3.1909 \(\int (a+b x) (d+e x)^2 (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=65 \[ \frac{2 e (a+b x)^7 (b d-a e)}{7 b^3}+\frac{(a+b x)^6 (b d-a e)^2}{6 b^3}+\frac{e^2 (a+b x)^8}{8 b^3} \]

[Out]

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

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Rubi [A]  time = 0.105298, antiderivative size = 65, 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} \[ \frac{2 e (a+b x)^7 (b d-a e)}{7 b^3}+\frac{(a+b x)^6 (b d-a e)^2}{6 b^3}+\frac{e^2 (a+b x)^8}{8 b^3} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [B]  time = 0.0329928, size = 189, normalized size = 2.91 \[ \frac{1}{6} b^3 x^6 \left (10 a^2 e^2+10 a b d e+b^2 d^2\right )+a b^2 x^5 \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+\frac{5}{4} a^2 b x^4 \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )+\frac{1}{3} a^3 x^3 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )+\frac{1}{2} a^4 d x^2 (2 a e+5 b d)+a^5 d^2 x+\frac{1}{7} b^4 e x^7 (5 a e+2 b d)+\frac{1}{8} b^5 e^2 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.001, size = 301, normalized size = 4.6 \begin{align*}{\frac{{b}^{5}{e}^{2}{x}^{8}}{8}}+{\frac{ \left ( \left ( a{e}^{2}+2\,bde \right ){b}^{4}+4\,{b}^{4}{e}^{2}a \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,ade+b{d}^{2} \right ){b}^{4}+4\, \left ( a{e}^{2}+2\,bde \right ) a{b}^{3}+6\,{b}^{3}{e}^{2}{a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( a{d}^{2}{b}^{4}+4\, \left ( 2\,ade+b{d}^{2} \right ) a{b}^{3}+6\, \left ( a{e}^{2}+2\,bde \right ){a}^{2}{b}^{2}+4\,{b}^{2}{e}^{2}{a}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{2}{d}^{2}{b}^{3}+6\, \left ( 2\,ade+b{d}^{2} \right ){a}^{2}{b}^{2}+4\, \left ( a{e}^{2}+2\,bde \right ){a}^{3}b+b{e}^{2}{a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{a}^{3}{d}^{2}{b}^{2}+4\, \left ( 2\,ade+b{d}^{2} \right ){a}^{3}b+ \left ( a{e}^{2}+2\,bde \right ){a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{a}^{4}{d}^{2}b+ \left ( 2\,ade+b{d}^{2} \right ){a}^{4} \right ){x}^{2}}{2}}+{a}^{5}{d}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.08985, size = 266, normalized size = 4.09 \begin{align*} \frac{1}{8} \, b^{5} e^{2} x^{8} + a^{5} d^{2} x + \frac{1}{7} \,{\left (2 \, b^{5} d e + 5 \, a b^{4} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (b^{5} d^{2} + 10 \, a b^{4} d e + 10 \, a^{2} b^{3} e^{2}\right )} x^{6} +{\left (a b^{4} d^{2} + 4 \, a^{2} b^{3} d e + 2 \, a^{3} b^{2} e^{2}\right )} x^{5} + \frac{5}{4} \,{\left (2 \, a^{2} b^{3} d^{2} + 4 \, a^{3} b^{2} d e + a^{4} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} b^{2} d^{2} + 10 \, a^{4} b d e + a^{5} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d^{2} + 2 \, a^{5} d e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.28808, size = 460, normalized size = 7.08 \begin{align*} \frac{1}{8} x^{8} e^{2} b^{5} + \frac{2}{7} x^{7} e d b^{5} + \frac{5}{7} x^{7} e^{2} b^{4} a + \frac{1}{6} x^{6} d^{2} b^{5} + \frac{5}{3} x^{6} e d b^{4} a + \frac{5}{3} x^{6} e^{2} b^{3} a^{2} + x^{5} d^{2} b^{4} a + 4 x^{5} e d b^{3} a^{2} + 2 x^{5} e^{2} b^{2} a^{3} + \frac{5}{2} x^{4} d^{2} b^{3} a^{2} + 5 x^{4} e d b^{2} a^{3} + \frac{5}{4} x^{4} e^{2} b a^{4} + \frac{10}{3} x^{3} d^{2} b^{2} a^{3} + \frac{10}{3} x^{3} e d b a^{4} + \frac{1}{3} x^{3} e^{2} a^{5} + \frac{5}{2} x^{2} d^{2} b a^{4} + x^{2} e d a^{5} + x d^{2} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.09642, size = 218, normalized size = 3.35 \begin{align*} a^{5} d^{2} x + \frac{b^{5} e^{2} x^{8}}{8} + x^{7} \left (\frac{5 a b^{4} e^{2}}{7} + \frac{2 b^{5} d e}{7}\right ) + x^{6} \left (\frac{5 a^{2} b^{3} e^{2}}{3} + \frac{5 a b^{4} d e}{3} + \frac{b^{5} d^{2}}{6}\right ) + x^{5} \left (2 a^{3} b^{2} e^{2} + 4 a^{2} b^{3} d e + a b^{4} d^{2}\right ) + x^{4} \left (\frac{5 a^{4} b e^{2}}{4} + 5 a^{3} b^{2} d e + \frac{5 a^{2} b^{3} d^{2}}{2}\right ) + x^{3} \left (\frac{a^{5} e^{2}}{3} + \frac{10 a^{4} b d e}{3} + \frac{10 a^{3} b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{5} d e + \frac{5 a^{4} b d^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.10483, size = 286, normalized size = 4.4 \begin{align*} \frac{1}{8} \, b^{5} x^{8} e^{2} + \frac{2}{7} \, b^{5} d x^{7} e + \frac{1}{6} \, b^{5} d^{2} x^{6} + \frac{5}{7} \, a b^{4} x^{7} e^{2} + \frac{5}{3} \, a b^{4} d x^{6} e + a b^{4} d^{2} x^{5} + \frac{5}{3} \, a^{2} b^{3} x^{6} e^{2} + 4 \, a^{2} b^{3} d x^{5} e + \frac{5}{2} \, a^{2} b^{3} d^{2} x^{4} + 2 \, a^{3} b^{2} x^{5} e^{2} + 5 \, a^{3} b^{2} d x^{4} e + \frac{10}{3} \, a^{3} b^{2} d^{2} x^{3} + \frac{5}{4} \, a^{4} b x^{4} e^{2} + \frac{10}{3} \, a^{4} b d x^{3} e + \frac{5}{2} \, a^{4} b d^{2} x^{2} + \frac{1}{3} \, a^{5} x^{3} e^{2} + a^{5} d x^{2} e + a^{5} d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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