\(\int (d+e x)^2 (b x+c x^2)^2 \, dx\) [233]

Optimal result

Integrand size = 19, antiderivative size = 87 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} b^2 d^2 x^3+\frac {1}{2} b d (c d+b e) x^4+\frac {1}{5} \left (c^2 d^2+4 b c d e+b^2 e^2\right ) x^5+\frac {1}{3} c e (c d+b e) x^6+\frac {1}{7} c^2 e^2 x^7 \]

[Out]

1/3*b^2*d^2*x^3+1/2*b*d*(b*e+c*d)*x^4+1/5*(b^2*e^2+4*b*c*d*e+c^2*d^2)*x^5+1/3*c*e*(b*e+c*d)*x^6+1/7*c^2*e^2*x^
7

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{5} x^5 \left (b^2 e^2+4 b c d e+c^2 d^2\right )+\frac {1}{3} b^2 d^2 x^3+\frac {1}{3} c e x^6 (b e+c d)+\frac {1}{2} b d x^4 (b e+c d)+\frac {1}{7} c^2 e^2 x^7 \]

[In]

Int[(d + e*x)^2*(b*x + c*x^2)^2,x]

[Out]

(b^2*d^2*x^3)/3 + (b*d*(c*d + b*e)*x^4)/2 + ((c^2*d^2 + 4*b*c*d*e + b^2*e^2)*x^5)/5 + (c*e*(c*d + b*e)*x^6)/3
+ (c^2*e^2*x^7)/7

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (b^2 d^2 x^2+2 b d (c d+b e) x^3+\left (c^2 d^2+4 b c d e+b^2 e^2\right ) x^4+2 c e (c d+b e) x^5+c^2 e^2 x^6\right ) \, dx \\ & = \frac {1}{3} b^2 d^2 x^3+\frac {1}{2} b d (c d+b e) x^4+\frac {1}{5} \left (c^2 d^2+4 b c d e+b^2 e^2\right ) x^5+\frac {1}{3} c e (c d+b e) x^6+\frac {1}{7} c^2 e^2 x^7 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} b^2 d^2 x^3+\frac {1}{2} b d (c d+b e) x^4+\frac {1}{5} \left (c^2 d^2+4 b c d e+b^2 e^2\right ) x^5+\frac {1}{3} c e (c d+b e) x^6+\frac {1}{7} c^2 e^2 x^7 \]

[In]

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

[Out]

(b^2*d^2*x^3)/3 + (b*d*(c*d + b*e)*x^4)/2 + ((c^2*d^2 + 4*b*c*d*e + b^2*e^2)*x^5)/5 + (c*e*(c*d + b*e)*x^6)/3
+ (c^2*e^2*x^7)/7

Maple [A] (verified)

Time = 1.96 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02

method	result	size
norman	\(\frac {c^{2} e^{2} x^{7}}{7}+\left (\frac {1}{3} b c \,e^{2}+\frac {1}{3} d e \,c^{2}\right ) x^{6}+\left (\frac {1}{5} b^{2} e^{2}+\frac {4}{5} b c d e +\frac {1}{5} c^{2} d^{2}\right ) x^{5}+\left (\frac {1}{2} b^{2} d e +\frac {1}{2} b c \,d^{2}\right ) x^{4}+\frac {d^{2} x^{3} b^{2}}{3}\)	\(89\)
default	\(\frac {c^{2} e^{2} x^{7}}{7}+\frac {\left (2 b c \,e^{2}+2 d e \,c^{2}\right ) x^{6}}{6}+\frac {\left (b^{2} e^{2}+4 b c d e +c^{2} d^{2}\right ) x^{5}}{5}+\frac {\left (2 b^{2} d e +2 b c \,d^{2}\right ) x^{4}}{4}+\frac {d^{2} x^{3} b^{2}}{3}\)	\(90\)
gosper	\(\frac {x^{3} \left (30 c^{2} e^{2} x^{4}+70 x^{3} b c \,e^{2}+70 x^{3} d e \,c^{2}+42 x^{2} b^{2} e^{2}+168 x^{2} b c d e +42 c^{2} d^{2} x^{2}+105 b^{2} d e x +105 b c \,d^{2} x +70 b^{2} d^{2}\right )}{210}\)	\(93\)
risch	\(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} b c \,e^{2}+\frac {1}{3} x^{6} d e \,c^{2}+\frac {1}{5} x^{5} b^{2} e^{2}+\frac {4}{5} x^{5} b c d e +\frac {1}{5} c^{2} d^{2} x^{5}+\frac {1}{2} x^{4} b^{2} d e +\frac {1}{2} x^{4} b c \,d^{2}+\frac {1}{3} d^{2} x^{3} b^{2}\)	\(95\)
parallelrisch	\(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} b c \,e^{2}+\frac {1}{3} x^{6} d e \,c^{2}+\frac {1}{5} x^{5} b^{2} e^{2}+\frac {4}{5} x^{5} b c d e +\frac {1}{5} c^{2} d^{2} x^{5}+\frac {1}{2} x^{4} b^{2} d e +\frac {1}{2} x^{4} b c \,d^{2}+\frac {1}{3} d^{2} x^{3} b^{2}\)	\(95\)

[In]

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

[Out]

1/7*c^2*e^2*x^7+(1/3*b*c*e^2+1/3*d*e*c^2)*x^6+(1/5*b^2*e^2+4/5*b*c*d*e+1/5*c^2*d^2)*x^5+(1/2*b^2*d*e+1/2*b*c*d
^2)*x^4+1/3*d^2*x^3*b^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, b^{2} d^{2} x^{3} + \frac {1}{3} \, {\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{2} + b^{2} d e\right )} x^{4} \]

[In]

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

[Out]

1/7*c^2*e^2*x^7 + 1/3*b^2*d^2*x^3 + 1/3*(c^2*d*e + b*c*e^2)*x^6 + 1/5*(c^2*d^2 + 4*b*c*d*e + b^2*e^2)*x^5 + 1/
2*(b*c*d^2 + b^2*d*e)*x^4

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {b^{2} d^{2} x^{3}}{3} + \frac {c^{2} e^{2} x^{7}}{7} + x^{6} \left (\frac {b c e^{2}}{3} + \frac {c^{2} d e}{3}\right ) + x^{5} \left (\frac {b^{2} e^{2}}{5} + \frac {4 b c d e}{5} + \frac {c^{2} d^{2}}{5}\right ) + x^{4} \left (\frac {b^{2} d e}{2} + \frac {b c d^{2}}{2}\right ) \]

[In]

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

[Out]

b**2*d**2*x**3/3 + c**2*e**2*x**7/7 + x**6*(b*c*e**2/3 + c**2*d*e/3) + x**5*(b**2*e**2/5 + 4*b*c*d*e/5 + c**2*
d**2/5) + x**4*(b**2*d*e/2 + b*c*d**2/2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, b^{2} d^{2} x^{3} + \frac {1}{3} \, {\left (c^{2} d e + b c e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{2} + b^{2} d e\right )} x^{4} \]

[In]

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

[Out]

1/7*c^2*e^2*x^7 + 1/3*b^2*d^2*x^3 + 1/3*(c^2*d*e + b*c*e^2)*x^6 + 1/5*(c^2*d^2 + 4*b*c*d*e + b^2*e^2)*x^5 + 1/
2*(b*c*d^2 + b^2*d*e)*x^4

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, c^{2} d e x^{6} + \frac {1}{3} \, b c e^{2} x^{6} + \frac {1}{5} \, c^{2} d^{2} x^{5} + \frac {4}{5} \, b c d e x^{5} + \frac {1}{5} \, b^{2} e^{2} x^{5} + \frac {1}{2} \, b c d^{2} x^{4} + \frac {1}{2} \, b^{2} d e x^{4} + \frac {1}{3} \, b^{2} d^{2} x^{3} \]

[In]

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

[Out]

1/7*c^2*e^2*x^7 + 1/3*c^2*d*e*x^6 + 1/3*b*c*e^2*x^6 + 1/5*c^2*d^2*x^5 + 4/5*b*c*d*e*x^5 + 1/5*b^2*e^2*x^5 + 1/
2*b*c*d^2*x^4 + 1/2*b^2*d*e*x^4 + 1/3*b^2*d^2*x^3

Mupad [B] (verification not implemented)

Time = 9.56 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int (d+e x)^2 \left (b x+c x^2\right )^2 \, dx=x^5\,\left (\frac {b^2\,e^2}{5}+\frac {4\,b\,c\,d\,e}{5}+\frac {c^2\,d^2}{5}\right )+\frac {b^2\,d^2\,x^3}{3}+\frac {c^2\,e^2\,x^7}{7}+\frac {b\,d\,x^4\,\left (b\,e+c\,d\right )}{2}+\frac {c\,e\,x^6\,\left (b\,e+c\,d\right )}{3} \]

[In]

int((b*x + c*x^2)^2*(d + e*x)^2,x)

[Out]

x^5*((b^2*e^2)/5 + (c^2*d^2)/5 + (4*b*c*d*e)/5) + (b^2*d^2*x^3)/3 + (c^2*e^2*x^7)/7 + (b*d*x^4*(b*e + c*d))/2
+ (c*e*x^6*(b*e + c*d))/3