3.12.67 \(\int (d+e x)^4 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=119 \[ \frac {e^3 (a+b x)^8 (b d-a e)}{2 b^5}+\frac {6 e^2 (a+b x)^7 (b d-a e)^2}{7 b^5}+\frac {2 e (a+b x)^6 (b d-a e)^3}{3 b^5}+\frac {(a+b x)^5 (b d-a e)^4}{5 b^5}+\frac {e^4 (a+b x)^9}{9 b^5} \]

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Rubi [A]  time = 0.16, antiderivative size = 119, 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^3 (a+b x)^8 (b d-a e)}{2 b^5}+\frac {6 e^2 (a+b x)^7 (b d-a e)^2}{7 b^5}+\frac {2 e (a+b x)^6 (b d-a e)^3}{3 b^5}+\frac {(a+b x)^5 (b d-a e)^4}{5 b^5}+\frac {e^4 (a+b x)^9}{9 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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)^4 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (d+e x)^4 \, dx\\ &=\int \left (\frac {(b d-a e)^4 (a+b x)^4}{b^4}+\frac {4 e (b d-a e)^3 (a+b x)^5}{b^4}+\frac {6 e^2 (b d-a e)^2 (a+b x)^6}{b^4}+\frac {4 e^3 (b d-a e) (a+b x)^7}{b^4}+\frac {e^4 (a+b x)^8}{b^4}\right ) \, dx\\ &=\frac {(b d-a e)^4 (a+b x)^5}{5 b^5}+\frac {2 e (b d-a e)^3 (a+b x)^6}{3 b^5}+\frac {6 e^2 (b d-a e)^2 (a+b x)^7}{7 b^5}+\frac {e^3 (b d-a e) (a+b x)^8}{2 b^5}+\frac {e^4 (a+b x)^9}{9 b^5}\\ \end {align*}

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Mathematica [B]  time = 0.04, size = 273, normalized size = 2.29 \begin {gather*} a^4 d^4 x+2 a^3 d^3 x^2 (a e+b d)+\frac {2}{7} b^2 e^2 x^7 \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )+\frac {2}{3} a^2 d^2 x^3 \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )+\frac {2}{3} b e x^6 \left (a^3 e^3+6 a^2 b d e^2+6 a b^2 d^2 e+b^3 d^3\right )+a d x^4 \left (a^3 e^3+6 a^2 b d e^2+6 a b^2 d^2 e+b^3 d^3\right )+\frac {1}{5} x^5 \left (a^4 e^4+16 a^3 b d e^3+36 a^2 b^2 d^2 e^2+16 a b^3 d^3 e+b^4 d^4\right )+\frac {1}{2} b^3 e^3 x^8 (a e+b d)+\frac {1}{9} b^4 e^4 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^4*d^4*x + 2*a^3*d^3*(b*d + a*e)*x^2 + (2*a^2*d^2*(3*b^2*d^2 + 8*a*b*d*e + 3*a^2*e^2)*x^3)/3 + a*d*(b^3*d^3 +
6*a*b^2*d^2*e + 6*a^2*b*d*e^2 + a^3*e^3)*x^4 + ((b^4*d^4 + 16*a*b^3*d^3*e + 36*a^2*b^2*d^2*e^2 + 16*a^3*b*d*e
^3 + a^4*e^4)*x^5)/5 + (2*b*e*(b^3*d^3 + 6*a*b^2*d^2*e + 6*a^2*b*d*e^2 + a^3*e^3)*x^6)/3 + (2*b^2*e^2*(3*b^2*d
^2 + 8*a*b*d*e + 3*a^2*e^2)*x^7)/7 + (b^3*e^3*(b*d + a*e)*x^8)/2 + (b^4*e^4*x^9)/9

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

[Out]

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

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fricas [B]  time = 0.34, size = 321, normalized size = 2.70 \begin {gather*} \frac {1}{9} x^{9} e^{4} b^{4} + \frac {1}{2} x^{8} e^{3} d b^{4} + \frac {1}{2} x^{8} e^{4} b^{3} a + \frac {6}{7} x^{7} e^{2} d^{2} b^{4} + \frac {16}{7} x^{7} e^{3} d b^{3} a + \frac {6}{7} x^{7} e^{4} b^{2} a^{2} + \frac {2}{3} x^{6} e d^{3} b^{4} + 4 x^{6} e^{2} d^{2} b^{3} a + 4 x^{6} e^{3} d b^{2} a^{2} + \frac {2}{3} x^{6} e^{4} b a^{3} + \frac {1}{5} x^{5} d^{4} b^{4} + \frac {16}{5} x^{5} e d^{3} b^{3} a + \frac {36}{5} x^{5} e^{2} d^{2} b^{2} a^{2} + \frac {16}{5} x^{5} e^{3} d b a^{3} + \frac {1}{5} x^{5} e^{4} a^{4} + x^{4} d^{4} b^{3} a + 6 x^{4} e d^{3} b^{2} a^{2} + 6 x^{4} e^{2} d^{2} b a^{3} + x^{4} e^{3} d a^{4} + 2 x^{3} d^{4} b^{2} a^{2} + \frac {16}{3} x^{3} e d^{3} b a^{3} + 2 x^{3} e^{2} d^{2} a^{4} + 2 x^{2} d^{4} b a^{3} + 2 x^{2} e d^{3} a^{4} + x d^{4} a^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*e^4*b^4 + 1/2*x^8*e^3*d*b^4 + 1/2*x^8*e^4*b^3*a + 6/7*x^7*e^2*d^2*b^4 + 16/7*x^7*e^3*d*b^3*a + 6/7*x^7
*e^4*b^2*a^2 + 2/3*x^6*e*d^3*b^4 + 4*x^6*e^2*d^2*b^3*a + 4*x^6*e^3*d*b^2*a^2 + 2/3*x^6*e^4*b*a^3 + 1/5*x^5*d^4
*b^4 + 16/5*x^5*e*d^3*b^3*a + 36/5*x^5*e^2*d^2*b^2*a^2 + 16/5*x^5*e^3*d*b*a^3 + 1/5*x^5*e^4*a^4 + x^4*d^4*b^3*
a + 6*x^4*e*d^3*b^2*a^2 + 6*x^4*e^2*d^2*b*a^3 + x^4*e^3*d*a^4 + 2*x^3*d^4*b^2*a^2 + 16/3*x^3*e*d^3*b*a^3 + 2*x
^3*e^2*d^2*a^4 + 2*x^2*d^4*b*a^3 + 2*x^2*e*d^3*a^4 + x*d^4*a^4

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giac [B]  time = 0.16, size = 311, normalized size = 2.61 \begin {gather*} \frac {1}{9} \, b^{4} x^{9} e^{4} + \frac {1}{2} \, b^{4} d x^{8} e^{3} + \frac {6}{7} \, b^{4} d^{2} x^{7} e^{2} + \frac {2}{3} \, b^{4} d^{3} x^{6} e + \frac {1}{5} \, b^{4} d^{4} x^{5} + \frac {1}{2} \, a b^{3} x^{8} e^{4} + \frac {16}{7} \, a b^{3} d x^{7} e^{3} + 4 \, a b^{3} d^{2} x^{6} e^{2} + \frac {16}{5} \, a b^{3} d^{3} x^{5} e + a b^{3} d^{4} x^{4} + \frac {6}{7} \, a^{2} b^{2} x^{7} e^{4} + 4 \, a^{2} b^{2} d x^{6} e^{3} + \frac {36}{5} \, a^{2} b^{2} d^{2} x^{5} e^{2} + 6 \, a^{2} b^{2} d^{3} x^{4} e + 2 \, a^{2} b^{2} d^{4} x^{3} + \frac {2}{3} \, a^{3} b x^{6} e^{4} + \frac {16}{5} \, a^{3} b d x^{5} e^{3} + 6 \, a^{3} b d^{2} x^{4} e^{2} + \frac {16}{3} \, a^{3} b d^{3} x^{3} e + 2 \, a^{3} b d^{4} x^{2} + \frac {1}{5} \, a^{4} x^{5} e^{4} + a^{4} d x^{4} e^{3} + 2 \, a^{4} d^{2} x^{3} e^{2} + 2 \, a^{4} d^{3} x^{2} e + a^{4} d^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^4*x^9*e^4 + 1/2*b^4*d*x^8*e^3 + 6/7*b^4*d^2*x^7*e^2 + 2/3*b^4*d^3*x^6*e + 1/5*b^4*d^4*x^5 + 1/2*a*b^3*x^
8*e^4 + 16/7*a*b^3*d*x^7*e^3 + 4*a*b^3*d^2*x^6*e^2 + 16/5*a*b^3*d^3*x^5*e + a*b^3*d^4*x^4 + 6/7*a^2*b^2*x^7*e^
4 + 4*a^2*b^2*d*x^6*e^3 + 36/5*a^2*b^2*d^2*x^5*e^2 + 6*a^2*b^2*d^3*x^4*e + 2*a^2*b^2*d^4*x^3 + 2/3*a^3*b*x^6*e
^4 + 16/5*a^3*b*d*x^5*e^3 + 6*a^3*b*d^2*x^4*e^2 + 16/3*a^3*b*d^3*x^3*e + 2*a^3*b*d^4*x^2 + 1/5*a^4*x^5*e^4 + a
^4*d*x^4*e^3 + 2*a^4*d^2*x^3*e^2 + 2*a^4*d^3*x^2*e + a^4*d^4*x

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maple [B]  time = 0.04, size = 295, normalized size = 2.48 \begin {gather*} \frac {b^{4} e^{4} x^{9}}{9}+a^{4} d^{4} x +\frac {\left (4 e^{4} a \,b^{3}+4 b^{4} d \,e^{3}\right ) x^{8}}{8}+\frac {\left (6 e^{4} b^{2} a^{2}+16 d \,e^{3} a \,b^{3}+6 d^{2} e^{2} b^{4}\right ) x^{7}}{7}+\frac {\left (4 e^{4} a^{3} b +24 d \,e^{3} b^{2} a^{2}+24 d^{2} e^{2} a \,b^{3}+4 d^{3} e \,b^{4}\right ) x^{6}}{6}+\frac {\left (e^{4} a^{4}+16 d \,e^{3} a^{3} b +36 d^{2} e^{2} b^{2} a^{2}+16 d^{3} e a \,b^{3}+d^{4} b^{4}\right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a^{4}+24 d^{2} e^{2} a^{3} b +24 d^{3} e \,b^{2} a^{2}+4 d^{4} a \,b^{3}\right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a^{4}+16 d^{3} e \,a^{3} b +6 d^{4} b^{2} a^{2}\right ) x^{3}}{3}+\frac {\left (4 d^{3} e \,a^{4}+4 d^{4} a^{3} b \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^4*e^4*x^9+1/8*(4*a*b^3*e^4+4*b^4*d*e^3)*x^8+1/7*(6*a^2*b^2*e^4+16*a*b^3*d*e^3+6*b^4*d^2*e^2)*x^7+1/6*(4*
a^3*b*e^4+24*a^2*b^2*d*e^3+24*a*b^3*d^2*e^2+4*b^4*d^3*e)*x^6+1/5*(a^4*e^4+16*a^3*b*d*e^3+36*a^2*b^2*d^2*e^2+16
*a*b^3*d^3*e+b^4*d^4)*x^5+1/4*(4*a^4*d*e^3+24*a^3*b*d^2*e^2+24*a^2*b^2*d^3*e+4*a*b^3*d^4)*x^4+1/3*(6*a^4*d^2*e
^2+16*a^3*b*d^3*e+6*a^2*b^2*d^4)*x^3+1/2*(4*a^4*d^3*e+4*a^3*b*d^4)*x^2+a^4*d^4*x

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maxima [B]  time = 1.30, size = 285, normalized size = 2.39 \begin {gather*} \frac {1}{9} \, b^{4} e^{4} x^{9} + a^{4} d^{4} x + \frac {1}{2} \, {\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{8} + \frac {2}{7} \, {\left (3 \, b^{4} d^{2} e^{2} + 8 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{7} + \frac {2}{3} \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{4} + 16 \, a b^{3} d^{3} e + 36 \, a^{2} b^{2} d^{2} e^{2} + 16 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} x^{5} + {\left (a b^{3} d^{4} + 6 \, a^{2} b^{2} d^{3} e + 6 \, a^{3} b d^{2} e^{2} + a^{4} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} d^{4} + 8 \, a^{3} b d^{3} e + 3 \, a^{4} d^{2} e^{2}\right )} x^{3} + 2 \, {\left (a^{3} b d^{4} + a^{4} d^{3} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^4*e^4*x^9 + a^4*d^4*x + 1/2*(b^4*d*e^3 + a*b^3*e^4)*x^8 + 2/7*(3*b^4*d^2*e^2 + 8*a*b^3*d*e^3 + 3*a^2*b^2
*e^4)*x^7 + 2/3*(b^4*d^3*e + 6*a*b^3*d^2*e^2 + 6*a^2*b^2*d*e^3 + a^3*b*e^4)*x^6 + 1/5*(b^4*d^4 + 16*a*b^3*d^3*
e + 36*a^2*b^2*d^2*e^2 + 16*a^3*b*d*e^3 + a^4*e^4)*x^5 + (a*b^3*d^4 + 6*a^2*b^2*d^3*e + 6*a^3*b*d^2*e^2 + a^4*
d*e^3)*x^4 + 2/3*(3*a^2*b^2*d^4 + 8*a^3*b*d^3*e + 3*a^4*d^2*e^2)*x^3 + 2*(a^3*b*d^4 + a^4*d^3*e)*x^2

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mupad [B]  time = 0.09, size = 271, normalized size = 2.28 \begin {gather*} x^5\,\left (\frac {a^4\,e^4}{5}+\frac {16\,a^3\,b\,d\,e^3}{5}+\frac {36\,a^2\,b^2\,d^2\,e^2}{5}+\frac {16\,a\,b^3\,d^3\,e}{5}+\frac {b^4\,d^4}{5}\right )+x^4\,\left (a^4\,d\,e^3+6\,a^3\,b\,d^2\,e^2+6\,a^2\,b^2\,d^3\,e+a\,b^3\,d^4\right )+x^6\,\left (\frac {2\,a^3\,b\,e^4}{3}+4\,a^2\,b^2\,d\,e^3+4\,a\,b^3\,d^2\,e^2+\frac {2\,b^4\,d^3\,e}{3}\right )+a^4\,d^4\,x+\frac {b^4\,e^4\,x^9}{9}+2\,a^3\,d^3\,x^2\,\left (a\,e+b\,d\right )+\frac {b^3\,e^3\,x^8\,\left (a\,e+b\,d\right )}{2}+\frac {2\,a^2\,d^2\,x^3\,\left (3\,a^2\,e^2+8\,a\,b\,d\,e+3\,b^2\,d^2\right )}{3}+\frac {2\,b^2\,e^2\,x^7\,\left (3\,a^2\,e^2+8\,a\,b\,d\,e+3\,b^2\,d^2\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^5*((a^4*e^4)/5 + (b^4*d^4)/5 + (36*a^2*b^2*d^2*e^2)/5 + (16*a*b^3*d^3*e)/5 + (16*a^3*b*d*e^3)/5) + x^4*(a*b^
3*d^4 + a^4*d*e^3 + 6*a^2*b^2*d^3*e + 6*a^3*b*d^2*e^2) + x^6*((2*a^3*b*e^4)/3 + (2*b^4*d^3*e)/3 + 4*a*b^3*d^2*
e^2 + 4*a^2*b^2*d*e^3) + a^4*d^4*x + (b^4*e^4*x^9)/9 + 2*a^3*d^3*x^2*(a*e + b*d) + (b^3*e^3*x^8*(a*e + b*d))/2
+ (2*a^2*d^2*x^3*(3*a^2*e^2 + 3*b^2*d^2 + 8*a*b*d*e))/3 + (2*b^2*e^2*x^7*(3*a^2*e^2 + 3*b^2*d^2 + 8*a*b*d*e))
/7

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sympy [B]  time = 0.12, size = 318, normalized size = 2.67 \begin {gather*} a^{4} d^{4} x + \frac {b^{4} e^{4} x^{9}}{9} + x^{8} \left (\frac {a b^{3} e^{4}}{2} + \frac {b^{4} d e^{3}}{2}\right ) + x^{7} \left (\frac {6 a^{2} b^{2} e^{4}}{7} + \frac {16 a b^{3} d e^{3}}{7} + \frac {6 b^{4} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac {2 a^{3} b e^{4}}{3} + 4 a^{2} b^{2} d e^{3} + 4 a b^{3} d^{2} e^{2} + \frac {2 b^{4} d^{3} e}{3}\right ) + x^{5} \left (\frac {a^{4} e^{4}}{5} + \frac {16 a^{3} b d e^{3}}{5} + \frac {36 a^{2} b^{2} d^{2} e^{2}}{5} + \frac {16 a b^{3} d^{3} e}{5} + \frac {b^{4} d^{4}}{5}\right ) + x^{4} \left (a^{4} d e^{3} + 6 a^{3} b d^{2} e^{2} + 6 a^{2} b^{2} d^{3} e + a b^{3} d^{4}\right ) + x^{3} \left (2 a^{4} d^{2} e^{2} + \frac {16 a^{3} b d^{3} e}{3} + 2 a^{2} b^{2} d^{4}\right ) + x^{2} \left (2 a^{4} d^{3} e + 2 a^{3} b d^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*d**4*x + b**4*e**4*x**9/9 + x**8*(a*b**3*e**4/2 + b**4*d*e**3/2) + x**7*(6*a**2*b**2*e**4/7 + 16*a*b**3*d
*e**3/7 + 6*b**4*d**2*e**2/7) + x**6*(2*a**3*b*e**4/3 + 4*a**2*b**2*d*e**3 + 4*a*b**3*d**2*e**2 + 2*b**4*d**3*
e/3) + x**5*(a**4*e**4/5 + 16*a**3*b*d*e**3/5 + 36*a**2*b**2*d**2*e**2/5 + 16*a*b**3*d**3*e/5 + b**4*d**4/5) +
x**4*(a**4*d*e**3 + 6*a**3*b*d**2*e**2 + 6*a**2*b**2*d**3*e + a*b**3*d**4) + x**3*(2*a**4*d**2*e**2 + 16*a**3
*b*d**3*e/3 + 2*a**2*b**2*d**4) + x**2*(2*a**4*d**3*e + 2*a**3*b*d**4)

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