3.3.22 \(\int (d+e x)^3 (b x+c x^2) \, dx\) [222]

Optimal. Leaf size=62 \[ \frac {d (c d-b e) (d+e x)^4}{4 e^3}-\frac {(2 c d-b e) (d+e x)^5}{5 e^3}+\frac {c (d+e x)^6}{6 e^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {712} \begin {gather*} -\frac {(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac {d (d+e x)^4 (c d-b e)}{4 e^3}+\frac {c (d+e x)^6}{6 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(b*x + c*x^2),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 67, normalized size = 1.08 \begin {gather*} \frac {1}{60} x^2 \left (30 b d^3+20 d^2 (c d+3 b e) x+45 d e (c d+b e) x^2+12 e^2 (3 c d+b e) x^3+10 c e^3 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(b*x + c*x^2),x]

[Out]

(x^2*(30*b*d^3 + 20*d^2*(c*d + 3*b*e)*x + 45*d*e*(c*d + b*e)*x^2 + 12*e^2*(3*c*d + b*e)*x^3 + 10*c*e^3*x^4))/6
0

________________________________________________________________________________________

Maple [A]
time = 0.38, size = 76, normalized size = 1.23

method	result	size
norman	\(\frac {e^{3} c \,x^{6}}{6}+\left (\frac {1}{5} e^{3} b +\frac {3}{5} d \,e^{2} c \right ) x^{5}+\left (\frac {3}{4} b d \,e^{2}+\frac {3}{4} d^{2} e c \right ) x^{4}+\left (b \,d^{2} e +\frac {1}{3} d^{3} c \right ) x^{3}+\frac {b \,d^{3} x^{2}}{2}\)	\(74\)
gosper	\(\frac {x^{2} \left (10 e^{3} c \,x^{4}+12 x^{3} e^{3} b +36 x^{3} d \,e^{2} c +45 x^{2} b d \,e^{2}+45 x^{2} d^{2} e c +60 x b \,d^{2} e +20 x \,d^{3} c +30 b \,d^{3}\right )}{60}\)	\(76\)
default	\(\frac {e^{3} c \,x^{6}}{6}+\frac {\left (e^{3} b +3 d \,e^{2} c \right ) x^{5}}{5}+\frac {\left (3 b d \,e^{2}+3 d^{2} e c \right ) x^{4}}{4}+\frac {\left (3 b \,d^{2} e +d^{3} c \right ) x^{3}}{3}+\frac {b \,d^{3} x^{2}}{2}\)	\(76\)
risch	\(\frac {1}{6} e^{3} c \,x^{6}+\frac {1}{5} x^{5} e^{3} b +\frac {3}{5} x^{5} d \,e^{2} c +\frac {3}{4} x^{4} b d \,e^{2}+\frac {3}{4} x^{4} d^{2} e c +x^{3} b \,d^{2} e +\frac {1}{3} x^{3} d^{3} c +\frac {1}{2} b \,d^{3} x^{2}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(c*x^2+b*x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 71, normalized size = 1.15 \begin {gather*} \frac {1}{6} \, c x^{6} e^{3} + \frac {1}{2} \, b d^{3} x^{2} + \frac {1}{5} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + \frac {3}{4} \, {\left (c d^{2} e + b d e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{3} + 3 \, b d^{2} e\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 1.82, size = 78, normalized size = 1.26 \begin {gather*} \frac {1}{3} \, c d^{3} x^{3} + \frac {1}{2} \, b d^{3} x^{2} + \frac {1}{30} \, {\left (5 \, c x^{6} + 6 \, b x^{5}\right )} e^{3} + \frac {3}{20} \, {\left (4 \, c d x^{5} + 5 \, b d x^{4}\right )} e^{2} + \frac {1}{4} \, {\left (3 \, c d^{2} x^{4} + 4 \, b d^{2} x^{3}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x),x, algorithm="fricas")

[Out]

1/3*c*d^3*x^3 + 1/2*b*d^3*x^2 + 1/30*(5*c*x^6 + 6*b*x^5)*e^3 + 3/20*(4*c*d*x^5 + 5*b*d*x^4)*e^2 + 1/4*(3*c*d^2
*x^4 + 4*b*d^2*x^3)*e

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 80, normalized size = 1.29 \begin {gather*} \frac {b d^{3} x^{2}}{2} + \frac {c e^{3} x^{6}}{6} + x^{5} \left (\frac {b e^{3}}{5} + \frac {3 c d e^{2}}{5}\right ) + x^{4} \cdot \left (\frac {3 b d e^{2}}{4} + \frac {3 c d^{2} e}{4}\right ) + x^{3} \left (b d^{2} e + \frac {c d^{3}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(c*x**2+b*x),x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 1.30, size = 74, normalized size = 1.19 \begin {gather*} \frac {1}{6} \, c x^{6} e^{3} + \frac {3}{5} \, c d x^{5} e^{2} + \frac {3}{4} \, c d^{2} x^{4} e + \frac {1}{3} \, c d^{3} x^{3} + \frac {1}{5} \, b x^{5} e^{3} + \frac {3}{4} \, b d x^{4} e^{2} + b d^{2} x^{3} e + \frac {1}{2} \, b d^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+b*x),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.15, size = 68, normalized size = 1.10 \begin {gather*} x^3\,\left (\frac {c\,d^3}{3}+b\,e\,d^2\right )+x^5\,\left (\frac {b\,e^3}{5}+\frac {3\,c\,d\,e^2}{5}\right )+\frac {b\,d^3\,x^2}{2}+\frac {c\,e^3\,x^6}{6}+\frac {3\,d\,e\,x^4\,\left (b\,e+c\,d\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)*(d + e*x)^3,x)

[Out]

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

________________________________________________________________________________________