\(\int (d+e x)^4 (b x+c x^2) \, dx\) [221]

Optimal result

Integrand size = 17, antiderivative size = 62 \[ \int (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {d (c d-b e) (d+e x)^5}{5 e^3}-\frac {(2 c d-b e) (d+e x)^6}{6 e^3}+\frac {c (d+e x)^7}{7 e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {712} \[ \int (d+e x)^4 \left (b x+c x^2\right ) \, dx=-\frac {(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac {d (d+e x)^5 (c d-b e)}{5 e^3}+\frac {c (d+e x)^7}{7 e^3} \]

[In]

Int[(d + e*x)^4*(b*x + c*x^2),x]

[Out]

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

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 (\frac {d (c d-b e) (d+e x)^4}{e^2}+\frac {(-2 c d+b e) (d+e x)^5}{e^2}+\frac {c (d+e x)^6}{e^2}\right ) \, dx \\ & = \frac {d (c d-b e) (d+e x)^5}{5 e^3}-\frac {(2 c d-b e) (d+e x)^6}{6 e^3}+\frac {c (d+e x)^7}{7 e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.60 \[ \int (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {1}{2} b d^4 x^2+\frac {1}{3} d^3 (c d+4 b e) x^3+\frac {1}{2} d^2 e (2 c d+3 b e) x^4+\frac {2}{5} d e^2 (3 c d+2 b e) x^5+\frac {1}{6} e^3 (4 c d+b e) x^6+\frac {1}{7} c e^4 x^7 \]

[In]

Integrate[(d + e*x)^4*(b*x + c*x^2),x]

[Out]

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

Maple [A] (verified)

Time = 1.98 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.56

method	result	size
norman	\(\frac {e^{4} c \,x^{7}}{7}+\left (\frac {1}{6} e^{4} b +\frac {2}{3} d \,e^{3} c \right ) x^{6}+\left (\frac {4}{5} b d \,e^{3}+\frac {6}{5} d^{2} e^{2} c \right ) x^{5}+\left (\frac {3}{2} b \,d^{2} e^{2}+d^{3} e c \right ) x^{4}+\left (\frac {4}{3} b \,d^{3} e +\frac {1}{3} c \,d^{4}\right ) x^{3}+\frac {b \,d^{4} x^{2}}{2}\)	\(97\)
gosper	\(\frac {x^{2} \left (30 e^{4} c \,x^{5}+35 x^{4} e^{4} b +140 x^{4} d \,e^{3} c +168 x^{3} b d \,e^{3}+252 x^{3} d^{2} e^{2} c +315 x^{2} b \,d^{2} e^{2}+210 x^{2} d^{3} e c +280 x b \,d^{3} e +70 x c \,d^{4}+105 b \,d^{4}\right )}{210}\)	\(100\)
default	\(\frac {e^{4} c \,x^{7}}{7}+\frac {\left (e^{4} b +4 d \,e^{3} c \right ) x^{6}}{6}+\frac {\left (4 b d \,e^{3}+6 d^{2} e^{2} c \right ) x^{5}}{5}+\frac {\left (6 b \,d^{2} e^{2}+4 d^{3} e c \right ) x^{4}}{4}+\frac {\left (4 b \,d^{3} e +c \,d^{4}\right ) x^{3}}{3}+\frac {b \,d^{4} x^{2}}{2}\)	\(100\)
risch	\(\frac {1}{7} e^{4} c \,x^{7}+\frac {1}{6} x^{6} e^{4} b +\frac {2}{3} x^{6} d \,e^{3} c +\frac {4}{5} x^{5} b d \,e^{3}+\frac {6}{5} x^{5} d^{2} e^{2} c +\frac {3}{2} x^{4} b \,d^{2} e^{2}+x^{4} d^{3} e c +\frac {4}{3} x^{3} b \,d^{3} e +\frac {1}{3} c \,d^{4} x^{3}+\frac {1}{2} b \,d^{4} x^{2}\)	\(101\)
parallelrisch	\(\frac {1}{7} e^{4} c \,x^{7}+\frac {1}{6} x^{6} e^{4} b +\frac {2}{3} x^{6} d \,e^{3} c +\frac {4}{5} x^{5} b d \,e^{3}+\frac {6}{5} x^{5} d^{2} e^{2} c +\frac {3}{2} x^{4} b \,d^{2} e^{2}+x^{4} d^{3} e c +\frac {4}{3} x^{3} b \,d^{3} e +\frac {1}{3} c \,d^{4} x^{3}+\frac {1}{2} b \,d^{4} x^{2}\)	\(101\)

[In]

int((e*x+d)^4*(c*x^2+b*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.60 \[ \int (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {1}{7} \, c e^{4} x^{7} + \frac {1}{2} \, b d^{4} x^{2} + \frac {1}{6} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{6} + \frac {2}{5} \, {\left (3 \, c d^{2} e^{2} + 2 \, b d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{4} + 4 \, b d^{3} e\right )} x^{3} \]

[In]

integrate((e*x+d)^4*(c*x^2+b*x),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (53) = 106\).

Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.73 \[ \int (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {b d^{4} x^{2}}{2} + \frac {c e^{4} x^{7}}{7} + x^{6} \left (\frac {b e^{4}}{6} + \frac {2 c d e^{3}}{3}\right ) + x^{5} \cdot \left (\frac {4 b d e^{3}}{5} + \frac {6 c d^{2} e^{2}}{5}\right ) + x^{4} \cdot \left (\frac {3 b d^{2} e^{2}}{2} + c d^{3} e\right ) + x^{3} \cdot \left (\frac {4 b d^{3} e}{3} + \frac {c d^{4}}{3}\right ) \]

[In]

integrate((e*x+d)**4*(c*x**2+b*x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.60 \[ \int (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {1}{7} \, c e^{4} x^{7} + \frac {1}{2} \, b d^{4} x^{2} + \frac {1}{6} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{6} + \frac {2}{5} \, {\left (3 \, c d^{2} e^{2} + 2 \, b d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{4} + 4 \, b d^{3} e\right )} x^{3} \]

[In]

integrate((e*x+d)^4*(c*x^2+b*x),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.61 \[ \int (d+e x)^4 \left (b x+c x^2\right ) \, dx=\frac {1}{7} \, c e^{4} x^{7} + \frac {2}{3} \, c d e^{3} x^{6} + \frac {1}{6} \, b e^{4} x^{6} + \frac {6}{5} \, c d^{2} e^{2} x^{5} + \frac {4}{5} \, b d e^{3} x^{5} + c d^{3} e x^{4} + \frac {3}{2} \, b d^{2} e^{2} x^{4} + \frac {1}{3} \, c d^{4} x^{3} + \frac {4}{3} \, b d^{3} e x^{3} + \frac {1}{2} \, b d^{4} x^{2} \]

[In]

integrate((e*x+d)^4*(c*x^2+b*x),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.47 \[ \int (d+e x)^4 \left (b x+c x^2\right ) \, dx=x^3\,\left (\frac {c\,d^4}{3}+\frac {4\,b\,e\,d^3}{3}\right )+x^6\,\left (\frac {b\,e^4}{6}+\frac {2\,c\,d\,e^3}{3}\right )+\frac {b\,d^4\,x^2}{2}+\frac {c\,e^4\,x^7}{7}+\frac {d^2\,e\,x^4\,\left (3\,b\,e+2\,c\,d\right )}{2}+\frac {2\,d\,e^2\,x^5\,\left (2\,b\,e+3\,c\,d\right )}{5} \]

[In]

int((b*x + c*x^2)*(d + e*x)^4,x)

[Out]

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