Optimal. Leaf size=121 \[ \frac {2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{p+1}}{(p+2) (p+3)}+\frac {2 d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{p+1}}{(p+1) (p+2) (p+3)}+\frac {d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{p+1}}{p+3} \]
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Rubi [A] time = 0.07, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {692, 629} \[ \frac {2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{p+1}}{(p+2) (p+3)}+\frac {2 d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{p+1}}{(p+1) (p+2) (p+3)}+\frac {d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{p+1}}{p+3} \]
Antiderivative was successfully verified.
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Rule 629
Rule 692
Rubi steps
\begin {align*} \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^p \, dx &=\frac {d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{1+p}}{3+p}+\frac {\left (2 \left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^p \, dx}{3+p}\\ &=\frac {2 \left (b^2-4 a c\right ) d^5 (b+2 c x)^2 \left (a+b x+c x^2\right )^{1+p}}{(2+p) (3+p)}+\frac {d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{1+p}}{3+p}+\frac {\left (2 \left (b^2-4 a c\right )^2 d^4\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^p \, dx}{(2+p) (3+p)}\\ &=\frac {2 \left (b^2-4 a c\right )^2 d^5 \left (a+b x+c x^2\right )^{1+p}}{(1+p) (2+p) (3+p)}+\frac {2 \left (b^2-4 a c\right ) d^5 (b+2 c x)^2 \left (a+b x+c x^2\right )^{1+p}}{(2+p) (3+p)}+\frac {d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{1+p}}{3+p}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 145, normalized size = 1.20 \[ \frac {d^5 (a+x (b+c x))^{p+1} \left (16 c^2 \left (2 a^2-2 a c (p+1) x^2+c^2 \left (p^2+3 p+2\right ) x^4\right )-8 b^2 c \left (a (p+3)-c \left (3 p^2+10 p+7\right ) x^2\right )-32 b c^2 (p+1) x \left (a-c (p+2) x^2\right )+b^4 \left (p^2+5 p+6\right )+8 b^3 c \left (p^2+4 p+3\right ) x\right )}{(p+1) (p+2) (p+3)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.09, size = 401, normalized size = 3.31 \[ \frac {{\left (a b^{4} d^{5} p^{2} + {\left (5 \, a b^{4} - 8 \, a^{2} b^{2} c\right )} d^{5} p + 16 \, {\left (c^{5} d^{5} p^{2} + 3 \, c^{5} d^{5} p + 2 \, c^{5} d^{5}\right )} x^{6} + 2 \, {\left (3 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} d^{5} + 48 \, {\left (b c^{4} d^{5} p^{2} + 3 \, b c^{4} d^{5} p + 2 \, b c^{4} d^{5}\right )} x^{5} + 8 \, {\left (15 \, b^{2} c^{3} d^{5} + {\left (7 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d^{5} p^{2} + 2 \, {\left (11 \, b^{2} c^{3} + a c^{4}\right )} d^{5} p\right )} x^{4} + 16 \, {\left (5 \, b^{3} c^{2} d^{5} + 2 \, {\left (b^{3} c^{2} + a b c^{3}\right )} d^{5} p^{2} + {\left (7 \, b^{3} c^{2} + 2 \, a b c^{3}\right )} d^{5} p\right )} x^{3} + {\left (30 \, b^{4} c d^{5} + 3 \, {\left (3 \, b^{4} c + 8 \, a b^{2} c^{2}\right )} d^{5} p^{2} + {\left (37 \, b^{4} c + 40 \, a b^{2} c^{2} - 32 \, a^{2} c^{3}\right )} d^{5} p\right )} x^{2} + {\left (6 \, b^{5} d^{5} + {\left (b^{5} + 8 \, a b^{3} c\right )} d^{5} p^{2} + {\left (5 \, b^{5} + 24 \, a b^{3} c - 32 \, a^{2} b c^{2}\right )} d^{5} p\right )} x\right )} {\left (c x^{2} + b x + a\right )}^{p}}{p^{3} + 6 \, p^{2} + 11 \, p + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 877, normalized size = 7.25 \[ \frac {16 \, {\left (c x^{2} + b x + a\right )}^{p} c^{5} d^{5} p^{2} x^{6} + 48 \, {\left (c x^{2} + b x + a\right )}^{p} b c^{4} d^{5} p^{2} x^{5} + 48 \, {\left (c x^{2} + b x + a\right )}^{p} c^{5} d^{5} p x^{6} + 56 \, {\left (c x^{2} + b x + a\right )}^{p} b^{2} c^{3} d^{5} p^{2} x^{4} + 16 \, {\left (c x^{2} + b x + a\right )}^{p} a c^{4} d^{5} p^{2} x^{4} + 144 \, {\left (c x^{2} + b x + a\right )}^{p} b c^{4} d^{5} p x^{5} + 32 \, {\left (c x^{2} + b x + a\right )}^{p} c^{5} d^{5} x^{6} + 32 \, {\left (c x^{2} + b x + a\right )}^{p} b^{3} c^{2} d^{5} p^{2} x^{3} + 32 \, {\left (c x^{2} + b x + a\right )}^{p} a b c^{3} d^{5} p^{2} x^{3} + 176 \, {\left (c x^{2} + b x + a\right )}^{p} b^{2} c^{3} d^{5} p x^{4} + 16 \, {\left (c x^{2} + b x + a\right )}^{p} a c^{4} d^{5} p x^{4} + 96 \, {\left (c x^{2} + b x + a\right )}^{p} b c^{4} d^{5} x^{5} + 9 \, {\left (c x^{2} + b x + a\right )}^{p} b^{4} c d^{5} p^{2} x^{2} + 24 \, {\left (c x^{2} + b x + a\right )}^{p} a b^{2} c^{2} d^{5} p^{2} x^{2} + 112 \, {\left (c x^{2} + b x + a\right )}^{p} b^{3} c^{2} d^{5} p x^{3} + 32 \, {\left (c x^{2} + b x + a\right )}^{p} a b c^{3} d^{5} p x^{3} + 120 \, {\left (c x^{2} + b x + a\right )}^{p} b^{2} c^{3} d^{5} x^{4} + {\left (c x^{2} + b x + a\right )}^{p} b^{5} d^{5} p^{2} x + 8 \, {\left (c x^{2} + b x + a\right )}^{p} a b^{3} c d^{5} p^{2} x + 37 \, {\left (c x^{2} + b x + a\right )}^{p} b^{4} c d^{5} p x^{2} + 40 \, {\left (c x^{2} + b x + a\right )}^{p} a b^{2} c^{2} d^{5} p x^{2} - 32 \, {\left (c x^{2} + b x + a\right )}^{p} a^{2} c^{3} d^{5} p x^{2} + 80 \, {\left (c x^{2} + b x + a\right )}^{p} b^{3} c^{2} d^{5} x^{3} + {\left (c x^{2} + b x + a\right )}^{p} a b^{4} d^{5} p^{2} + 5 \, {\left (c x^{2} + b x + a\right )}^{p} b^{5} d^{5} p x + 24 \, {\left (c x^{2} + b x + a\right )}^{p} a b^{3} c d^{5} p x - 32 \, {\left (c x^{2} + b x + a\right )}^{p} a^{2} b c^{2} d^{5} p x + 30 \, {\left (c x^{2} + b x + a\right )}^{p} b^{4} c d^{5} x^{2} + 5 \, {\left (c x^{2} + b x + a\right )}^{p} a b^{4} d^{5} p - 8 \, {\left (c x^{2} + b x + a\right )}^{p} a^{2} b^{2} c d^{5} p + 6 \, {\left (c x^{2} + b x + a\right )}^{p} b^{5} d^{5} x + 6 \, {\left (c x^{2} + b x + a\right )}^{p} a b^{4} d^{5} - 24 \, {\left (c x^{2} + b x + a\right )}^{p} a^{2} b^{2} c d^{5} + 32 \, {\left (c x^{2} + b x + a\right )}^{p} a^{3} c^{2} d^{5}}{p^{3} + 6 \, p^{2} + 11 \, p + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 233, normalized size = 1.93 \[ \frac {\left (16 c^{4} p^{2} x^{4}+32 b \,c^{3} p^{2} x^{3}+48 c^{4} p \,x^{4}+24 b^{2} c^{2} p^{2} x^{2}+96 b \,c^{3} p \,x^{3}+32 c^{4} x^{4}-32 a \,c^{3} p \,x^{2}+8 b^{3} c \,p^{2} x +80 b^{2} c^{2} p \,x^{2}+64 b \,c^{3} x^{3}-32 a b \,c^{2} p x -32 a \,c^{3} x^{2}+b^{4} p^{2}+32 b^{3} c p x +56 b^{2} c^{2} x^{2}-8 a \,b^{2} c p -32 a b \,c^{2} x +5 b^{4} p +24 b^{3} c x +32 a^{2} c^{2}-24 a \,b^{2} c +6 b^{4}\right ) d^{5} \left (c \,x^{2}+b x +a \right )^{p +1}}{p^{3}+6 p^{2}+11 p +6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.93, size = 295, normalized size = 2.44 \[ \frac {{\left (16 \, {\left (p^{2} + 3 \, p + 2\right )} c^{5} d^{5} x^{6} + 48 \, {\left (p^{2} + 3 \, p + 2\right )} b c^{4} d^{5} x^{5} + {\left (p^{2} + 5 \, p + 6\right )} a b^{4} d^{5} - 8 \, a^{2} b^{2} c d^{5} {\left (p + 3\right )} + 32 \, a^{3} c^{2} d^{5} + 8 \, {\left ({\left (7 \, p^{2} + 22 \, p + 15\right )} b^{2} c^{3} d^{5} + 2 \, {\left (p^{2} + p\right )} a c^{4} d^{5}\right )} x^{4} + 16 \, {\left ({\left (2 \, p^{2} + 7 \, p + 5\right )} b^{3} c^{2} d^{5} + 2 \, {\left (p^{2} + p\right )} a b c^{3} d^{5}\right )} x^{3} + {\left ({\left (9 \, p^{2} + 37 \, p + 30\right )} b^{4} c d^{5} + 8 \, {\left (3 \, p^{2} + 5 \, p\right )} a b^{2} c^{2} d^{5} - 32 \, a^{2} c^{3} d^{5} p\right )} x^{2} + {\left ({\left (p^{2} + 5 \, p + 6\right )} b^{5} d^{5} + 8 \, {\left (p^{2} + 3 \, p\right )} a b^{3} c d^{5} - 32 \, a^{2} b c^{2} d^{5} p\right )} x\right )} {\left (c x^{2} + b x + a\right )}^{p}}{p^{3} + 6 \, p^{2} + 11 \, p + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 375, normalized size = 3.10 \[ {\left (c\,x^2+b\,x+a\right )}^p\,\left (\frac {a\,d^5\,\left (32\,a^2\,c^2-8\,a\,b^2\,c\,p-24\,a\,b^2\,c+b^4\,p^2+5\,b^4\,p+6\,b^4\right )}{p^3+6\,p^2+11\,p+6}+\frac {c\,d^5\,x^2\,\left (-32\,a^2\,c^2\,p+24\,a\,b^2\,c\,p^2+40\,a\,b^2\,c\,p+9\,b^4\,p^2+37\,b^4\,p+30\,b^4\right )}{p^3+6\,p^2+11\,p+6}+\frac {16\,c^5\,d^5\,x^6\,\left (p^2+3\,p+2\right )}{p^3+6\,p^2+11\,p+6}+\frac {b\,d^5\,x\,\left (-32\,a^2\,c^2\,p+8\,a\,b^2\,c\,p^2+24\,a\,b^2\,c\,p+b^4\,p^2+5\,b^4\,p+6\,b^4\right )}{p^3+6\,p^2+11\,p+6}+\frac {48\,b\,c^4\,d^5\,x^5\,\left (p^2+3\,p+2\right )}{p^3+6\,p^2+11\,p+6}+\frac {8\,c^3\,d^5\,x^4\,\left (p+1\right )\,\left (7\,b^2\,p+15\,b^2+2\,a\,c\,p\right )}{p^3+6\,p^2+11\,p+6}+\frac {16\,b\,c^2\,d^5\,x^3\,\left (p+1\right )\,\left (2\,b^2\,p+5\,b^2+2\,a\,c\,p\right )}{p^3+6\,p^2+11\,p+6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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