3.12.89 \(\int (d+e x)^2 (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=65 \[ \frac {e (a+b x)^8 (b d-a e)}{4 b^3}+\frac {(a+b x)^7 (b d-a e)^2}{7 b^3}+\frac {e^2 (a+b x)^9}{9 b^3} \]

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Rubi [A]  time = 0.14, 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)^8 (b d-a e)}{4 b^3}+\frac {(a+b x)^7 (b d-a e)^2}{7 b^3}+\frac {e^2 (a+b x)^9}{9 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((b*d - a*e)^2*(a + b*x)^7)/(7*b^3) + (e*(b*d - a*e)*(a + b*x)^8)/(4*b^3) + (e^2*(a + b*x)^9)/(9*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 )^3 \, dx &=\int (a+b x)^6 (d+e x)^2 \, dx\\ &=\int \left (\frac {(b d-a e)^2 (a+b x)^6}{b^2}+\frac {2 e (b d-a e) (a+b x)^7}{b^2}+\frac {e^2 (a+b x)^8}{b^2}\right ) \, dx\\ &=\frac {(b d-a e)^2 (a+b x)^7}{7 b^3}+\frac {e (b d-a e) (a+b x)^8}{4 b^3}+\frac {e^2 (a+b x)^9}{9 b^3}\\ \end {align*}

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Mathematica [B]  time = 0.07, size = 199, normalized size = 3.06 \begin {gather*} \frac {1}{252} x \left (84 a^6 \left (3 d^2+3 d e x+e^2 x^2\right )+126 a^5 b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+126 a^4 b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )+84 a^3 b^3 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )+36 a^2 b^4 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )+9 a b^5 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+b^6 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(84*a^6*(3*d^2 + 3*d*e*x + e^2*x^2) + 126*a^5*b*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 126*a^4*b^2*x^2*(10*d^2 +
15*d*e*x + 6*e^2*x^2) + 84*a^3*b^3*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 36*a^2*b^4*x^4*(21*d^2 + 35*d*e*x +
15*e^2*x^2) + 9*a*b^5*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + b^6*x^6*(36*d^2 + 63*d*e*x + 28*e^2*x^2)))/252

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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 )^3 \, 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)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^3, x]

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fricas [B]  time = 0.35, size = 253, normalized size = 3.89 \begin {gather*} \frac {1}{9} x^{9} e^{2} b^{6} + \frac {1}{4} x^{8} e d b^{6} + \frac {3}{4} x^{8} e^{2} b^{5} a + \frac {1}{7} x^{7} d^{2} b^{6} + \frac {12}{7} x^{7} e d b^{5} a + \frac {15}{7} x^{7} e^{2} b^{4} a^{2} + x^{6} d^{2} b^{5} a + 5 x^{6} e d b^{4} a^{2} + \frac {10}{3} x^{6} e^{2} b^{3} a^{3} + 3 x^{5} d^{2} b^{4} a^{2} + 8 x^{5} e d b^{3} a^{3} + 3 x^{5} e^{2} b^{2} a^{4} + 5 x^{4} d^{2} b^{3} a^{3} + \frac {15}{2} x^{4} e d b^{2} a^{4} + \frac {3}{2} x^{4} e^{2} b a^{5} + 5 x^{3} d^{2} b^{2} a^{4} + 4 x^{3} e d b a^{5} + \frac {1}{3} x^{3} e^{2} a^{6} + 3 x^{2} d^{2} b a^{5} + x^{2} e d a^{6} + x d^{2} a^{6} \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)^3,x, algorithm="fricas")

[Out]

1/9*x^9*e^2*b^6 + 1/4*x^8*e*d*b^6 + 3/4*x^8*e^2*b^5*a + 1/7*x^7*d^2*b^6 + 12/7*x^7*e*d*b^5*a + 15/7*x^7*e^2*b^
4*a^2 + x^6*d^2*b^5*a + 5*x^6*e*d*b^4*a^2 + 10/3*x^6*e^2*b^3*a^3 + 3*x^5*d^2*b^4*a^2 + 8*x^5*e*d*b^3*a^3 + 3*x
^5*e^2*b^2*a^4 + 5*x^4*d^2*b^3*a^3 + 15/2*x^4*e*d*b^2*a^4 + 3/2*x^4*e^2*b*a^5 + 5*x^3*d^2*b^2*a^4 + 4*x^3*e*d*
b*a^5 + 1/3*x^3*e^2*a^6 + 3*x^2*d^2*b*a^5 + x^2*e*d*a^6 + x*d^2*a^6

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giac [B]  time = 0.18, size = 253, normalized size = 3.89 \begin {gather*} \frac {1}{9} \, b^{6} x^{9} e^{2} + \frac {1}{4} \, b^{6} d x^{8} e + \frac {1}{7} \, b^{6} d^{2} x^{7} + \frac {3}{4} \, a b^{5} x^{8} e^{2} + \frac {12}{7} \, a b^{5} d x^{7} e + a b^{5} d^{2} x^{6} + \frac {15}{7} \, a^{2} b^{4} x^{7} e^{2} + 5 \, a^{2} b^{4} d x^{6} e + 3 \, a^{2} b^{4} d^{2} x^{5} + \frac {10}{3} \, a^{3} b^{3} x^{6} e^{2} + 8 \, a^{3} b^{3} d x^{5} e + 5 \, a^{3} b^{3} d^{2} x^{4} + 3 \, a^{4} b^{2} x^{5} e^{2} + \frac {15}{2} \, a^{4} b^{2} d x^{4} e + 5 \, a^{4} b^{2} d^{2} x^{3} + \frac {3}{2} \, a^{5} b x^{4} e^{2} + 4 \, a^{5} b d x^{3} e + 3 \, a^{5} b d^{2} x^{2} + \frac {1}{3} \, a^{6} x^{3} e^{2} + a^{6} d x^{2} e + a^{6} 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)^3,x, algorithm="giac")

[Out]

1/9*b^6*x^9*e^2 + 1/4*b^6*d*x^8*e + 1/7*b^6*d^2*x^7 + 3/4*a*b^5*x^8*e^2 + 12/7*a*b^5*d*x^7*e + a*b^5*d^2*x^6 +
15/7*a^2*b^4*x^7*e^2 + 5*a^2*b^4*d*x^6*e + 3*a^2*b^4*d^2*x^5 + 10/3*a^3*b^3*x^6*e^2 + 8*a^3*b^3*d*x^5*e + 5*a
^3*b^3*d^2*x^4 + 3*a^4*b^2*x^5*e^2 + 15/2*a^4*b^2*d*x^4*e + 5*a^4*b^2*d^2*x^3 + 3/2*a^5*b*x^4*e^2 + 4*a^5*b*d*
x^3*e + 3*a^5*b*d^2*x^2 + 1/3*a^6*x^3*e^2 + a^6*d*x^2*e + a^6*d^2*x

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maple [B]  time = 0.04, size = 239, normalized size = 3.68 \begin {gather*} \frac {b^{6} e^{2} x^{9}}{9}+a^{6} d^{2} x +\frac {\left (6 e^{2} a \,b^{5}+2 d e \,b^{6}\right ) x^{8}}{8}+\frac {\left (15 e^{2} a^{2} b^{4}+12 d e a \,b^{5}+b^{6} d^{2}\right ) x^{7}}{7}+\frac {\left (20 e^{2} a^{3} b^{3}+30 d e \,a^{2} b^{4}+6 d^{2} a \,b^{5}\right ) x^{6}}{6}+\frac {\left (15 e^{2} a^{4} b^{2}+40 d e \,a^{3} b^{3}+15 d^{2} a^{2} b^{4}\right ) x^{5}}{5}+\frac {\left (6 e^{2} a^{5} b +30 d e \,a^{4} b^{2}+20 d^{2} a^{3} b^{3}\right ) x^{4}}{4}+\frac {\left (e^{2} a^{6}+12 d e \,a^{5} b +15 d^{2} a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (2 d e \,a^{6}+6 d^{2} a^{5} 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)^3,x)

[Out]

1/9*e^2*b^6*x^9+1/8*(6*a*b^5*e^2+2*b^6*d*e)*x^8+1/7*(15*a^2*b^4*e^2+12*a*b^5*d*e+b^6*d^2)*x^7+1/6*(20*a^3*b^3*
e^2+30*a^2*b^4*d*e+6*a*b^5*d^2)*x^6+1/5*(15*a^4*b^2*e^2+40*a^3*b^3*d*e+15*a^2*b^4*d^2)*x^5+1/4*(6*a^5*b*e^2+30
*a^4*b^2*d*e+20*a^3*b^3*d^2)*x^4+1/3*(a^6*e^2+12*a^5*b*d*e+15*a^4*b^2*d^2)*x^3+1/2*(2*a^6*d*e+6*a^5*b*d^2)*x^2
+d^2*a^6*x

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maxima [B]  time = 1.10, size = 234, normalized size = 3.60 \begin {gather*} \frac {1}{9} \, b^{6} e^{2} x^{9} + a^{6} d^{2} x + \frac {1}{4} \, {\left (b^{6} d e + 3 \, a b^{5} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{2} + 12 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, a b^{5} d^{2} + 15 \, a^{2} b^{4} d e + 10 \, a^{3} b^{3} e^{2}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{2} + 8 \, a^{3} b^{3} d e + 3 \, a^{4} b^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (10 \, a^{3} b^{3} d^{2} + 15 \, a^{4} b^{2} d e + 3 \, a^{5} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (15 \, a^{4} b^{2} d^{2} + 12 \, a^{5} b d e + a^{6} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{2} + a^{6} 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)^3,x, algorithm="maxima")

[Out]

1/9*b^6*e^2*x^9 + a^6*d^2*x + 1/4*(b^6*d*e + 3*a*b^5*e^2)*x^8 + 1/7*(b^6*d^2 + 12*a*b^5*d*e + 15*a^2*b^4*e^2)*
x^7 + 1/3*(3*a*b^5*d^2 + 15*a^2*b^4*d*e + 10*a^3*b^3*e^2)*x^6 + (3*a^2*b^4*d^2 + 8*a^3*b^3*d*e + 3*a^4*b^2*e^2
)*x^5 + 1/2*(10*a^3*b^3*d^2 + 15*a^4*b^2*d*e + 3*a^5*b*e^2)*x^4 + 1/3*(15*a^4*b^2*d^2 + 12*a^5*b*d*e + a^6*e^2
)*x^3 + (3*a^5*b*d^2 + a^6*d*e)*x^2

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mupad [B]  time = 0.09, size = 214, normalized size = 3.29 \begin {gather*} x^3\,\left (\frac {a^6\,e^2}{3}+4\,a^5\,b\,d\,e+5\,a^4\,b^2\,d^2\right )+x^7\,\left (\frac {15\,a^2\,b^4\,e^2}{7}+\frac {12\,a\,b^5\,d\,e}{7}+\frac {b^6\,d^2}{7}\right )+a^6\,d^2\,x+\frac {b^6\,e^2\,x^9}{9}+a^5\,d\,x^2\,\left (a\,e+3\,b\,d\right )+\frac {b^5\,e\,x^8\,\left (3\,a\,e+b\,d\right )}{4}+\frac {a^3\,b\,x^4\,\left (3\,a^2\,e^2+15\,a\,b\,d\,e+10\,b^2\,d^2\right )}{2}+\frac {a\,b^3\,x^6\,\left (10\,a^2\,e^2+15\,a\,b\,d\,e+3\,b^2\,d^2\right )}{3}+a^2\,b^2\,x^5\,\left (3\,a^2\,e^2+8\,a\,b\,d\,e+3\,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)^3,x)

[Out]

x^3*((a^6*e^2)/3 + 5*a^4*b^2*d^2 + 4*a^5*b*d*e) + x^7*((b^6*d^2)/7 + (15*a^2*b^4*e^2)/7 + (12*a*b^5*d*e)/7) +
a^6*d^2*x + (b^6*e^2*x^9)/9 + a^5*d*x^2*(a*e + 3*b*d) + (b^5*e*x^8*(3*a*e + b*d))/4 + (a^3*b*x^4*(3*a^2*e^2 +
10*b^2*d^2 + 15*a*b*d*e))/2 + (a*b^3*x^6*(10*a^2*e^2 + 3*b^2*d^2 + 15*a*b*d*e))/3 + a^2*b^2*x^5*(3*a^2*e^2 + 3
*b^2*d^2 + 8*a*b*d*e)

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sympy [B]  time = 0.11, size = 252, normalized size = 3.88 \begin {gather*} a^{6} d^{2} x + \frac {b^{6} e^{2} x^{9}}{9} + x^{8} \left (\frac {3 a b^{5} e^{2}}{4} + \frac {b^{6} d e}{4}\right ) + x^{7} \left (\frac {15 a^{2} b^{4} e^{2}}{7} + \frac {12 a b^{5} d e}{7} + \frac {b^{6} d^{2}}{7}\right ) + x^{6} \left (\frac {10 a^{3} b^{3} e^{2}}{3} + 5 a^{2} b^{4} d e + a b^{5} d^{2}\right ) + x^{5} \left (3 a^{4} b^{2} e^{2} + 8 a^{3} b^{3} d e + 3 a^{2} b^{4} d^{2}\right ) + x^{4} \left (\frac {3 a^{5} b e^{2}}{2} + \frac {15 a^{4} b^{2} d e}{2} + 5 a^{3} b^{3} d^{2}\right ) + x^{3} \left (\frac {a^{6} e^{2}}{3} + 4 a^{5} b d e + 5 a^{4} b^{2} d^{2}\right ) + x^{2} \left (a^{6} d e + 3 a^{5} 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)**3,x)

[Out]

a**6*d**2*x + b**6*e**2*x**9/9 + x**8*(3*a*b**5*e**2/4 + b**6*d*e/4) + x**7*(15*a**2*b**4*e**2/7 + 12*a*b**5*d
*e/7 + b**6*d**2/7) + x**6*(10*a**3*b**3*e**2/3 + 5*a**2*b**4*d*e + a*b**5*d**2) + x**5*(3*a**4*b**2*e**2 + 8*
a**3*b**3*d*e + 3*a**2*b**4*d**2) + x**4*(3*a**5*b*e**2/2 + 15*a**4*b**2*d*e/2 + 5*a**3*b**3*d**2) + x**3*(a**
6*e**2/3 + 4*a**5*b*d*e + 5*a**4*b**2*d**2) + x**2*(a**6*d*e + 3*a**5*b*d**2)

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