3.3.98 \(\int (d+e x) (3+2 x+5 x^2)^2 (2+x+3 x^2-5 x^3+4 x^4) \, dx\) [298]

Optimal. Leaf size=121 \[ 18 d x+\frac {3}{2} (11 d+6 e) x^2+\frac {1}{3} (107 d+33 e) x^3+\frac {1}{4} (65 d+107 e) x^4+\frac {1}{5} (148 d+65 e) x^5-\frac {37}{6} (d-4 e) x^6+\frac {37}{7} (3 d-e) x^7-\frac {3}{8} (15 d-37 e) x^8+\frac {5}{9} (20 d-9 e) x^9+10 e x^{10} \]

[Out]

18*d*x+3/2*(11*d+6*e)*x^2+1/3*(107*d+33*e)*x^3+1/4*(65*d+107*e)*x^4+1/5*(148*d+65*e)*x^5-37/6*(d-4*e)*x^6+37/7
*(3*d-e)*x^7-3/8*(15*d-37*e)*x^8+5/9*(20*d-9*e)*x^9+10*e*x^10

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Rubi [A]
time = 0.10, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1642} \begin {gather*} \frac {5}{9} x^9 (20 d-9 e)-\frac {3}{8} x^8 (15 d-37 e)+\frac {37}{7} x^7 (3 d-e)-\frac {37}{6} x^6 (d-4 e)+\frac {1}{5} x^5 (148 d+65 e)+\frac {1}{4} x^4 (65 d+107 e)+\frac {1}{3} x^3 (107 d+33 e)+\frac {3}{2} x^2 (11 d+6 e)+18 d x+10 e x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

18*d*x + (3*(11*d + 6*e)*x^2)/2 + ((107*d + 33*e)*x^3)/3 + ((65*d + 107*e)*x^4)/4 + ((148*d + 65*e)*x^5)/5 - (
37*(d - 4*e)*x^6)/6 + (37*(3*d - e)*x^7)/7 - (3*(15*d - 37*e)*x^8)/8 + (5*(20*d - 9*e)*x^9)/9 + 10*e*x^10

Rule 1642

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) \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+3 (11 d+6 e) x+(107 d+33 e) x^2+(65 d+107 e) x^3+(148 d+65 e) x^4-37 (d-4 e) x^5+37 (3 d-e) x^6-3 (15 d-37 e) x^7+5 (20 d-9 e) x^8+100 e x^9\right ) \, dx\\ &=18 d x+\frac {3}{2} (11 d+6 e) x^2+\frac {1}{3} (107 d+33 e) x^3+\frac {1}{4} (65 d+107 e) x^4+\frac {1}{5} (148 d+65 e) x^5-\frac {37}{6} (d-4 e) x^6+\frac {37}{7} (3 d-e) x^7-\frac {3}{8} (15 d-37 e) x^8+\frac {5}{9} (20 d-9 e) x^9+10 e x^{10}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 121, normalized size = 1.00 \begin {gather*} 18 d x+\frac {3}{2} (11 d+6 e) x^2+\frac {1}{3} (107 d+33 e) x^3+\frac {1}{4} (65 d+107 e) x^4+\frac {1}{5} (148 d+65 e) x^5-\frac {37}{6} (d-4 e) x^6+\frac {37}{7} (3 d-e) x^7-\frac {3}{8} (15 d-37 e) x^8+\frac {5}{9} (20 d-9 e) x^9+10 e x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

18*d*x + (3*(11*d + 6*e)*x^2)/2 + ((107*d + 33*e)*x^3)/3 + ((65*d + 107*e)*x^4)/4 + ((148*d + 65*e)*x^5)/5 - (
37*(d - 4*e)*x^6)/6 + (37*(3*d - e)*x^7)/7 - (3*(15*d - 37*e)*x^8)/8 + (5*(20*d - 9*e)*x^9)/9 + 10*e*x^10

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Maple [A]
time = 0.11, size = 108, normalized size = 0.89

method	result	size
norman	\(10 e \,x^{10}+\left (\frac {100 d}{9}-5 e \right ) x^{9}+\left (-\frac {45 d}{8}+\frac {111 e}{8}\right ) x^{8}+\left (\frac {111 d}{7}-\frac {37 e}{7}\right ) x^{7}+\left (-\frac {37 d}{6}+\frac {74 e}{3}\right ) x^{6}+\left (\frac {148 d}{5}+13 e \right ) x^{5}+\left (\frac {65 d}{4}+\frac {107 e}{4}\right ) x^{4}+\left (\frac {107 d}{3}+11 e \right ) x^{3}+\left (\frac {33 d}{2}+9 e \right ) x^{2}+18 d x\)	\(100\)
gosper	\(10 e \,x^{10}+\frac {100}{9} x^{9} d -5 x^{9} e -\frac {45}{8} x^{8} d +\frac {111}{8} e \,x^{8}+\frac {111}{7} x^{7} d -\frac {37}{7} x^{7} e -\frac {37}{6} x^{6} d +\frac {74}{3} e \,x^{6}+\frac {148}{5} d \,x^{5}+13 e \,x^{5}+\frac {65}{4} d \,x^{4}+\frac {107}{4} e \,x^{4}+\frac {107}{3} d \,x^{3}+11 e \,x^{3}+\frac {33}{2} d \,x^{2}+9 e \,x^{2}+18 d x\)	\(108\)
default	\(10 e \,x^{10}+\frac {\left (100 d -45 e \right ) x^{9}}{9}+\frac {\left (-45 d +111 e \right ) x^{8}}{8}+\frac {\left (111 d -37 e \right ) x^{7}}{7}+\frac {\left (-37 d +148 e \right ) x^{6}}{6}+\frac {\left (148 d +65 e \right ) x^{5}}{5}+\frac {\left (65 d +107 e \right ) x^{4}}{4}+\frac {\left (107 d +33 e \right ) x^{3}}{3}+\frac {\left (33 d +18 e \right ) x^{2}}{2}+18 d x\)	\(108\)
risch	\(10 e \,x^{10}+\frac {100}{9} x^{9} d -5 x^{9} e -\frac {45}{8} x^{8} d +\frac {111}{8} e \,x^{8}+\frac {111}{7} x^{7} d -\frac {37}{7} x^{7} e -\frac {37}{6} x^{6} d +\frac {74}{3} e \,x^{6}+\frac {148}{5} d \,x^{5}+13 e \,x^{5}+\frac {65}{4} d \,x^{4}+\frac {107}{4} e \,x^{4}+\frac {107}{3} d \,x^{3}+11 e \,x^{3}+\frac {33}{2} d \,x^{2}+9 e \,x^{2}+18 d x\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*e*x^10+1/9*(100*d-45*e)*x^9+1/8*(-45*d+111*e)*x^8+1/7*(111*d-37*e)*x^7+1/6*(-37*d+148*e)*x^6+1/5*(148*d+65*
e)*x^5+1/4*(65*d+107*e)*x^4+1/3*(107*d+33*e)*x^3+1/2*(33*d+18*e)*x^2+18*d*x

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Maxima [A]
time = 0.31, size = 114, normalized size = 0.94 \begin {gather*} 10 \, x^{10} e + \frac {5}{9} \, {\left (20 \, d - 9 \, e\right )} x^{9} - \frac {3}{8} \, {\left (15 \, d - 37 \, e\right )} x^{8} + \frac {37}{7} \, {\left (3 \, d - e\right )} x^{7} - \frac {37}{6} \, {\left (d - 4 \, e\right )} x^{6} + \frac {1}{5} \, {\left (148 \, d + 65 \, e\right )} x^{5} + \frac {1}{4} \, {\left (65 \, d + 107 \, e\right )} x^{4} + \frac {1}{3} \, {\left (107 \, d + 33 \, e\right )} x^{3} + \frac {3}{2} \, {\left (11 \, d + 6 \, e\right )} x^{2} + 18 \, d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*x^10*e + 5/9*(20*d - 9*e)*x^9 - 3/8*(15*d - 37*e)*x^8 + 37/7*(3*d - e)*x^7 - 37/6*(d - 4*e)*x^6 + 1/5*(148*
d + 65*e)*x^5 + 1/4*(65*d + 107*e)*x^4 + 1/3*(107*d + 33*e)*x^3 + 3/2*(11*d + 6*e)*x^2 + 18*d*x

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Fricas [A]
time = 0.40, size = 103, normalized size = 0.85 \begin {gather*} \frac {100}{9} \, d x^{9} - \frac {45}{8} \, d x^{8} + \frac {111}{7} \, d x^{7} - \frac {37}{6} \, d x^{6} + \frac {148}{5} \, d x^{5} + \frac {65}{4} \, d x^{4} + \frac {107}{3} \, d x^{3} + \frac {33}{2} \, d x^{2} + 18 \, d x + \frac {1}{168} \, {\left (1680 \, x^{10} - 840 \, x^{9} + 2331 \, x^{8} - 888 \, x^{7} + 4144 \, x^{6} + 2184 \, x^{5} + 4494 \, x^{4} + 1848 \, x^{3} + 1512 \, x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

100/9*d*x^9 - 45/8*d*x^8 + 111/7*d*x^7 - 37/6*d*x^6 + 148/5*d*x^5 + 65/4*d*x^4 + 107/3*d*x^3 + 33/2*d*x^2 + 18
*d*x + 1/168*(1680*x^10 - 840*x^9 + 2331*x^8 - 888*x^7 + 4144*x^6 + 2184*x^5 + 4494*x^4 + 1848*x^3 + 1512*x^2)
*e

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Sympy [A]
time = 0.02, size = 112, normalized size = 0.93 \begin {gather*} 18 d x + 10 e x^{10} + x^{9} \cdot \left (\frac {100 d}{9} - 5 e\right ) + x^{8} \left (- \frac {45 d}{8} + \frac {111 e}{8}\right ) + x^{7} \cdot \left (\frac {111 d}{7} - \frac {37 e}{7}\right ) + x^{6} \left (- \frac {37 d}{6} + \frac {74 e}{3}\right ) + x^{5} \cdot \left (\frac {148 d}{5} + 13 e\right ) + x^{4} \cdot \left (\frac {65 d}{4} + \frac {107 e}{4}\right ) + x^{3} \cdot \left (\frac {107 d}{3} + 11 e\right ) + x^{2} \cdot \left (\frac {33 d}{2} + 9 e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18*d*x + 10*e*x**10 + x**9*(100*d/9 - 5*e) + x**8*(-45*d/8 + 111*e/8) + x**7*(111*d/7 - 37*e/7) + x**6*(-37*d/
6 + 74*e/3) + x**5*(148*d/5 + 13*e) + x**4*(65*d/4 + 107*e/4) + x**3*(107*d/3 + 11*e) + x**2*(33*d/2 + 9*e)

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Giac [A]
time = 3.96, size = 116, normalized size = 0.96 \begin {gather*} 10 \, x^{10} e + \frac {100}{9} \, d x^{9} - 5 \, x^{9} e - \frac {45}{8} \, d x^{8} + \frac {111}{8} \, x^{8} e + \frac {111}{7} \, d x^{7} - \frac {37}{7} \, x^{7} e - \frac {37}{6} \, d x^{6} + \frac {74}{3} \, x^{6} e + \frac {148}{5} \, d x^{5} + 13 \, x^{5} e + \frac {65}{4} \, d x^{4} + \frac {107}{4} \, x^{4} e + \frac {107}{3} \, d x^{3} + 11 \, x^{3} e + \frac {33}{2} \, d x^{2} + 9 \, x^{2} e + 18 \, d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*x^10*e + 100/9*d*x^9 - 5*x^9*e - 45/8*d*x^8 + 111/8*x^8*e + 111/7*d*x^7 - 37/7*x^7*e - 37/6*d*x^6 + 74/3*x^
6*e + 148/5*d*x^5 + 13*x^5*e + 65/4*d*x^4 + 107/4*x^4*e + 107/3*d*x^3 + 11*x^3*e + 33/2*d*x^2 + 9*x^2*e + 18*d
*x

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Mupad [B]
time = 4.17, size = 101, normalized size = 0.83 \begin {gather*} 10\,e\,x^{10}+\left (\frac {100\,d}{9}-5\,e\right )\,x^9+\left (\frac {111\,e}{8}-\frac {45\,d}{8}\right )\,x^8+\left (\frac {111\,d}{7}-\frac {37\,e}{7}\right )\,x^7+\left (\frac {74\,e}{3}-\frac {37\,d}{6}\right )\,x^6+\left (\frac {148\,d}{5}+13\,e\right )\,x^5+\left (\frac {65\,d}{4}+\frac {107\,e}{4}\right )\,x^4+\left (\frac {107\,d}{3}+11\,e\right )\,x^3+\left (\frac {33\,d}{2}+9\,e\right )\,x^2+18\,d\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*((33*d)/2 + 9*e) + x^9*((100*d)/9 - 5*e) + x^3*((107*d)/3 + 11*e) - x^6*((37*d)/6 - (74*e)/3) + x^7*((111*
d)/7 - (37*e)/7) + x^5*((148*d)/5 + 13*e) - x^8*((45*d)/8 - (111*e)/8) + x^4*((65*d)/4 + (107*e)/4) + 18*d*x +
10*e*x^10

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