3.297 \(\int (d+e x)^2 (3+2 x+5 x^2)^2 (2+x+3 x^2-5 x^3+4 x^4) \, dx\)

Optimal. Leaf size=201 \[ \frac{1}{9} x^9 \left (100 d^2-90 d e+111 e^2\right )-\frac{1}{8} x^8 \left (45 d^2-222 d e+37 e^2\right )+\frac{37}{7} x^7 \left (3 d^2-2 d e+4 e^2\right )-\frac{1}{6} x^6 \left (37 d^2-296 d e-65 e^2\right )+\frac{1}{5} x^5 \left (148 d^2+130 d e+107 e^2\right )+\frac{1}{4} x^4 \left (65 d^2+214 d e+33 e^2\right )+\frac{1}{3} x^3 \left (107 d^2+66 d e+18 e^2\right )+18 d^2 x+\frac{1}{2} e x^{10} (40 d-9 e)+\frac{3}{2} d x^2 (11 d+12 e)+\frac{100 e^2 x^{11}}{11} \]

[Out]

18*d^2*x + (3*d*(11*d + 12*e)*x^2)/2 + ((107*d^2 + 66*d*e + 18*e^2)*x^3)/3 + ((65*d^2 + 214*d*e + 33*e^2)*x^4)
/4 + ((148*d^2 + 130*d*e + 107*e^2)*x^5)/5 - ((37*d^2 - 296*d*e - 65*e^2)*x^6)/6 + (37*(3*d^2 - 2*d*e + 4*e^2)
*x^7)/7 - ((45*d^2 - 222*d*e + 37*e^2)*x^8)/8 + ((100*d^2 - 90*d*e + 111*e^2)*x^9)/9 + ((40*d - 9*e)*e*x^10)/2
+ (100*e^2*x^11)/11

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Rubi [A]  time = 0.240234, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {1628} \[ \frac{1}{9} x^9 \left (100 d^2-90 d e+111 e^2\right )-\frac{1}{8} x^8 \left (45 d^2-222 d e+37 e^2\right )+\frac{37}{7} x^7 \left (3 d^2-2 d e+4 e^2\right )-\frac{1}{6} x^6 \left (37 d^2-296 d e-65 e^2\right )+\frac{1}{5} x^5 \left (148 d^2+130 d e+107 e^2\right )+\frac{1}{4} x^4 \left (65 d^2+214 d e+33 e^2\right )+\frac{1}{3} x^3 \left (107 d^2+66 d e+18 e^2\right )+18 d^2 x+\frac{1}{2} e x^{10} (40 d-9 e)+\frac{3}{2} d x^2 (11 d+12 e)+\frac{100 e^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]

[Out]

18*d^2*x + (3*d*(11*d + 12*e)*x^2)/2 + ((107*d^2 + 66*d*e + 18*e^2)*x^3)/3 + ((65*d^2 + 214*d*e + 33*e^2)*x^4)
/4 + ((148*d^2 + 130*d*e + 107*e^2)*x^5)/5 - ((37*d^2 - 296*d*e - 65*e^2)*x^6)/6 + (37*(3*d^2 - 2*d*e + 4*e^2)
*x^7)/7 - ((45*d^2 - 222*d*e + 37*e^2)*x^8)/8 + ((100*d^2 - 90*d*e + 111*e^2)*x^9)/9 + ((40*d - 9*e)*e*x^10)/2
+ (100*e^2*x^11)/11

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int (d+e x)^2 \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx &=\int \left (18 d^2+3 d (11 d+12 e) x+\left (107 d^2+66 d e+18 e^2\right ) x^2+\left (65 d^2+214 d e+33 e^2\right ) x^3+\left (148 d^2+130 d e+107 e^2\right ) x^4-\left (37 d^2-296 d e-65 e^2\right ) x^5+37 \left (3 d^2-2 d e+4 e^2\right ) x^6-\left (45 d^2-222 d e+37 e^2\right ) x^7+\left (100 d^2-90 d e+111 e^2\right ) x^8+5 (40 d-9 e) e x^9+100 e^2 x^{10}\right ) \, dx\\ &=18 d^2 x+\frac{3}{2} d (11 d+12 e) x^2+\frac{1}{3} \left (107 d^2+66 d e+18 e^2\right ) x^3+\frac{1}{4} \left (65 d^2+214 d e+33 e^2\right ) x^4+\frac{1}{5} \left (148 d^2+130 d e+107 e^2\right ) x^5-\frac{1}{6} \left (37 d^2-296 d e-65 e^2\right ) x^6+\frac{37}{7} \left (3 d^2-2 d e+4 e^2\right ) x^7-\frac{1}{8} \left (45 d^2-222 d e+37 e^2\right ) x^8+\frac{1}{9} \left (100 d^2-90 d e+111 e^2\right ) x^9+\frac{1}{2} (40 d-9 e) e x^{10}+\frac{100 e^2 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0267461, size = 201, normalized size = 1. \[ \frac{1}{9} x^9 \left (100 d^2-90 d e+111 e^2\right )+\frac{1}{8} x^8 \left (-45 d^2+222 d e-37 e^2\right )+\frac{37}{7} x^7 \left (3 d^2-2 d e+4 e^2\right )+\frac{1}{6} x^6 \left (-37 d^2+296 d e+65 e^2\right )+\frac{1}{5} x^5 \left (148 d^2+130 d e+107 e^2\right )+\frac{1}{4} x^4 \left (65 d^2+214 d e+33 e^2\right )+\frac{1}{3} x^3 \left (107 d^2+66 d e+18 e^2\right )+18 d^2 x+\frac{1}{2} e x^{10} (40 d-9 e)+\frac{3}{2} d x^2 (11 d+12 e)+\frac{100 e^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(3 + 2*x + 5*x^2)^2*(2 + x + 3*x^2 - 5*x^3 + 4*x^4),x]

[Out]

18*d^2*x + (3*d*(11*d + 12*e)*x^2)/2 + ((107*d^2 + 66*d*e + 18*e^2)*x^3)/3 + ((65*d^2 + 214*d*e + 33*e^2)*x^4)
/4 + ((148*d^2 + 130*d*e + 107*e^2)*x^5)/5 + ((-37*d^2 + 296*d*e + 65*e^2)*x^6)/6 + (37*(3*d^2 - 2*d*e + 4*e^2
)*x^7)/7 + ((-45*d^2 + 222*d*e - 37*e^2)*x^8)/8 + ((100*d^2 - 90*d*e + 111*e^2)*x^9)/9 + ((40*d - 9*e)*e*x^10)
/2 + (100*e^2*x^11)/11

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Maple [A]  time = 0.041, size = 186, normalized size = 0.9 \begin{align*}{\frac{100\,{e}^{2}{x}^{11}}{11}}+{\frac{ \left ( 200\,de-45\,{e}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ( 100\,{d}^{2}-90\,de+111\,{e}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( -45\,{d}^{2}+222\,de-37\,{e}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 111\,{d}^{2}-74\,de+148\,{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( -37\,{d}^{2}+296\,de+65\,{e}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 148\,{d}^{2}+130\,de+107\,{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 65\,{d}^{2}+214\,de+33\,{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 107\,{d}^{2}+66\,de+18\,{e}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 33\,{d}^{2}+36\,de \right ){x}^{2}}{2}}+18\,{d}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2),x)

[Out]

100/11*e^2*x^11+1/10*(200*d*e-45*e^2)*x^10+1/9*(100*d^2-90*d*e+111*e^2)*x^9+1/8*(-45*d^2+222*d*e-37*e^2)*x^8+1
/7*(111*d^2-74*d*e+148*e^2)*x^7+1/6*(-37*d^2+296*d*e+65*e^2)*x^6+1/5*(148*d^2+130*d*e+107*e^2)*x^5+1/4*(65*d^2
+214*d*e+33*e^2)*x^4+1/3*(107*d^2+66*d*e+18*e^2)*x^3+1/2*(33*d^2+36*d*e)*x^2+18*d^2*x

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Maxima [A]  time = 0.995225, size = 250, normalized size = 1.24 \begin{align*} \frac{100}{11} \, e^{2} x^{11} + \frac{1}{2} \,{\left (40 \, d e - 9 \, e^{2}\right )} x^{10} + \frac{1}{9} \,{\left (100 \, d^{2} - 90 \, d e + 111 \, e^{2}\right )} x^{9} - \frac{1}{8} \,{\left (45 \, d^{2} - 222 \, d e + 37 \, e^{2}\right )} x^{8} + \frac{37}{7} \,{\left (3 \, d^{2} - 2 \, d e + 4 \, e^{2}\right )} x^{7} - \frac{1}{6} \,{\left (37 \, d^{2} - 296 \, d e - 65 \, e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (148 \, d^{2} + 130 \, d e + 107 \, e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (65 \, d^{2} + 214 \, d e + 33 \, e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (107 \, d^{2} + 66 \, d e + 18 \, e^{2}\right )} x^{3} + 18 \, d^{2} x + \frac{3}{2} \,{\left (11 \, d^{2} + 12 \, d e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="maxima")

[Out]

100/11*e^2*x^11 + 1/2*(40*d*e - 9*e^2)*x^10 + 1/9*(100*d^2 - 90*d*e + 111*e^2)*x^9 - 1/8*(45*d^2 - 222*d*e + 3
7*e^2)*x^8 + 37/7*(3*d^2 - 2*d*e + 4*e^2)*x^7 - 1/6*(37*d^2 - 296*d*e - 65*e^2)*x^6 + 1/5*(148*d^2 + 130*d*e +
107*e^2)*x^5 + 1/4*(65*d^2 + 214*d*e + 33*e^2)*x^4 + 1/3*(107*d^2 + 66*d*e + 18*e^2)*x^3 + 18*d^2*x + 3/2*(11
*d^2 + 12*d*e)*x^2

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Fricas [A]  time = 0.825465, size = 540, normalized size = 2.69 \begin{align*} \frac{100}{11} x^{11} e^{2} - \frac{9}{2} x^{10} e^{2} + 20 x^{10} e d + \frac{37}{3} x^{9} e^{2} - 10 x^{9} e d + \frac{100}{9} x^{9} d^{2} - \frac{37}{8} x^{8} e^{2} + \frac{111}{4} x^{8} e d - \frac{45}{8} x^{8} d^{2} + \frac{148}{7} x^{7} e^{2} - \frac{74}{7} x^{7} e d + \frac{111}{7} x^{7} d^{2} + \frac{65}{6} x^{6} e^{2} + \frac{148}{3} x^{6} e d - \frac{37}{6} x^{6} d^{2} + \frac{107}{5} x^{5} e^{2} + 26 x^{5} e d + \frac{148}{5} x^{5} d^{2} + \frac{33}{4} x^{4} e^{2} + \frac{107}{2} x^{4} e d + \frac{65}{4} x^{4} d^{2} + 6 x^{3} e^{2} + 22 x^{3} e d + \frac{107}{3} x^{3} d^{2} + 18 x^{2} e d + \frac{33}{2} x^{2} d^{2} + 18 x d^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="fricas")

[Out]

100/11*x^11*e^2 - 9/2*x^10*e^2 + 20*x^10*e*d + 37/3*x^9*e^2 - 10*x^9*e*d + 100/9*x^9*d^2 - 37/8*x^8*e^2 + 111/
4*x^8*e*d - 45/8*x^8*d^2 + 148/7*x^7*e^2 - 74/7*x^7*e*d + 111/7*x^7*d^2 + 65/6*x^6*e^2 + 148/3*x^6*e*d - 37/6*
x^6*d^2 + 107/5*x^5*e^2 + 26*x^5*e*d + 148/5*x^5*d^2 + 33/4*x^4*e^2 + 107/2*x^4*e*d + 65/4*x^4*d^2 + 6*x^3*e^2
+ 22*x^3*e*d + 107/3*x^3*d^2 + 18*x^2*e*d + 33/2*x^2*d^2 + 18*x*d^2

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Sympy [A]  time = 0.104831, size = 206, normalized size = 1.02 \begin{align*} 18 d^{2} x + \frac{100 e^{2} x^{11}}{11} + x^{10} \left (20 d e - \frac{9 e^{2}}{2}\right ) + x^{9} \left (\frac{100 d^{2}}{9} - 10 d e + \frac{37 e^{2}}{3}\right ) + x^{8} \left (- \frac{45 d^{2}}{8} + \frac{111 d e}{4} - \frac{37 e^{2}}{8}\right ) + x^{7} \left (\frac{111 d^{2}}{7} - \frac{74 d e}{7} + \frac{148 e^{2}}{7}\right ) + x^{6} \left (- \frac{37 d^{2}}{6} + \frac{148 d e}{3} + \frac{65 e^{2}}{6}\right ) + x^{5} \left (\frac{148 d^{2}}{5} + 26 d e + \frac{107 e^{2}}{5}\right ) + x^{4} \left (\frac{65 d^{2}}{4} + \frac{107 d e}{2} + \frac{33 e^{2}}{4}\right ) + x^{3} \left (\frac{107 d^{2}}{3} + 22 d e + 6 e^{2}\right ) + x^{2} \left (\frac{33 d^{2}}{2} + 18 d e\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(5*x**2+2*x+3)**2*(4*x**4-5*x**3+3*x**2+x+2),x)

[Out]

18*d**2*x + 100*e**2*x**11/11 + x**10*(20*d*e - 9*e**2/2) + x**9*(100*d**2/9 - 10*d*e + 37*e**2/3) + x**8*(-45
*d**2/8 + 111*d*e/4 - 37*e**2/8) + x**7*(111*d**2/7 - 74*d*e/7 + 148*e**2/7) + x**6*(-37*d**2/6 + 148*d*e/3 +
65*e**2/6) + x**5*(148*d**2/5 + 26*d*e + 107*e**2/5) + x**4*(65*d**2/4 + 107*d*e/2 + 33*e**2/4) + x**3*(107*d*
*2/3 + 22*d*e + 6*e**2) + x**2*(33*d**2/2 + 18*d*e)

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Giac [A]  time = 1.153, size = 278, normalized size = 1.38 \begin{align*} \frac{100}{11} \, x^{11} e^{2} + 20 \, d x^{10} e + \frac{100}{9} \, d^{2} x^{9} - \frac{9}{2} \, x^{10} e^{2} - 10 \, d x^{9} e - \frac{45}{8} \, d^{2} x^{8} + \frac{37}{3} \, x^{9} e^{2} + \frac{111}{4} \, d x^{8} e + \frac{111}{7} \, d^{2} x^{7} - \frac{37}{8} \, x^{8} e^{2} - \frac{74}{7} \, d x^{7} e - \frac{37}{6} \, d^{2} x^{6} + \frac{148}{7} \, x^{7} e^{2} + \frac{148}{3} \, d x^{6} e + \frac{148}{5} \, d^{2} x^{5} + \frac{65}{6} \, x^{6} e^{2} + 26 \, d x^{5} e + \frac{65}{4} \, d^{2} x^{4} + \frac{107}{5} \, x^{5} e^{2} + \frac{107}{2} \, d x^{4} e + \frac{107}{3} \, d^{2} x^{3} + \frac{33}{4} \, x^{4} e^{2} + 22 \, d x^{3} e + \frac{33}{2} \, d^{2} x^{2} + 6 \, x^{3} e^{2} + 18 \, d x^{2} e + 18 \, d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(5*x^2+2*x+3)^2*(4*x^4-5*x^3+3*x^2+x+2),x, algorithm="giac")

[Out]

100/11*x^11*e^2 + 20*d*x^10*e + 100/9*d^2*x^9 - 9/2*x^10*e^2 - 10*d*x^9*e - 45/8*d^2*x^8 + 37/3*x^9*e^2 + 111/
4*d*x^8*e + 111/7*d^2*x^7 - 37/8*x^8*e^2 - 74/7*d*x^7*e - 37/6*d^2*x^6 + 148/7*x^7*e^2 + 148/3*d*x^6*e + 148/5
*d^2*x^5 + 65/6*x^6*e^2 + 26*d*x^5*e + 65/4*d^2*x^4 + 107/5*x^5*e^2 + 107/2*d*x^4*e + 107/3*d^2*x^3 + 33/4*x^4
*e^2 + 22*d*x^3*e + 33/2*d^2*x^2 + 6*x^3*e^2 + 18*d*x^2*e + 18*d^2*x