a c^2 d^8+858 b c^2 d^{10}+455 c^3 d^{12}\right ) e^3 x^4+\frac {3}{5} d \left (5 a^2 b+35 a b^2 d^2+35 a^2 c d^2+42 b^3 d^4+252 a b c d^4+330 b^2 c d^6+330 a c^2 d^6+715 b c^2 d^8+455 c^3 d^{10}\right ) e^4 x^5+\frac {1}{2} \left (a^2 b+21 a b^2 d^2+21 a^2 c d^2+42 b^3 d^4+252 a b c d^4+462 b^2 c d^6+462 a c^2 d^6+1287 b c^2 d^8+1001 c^3 d^{10}\right ) e^5 x^6+\frac {1}{7} d \left (21 a b^2+21 a^2 c+84 b^3 d^2+504 a b c d^2+1386 b^2 c d^4+1386 a c^2 d^4+5148 b c^2 d^6+5005 c^3 d^8\right ) e^6 x^7+\frac {3}{8} \left (a b^2+a^2 c+12 b^3 d^2+72 a b c d^2+330 b^2 c d^4+330 a c^2 d^4+1716 b c^2 d^6+2145 c^3 d^8\right ) e^7 x^8+d \left (b^3+6 a b c+55 b^2 c d^2+55 a c^2 d^2+429 b c^2 d^4+715 c^3 d^6\right ) e^8 x^9+\frac {1}{10} \left (b^3+6 a b c+165 b^2 c d^2+165 a c^2 d^2+2145 b c^2 d^4+5005 c^3 d^6\right ) e^9 x^{10}+3 c d \left (b^2+a c+26 b c d^2+91 c^2 d^4\right ) e^{10} x^{11}+\frac {1}{4} c \left (b^2+a c+78 b c d^2+455 c^2 d^4\right ) e^{11} x^{12}+c^2 d \left (3 b+35 c d^2\right ) e^{12} x^{13}+\frac {3}{14} c^2 \left (b+35 c d^2\right ) e^{13} x^{14}+c^3 d e^{14} x^{15}+\frac {1}{16} c^3 e^{15} x^{16} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(7549\) vs.
\(2(124)=248\).
time = 0.24, size = 7550, normalized size = 54.71 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1019 vs.
\(2 (124) = 248\).
time = 0.30, size = 1019, normalized size = 7.38 \begin {gather*} \frac {1}{16} \, c^{3} x^{16} e^{15} + c^{3} d x^{15} e^{14} + \frac {3}{14} \, {\left (35 \, c^{3} d^{2} e^{13} + b c^{2} e^{13}\right )} x^{14} + {\left (35 \, c^{3} d^{3} e^{12} + 3 \, b c^{2} d e^{12}\right )} x^{13} + \frac {1}{4} \, {\left (455 \, c^{3} d^{4} e^{11} + 78 \, b c^{2} d^{2} e^{11} + b^{2} c e^{11} + a c^{2} e^{11}\right )} x^{12} + 3 \, {\left (91 \, c^{3} d^{5} e^{10} + 26 \, b c^{2} d^{3} e^{10} + {\left (b^{2} c e^{10} + a c^{2} e^{10}\right )} d\right )} x^{11} + \frac {1}{10} \, {\left (5005 \, c^{3} d^{6} e^{9} + 2145 \, b c^{2} d^{4} e^{9} + b^{3} e^{9} + 6 \, a b c e^{9} + 165 \, {\left (b^{2} c e^{9} + a c^{2} e^{9}\right )} d^{2}\right )} x^{10} + {\left (715 \, c^{3} d^{7} e^{8} + 429 \, b c^{2} d^{5} e^{8} + 55 \, {\left (b^{2} c e^{8} + a c^{2} e^{8}\right )} d^{3} + {\left (b^{3} e^{8} + 6 \, a b c e^{8}\right )} d\right )} x^{9} + \frac {3}{8} \, {\left (2145 \, c^{3} d^{8} e^{7} + 1716 \, b c^{2} d^{6} e^{7} + 330 \, {\left (b^{2} c e^{7} + a c^{2} e^{7}\right )} d^{4} + a b^{2} e^{7} + a^{2} c e^{7} + 12 \, {\left (b^{3} e^{7} + 6 \, a b c e^{7}\right )} d^{2}\right )} x^{8} + \frac {1}{7} \, {\left (5005 \, c^{3} d^{9} e^{6} + 5148 \, b c^{2} d^{7} e^{6} + 1386 \, {\left (b^{2} c e^{6} + a c^{2} e^{6}\right )} d^{5} + 84 \, {\left (b^{3} e^{6} + 6 \, a b c e^{6}\right )} d^{3} + 21 \, {\left (a b^{2} e^{6} + a^{2} c e^{6}\right )} d\right )} x^{7} + \frac {1}{2} \, {\left (1001 \, c^{3} d^{10} e^{5} + 1287 \, b c^{2} d^{8} e^{5} + 462 \, {\left (b^{2} c e^{5} + a c^{2} e^{5}\right )} d^{6} + 42 \, {\left (b^{3} e^{5} + 6 \, a b c e^{5}\right )} d^{4} + a^{2} b e^{5} + 21 \, {\left (a b^{2} e^{5} + a^{2} c e^{5}\right )} d^{2}\right )} x^{6} + \frac {3}{5} \, {\left (455 \, c^{3} d^{11} e^{4} + 715 \, b c^{2} d^{9} e^{4} + 330 \, {\left (b^{2} c e^{4} + a c^{2} e^{4}\right )} d^{7} + 42 \, {\left (b^{3} e^{4} + 6 \, a b c e^{4}\right )} d^{5} + 5 \, a^{2} b d e^{4} + 35 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (455 \, c^{3} d^{12} e^{3} + 858 \, b c^{2} d^{10} e^{3} + 495 \, {\left (b^{2} c e^{3} + a c^{2} e^{3}\right )} d^{8} + 84 \, {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{6} + 30 \, a^{2} b d^{2} e^{3} + 105 \, {\left (a b^{2} e^{3} + a^{2} c e^{3}\right )} d^{4} + a^{3} e^{3}\right )} x^{4} + {\left (35 \, c^{3} d^{13} e^{2} + 78 \, b c^{2} d^{11} e^{2} + 55 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{9} + 12 \, {\left (b^{3} e^{2} + 6 \, a b c e^{2}\right )} d^{7} + 10 \, a^{2} b d^{3} e^{2} + 21 \, {\left (a b^{2} e^{2} + a^{2} c e^{2}\right )} d^{5} + a^{3} d e^{2}\right )} x^{3} + \frac {3}{2} \, {\left (5 \, c^{3} d^{14} e + 13 \, b c^{2} d^{12} e + 11 \, {\left (b^{2} c e + a c^{2} e\right )} d^{10} + 3 \, {\left (b^{3} e + 6 \, a b c e\right )} d^{8} + 5 \, a^{2} b d^{4} e + 7 \, {\left (a b^{2} e + a^{2} c e\right )} d^{6} + a^{3} d^{2} e\right )} x^{2} + {\left (c^{3} d^{15} + 3 \, b c^{2} d^{13} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{11} + {\left (b^{3} + 6 \, a b c\right )} d^{9} + 3 \, a^{2} b d^{5} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{7} + a^{3} d^{3}\right )} x \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. 859 vs.
\(2 (124) = 248\).
time = 0.34, size = 859, normalized size = 6.22 \begin {gather*} \frac {1}{16} \, c^{3} x^{16} e^{15} + c^{3} d x^{15} e^{14} + \frac {3}{14} \, {\left (35 \, c^{3} d^{2} + b c^{2}\right )} x^{14} e^{13} + {\left (35 \, c^{3} d^{3} + 3 \, b c^{2} d\right )} x^{13} e^{12} + \frac {1}{4} \, {\left (455 \, c^{3} d^{4} + 78 \, b c^{2} d^{2} + b^{2} c + a c^{2}\right )} x^{12} e^{11} + 3 \, {\left (91 \, c^{3} d^{5} + 26 \, b c^{2} d^{3} + {\left (b^{2} c + a c^{2}\right )} d\right )} x^{11} e^{10} + \frac {1}{10} \, {\left (5005 \, c^{3} d^{6} + 2145 \, b c^{2} d^{4} + b^{3} + 6 \, a b c + 165 \, {\left (b^{2} c + a c^{2}\right )} d^{2}\right )} x^{10} e^{9} + {\left (715 \, c^{3} d^{7} + 429 \, b c^{2} d^{5} + 55 \, {\left (b^{2} c + a c^{2}\right )} d^{3} + {\left (b^{3} + 6 \, a b c\right )} d\right )} x^{9} e^{8} + \frac {3}{8} \, {\left (2145 \, c^{3} d^{8} + 1716 \, b c^{2} d^{6} + 330 \, {\left (b^{2} c + a c^{2}\right )} d^{4} + a b^{2} + a^{2} c + 12 \, {\left (b^{3} + 6 \, a b c\right )} d^{2}\right )} x^{8} e^{7} + \frac {1}{7} \, {\left (5005 \, c^{3} d^{9} + 5148 \, b c^{2} d^{7} + 1386 \, {\left (b^{2} c + a c^{2}\right )} d^{5} + 84 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} + 21 \, {\left (a b^{2} + a^{2} c\right )} d\right )} x^{7} e^{6} + \frac {1}{2} \, {\left (1001 \, c^{3} d^{10} + 1287 \, b c^{2} d^{8} + 462 \, {\left (b^{2} c + a c^{2}\right )} d^{6} + 42 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} + a^{2} b + 21 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} x^{6} e^{5} + \frac {3}{5} \, {\left (455 \, c^{3} d^{11} + 715 \, b c^{2} d^{9} + 330 \, {\left (b^{2} c + a c^{2}\right )} d^{7} + 42 \, {\left (b^{3} + 6 \, a b c\right )} d^{5} + 5 \, a^{2} b d + 35 \, {\left (a b^{2} + a^{2} c\right )} d^{3}\right )} x^{5} e^{4} + \frac {1}{4} \, {\left (455 \, c^{3} d^{12} + 858 \, b c^{2} d^{10} + 495 \, {\left (b^{2} c + a c^{2}\right )} d^{8} + 84 \, {\left (b^{3} + 6 \, a b c\right )} d^{6} + 30 \, a^{2} b d^{2} + 105 \, {\left (a b^{2} + a^{2} c\right )} d^{4} + a^{3}\right )} x^{4} e^{3} + {\left (35 \, c^{3} d^{13} + 78 \, b c^{2} d^{11} + 55 \, {\left (b^{2} c + a c^{2}\right )} d^{9} + 12 \, {\left (b^{3} + 6 \, a b c\right )} d^{7} + 10 \, a^{2} b d^{3} + 21 \, {\left (a b^{2} + a^{2} c\right )} d^{5} + a^{3} d\right )} x^{3} e^{2} + \frac {3}{2} \, {\left (5 \, c^{3} d^{14} + 13 \, b c^{2} d^{12} + 11 \, {\left (b^{2} c + a c^{2}\right )} d^{10} + 3 \, {\left (b^{3} + 6 \, a b c\right )} d^{8} + 5 \, a^{2} b d^{4} + 7 \, {\left (a b^{2} + a^{2} c\right )} d^{6} + a^{3} d^{2}\right )} x^{2} e + {\left (c^{3} d^{15} + 3 \, b c^{2} d^{13} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{11} + {\left (b^{3} + 6 \, a b c\right )} d^{9} + 3 \, a^{2} b d^{5} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{7} + a^{3} d^{3}\right )} x \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. 1314 vs.
\(2 (117) = 234\).
time = 0.13, size = 1314, normalized size = 9.52 \begin {gather*} c^{3} d e^{14} x^{15} + \frac {c^{3} e^{15} x^{16}}{16} + x^{14} \cdot \left (\frac {3 b c^{2} e^{13}}{14} + \frac {15 c^{3} d^{2} e^{13}}{2}\right ) + x^{13} \cdot \left (3 b c^{2} d e^{12} + 35 c^{3} d^{3} e^{12}\right ) + x^{12} \left (\frac {a c^{2} e^{11}}{4} + \frac {b^{2} c e^{11}}{4} + \frac {39 b c^{2} d^{2} e^{11}}{2} + \frac {455 c^{3} d^{4} e^{11}}{4}\right ) + x^{11} \cdot \left (3 a c^{2} d e^{10} + 3 b^{2} c d e^{10} + 78 b c^{2} d^{3} e^{10} + 273 c^{3} d^{5} e^{10}\right ) + x^{10} \cdot \left (\frac {3 a b c e^{9}}{5} + \frac {33 a c^{2} d^{2} e^{9}}{2} + \frac {b^{3} e^{9}}{10} + \frac {33 b^{2} c d^{2} e^{9}}{2} + \frac {429 b c^{2} d^{4} e^{9}}{2} + \frac {1001 c^{3} d^{6} e^{9}}{2}\right ) + x^{9} \cdot \left (6 a b c d e^{8} + 55 a c^{2} d^{3} e^{8} + b^{3} d e^{8} + 55 b^{2} c d^{3} e^{8} + 429 b c^{2} d^{5} e^{8} + 715 c^{3} d^{7} e^{8}\right ) + x^{8} \cdot \left (\frac {3 a^{2} c e^{7}}{8} + \frac {3 a b^{2} e^{7}}{8} + 27 a b c d^{2} e^{7} + \frac {495 a c^{2} d^{4} e^{7}}{4} + \frac {9 b^{3} d^{2} e^{7}}{2} + \frac {495 b^{2} c d^{4} e^{7}}{4} + \frac {1287 b c^{2} d^{6} e^{7}}{2} + \frac {6435 c^{3} d^{8} e^{7}}{8}\right ) + x^{7} \cdot \left (3 a^{2} c d e^{6} + 3 a b^{2} d e^{6} + 72 a b c d^{3} e^{6} + 198 a c^{2} d^{5} e^{6} + 12 b^{3} d^{3} e^{6} + 198 b^{2} c d^{5} e^{6} + \frac {5148 b c^{2} d^{7} e^{6}}{7} + 715 c^{3} d^{9} e^{6}\right ) + x^{6} \left (\frac {a^{2} b e^{5}}{2} + \frac {21 a^{2} c d^{2} e^{5}}{2} + \frac {21 a b^{2} d^{2} e^{5}}{2} + 126 a b c d^{4} e^{5} + 231 a c^{2} d^{6} e^{5} + 21 b^{3} d^{4} e^{5} + 231 b^{2} c d^{6} e^{5} + \frac {1287 b c^{2} d^{8} e^{5}}{2} + \frac {1001 c^{3} d^{10} e^{5}}{2}\right ) + x^{5} \cdot \left (3 a^{2} b d e^{4} + 21 a^{2} c d^{3} e^{4} + 21 a b^{2} d^{3} e^{4} + \frac {756 a b c d^{5} e^{4}}{5} + 198 a c^{2} d^{7} e^{4} + \frac {126 b^{3} d^{5} e^{4}}{5} + 198 b^{2} c d^{7} e^{4} + 429 b c^{2} d^{9} e^{4} + 273 c^{3} d^{11} e^{4}\right ) + x^{4} \left (\frac {a^{3} e^{3}}{4} + \frac {15 a^{2} b d^{2} e^{3}}{2} + \frac {105 a^{2} c d^{4} e^{3}}{4} + \frac {105 a b^{2} d^{4} e^{3}}{4} + 126 a b c d^{6} e^{3} + \frac {495 a c^{2} d^{8} e^{3}}{4} + 21 b^{3} d^{6} e^{3} + \frac {495 b^{2} c d^{8} e^{3}}{4} + \frac {429 b c^{2} d^{10} e^{3}}{2} + \frac {455 c^{3} d^{12} e^{3}}{4}\right ) + x^{3} \left (a^{3} d e^{2} + 10 a^{2} b d^{3} e^{2} + 21 a^{2} c d^{5} e^{2} + 21 a b^{2} d^{5} e^{2} + 72 a b c d^{7} e^{2} + 55 a c^{2} d^{9} e^{2} + 12 b^{3} d^{7} e^{2} + 55 b^{2} c d^{9} e^{2} + 78 b c^{2} d^{11} e^{2} + 35 c^{3} d^{13} e^{2}\right ) + x^{2} \cdot \left (\frac {3 a^{3} d^{2} e}{2} + \frac {15 a^{2} b d^{4} e}{2} + \frac {21 a^{2} c d^{6} e}{2} + \frac {21 a b^{2} d^{6} e}{2} + 27 a b c d^{8} e + \frac {33 a c^{2} d^{10} e}{2} + \frac {9 b^{3} d^{8} e}{2} + \frac {33 b^{2} c d^{10} e}{2} + \frac {39 b c^{2} d^{12} e}{2} + \frac {15 c^{3} d^{14} e}{2}\right ) + x \left (a^{3} d^{3} + 3 a^{2} b d^{5} + 3 a^{2} c d^{7} + 3 a b^{2} d^{7} + 6 a b c d^{9} + 3 a c^{2} d^{11} + b^{3} d^{9} + 3 b^{2} c d^{11} + 3 b c^{2} d^{13} + c^{3} d^{15}\right ) \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. 1109 vs.
\(2 (124) = 248\).
time = 4.05, size = 1109, normalized size = 8.04 \begin {gather*} \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} c^{3} d^{14} + \frac {7}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} c^{3} d^{12} e + \frac {7}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{3} c^{3} d^{10} e^{2} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )} b c^{2} d^{12} + \frac {35}{8} \, {\left (x^{2} e + 2 \, d x\right )}^{4} c^{3} d^{8} e^{3} + \frac {9}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{2} b c^{2} d^{10} e + \frac {7}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{5} c^{3} d^{6} e^{4} + \frac {15}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{3} b c^{2} d^{8} e^{2} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )} b^{2} c d^{10} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )} a c^{2} d^{10} + \frac {7}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{6} c^{3} d^{4} e^{5} + \frac {15}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{4} b c^{2} d^{6} e^{3} + \frac {15}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} b^{2} c d^{8} e + \frac {15}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} a c^{2} d^{8} e + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{7} c^{3} d^{2} e^{6} + \frac {9}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{5} b c^{2} d^{4} e^{4} + 5 \, {\left (x^{2} e + 2 \, d x\right )}^{3} b^{2} c d^{6} e^{2} + 5 \, {\left (x^{2} e + 2 \, d x\right )}^{3} a c^{2} d^{6} e^{2} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} b^{3} d^{8} + 3 \, {\left (x^{2} e + 2 \, d x\right )} a b c d^{8} + \frac {1}{16} \, {\left (x^{2} e + 2 \, d x\right )}^{8} c^{3} e^{7} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{6} b c^{2} d^{2} e^{5} + \frac {15}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{4} b^{2} c d^{4} e^{3} + \frac {15}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{4} a c^{2} d^{4} e^{3} + {\left (x^{2} e + 2 \, d x\right )}^{2} b^{3} d^{6} e + 6 \, {\left (x^{2} e + 2 \, d x\right )}^{2} a b c d^{6} e + \frac {3}{14} \, {\left (x^{2} e + 2 \, d x\right )}^{7} b c^{2} e^{6} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{5} b^{2} c d^{2} e^{4} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{5} a c^{2} d^{2} e^{4} + {\left (x^{2} e + 2 \, d x\right )}^{3} b^{3} d^{4} e^{2} + 6 \, {\left (x^{2} e + 2 \, d x\right )}^{3} a b c d^{4} e^{2} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )} a b^{2} d^{6} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )} a^{2} c d^{6} + \frac {1}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{6} b^{2} c e^{5} + \frac {1}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{6} a c^{2} e^{5} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{4} b^{3} d^{2} e^{3} + 3 \, {\left (x^{2} e + 2 \, d x\right )}^{4} a b c d^{2} e^{3} + \frac {9}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} a b^{2} d^{4} e + \frac {9}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} a^{2} c d^{4} e + \frac {1}{10} \, {\left (x^{2} e + 2 \, d x\right )}^{5} b^{3} e^{4} + \frac {3}{5} \, {\left (x^{2} e + 2 \, d x\right )}^{5} a b c e^{4} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{3} a b^{2} d^{2} e^{2} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{3} a^{2} c d^{2} e^{2} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )} a^{2} b d^{4} + \frac {3}{8} \, {\left (x^{2} e + 2 \, d x\right )}^{4} a b^{2} e^{3} + \frac {3}{8} \, {\left (x^{2} e + 2 \, d x\right )}^{4} a^{2} c e^{3} + \frac {3}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{2} a^{2} b d^{2} e + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )}^{3} a^{2} b e^{2} + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} a^{3} d^{2} + \frac {1}{4} \, {\left (x^{2} e + 2 \, d x\right )}^{2} a^{3} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.66, size = 777, normalized size = 5.63 \begin {gather*} \frac {3\,e^7\,x^8\,\left (a^2\,c+a\,b^2+72\,a\,b\,c\,d^2+330\,a\,c^2\,d^4+12\,b^3\,d^2+330\,b^2\,c\,d^4+1716\,b\,c^2\,d^6+2145\,c^3\,d^8\right )}{8}+\frac {e^5\,x^6\,\left (a^2\,b+21\,a^2\,c\,d^2+21\,a\,b^2\,d^2+252\,a\,b\,c\,d^4+462\,a\,c^2\,d^6+42\,b^3\,d^4+462\,b^2\,c\,d^6+1287\,b\,c^2\,d^8+1001\,c^3\,d^{10}\right )}{2}+\frac {e^9\,x^{10}\,\left (b^3+165\,b^2\,c\,d^2+2145\,b\,c^2\,d^4+6\,a\,b\,c+5005\,c^3\,d^6+165\,a\,c^2\,d^2\right )}{10}+\frac {c^3\,e^{15}\,x^{16}}{16}+d^3\,x\,{\left (c\,d^4+b\,d^2+a\right )}^3+\frac {e^3\,x^4\,\left (a^3+30\,a^2\,b\,d^2+105\,a^2\,c\,d^4+105\,a\,b^2\,d^4+504\,a\,b\,c\,d^6+495\,a\,c^2\,d^8+84\,b^3\,d^6+495\,b^2\,c\,d^8+858\,b\,c^2\,d^{10}+455\,c^3\,d^{12}\right )}{4}+\frac {3\,c^2\,e^{13}\,x^{14}\,\left (35\,c\,d^2+b\right )}{14}+c^3\,d\,e^{14}\,x^{15}+d\,e^2\,x^3\,\left (a^3+10\,a^2\,b\,d^2+21\,a^2\,c\,d^4+21\,a\,b^2\,d^4+72\,a\,b\,c\,d^6+55\,a\,c^2\,d^8+12\,b^3\,d^6+55\,b^2\,c\,d^8+78\,b\,c^2\,d^{10}+35\,c^3\,d^{12}\right )+\frac {c\,e^{11}\,x^{12}\,\left (b^2+78\,b\,c\,d^2+455\,c^2\,d^4+a\,c\right )}{4}+\frac {d\,e^6\,x^7\,\left (21\,a^2\,c+21\,a\,b^2+504\,a\,b\,c\,d^2+1386\,a\,c^2\,d^4+84\,b^3\,d^2+1386\,b^2\,c\,d^4+5148\,b\,c^2\,d^6+5005\,c^3\,d^8\right )}{7}+\frac {3\,d\,e^4\,x^5\,\left (5\,a^2\,b+35\,a^2\,c\,d^2+35\,a\,b^2\,d^2+252\,a\,b\,c\,d^4+330\,a\,c^2\,d^6+42\,b^3\,d^4+330\,b^2\,c\,d^6+715\,b\,c^2\,d^8+455\,c^3\,d^{10}\right )}{5}+d\,e^8\,x^9\,\left (b^3+55\,b^2\,c\,d^2+429\,b\,c^2\,d^4+6\,a\,b\,c+715\,c^3\,d^6+55\,a\,c^2\,d^2\right )+\frac {3\,d^2\,e\,x^2\,{\left (c\,d^4+b\,d^2+a\right )}^2\,\left (5\,c\,d^4+3\,b\,d^2+a\right )}{2}+c^2\,d\,e^{12}\,x^{13}\,\left (35\,c\,d^2+3\,b\right )+3\,c\,d\,e^{10}\,x^{11}\,\left (b^2+26\,b\,c\,d^2+91\,c^2\,d^4+a\,c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________