### 3.1465 $$\int (d+e x)^4 (a^2+2 a b x+b^2 x^2)^2 \, dx$$

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

[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)

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Rubi [A]  time = 0.164931, antiderivative size = 119, normalized size of antiderivative = 1., 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} $\frac{e^3 (a+b x)^8 (b d-a e)}{2 b^5}+\frac{6 e^2 (a+b x)^7 (b d-a e)^2}{7 b^5}+\frac{2 e (a+b x)^6 (b d-a e)^3}{3 b^5}+\frac{(a+b x)^5 (b d-a e)^4}{5 b^5}+\frac{e^4 (a+b x)^9}{9 b^5}$

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*}

Mathematica [B]  time = 0.0385267, size = 273, normalized size = 2.29 $\frac{2}{7} b^2 e^2 x^7 \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )+\frac{2}{3} b e x^6 \left (6 a^2 b d e^2+a^3 e^3+6 a b^2 d^2 e+b^3 d^3\right )+\frac{1}{5} x^5 \left (36 a^2 b^2 d^2 e^2+16 a^3 b d e^3+a^4 e^4+16 a b^3 d^3 e+b^4 d^4\right )+a d x^4 \left (6 a^2 b d e^2+a^3 e^3+6 a b^2 d^2 e+b^3 d^3\right )+\frac{2}{3} a^2 d^2 x^3 \left (3 a^2 e^2+8 a b d e+3 b^2 d^2\right )+2 a^3 d^3 x^2 (a e+b d)+a^4 d^4 x+\frac{1}{2} b^3 e^3 x^8 (a e+b d)+\frac{1}{9} b^4 e^4 x^9$

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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Maple [B]  time = 0.04, size = 295, normalized size = 2.5 \begin{align*}{\frac{{b}^{4}{e}^{4}{x}^{9}}{9}}+{\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}ea{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}}+{a}^{4}{d}^{4}x \end{align*}

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 = 1.09669, size = 385, normalized size = 3.24 \begin{align*} \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{align*}

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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Fricas [B]  time = 1.51175, size = 680, normalized size = 5.71 \begin{align*} \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{align*}

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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Sympy [B]  time = 0.115406, size = 318, normalized size = 2.67 \begin{align*} 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{align*}

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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Giac [B]  time = 1.1317, size = 420, normalized size = 3.53 \begin{align*} \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{align*}

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