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 = 0.182108, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{p+1}}{(p+2) (p+3)}+\frac{2 d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{p+1}}{(p+1) (p+2) (p+3)}+\frac{d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{p+1}}{p+3} \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 41.9559, size = 109, normalized size = 0.9 \[ \frac{d^{5} \left (b + 2 c x\right )^{4} \left (a + b x + c x^{2}\right )^{p + 1}}{p + 3} + \frac{2 d^{5} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{p + 1}}{\left (p + 2\right ) \left (p + 3\right )} + \frac{2 d^{5} \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{p + 1}}{\left (p + 1\right ) \left (p + 2\right ) \left (p + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**5*(c*x**2+b*x+a)**p,x)
[Out]
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Mathematica [A] time = 0.118788, size = 132, normalized size = 1.09 \[ \frac{d^5 (a+x (b+c x))^p \left (-\frac{p \left (b^2-4 a c\right ) (b+2 c x)^4}{2 (p+2) (p+3)}-\frac{p \left (b^2-4 a c\right )^2 (b+2 c x)^2}{(p+1) (p+2) (p+3)}-\frac{\left (b^2-4 a c\right )^3}{(p+1) (p+2) (p+3)}+\frac{(b+2 c x)^6}{2 p+6}\right )}{2 c} \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^p,x]
[Out]
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Maple [A] time = 0.019, size = 233, normalized size = 1.9 \[{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{1+p}{d}^{5} \left ( 16\,{c}^{4}{p}^{2}{x}^{4}+32\,b{c}^{3}{p}^{2}{x}^{3}+48\,{c}^{4}p{x}^{4}+24\,{b}^{2}{c}^{2}{p}^{2}{x}^{2}+96\,b{c}^{3}p{x}^{3}+32\,{c}^{4}{x}^{4}-32\,a{c}^{3}p{x}^{2}+8\,{b}^{3}c{p}^{2}x+80\,{b}^{2}{c}^{2}p{x}^{2}+64\,b{c}^{3}{x}^{3}-32\,ab{c}^{2}px-32\,{x}^{2}a{c}^{3}+{b}^{4}{p}^{2}+32\,{b}^{3}cpx+56\,{x}^{2}{b}^{2}{c}^{2}-8\,a{b}^{2}cp-32\,xab{c}^{2}+5\,{b}^{4}p+24\,{b}^{3}cx+32\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+6\,{b}^{4} \right ) }{{p}^{3}+6\,{p}^{2}+11\,p+6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^p,x)
[Out]
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Maxima [A] time = 0.751076, size = 398, normalized size = 3.29 \[ \frac{{\left (16 \,{\left (p^{2} + 3 \, p + 2\right )} c^{5} d^{5} x^{6} + 48 \,{\left (p^{2} + 3 \, p + 2\right )} b c^{4} d^{5} x^{5} +{\left (p^{2} + 5 \, p + 6\right )} a b^{4} d^{5} - 8 \, a^{2} b^{2} c d^{5}{\left (p + 3\right )} + 32 \, a^{3} c^{2} d^{5} + 8 \,{\left ({\left (7 \, p^{2} + 22 \, p + 15\right )} b^{2} c^{3} d^{5} + 2 \,{\left (p^{2} + p\right )} a c^{4} d^{5}\right )} x^{4} + 16 \,{\left ({\left (2 \, p^{2} + 7 \, p + 5\right )} b^{3} c^{2} d^{5} + 2 \,{\left (p^{2} + p\right )} a b c^{3} d^{5}\right )} x^{3} +{\left ({\left (9 \, p^{2} + 37 \, p + 30\right )} b^{4} c d^{5} + 8 \,{\left (3 \, p^{2} + 5 \, p\right )} a b^{2} c^{2} d^{5} - 32 \, a^{2} c^{3} d^{5} p\right )} x^{2} +{\left ({\left (p^{2} + 5 \, p + 6\right )} b^{5} d^{5} + 8 \,{\left (p^{2} + 3 \, p\right )} a b^{3} c d^{5} - 32 \, a^{2} b c^{2} d^{5} p\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{p}}{p^{3} + 6 \, p^{2} + 11 \, p + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^5*(c*x^2 + b*x + a)^p,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23213, size = 541, normalized size = 4.47 \[ \frac{{\left (a b^{4} d^{5} p^{2} +{\left (5 \, a b^{4} - 8 \, a^{2} b^{2} c\right )} d^{5} p + 16 \,{\left (c^{5} d^{5} p^{2} + 3 \, c^{5} d^{5} p + 2 \, c^{5} d^{5}\right )} x^{6} + 2 \,{\left (3 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} d^{5} + 48 \,{\left (b c^{4} d^{5} p^{2} + 3 \, b c^{4} d^{5} p + 2 \, b c^{4} d^{5}\right )} x^{5} + 8 \,{\left (15 \, b^{2} c^{3} d^{5} +{\left (7 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d^{5} p^{2} + 2 \,{\left (11 \, b^{2} c^{3} + a c^{4}\right )} d^{5} p\right )} x^{4} + 16 \,{\left (5 \, b^{3} c^{2} d^{5} + 2 \,{\left (b^{3} c^{2} + a b c^{3}\right )} d^{5} p^{2} +{\left (7 \, b^{3} c^{2} + 2 \, a b c^{3}\right )} d^{5} p\right )} x^{3} +{\left (30 \, b^{4} c d^{5} + 3 \,{\left (3 \, b^{4} c + 8 \, a b^{2} c^{2}\right )} d^{5} p^{2} +{\left (37 \, b^{4} c + 40 \, a b^{2} c^{2} - 32 \, a^{2} c^{3}\right )} d^{5} p\right )} x^{2} +{\left (6 \, b^{5} d^{5} +{\left (b^{5} + 8 \, a b^{3} c\right )} d^{5} p^{2} +{\left (5 \, b^{5} + 24 \, a b^{3} c - 32 \, a^{2} b c^{2}\right )} d^{5} p\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{p}}{p^{3} + 6 \, p^{2} + 11 \, p + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^5*(c*x^2 + b*x + a)^p,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**5*(c*x**2+b*x+a)**p,x)
[Out]
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GIAC/XCAS [A] time = 0.235653, size = 1276, normalized size = 10.55 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^5*(c*x^2 + b*x + a)^p,x, algorithm="giac")
[Out]