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

Optimal. Leaf size=65 \[ \frac {e (a+b x)^6 (b d-a e)}{3 b^3}+\frac {(a+b x)^5 (b d-a e)^2}{5 b^3}+\frac {e^2 (a+b x)^7}{7 b^3} \]

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Rubi [A]  time = 0.09, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} \frac {e (a+b x)^6 (b d-a e)}{3 b^3}+\frac {(a+b x)^5 (b d-a e)^2}{5 b^3}+\frac {e^2 (a+b x)^7}{7 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [B]  time = 0.02, size = 148, normalized size = 2.28 \begin {gather*} a^4 d^2 x+a^3 d x^2 (a e+2 b d)+\frac {1}{5} b^2 x^5 \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+a b x^4 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+\frac {1}{3} a^2 x^3 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+\frac {1}{3} b^3 e x^6 (2 a e+b d)+\frac {1}{7} b^4 e^2 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^4*d^2*x + a^3*d*(2*b*d + a*e)*x^2 + (a^2*(6*b^2*d^2 + 8*a*b*d*e + a^2*e^2)*x^3)/3 + a*b*(b^2*d^2 + 3*a*b*d*e
+ a^2*e^2)*x^4 + (b^2*(b^2*d^2 + 8*a*b*d*e + 6*a^2*e^2)*x^5)/5 + (b^3*e*(b*d + 2*a*e)*x^6)/3 + (b^4*e^2*x^7)/
7

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.35, size = 170, normalized size = 2.62 \begin {gather*} \frac {1}{7} x^{7} e^{2} b^{4} + \frac {1}{3} x^{6} e d b^{4} + \frac {2}{3} x^{6} e^{2} b^{3} a + \frac {1}{5} x^{5} d^{2} b^{4} + \frac {8}{5} x^{5} e d b^{3} a + \frac {6}{5} x^{5} e^{2} b^{2} a^{2} + x^{4} d^{2} b^{3} a + 3 x^{4} e d b^{2} a^{2} + x^{4} e^{2} b a^{3} + 2 x^{3} d^{2} b^{2} a^{2} + \frac {8}{3} x^{3} e d b a^{3} + \frac {1}{3} x^{3} e^{2} a^{4} + 2 x^{2} d^{2} b a^{3} + x^{2} e d a^{4} + x d^{2} a^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*e^2*b^4 + 1/3*x^6*e*d*b^4 + 2/3*x^6*e^2*b^3*a + 1/5*x^5*d^2*b^4 + 8/5*x^5*e*d*b^3*a + 6/5*x^5*e^2*b^2*
a^2 + x^4*d^2*b^3*a + 3*x^4*e*d*b^2*a^2 + x^4*e^2*b*a^3 + 2*x^3*d^2*b^2*a^2 + 8/3*x^3*e*d*b*a^3 + 1/3*x^3*e^2*
a^4 + 2*x^2*d^2*b*a^3 + x^2*e*d*a^4 + x*d^2*a^4

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giac [B]  time = 0.15, size = 170, normalized size = 2.62 \begin {gather*} \frac {1}{7} \, b^{4} x^{7} e^{2} + \frac {1}{3} \, b^{4} d x^{6} e + \frac {1}{5} \, b^{4} d^{2} x^{5} + \frac {2}{3} \, a b^{3} x^{6} e^{2} + \frac {8}{5} \, a b^{3} d x^{5} e + a b^{3} d^{2} x^{4} + \frac {6}{5} \, a^{2} b^{2} x^{5} e^{2} + 3 \, a^{2} b^{2} d x^{4} e + 2 \, a^{2} b^{2} d^{2} x^{3} + a^{3} b x^{4} e^{2} + \frac {8}{3} \, a^{3} b d x^{3} e + 2 \, a^{3} b d^{2} x^{2} + \frac {1}{3} \, a^{4} x^{3} e^{2} + a^{4} d x^{2} e + a^{4} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^4*x^7*e^2 + 1/3*b^4*d*x^6*e + 1/5*b^4*d^2*x^5 + 2/3*a*b^3*x^6*e^2 + 8/5*a*b^3*d*x^5*e + a*b^3*d^2*x^4 +
6/5*a^2*b^2*x^5*e^2 + 3*a^2*b^2*d*x^4*e + 2*a^2*b^2*d^2*x^3 + a^3*b*x^4*e^2 + 8/3*a^3*b*d*x^3*e + 2*a^3*b*d^2*
x^2 + 1/3*a^4*x^3*e^2 + a^4*d*x^2*e + a^4*d^2*x

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maple [B]  time = 0.04, size = 163, normalized size = 2.51 \begin {gather*} \frac {b^{4} e^{2} x^{7}}{7}+a^{4} d^{2} x +\frac {\left (4 e^{2} a \,b^{3}+2 b^{4} d e \right ) x^{6}}{6}+\frac {\left (6 a^{2} b^{2} e^{2}+8 d e a \,b^{3}+b^{4} d^{2}\right ) x^{5}}{5}+\frac {\left (4 e^{2} a^{3} b +12 d e \,b^{2} a^{2}+4 d^{2} a \,b^{3}\right ) x^{4}}{4}+\frac {\left (e^{2} a^{4}+8 d e \,a^{3} b +6 d^{2} b^{2} a^{2}\right ) x^{3}}{3}+\frac {\left (2 d e \,a^{4}+4 d^{2} a^{3} b \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*e^2*b^4*x^7+1/6*(4*a*b^3*e^2+2*b^4*d*e)*x^6+1/5*(6*a^2*b^2*e^2+8*a*b^3*d*e+b^4*d^2)*x^5+1/4*(4*a^3*b*e^2+1
2*a^2*b^2*d*e+4*a*b^3*d^2)*x^4+1/3*(a^4*e^2+8*a^3*b*d*e+6*a^2*b^2*d^2)*x^3+1/2*(2*a^4*d*e+4*a^3*b*d^2)*x^2+a^4
*d^2*x

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maxima [B]  time = 1.38, size = 156, normalized size = 2.40 \begin {gather*} \frac {1}{7} \, b^{4} e^{2} x^{7} + a^{4} d^{2} x + \frac {1}{3} \, {\left (b^{4} d e + 2 \, a b^{3} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{2} + 8 \, a b^{3} d e + 6 \, a^{2} b^{2} e^{2}\right )} x^{5} + {\left (a b^{3} d^{2} + 3 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} d^{2} + 8 \, a^{3} b d e + a^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^4*e^2*x^7 + a^4*d^2*x + 1/3*(b^4*d*e + 2*a*b^3*e^2)*x^6 + 1/5*(b^4*d^2 + 8*a*b^3*d*e + 6*a^2*b^2*e^2)*x^
5 + (a*b^3*d^2 + 3*a^2*b^2*d*e + a^3*b*e^2)*x^4 + 1/3*(6*a^2*b^2*d^2 + 8*a^3*b*d*e + a^4*e^2)*x^3 + (2*a^3*b*d
^2 + a^4*d*e)*x^2

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mupad [B]  time = 0.52, size = 144, normalized size = 2.22 \begin {gather*} x^3\,\left (\frac {a^4\,e^2}{3}+\frac {8\,a^3\,b\,d\,e}{3}+2\,a^2\,b^2\,d^2\right )+x^5\,\left (\frac {6\,a^2\,b^2\,e^2}{5}+\frac {8\,a\,b^3\,d\,e}{5}+\frac {b^4\,d^2}{5}\right )+a^4\,d^2\,x+\frac {b^4\,e^2\,x^7}{7}+a^3\,d\,x^2\,\left (a\,e+2\,b\,d\right )+\frac {b^3\,e\,x^6\,\left (2\,a\,e+b\,d\right )}{3}+a\,b\,x^4\,\left (a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*((a^4*e^2)/3 + 2*a^2*b^2*d^2 + (8*a^3*b*d*e)/3) + x^5*((b^4*d^2)/5 + (6*a^2*b^2*e^2)/5 + (8*a*b^3*d*e)/5)
+ a^4*d^2*x + (b^4*e^2*x^7)/7 + a^3*d*x^2*(a*e + 2*b*d) + (b^3*e*x^6*(2*a*e + b*d))/3 + a*b*x^4*(a^2*e^2 + b^2
*d^2 + 3*a*b*d*e)

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sympy [B]  time = 0.10, size = 168, normalized size = 2.58 \begin {gather*} a^{4} d^{2} x + \frac {b^{4} e^{2} x^{7}}{7} + x^{6} \left (\frac {2 a b^{3} e^{2}}{3} + \frac {b^{4} d e}{3}\right ) + x^{5} \left (\frac {6 a^{2} b^{2} e^{2}}{5} + \frac {8 a b^{3} d e}{5} + \frac {b^{4} d^{2}}{5}\right ) + x^{4} \left (a^{3} b e^{2} + 3 a^{2} b^{2} d e + a b^{3} d^{2}\right ) + x^{3} \left (\frac {a^{4} e^{2}}{3} + \frac {8 a^{3} b d e}{3} + 2 a^{2} b^{2} d^{2}\right ) + x^{2} \left (a^{4} d e + 2 a^{3} b d^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*d**2*x + b**4*e**2*x**7/7 + x**6*(2*a*b**3*e**2/3 + b**4*d*e/3) + x**5*(6*a**2*b**2*e**2/5 + 8*a*b**3*d*e
/5 + b**4*d**2/5) + x**4*(a**3*b*e**2 + 3*a**2*b**2*d*e + a*b**3*d**2) + x**3*(a**4*e**2/3 + 8*a**3*b*d*e/3 +
2*a**2*b**2*d**2) + x**2*(a**4*d*e + 2*a**3*b*d**2)

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