Optimal. Leaf size=73 \[ -\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^7}{112 c^3}+\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{160 c^3}+\frac {d^4 (b+2 c x)^9}{288 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {683} \[ -\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^7}{112 c^3}+\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{160 c^3}+\frac {d^4 (b+2 c x)^9}{288 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 683
Rubi steps
\begin {align*} \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2 (b d+2 c d x)^4}{16 c^2}+\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^6}{8 c^2 d^2}+\frac {(b d+2 c d x)^8}{16 c^2 d^4}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5}{160 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^7}{112 c^3}+\frac {d^4 (b+2 c x)^9}{288 c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.03, size = 179, normalized size = 2.45 \[ d^4 \left (a^2 b^4 x+\frac {1}{5} c^2 x^5 \left (16 a^2 c^2+112 a b^2 c+41 b^4\right )+\frac {1}{2} b c x^4 \left (16 a^2 c^2+32 a b^2 c+5 b^4\right )+\frac {1}{3} b^2 x^3 \left (24 a^2 c^2+18 a b^2 c+b^4\right )+\frac {8}{7} c^4 x^7 \left (4 a c+13 b^2\right )+\frac {4}{3} b c^3 x^6 \left (12 a c+11 b^2\right )+a b^3 x^2 \left (4 a c+b^2\right )+8 b c^5 x^8+\frac {16 c^6 x^9}{9}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.89, size = 240, normalized size = 3.29 \[ \frac {16}{9} x^{9} d^{4} c^{6} + 8 x^{8} d^{4} c^{5} b + \frac {104}{7} x^{7} d^{4} c^{4} b^{2} + \frac {32}{7} x^{7} d^{4} c^{5} a + \frac {44}{3} x^{6} d^{4} c^{3} b^{3} + 16 x^{6} d^{4} c^{4} b a + \frac {41}{5} x^{5} d^{4} c^{2} b^{4} + \frac {112}{5} x^{5} d^{4} c^{3} b^{2} a + \frac {16}{5} x^{5} d^{4} c^{4} a^{2} + \frac {5}{2} x^{4} d^{4} c b^{5} + 16 x^{4} d^{4} c^{2} b^{3} a + 8 x^{4} d^{4} c^{3} b a^{2} + \frac {1}{3} x^{3} d^{4} b^{6} + 6 x^{3} d^{4} c b^{4} a + 8 x^{3} d^{4} c^{2} b^{2} a^{2} + x^{2} d^{4} b^{5} a + 4 x^{2} d^{4} c b^{3} a^{2} + x d^{4} b^{4} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 240, normalized size = 3.29 \[ \frac {16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac {104}{7} \, b^{2} c^{4} d^{4} x^{7} + \frac {32}{7} \, a c^{5} d^{4} x^{7} + \frac {44}{3} \, b^{3} c^{3} d^{4} x^{6} + 16 \, a b c^{4} d^{4} x^{6} + \frac {41}{5} \, b^{4} c^{2} d^{4} x^{5} + \frac {112}{5} \, a b^{2} c^{3} d^{4} x^{5} + \frac {16}{5} \, a^{2} c^{4} d^{4} x^{5} + \frac {5}{2} \, b^{5} c d^{4} x^{4} + 16 \, a b^{3} c^{2} d^{4} x^{4} + 8 \, a^{2} b c^{3} d^{4} x^{4} + \frac {1}{3} \, b^{6} d^{4} x^{3} + 6 \, a b^{4} c d^{4} x^{3} + 8 \, a^{2} b^{2} c^{2} d^{4} x^{3} + a b^{5} d^{4} x^{2} + 4 \, a^{2} b^{3} c d^{4} x^{2} + a^{2} b^{4} d^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 300, normalized size = 4.11 \[ \frac {16 c^{6} d^{4} x^{9}}{9}+8 b \,c^{5} d^{4} x^{8}+a^{2} b^{4} d^{4} x +\frac {\left (88 b^{2} c^{4} d^{4}+16 \left (2 a c +b^{2}\right ) c^{4} d^{4}\right ) x^{7}}{7}+\frac {\left (32 a b \,c^{4} d^{4}+56 b^{3} c^{3} d^{4}+32 \left (2 a c +b^{2}\right ) b \,c^{3} d^{4}\right ) x^{6}}{6}+\frac {\left (16 a^{2} c^{4} d^{4}+64 a \,b^{2} c^{3} d^{4}+17 b^{4} c^{2} d^{4}+24 \left (2 a c +b^{2}\right ) b^{2} c^{2} d^{4}\right ) x^{5}}{5}+\frac {\left (32 a^{2} b \,c^{3} d^{4}+48 a \,b^{3} c^{2} d^{4}+2 b^{5} c \,d^{4}+8 \left (2 a c +b^{2}\right ) b^{3} c \,d^{4}\right ) x^{4}}{4}+\frac {\left (24 a^{2} b^{2} c^{2} d^{4}+16 a \,b^{4} c \,d^{4}+\left (2 a c +b^{2}\right ) b^{4} d^{4}\right ) x^{3}}{3}+\frac {\left (8 b^{3} d^{4} c \,a^{2}+2 b^{5} d^{4} a \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.42, size = 201, normalized size = 2.75 \[ \frac {16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac {8}{7} \, {\left (13 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{7} + a^{2} b^{4} d^{4} x + \frac {4}{3} \, {\left (11 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (41 \, b^{4} c^{2} + 112 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{4} x^{5} + \frac {1}{2} \, {\left (5 \, b^{5} c + 32 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{4} x^{4} + \frac {1}{3} \, {\left (b^{6} + 18 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2}\right )} d^{4} x^{3} + {\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d^{4} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.46, size = 190, normalized size = 2.60 \[ \frac {16\,c^6\,d^4\,x^9}{9}+\frac {c^2\,d^4\,x^5\,\left (16\,a^2\,c^2+112\,a\,b^2\,c+41\,b^4\right )}{5}+a^2\,b^4\,d^4\,x+8\,b\,c^5\,d^4\,x^8+\frac {8\,c^4\,d^4\,x^7\,\left (13\,b^2+4\,a\,c\right )}{7}+\frac {b^2\,d^4\,x^3\,\left (24\,a^2\,c^2+18\,a\,b^2\,c+b^4\right )}{3}+\frac {b\,c\,d^4\,x^4\,\left (16\,a^2\,c^2+32\,a\,b^2\,c+5\,b^4\right )}{2}+a\,b^3\,d^4\,x^2\,\left (b^2+4\,a\,c\right )+\frac {4\,b\,c^3\,d^4\,x^6\,\left (11\,b^2+12\,a\,c\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.11, size = 248, normalized size = 3.40 \[ a^{2} b^{4} d^{4} x + 8 b c^{5} d^{4} x^{8} + \frac {16 c^{6} d^{4} x^{9}}{9} + x^{7} \left (\frac {32 a c^{5} d^{4}}{7} + \frac {104 b^{2} c^{4} d^{4}}{7}\right ) + x^{6} \left (16 a b c^{4} d^{4} + \frac {44 b^{3} c^{3} d^{4}}{3}\right ) + x^{5} \left (\frac {16 a^{2} c^{4} d^{4}}{5} + \frac {112 a b^{2} c^{3} d^{4}}{5} + \frac {41 b^{4} c^{2} d^{4}}{5}\right ) + x^{4} \left (8 a^{2} b c^{3} d^{4} + 16 a b^{3} c^{2} d^{4} + \frac {5 b^{5} c d^{4}}{2}\right ) + x^{3} \left (8 a^{2} b^{2} c^{2} d^{4} + 6 a b^{4} c d^{4} + \frac {b^{6} d^{4}}{3}\right ) + x^{2} \left (4 a^{2} b^{3} c d^{4} + a b^{5} d^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________