\(\int (d+e x) (a+b x+c x^2)^2 \, dx\) [2121]

Optimal result

Integrand size = 18, antiderivative size = 96 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^2 d x+\frac {1}{2} a (2 b d+a e) x^2+\frac {1}{3} \left (b^2 d+2 a c d+2 a b e\right ) x^3+\frac {1}{4} \left (2 b c d+b^2 e+2 a c e\right ) x^4+\frac {1}{5} c (c d+2 b e) x^5+\frac {1}{6} c^2 e x^6 \]

[Out]

a^2*d*x+1/2*a*(a*e+2*b*d)*x^2+1/3*(2*a*b*e+2*a*c*d+b^2*d)*x^3+1/4*(2*a*c*e+b^2*e+2*b*c*d)*x^4+1/5*c*(2*b*e+c*d
)*x^5+1/6*c^2*e*x^6

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {645} \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^2 d x+\frac {1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac {1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac {1}{2} a x^2 (a e+2 b d)+\frac {1}{5} c x^5 (2 b e+c d)+\frac {1}{6} c^2 e x^6 \]

[In]

Int[(d + e*x)*(a + b*x + c*x^2)^2,x]

[Out]

a^2*d*x + (a*(2*b*d + a*e)*x^2)/2 + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + ((2*b*c*d + b^2*e + 2*a*c*e)*x^4)/4
+ (c*(c*d + 2*b*e)*x^5)/5 + (c^2*e*x^6)/6

Rule 645

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 d+a (2 b d+a e) x+\left (b^2 d+2 a c d+2 a b e\right ) x^2+\left (2 b c d+b^2 e+2 a c e\right ) x^3+c (c d+2 b e) x^4+c^2 e x^5\right ) \, dx \\ & = a^2 d x+\frac {1}{2} a (2 b d+a e) x^2+\frac {1}{3} \left (b^2 d+2 a c d+2 a b e\right ) x^3+\frac {1}{4} \left (2 b c d+b^2 e+2 a c e\right ) x^4+\frac {1}{5} c (c d+2 b e) x^5+\frac {1}{6} c^2 e x^6 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^2 d x+\frac {1}{2} a (2 b d+a e) x^2+\frac {1}{3} \left (b^2 d+2 a c d+2 a b e\right ) x^3+\frac {1}{4} \left (2 b c d+b^2 e+2 a c e\right ) x^4+\frac {1}{5} c (c d+2 b e) x^5+\frac {1}{6} c^2 e x^6 \]

[In]

Integrate[(d + e*x)*(a + b*x + c*x^2)^2,x]

[Out]

a^2*d*x + (a*(2*b*d + a*e)*x^2)/2 + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + ((2*b*c*d + b^2*e + 2*a*c*e)*x^4)/4
+ (c*(c*d + 2*b*e)*x^5)/5 + (c^2*e*x^6)/6

Maple [A] (verified)

Time = 2.86 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94

method	result	size
norman	\(\frac {c^{2} e \,x^{6}}{6}+\left (\frac {2}{5} b c e +\frac {1}{5} c^{2} d \right ) x^{5}+\left (\frac {1}{2} a c e +\frac {1}{4} b^{2} e +\frac {1}{2} b c d \right ) x^{4}+\left (\frac {2}{3} a e b +\frac {2}{3} a c d +\frac {1}{3} b^{2} d \right ) x^{3}+\left (\frac {1}{2} a^{2} e +a b d \right ) x^{2}+a^{2} d x\)	\(90\)
default	\(\frac {c^{2} e \,x^{6}}{6}+\frac {\left (2 b c e +c^{2} d \right ) x^{5}}{5}+\frac {\left (2 b c d +e \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (d \left (2 a c +b^{2}\right )+2 a e b \right ) x^{3}}{3}+\frac {\left (a^{2} e +2 a b d \right ) x^{2}}{2}+a^{2} d x\)	\(91\)
gosper	\(\frac {1}{6} c^{2} e \,x^{6}+\frac {2}{5} x^{5} b c e +\frac {1}{5} c^{2} d \,x^{5}+\frac {1}{2} a c e \,x^{4}+\frac {1}{4} b^{2} e \,x^{4}+\frac {1}{2} c b d \,x^{4}+\frac {2}{3} a b e \,x^{3}+\frac {2}{3} a c d \,x^{3}+\frac {1}{3} d \,x^{3} b^{2}+\frac {1}{2} a^{2} e \,x^{2}+x^{2} a b d +a^{2} d x\)	\(100\)
risch	\(\frac {1}{6} c^{2} e \,x^{6}+\frac {2}{5} x^{5} b c e +\frac {1}{5} c^{2} d \,x^{5}+\frac {1}{2} a c e \,x^{4}+\frac {1}{4} b^{2} e \,x^{4}+\frac {1}{2} c b d \,x^{4}+\frac {2}{3} a b e \,x^{3}+\frac {2}{3} a c d \,x^{3}+\frac {1}{3} d \,x^{3} b^{2}+\frac {1}{2} a^{2} e \,x^{2}+x^{2} a b d +a^{2} d x\)	\(100\)
parallelrisch	\(\frac {1}{6} c^{2} e \,x^{6}+\frac {2}{5} x^{5} b c e +\frac {1}{5} c^{2} d \,x^{5}+\frac {1}{2} a c e \,x^{4}+\frac {1}{4} b^{2} e \,x^{4}+\frac {1}{2} c b d \,x^{4}+\frac {2}{3} a b e \,x^{3}+\frac {2}{3} a c d \,x^{3}+\frac {1}{3} d \,x^{3} b^{2}+\frac {1}{2} a^{2} e \,x^{2}+x^{2} a b d +a^{2} d x\)	\(100\)

[In]

int((e*x+d)*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/6*c^2*e*x^6+(2/5*b*c*e+1/5*c^2*d)*x^5+(1/2*a*c*e+1/4*b^2*e+1/2*b*c*d)*x^4+(2/3*a*e*b+2/3*a*c*d+1/3*b^2*d)*x^
3+(1/2*a^2*e+a*b*d)*x^2+a^2*d*x

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} e x^{6} + \frac {1}{5} \, {\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{4} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d + a^{2} e\right )} x^{2} \]

[In]

integrate((e*x+d)*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

1/6*c^2*e*x^6 + 1/5*(c^2*d + 2*b*c*e)*x^5 + 1/4*(2*b*c*d + (b^2 + 2*a*c)*e)*x^4 + a^2*d*x + 1/3*(2*a*b*e + (b^
2 + 2*a*c)*d)*x^3 + 1/2*(2*a*b*d + a^2*e)*x^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.04 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^{2} d x + \frac {c^{2} e x^{6}}{6} + x^{5} \cdot \left (\frac {2 b c e}{5} + \frac {c^{2} d}{5}\right ) + x^{4} \left (\frac {a c e}{2} + \frac {b^{2} e}{4} + \frac {b c d}{2}\right ) + x^{3} \cdot \left (\frac {2 a b e}{3} + \frac {2 a c d}{3} + \frac {b^{2} d}{3}\right ) + x^{2} \left (\frac {a^{2} e}{2} + a b d\right ) \]

[In]

integrate((e*x+d)*(c*x**2+b*x+a)**2,x)

[Out]

a**2*d*x + c**2*e*x**6/6 + x**5*(2*b*c*e/5 + c**2*d/5) + x**4*(a*c*e/2 + b**2*e/4 + b*c*d/2) + x**3*(2*a*b*e/3
+ 2*a*c*d/3 + b**2*d/3) + x**2*(a**2*e/2 + a*b*d)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} e x^{6} + \frac {1}{5} \, {\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{4} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d + a^{2} e\right )} x^{2} \]

[In]

integrate((e*x+d)*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

1/6*c^2*e*x^6 + 1/5*(c^2*d + 2*b*c*e)*x^5 + 1/4*(2*b*c*d + (b^2 + 2*a*c)*e)*x^4 + a^2*d*x + 1/3*(2*a*b*e + (b^
2 + 2*a*c)*d)*x^3 + 1/2*(2*a*b*d + a^2*e)*x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} e x^{6} + \frac {1}{5} \, c^{2} d x^{5} + \frac {2}{5} \, b c e x^{5} + \frac {1}{2} \, b c d x^{4} + \frac {1}{4} \, b^{2} e x^{4} + \frac {1}{2} \, a c e x^{4} + \frac {1}{3} \, b^{2} d x^{3} + \frac {2}{3} \, a c d x^{3} + \frac {2}{3} \, a b e x^{3} + a b d x^{2} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x \]

[In]

integrate((e*x+d)*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

1/6*c^2*e*x^6 + 1/5*c^2*d*x^5 + 2/5*b*c*e*x^5 + 1/2*b*c*d*x^4 + 1/4*b^2*e*x^4 + 1/2*a*c*e*x^4 + 1/3*b^2*d*x^3
+ 2/3*a*c*d*x^3 + 2/3*a*b*e*x^3 + a*b*d*x^2 + 1/2*a^2*e*x^2 + a^2*d*x

Mupad [B] (verification not implemented)

Time = 9.75 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=x^3\,\left (\frac {d\,b^2}{3}+\frac {2\,a\,e\,b}{3}+\frac {2\,a\,c\,d}{3}\right )+x^4\,\left (\frac {e\,b^2}{4}+\frac {c\,d\,b}{2}+\frac {a\,c\,e}{2}\right )+x^2\,\left (\frac {e\,a^2}{2}+b\,d\,a\right )+x^5\,\left (\frac {d\,c^2}{5}+\frac {2\,b\,e\,c}{5}\right )+\frac {c^2\,e\,x^6}{6}+a^2\,d\,x \]

[In]

int((d + e*x)*(a + b*x + c*x^2)^2,x)

[Out]

x^3*((b^2*d)/3 + (2*a*b*e)/3 + (2*a*c*d)/3) + x^4*((b^2*e)/4 + (a*c*e)/2 + (b*c*d)/2) + x^2*((a^2*e)/2 + a*b*d
) + x^5*((c^2*d)/5 + (2*b*c*e)/5) + (c^2*e*x^6)/6 + a^2*d*x