3.221 \(\int (d+e x)^4 (b x+c x^2) \, dx\)

Optimal. Leaf size=62 \[ -\frac {(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac {d (d+e x)^5 (c d-b e)}{5 e^3}+\frac {c (d+e x)^7}{7 e^3} \]

[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

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Rubi [A]  time = 0.07, 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 = {698} \[ -\frac {(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac {d (d+e x)^5 (c d-b e)}{5 e^3}+\frac {c (d+e x)^7}{7 e^3} \]

Antiderivative was successfully verified.

[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 698

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)^4 \left (b x+c x^2\right ) \, dx &=\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*}

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Mathematica [A]  time = 0.02, size = 99, normalized size = 1.60 \[ \frac {1}{3} d^3 x^3 (4 b e+c d)+\frac {1}{2} d^2 e x^4 (3 b e+2 c d)+\frac {1}{6} e^3 x^6 (b e+4 c d)+\frac {2}{5} d e^2 x^5 (2 b e+3 c d)+\frac {1}{2} b d^4 x^2+\frac {1}{7} c e^4 x^7 \]

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.87, size = 100, normalized size = 1.61 \[ \frac {1}{7} x^{7} e^{4} c + \frac {2}{3} x^{6} e^{3} d c + \frac {1}{6} x^{6} e^{4} b + \frac {6}{5} x^{5} e^{2} d^{2} c + \frac {4}{5} x^{5} e^{3} d b + x^{4} e d^{3} c + \frac {3}{2} x^{4} e^{2} d^{2} b + \frac {1}{3} x^{3} d^{4} c + \frac {4}{3} x^{3} e d^{3} b + \frac {1}{2} x^{2} d^{4} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 96, normalized size = 1.55 \[ \frac {1}{7} \, c x^{7} e^{4} + \frac {2}{3} \, c d x^{6} e^{3} + \frac {6}{5} \, c d^{2} x^{5} e^{2} + c d^{3} x^{4} e + \frac {1}{3} \, c d^{4} x^{3} + \frac {1}{6} \, b x^{6} e^{4} + \frac {4}{5} \, b d x^{5} e^{3} + \frac {3}{2} \, b d^{2} x^{4} e^{2} + \frac {4}{3} \, b d^{3} x^{3} e + \frac {1}{2} \, b d^{4} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 100, normalized size = 1.61 \[ \frac {c \,e^{4} x^{7}}{7}+\frac {b \,d^{4} x^{2}}{2}+\frac {\left (e^{4} b +4 d \,e^{3} c \right ) x^{6}}{6}+\frac {\left (4 d \,e^{3} b +6 d^{2} e^{2} c \right ) x^{5}}{5}+\frac {\left (6 d^{2} e^{2} b +4 d^{3} e c \right ) x^{4}}{4}+\frac {\left (4 d^{3} e b +d^{4} c \right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 99, normalized size = 1.60 \[ \frac {1}{7} \, c e^{4} x^{7} + \frac {1}{2} \, b d^{4} x^{2} + \frac {1}{6} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{6} + \frac {2}{5} \, {\left (3 \, c d^{2} e^{2} + 2 \, b d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{4} + 4 \, b d^{3} e\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.17, size = 91, normalized size = 1.47 \[ x^3\,\left (\frac {c\,d^4}{3}+\frac {4\,b\,e\,d^3}{3}\right )+x^6\,\left (\frac {b\,e^4}{6}+\frac {2\,c\,d\,e^3}{3}\right )+\frac {b\,d^4\,x^2}{2}+\frac {c\,e^4\,x^7}{7}+\frac {d^2\,e\,x^4\,\left (3\,b\,e+2\,c\,d\right )}{2}+\frac {2\,d\,e^2\,x^5\,\left (2\,b\,e+3\,c\,d\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 0.09, size = 107, normalized size = 1.73 \[ \frac {b d^{4} x^{2}}{2} + \frac {c e^{4} x^{7}}{7} + x^{6} \left (\frac {b e^{4}}{6} + \frac {2 c d e^{3}}{3}\right ) + x^{5} \left (\frac {4 b d e^{3}}{5} + \frac {6 c d^{2} e^{2}}{5}\right ) + x^{4} \left (\frac {3 b d^{2} e^{2}}{2} + c d^{3} e\right ) + x^{3} \left (\frac {4 b d^{3} e}{3} + \frac {c d^{4}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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