3.16.14 \(\int (d+e x)^4 (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\)

Optimal. Leaf size=39 \[ \frac {1}{6} (d+e x)^6 \left (a-\frac {c d^2}{e^2}\right )+\frac {c d (d+e x)^7}{7 e^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {626, 43} \begin {gather*} \frac {1}{6} (d+e x)^6 \left (a-\frac {c d^2}{e^2}\right )+\frac {c d (d+e x)^7}{7 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

((a - (c*d^2)/e^2)*(d + e*x)^6)/6 + (c*d*(d + e*x)^7)/(7*e^2)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 626

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

Rubi steps

\begin {align*} \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx &=\int (a e+c d x) (d+e x)^5 \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^5}{e}+\frac {c d (d+e x)^6}{e}\right ) \, dx\\ &=\frac {1}{6} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^6+\frac {c d (d+e x)^7}{7 e^2}\\ \end {align*}

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Mathematica [B]  time = 0.03, size = 117, normalized size = 3.00 \begin {gather*} \frac {1}{42} x \left (7 a e \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+c d x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(7*a*e*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5) + c*d*x*(21*d^5 + 70*
d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5)))/42

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

IntegrateAlgebraic[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2), x]

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fricas [B]  time = 0.35, size = 127, normalized size = 3.26 \begin {gather*} \frac {1}{7} x^{7} e^{5} d c + \frac {5}{6} x^{6} e^{4} d^{2} c + \frac {1}{6} x^{6} e^{6} a + 2 x^{5} e^{3} d^{3} c + x^{5} e^{5} d a + \frac {5}{2} x^{4} e^{2} d^{4} c + \frac {5}{2} x^{4} e^{4} d^{2} a + \frac {5}{3} x^{3} e d^{5} c + \frac {10}{3} x^{3} e^{3} d^{3} a + \frac {1}{2} x^{2} d^{6} c + \frac {5}{2} x^{2} e^{2} d^{4} a + x e d^{5} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*e^5*d*c + 5/6*x^6*e^4*d^2*c + 1/6*x^6*e^6*a + 2*x^5*e^3*d^3*c + x^5*e^5*d*a + 5/2*x^4*e^2*d^4*c + 5/2*
x^4*e^4*d^2*a + 5/3*x^3*e*d^5*c + 10/3*x^3*e^3*d^3*a + 1/2*x^2*d^6*c + 5/2*x^2*e^2*d^4*a + x*e*d^5*a

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giac [B]  time = 0.15, size = 120, normalized size = 3.08 \begin {gather*} \frac {1}{7} \, c d x^{7} e^{5} + \frac {5}{6} \, c d^{2} x^{6} e^{4} + 2 \, c d^{3} x^{5} e^{3} + \frac {5}{2} \, c d^{4} x^{4} e^{2} + \frac {5}{3} \, c d^{5} x^{3} e + \frac {1}{2} \, c d^{6} x^{2} + \frac {1}{6} \, a x^{6} e^{6} + a d x^{5} e^{5} + \frac {5}{2} \, a d^{2} x^{4} e^{4} + \frac {10}{3} \, a d^{3} x^{3} e^{3} + \frac {5}{2} \, a d^{4} x^{2} e^{2} + a d^{5} x e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c*d*x^7*e^5 + 5/6*c*d^2*x^6*e^4 + 2*c*d^3*x^5*e^3 + 5/2*c*d^4*x^4*e^2 + 5/3*c*d^5*x^3*e + 1/2*c*d^6*x^2 +
1/6*a*x^6*e^6 + a*d*x^5*e^5 + 5/2*a*d^2*x^4*e^4 + 10/3*a*d^3*x^3*e^3 + 5/2*a*d^4*x^2*e^2 + a*d^5*x*e

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maple [B]  time = 0.05, size = 198, normalized size = 5.08 \begin {gather*} \frac {c d \,e^{5} x^{7}}{7}+a \,d^{5} e x +\frac {\left (4 c \,d^{2} e^{4}+\left (a \,e^{2}+c \,d^{2}\right ) e^{4}\right ) x^{6}}{6}+\frac {\left (a d \,e^{5}+6 c \,d^{3} e^{3}+4 \left (a \,e^{2}+c \,d^{2}\right ) d \,e^{3}\right ) x^{5}}{5}+\frac {\left (4 a \,d^{2} e^{4}+4 c \,d^{4} e^{2}+6 \left (a \,e^{2}+c \,d^{2}\right ) d^{2} e^{2}\right ) x^{4}}{4}+\frac {\left (6 a \,d^{3} e^{3}+c \,d^{5} e +4 \left (a \,e^{2}+c \,d^{2}\right ) d^{3} e \right ) x^{3}}{3}+\frac {\left (4 a \,d^{4} e^{2}+\left (a \,e^{2}+c \,d^{2}\right ) d^{4}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.98, size = 121, normalized size = 3.10 \begin {gather*} \frac {1}{7} \, c d e^{5} x^{7} + a d^{5} e x + \frac {1}{6} \, {\left (5 \, c d^{2} e^{4} + a e^{6}\right )} x^{6} + {\left (2 \, c d^{3} e^{3} + a d e^{5}\right )} x^{5} + \frac {5}{2} \, {\left (c d^{4} e^{2} + a d^{2} e^{4}\right )} x^{4} + \frac {5}{3} \, {\left (c d^{5} e + 2 \, a d^{3} e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{6} + 5 \, a d^{4} e^{2}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 122, normalized size = 3.13 \begin {gather*} x^4\,\left (\frac {5\,c\,d^4\,e^2}{2}+\frac {5\,a\,d^2\,e^4}{2}\right )+x^2\,\left (\frac {c\,d^6}{2}+\frac {5\,a\,d^4\,e^2}{2}\right )+x^6\,\left (\frac {5\,c\,d^2\,e^4}{6}+\frac {a\,e^6}{6}\right )+x^5\,\left (2\,c\,d^3\,e^3+a\,d\,e^5\right )+x^3\,\left (\frac {5\,c\,d^5\,e}{3}+\frac {10\,a\,d^3\,e^3}{3}\right )+a\,d^5\,e\,x+\frac {c\,d\,e^5\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*((5*a*d^2*e^4)/2 + (5*c*d^4*e^2)/2) + x^2*((c*d^6)/2 + (5*a*d^4*e^2)/2) + x^6*((a*e^6)/6 + (5*c*d^2*e^4)/6
) + x^5*(2*c*d^3*e^3 + a*d*e^5) + x^3*((10*a*d^3*e^3)/3 + (5*c*d^5*e)/3) + a*d^5*e*x + (c*d*e^5*x^7)/7

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sympy [B]  time = 0.09, size = 136, normalized size = 3.49 \begin {gather*} a d^{5} e x + \frac {c d e^{5} x^{7}}{7} + x^{6} \left (\frac {a e^{6}}{6} + \frac {5 c d^{2} e^{4}}{6}\right ) + x^{5} \left (a d e^{5} + 2 c d^{3} e^{3}\right ) + x^{4} \left (\frac {5 a d^{2} e^{4}}{2} + \frac {5 c d^{4} e^{2}}{2}\right ) + x^{3} \left (\frac {10 a d^{3} e^{3}}{3} + \frac {5 c d^{5} e}{3}\right ) + x^{2} \left (\frac {5 a d^{4} e^{2}}{2} + \frac {c d^{6}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d**5*e*x + c*d*e**5*x**7/7 + x**6*(a*e**6/6 + 5*c*d**2*e**4/6) + x**5*(a*d*e**5 + 2*c*d**3*e**3) + x**4*(5*a
*d**2*e**4/2 + 5*c*d**4*e**2/2) + x**3*(10*a*d**3*e**3/3 + 5*c*d**5*e/3) + x**2*(5*a*d**4*e**2/2 + c*d**6/2)

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