3.17.73 \(\int (a+b x) (d+e x)^2 (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=65 \[ \frac {2 e (a+b x)^5 (b d-a e)}{5 b^3}+\frac {(a+b x)^4 (b d-a e)^2}{4 b^3}+\frac {e^2 (a+b x)^6}{6 b^3} \]

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Rubi [A]  time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 43} \begin {gather*} \frac {2 e (a+b x)^5 (b d-a e)}{5 b^3}+\frac {(a+b x)^4 (b d-a e)^2}{4 b^3}+\frac {e^2 (a+b x)^6}{6 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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])

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 122, normalized size = 1.88 \begin {gather*} a^3 d^2 x+\frac {1}{4} b x^4 \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+\frac {1}{3} a x^3 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac {1}{2} a^2 d x^2 (2 a e+3 b d)+\frac {1}{5} b^2 e x^5 (3 a e+2 b d)+\frac {1}{6} b^3 e^2 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.39, size = 130, normalized size = 2.00 \begin {gather*} \frac {1}{6} x^{6} e^{2} b^{3} + \frac {2}{5} x^{5} e d b^{3} + \frac {3}{5} x^{5} e^{2} b^{2} a + \frac {1}{4} x^{4} d^{2} b^{3} + \frac {3}{2} x^{4} e d b^{2} a + \frac {3}{4} x^{4} e^{2} b a^{2} + x^{3} d^{2} b^{2} a + 2 x^{3} e d b a^{2} + \frac {1}{3} x^{3} e^{2} a^{3} + \frac {3}{2} x^{2} d^{2} b a^{2} + x^{2} e d a^{3} + x d^{2} a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.15, size = 130, normalized size = 2.00 \begin {gather*} \frac {1}{6} \, b^{3} x^{6} e^{2} + \frac {2}{5} \, b^{3} d x^{5} e + \frac {1}{4} \, b^{3} d^{2} x^{4} + \frac {3}{5} \, a b^{2} x^{5} e^{2} + \frac {3}{2} \, a b^{2} d x^{4} e + a b^{2} d^{2} x^{3} + \frac {3}{4} \, a^{2} b x^{4} e^{2} + 2 \, a^{2} b d x^{3} e + \frac {3}{2} \, a^{2} b d^{2} x^{2} + \frac {1}{3} \, a^{3} x^{3} e^{2} + a^{3} d x^{2} e + a^{3} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 169, normalized size = 2.60 \begin {gather*} \frac {b^{3} e^{2} x^{6}}{6}+a^{3} d^{2} x +\frac {\left (2 a \,b^{2} e^{2}+\left (a \,e^{2}+2 b d e \right ) b^{2}\right ) x^{5}}{5}+\frac {\left (a^{2} b \,e^{2}+2 \left (a \,e^{2}+2 b d e \right ) a b +\left (2 a d e +b \,d^{2}\right ) b^{2}\right ) x^{4}}{4}+\frac {\left (a \,b^{2} d^{2}+\left (a \,e^{2}+2 b d e \right ) a^{2}+2 \left (2 a d e +b \,d^{2}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 a^{2} b \,d^{2}+\left (2 a d e +b \,d^{2}\right ) a^{2}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.53, size = 124, normalized size = 1.91 \begin {gather*} \frac {1}{6} \, b^{3} e^{2} x^{6} + a^{3} d^{2} x + \frac {1}{5} \, {\left (2 \, b^{3} d e + 3 \, a b^{2} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{2} + 6 \, a b^{2} d e + 3 \, a^{2} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{2} d^{2} + 6 \, a^{2} b d e + a^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 115, normalized size = 1.77 \begin {gather*} x^3\,\left (\frac {a^3\,e^2}{3}+2\,a^2\,b\,d\,e+a\,b^2\,d^2\right )+x^4\,\left (\frac {3\,a^2\,b\,e^2}{4}+\frac {3\,a\,b^2\,d\,e}{2}+\frac {b^3\,d^2}{4}\right )+a^3\,d^2\,x+\frac {b^3\,e^2\,x^6}{6}+\frac {a^2\,d\,x^2\,\left (2\,a\,e+3\,b\,d\right )}{2}+\frac {b^2\,e\,x^5\,\left (3\,a\,e+2\,b\,d\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.09, size = 133, normalized size = 2.05 \begin {gather*} a^{3} d^{2} x + \frac {b^{3} e^{2} x^{6}}{6} + x^{5} \left (\frac {3 a b^{2} e^{2}}{5} + \frac {2 b^{3} d e}{5}\right ) + x^{4} \left (\frac {3 a^{2} b e^{2}}{4} + \frac {3 a b^{2} d e}{2} + \frac {b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac {a^{3} e^{2}}{3} + 2 a^{2} b d e + a b^{2} d^{2}\right ) + x^{2} \left (a^{3} d e + \frac {3 a^{2} b d^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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