3.1908 \(\int (a+b x) (d+e x)^3 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=92 \[ \frac{3 e^2 (a+b x)^8 (b d-a e)}{8 b^4}+\frac{3 e (a+b x)^7 (b d-a e)^2}{7 b^4}+\frac{(a+b x)^6 (b d-a e)^3}{6 b^4}+\frac{e^3 (a+b x)^9}{9 b^4} \]

[Out]

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

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Rubi [A]  time = 0.149966, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ \frac{3 e^2 (a+b x)^8 (b d-a e)}{8 b^4}+\frac{3 e (a+b x)^7 (b d-a e)^2}{7 b^4}+\frac{(a+b x)^6 (b d-a e)^3}{6 b^4}+\frac{e^3 (a+b x)^9}{9 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [B]  time = 0.0685835, size = 235, normalized size = 2.55 \[ \frac{1}{504} x \left (84 a^3 b^2 x^2 \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+36 a^2 b^3 x^3 \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )+126 a^4 b x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+126 a^5 \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+9 a b^4 x^4 \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )+b^5 x^5 \left (216 d^2 e x+84 d^3+189 d e^2 x^2+56 e^3 x^3\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(126*a^5*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 126*a^4*b*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*
e^3*x^3) + 84*a^3*b^2*x^2*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 36*a^2*b^3*x^3*(35*d^3 + 84*d^2*
e*x + 70*d*e^2*x^2 + 20*e^3*x^3) + 9*a*b^4*x^4*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + b^5*x^5*(
84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3)))/504

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Maple [B]  time = 0.001, size = 430, normalized size = 4.7 \begin{align*}{\frac{{b}^{5}{e}^{3}{x}^{9}}{9}}+{\frac{ \left ( \left ( a{e}^{3}+3\,bd{e}^{2} \right ){b}^{4}+4\,{b}^{4}{e}^{3}a \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){b}^{4}+4\, \left ( a{e}^{3}+3\,bd{e}^{2} \right ) a{b}^{3}+6\,{b}^{3}{e}^{3}{a}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){b}^{4}+4\, \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ) a{b}^{3}+6\, \left ( a{e}^{3}+3\,bd{e}^{2} \right ){a}^{2}{b}^{2}+4\,{b}^{2}{e}^{3}{a}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( a{d}^{3}{b}^{4}+4\, \left ( 3\,a{d}^{2}e+b{d}^{3} \right ) a{b}^{3}+6\, \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){a}^{2}{b}^{2}+4\, \left ( a{e}^{3}+3\,bd{e}^{2} \right ){a}^{3}b+b{e}^{3}{a}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{2}{d}^{3}{b}^{3}+6\, \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){a}^{2}{b}^{2}+4\, \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){a}^{3}b+ \left ( a{e}^{3}+3\,bd{e}^{2} \right ){a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{a}^{3}{d}^{3}{b}^{2}+4\, \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){a}^{3}b+ \left ( 3\,ad{e}^{2}+3\,b{d}^{2}e \right ){a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{a}^{4}{d}^{3}b+ \left ( 3\,a{d}^{2}e+b{d}^{3} \right ){a}^{4} \right ){x}^{2}}{2}}+{a}^{5}{d}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

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Maxima [B]  time = 1.07571, size = 374, normalized size = 4.07 \begin{align*} \frac{1}{9} \, b^{5} e^{3} x^{9} + a^{5} d^{3} x + \frac{1}{8} \,{\left (3 \, b^{5} d e^{2} + 5 \, a b^{4} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (3 \, b^{5} d^{2} e + 15 \, a b^{4} d e^{2} + 10 \, a^{2} b^{3} e^{3}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{3} + 15 \, a b^{4} d^{2} e + 30 \, a^{2} b^{3} d e^{2} + 10 \, a^{3} b^{2} e^{3}\right )} x^{6} +{\left (a b^{4} d^{3} + 6 \, a^{2} b^{3} d^{2} e + 6 \, a^{3} b^{2} d e^{2} + a^{4} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (10 \, a^{2} b^{3} d^{3} + 30 \, a^{3} b^{2} d^{2} e + 15 \, a^{4} b d e^{2} + a^{5} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} b^{2} d^{3} + 15 \, a^{4} b d^{2} e + 3 \, a^{5} d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d^{3} + 3 \, a^{5} d^{2} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/9*b^5*e^3*x^9 + a^5*d^3*x + 1/8*(3*b^5*d*e^2 + 5*a*b^4*e^3)*x^8 + 1/7*(3*b^5*d^2*e + 15*a*b^4*d*e^2 + 10*a^2
*b^3*e^3)*x^7 + 1/6*(b^5*d^3 + 15*a*b^4*d^2*e + 30*a^2*b^3*d*e^2 + 10*a^3*b^2*e^3)*x^6 + (a*b^4*d^3 + 6*a^2*b^
3*d^2*e + 6*a^3*b^2*d*e^2 + a^4*b*e^3)*x^5 + 1/4*(10*a^2*b^3*d^3 + 30*a^3*b^2*d^2*e + 15*a^4*b*d*e^2 + a^5*e^3
)*x^4 + 1/3*(10*a^3*b^2*d^3 + 15*a^4*b*d^2*e + 3*a^5*d*e^2)*x^3 + 1/2*(5*a^4*b*d^3 + 3*a^5*d^2*e)*x^2

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Fricas [B]  time = 1.31299, size = 651, normalized size = 7.08 \begin{align*} \frac{1}{9} x^{9} e^{3} b^{5} + \frac{3}{8} x^{8} e^{2} d b^{5} + \frac{5}{8} x^{8} e^{3} b^{4} a + \frac{3}{7} x^{7} e d^{2} b^{5} + \frac{15}{7} x^{7} e^{2} d b^{4} a + \frac{10}{7} x^{7} e^{3} b^{3} a^{2} + \frac{1}{6} x^{6} d^{3} b^{5} + \frac{5}{2} x^{6} e d^{2} b^{4} a + 5 x^{6} e^{2} d b^{3} a^{2} + \frac{5}{3} x^{6} e^{3} b^{2} a^{3} + x^{5} d^{3} b^{4} a + 6 x^{5} e d^{2} b^{3} a^{2} + 6 x^{5} e^{2} d b^{2} a^{3} + x^{5} e^{3} b a^{4} + \frac{5}{2} x^{4} d^{3} b^{3} a^{2} + \frac{15}{2} x^{4} e d^{2} b^{2} a^{3} + \frac{15}{4} x^{4} e^{2} d b a^{4} + \frac{1}{4} x^{4} e^{3} a^{5} + \frac{10}{3} x^{3} d^{3} b^{2} a^{3} + 5 x^{3} e d^{2} b a^{4} + x^{3} e^{2} d a^{5} + \frac{5}{2} x^{2} d^{3} b a^{4} + \frac{3}{2} x^{2} e d^{2} a^{5} + x d^{3} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.10843, size = 308, normalized size = 3.35 \begin{align*} a^{5} d^{3} x + \frac{b^{5} e^{3} x^{9}}{9} + x^{8} \left (\frac{5 a b^{4} e^{3}}{8} + \frac{3 b^{5} d e^{2}}{8}\right ) + x^{7} \left (\frac{10 a^{2} b^{3} e^{3}}{7} + \frac{15 a b^{4} d e^{2}}{7} + \frac{3 b^{5} d^{2} e}{7}\right ) + x^{6} \left (\frac{5 a^{3} b^{2} e^{3}}{3} + 5 a^{2} b^{3} d e^{2} + \frac{5 a b^{4} d^{2} e}{2} + \frac{b^{5} d^{3}}{6}\right ) + x^{5} \left (a^{4} b e^{3} + 6 a^{3} b^{2} d e^{2} + 6 a^{2} b^{3} d^{2} e + a b^{4} d^{3}\right ) + x^{4} \left (\frac{a^{5} e^{3}}{4} + \frac{15 a^{4} b d e^{2}}{4} + \frac{15 a^{3} b^{2} d^{2} e}{2} + \frac{5 a^{2} b^{3} d^{3}}{2}\right ) + x^{3} \left (a^{5} d e^{2} + 5 a^{4} b d^{2} e + \frac{10 a^{3} b^{2} d^{3}}{3}\right ) + x^{2} \left (\frac{3 a^{5} d^{2} e}{2} + \frac{5 a^{4} b d^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

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

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Giac [B]  time = 1.11298, size = 401, normalized size = 4.36 \begin{align*} \frac{1}{9} \, b^{5} x^{9} e^{3} + \frac{3}{8} \, b^{5} d x^{8} e^{2} + \frac{3}{7} \, b^{5} d^{2} x^{7} e + \frac{1}{6} \, b^{5} d^{3} x^{6} + \frac{5}{8} \, a b^{4} x^{8} e^{3} + \frac{15}{7} \, a b^{4} d x^{7} e^{2} + \frac{5}{2} \, a b^{4} d^{2} x^{6} e + a b^{4} d^{3} x^{5} + \frac{10}{7} \, a^{2} b^{3} x^{7} e^{3} + 5 \, a^{2} b^{3} d x^{6} e^{2} + 6 \, a^{2} b^{3} d^{2} x^{5} e + \frac{5}{2} \, a^{2} b^{3} d^{3} x^{4} + \frac{5}{3} \, a^{3} b^{2} x^{6} e^{3} + 6 \, a^{3} b^{2} d x^{5} e^{2} + \frac{15}{2} \, a^{3} b^{2} d^{2} x^{4} e + \frac{10}{3} \, a^{3} b^{2} d^{3} x^{3} + a^{4} b x^{5} e^{3} + \frac{15}{4} \, a^{4} b d x^{4} e^{2} + 5 \, a^{4} b d^{2} x^{3} e + \frac{5}{2} \, a^{4} b d^{3} x^{2} + \frac{1}{4} \, a^{5} x^{4} e^{3} + a^{5} d x^{3} e^{2} + \frac{3}{2} \, a^{5} d^{2} x^{2} e + a^{5} d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

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