3.1896 \(\int (a+b x) (d+e x)^4 (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=92 \[ -\frac{3 b^2 (d+e x)^7 (b d-a e)}{7 e^4}+\frac{b (d+e x)^6 (b d-a e)^2}{2 e^4}-\frac{(d+e x)^5 (b d-a e)^3}{5 e^4}+\frac{b^3 (d+e x)^8}{8 e^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.120025, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 43} \[ -\frac{3 b^2 (d+e x)^7 (b d-a e)}{7 e^4}+\frac{b (d+e x)^6 (b d-a e)^2}{2 e^4}-\frac{(d+e x)^5 (b d-a e)^3}{5 e^4}+\frac{b^3 (d+e x)^8}{8 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [B]  time = 0.0352289, size = 217, normalized size = 2.36 \[ \frac{1}{2} b e^2 x^6 \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )+\frac{1}{5} e x^5 \left (12 a^2 b d e^2+a^3 e^3+18 a b^2 d^2 e+4 b^3 d^3\right )+\frac{1}{4} d x^4 \left (18 a^2 b d e^2+4 a^3 e^3+12 a b^2 d^2 e+b^3 d^3\right )+a d^2 x^3 \left (2 a^2 e^2+4 a b d e+b^2 d^2\right )+\frac{1}{2} a^2 d^3 x^2 (4 a e+3 b d)+a^3 d^4 x+\frac{1}{7} b^2 e^3 x^7 (3 a e+4 b d)+\frac{1}{8} b^3 e^4 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*d^4*x + (a^2*d^3*(3*b*d + 4*a*e)*x^2)/2 + a*d^2*(b^2*d^2 + 4*a*b*d*e + 2*a^2*e^2)*x^3 + (d*(b^3*d^3 + 12*a
*b^2*d^2*e + 18*a^2*b*d*e^2 + 4*a^3*e^3)*x^4)/4 + (e*(4*b^3*d^3 + 18*a*b^2*d^2*e + 12*a^2*b*d*e^2 + a^3*e^3)*x
^5)/5 + (b*e^2*(2*b^2*d^2 + 4*a*b*d*e + a^2*e^2)*x^6)/2 + (b^2*e^3*(4*b*d + 3*a*e)*x^7)/7 + (b^3*e^4*x^8)/8

________________________________________________________________________________________

Maple [B]  time = 0.002, size = 319, normalized size = 3.5 \begin{align*}{\frac{{e}^{4}{b}^{3}{x}^{8}}{8}}+{\frac{ \left ( \left ( a{e}^{4}+4\,bd{e}^{3} \right ){b}^{2}+2\,a{b}^{2}{e}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,ad{e}^{3}+6\,b{d}^{2}{e}^{2} \right ){b}^{2}+2\, \left ( a{e}^{4}+4\,bd{e}^{3} \right ) ab+b{e}^{4}{a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 6\,a{d}^{2}{e}^{2}+4\,b{d}^{3}e \right ){b}^{2}+2\, \left ( 4\,ad{e}^{3}+6\,b{d}^{2}{e}^{2} \right ) ab+ \left ( a{e}^{4}+4\,bd{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 4\,a{d}^{3}e+b{d}^{4} \right ){b}^{2}+2\, \left ( 6\,a{d}^{2}{e}^{2}+4\,b{d}^{3}e \right ) ab+ \left ( 4\,ad{e}^{3}+6\,b{d}^{2}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{2}{d}^{4}a+2\, \left ( 4\,a{d}^{3}e+b{d}^{4} \right ) ab+ \left ( 6\,a{d}^{2}{e}^{2}+4\,b{d}^{3}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}{d}^{4}b+ \left ( 4\,a{d}^{3}e+b{d}^{4} \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 0.973833, size = 304, normalized size = 3.3 \begin{align*} \frac{1}{8} \, b^{3} e^{4} x^{8} + a^{3} d^{4} x + \frac{1}{7} \,{\left (4 \, b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, b^{3} d^{2} e^{2} + 4 \, a b^{2} d e^{3} + a^{2} b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, b^{3} d^{3} e + 18 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + a^{3} e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{4} + 12 \, a b^{2} d^{3} e + 18 \, a^{2} b d^{2} e^{2} + 4 \, a^{3} d e^{3}\right )} x^{4} +{\left (a b^{2} d^{4} + 4 \, a^{2} b d^{3} e + 2 \, a^{3} d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{4} + 4 \, a^{3} d^{3} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^3*e^4*x^8 + a^3*d^4*x + 1/7*(4*b^3*d*e^3 + 3*a*b^2*e^4)*x^7 + 1/2*(2*b^3*d^2*e^2 + 4*a*b^2*d*e^3 + a^2*b
*e^4)*x^6 + 1/5*(4*b^3*d^3*e + 18*a*b^2*d^2*e^2 + 12*a^2*b*d*e^3 + a^3*e^4)*x^5 + 1/4*(b^3*d^4 + 12*a*b^2*d^3*
e + 18*a^2*b*d^2*e^2 + 4*a^3*d*e^3)*x^4 + (a*b^2*d^4 + 4*a^2*b*d^3*e + 2*a^3*d^2*e^2)*x^3 + 1/2*(3*a^2*b*d^4 +
4*a^3*d^3*e)*x^2

________________________________________________________________________________________

Fricas [B]  time = 1.38826, size = 520, normalized size = 5.65 \begin{align*} \frac{1}{8} x^{8} e^{4} b^{3} + \frac{4}{7} x^{7} e^{3} d b^{3} + \frac{3}{7} x^{7} e^{4} b^{2} a + x^{6} e^{2} d^{2} b^{3} + 2 x^{6} e^{3} d b^{2} a + \frac{1}{2} x^{6} e^{4} b a^{2} + \frac{4}{5} x^{5} e d^{3} b^{3} + \frac{18}{5} x^{5} e^{2} d^{2} b^{2} a + \frac{12}{5} x^{5} e^{3} d b a^{2} + \frac{1}{5} x^{5} e^{4} a^{3} + \frac{1}{4} x^{4} d^{4} b^{3} + 3 x^{4} e d^{3} b^{2} a + \frac{9}{2} x^{4} e^{2} d^{2} b a^{2} + x^{4} e^{3} d a^{3} + x^{3} d^{4} b^{2} a + 4 x^{3} e d^{3} b a^{2} + 2 x^{3} e^{2} d^{2} a^{3} + \frac{3}{2} x^{2} d^{4} b a^{2} + 2 x^{2} e d^{3} a^{3} + x d^{4} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*e^4*b^3 + 4/7*x^7*e^3*d*b^3 + 3/7*x^7*e^4*b^2*a + x^6*e^2*d^2*b^3 + 2*x^6*e^3*d*b^2*a + 1/2*x^6*e^4*b*
a^2 + 4/5*x^5*e*d^3*b^3 + 18/5*x^5*e^2*d^2*b^2*a + 12/5*x^5*e^3*d*b*a^2 + 1/5*x^5*e^4*a^3 + 1/4*x^4*d^4*b^3 +
3*x^4*e*d^3*b^2*a + 9/2*x^4*e^2*d^2*b*a^2 + x^4*e^3*d*a^3 + x^3*d^4*b^2*a + 4*x^3*e*d^3*b*a^2 + 2*x^3*e^2*d^2*
a^3 + 3/2*x^2*d^4*b*a^2 + 2*x^2*e*d^3*a^3 + x*d^4*a^3

________________________________________________________________________________________

Sympy [B]  time = 0.094974, size = 243, normalized size = 2.64 \begin{align*} a^{3} d^{4} x + \frac{b^{3} e^{4} x^{8}}{8} + x^{7} \left (\frac{3 a b^{2} e^{4}}{7} + \frac{4 b^{3} d e^{3}}{7}\right ) + x^{6} \left (\frac{a^{2} b e^{4}}{2} + 2 a b^{2} d e^{3} + b^{3} d^{2} e^{2}\right ) + x^{5} \left (\frac{a^{3} e^{4}}{5} + \frac{12 a^{2} b d e^{3}}{5} + \frac{18 a b^{2} d^{2} e^{2}}{5} + \frac{4 b^{3} d^{3} e}{5}\right ) + x^{4} \left (a^{3} d e^{3} + \frac{9 a^{2} b d^{2} e^{2}}{2} + 3 a b^{2} d^{3} e + \frac{b^{3} d^{4}}{4}\right ) + x^{3} \left (2 a^{3} d^{2} e^{2} + 4 a^{2} b d^{3} e + a b^{2} d^{4}\right ) + x^{2} \left (2 a^{3} d^{3} e + \frac{3 a^{2} b d^{4}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*d**4*x + b**3*e**4*x**8/8 + x**7*(3*a*b**2*e**4/7 + 4*b**3*d*e**3/7) + x**6*(a**2*b*e**4/2 + 2*a*b**2*d*e
**3 + b**3*d**2*e**2) + x**5*(a**3*e**4/5 + 12*a**2*b*d*e**3/5 + 18*a*b**2*d**2*e**2/5 + 4*b**3*d**3*e/5) + x*
*4*(a**3*d*e**3 + 9*a**2*b*d**2*e**2/2 + 3*a*b**2*d**3*e + b**3*d**4/4) + x**3*(2*a**3*d**2*e**2 + 4*a**2*b*d*
*3*e + a*b**2*d**4) + x**2*(2*a**3*d**3*e + 3*a**2*b*d**4/2)

________________________________________________________________________________________

Giac [B]  time = 1.19078, size = 320, normalized size = 3.48 \begin{align*} \frac{1}{8} \, b^{3} x^{8} e^{4} + \frac{4}{7} \, b^{3} d x^{7} e^{3} + b^{3} d^{2} x^{6} e^{2} + \frac{4}{5} \, b^{3} d^{3} x^{5} e + \frac{1}{4} \, b^{3} d^{4} x^{4} + \frac{3}{7} \, a b^{2} x^{7} e^{4} + 2 \, a b^{2} d x^{6} e^{3} + \frac{18}{5} \, a b^{2} d^{2} x^{5} e^{2} + 3 \, a b^{2} d^{3} x^{4} e + a b^{2} d^{4} x^{3} + \frac{1}{2} \, a^{2} b x^{6} e^{4} + \frac{12}{5} \, a^{2} b d x^{5} e^{3} + \frac{9}{2} \, a^{2} b d^{2} x^{4} e^{2} + 4 \, a^{2} b d^{3} x^{3} e + \frac{3}{2} \, a^{2} b d^{4} x^{2} + \frac{1}{5} \, a^{3} x^{5} e^{4} + a^{3} d x^{4} e^{3} + 2 \, a^{3} d^{2} x^{3} e^{2} + 2 \, a^{3} d^{3} x^{2} e + a^{3} d^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^3*x^8*e^4 + 4/7*b^3*d*x^7*e^3 + b^3*d^2*x^6*e^2 + 4/5*b^3*d^3*x^5*e + 1/4*b^3*d^4*x^4 + 3/7*a*b^2*x^7*e^
4 + 2*a*b^2*d*x^6*e^3 + 18/5*a*b^2*d^2*x^5*e^2 + 3*a*b^2*d^3*x^4*e + a*b^2*d^4*x^3 + 1/2*a^2*b*x^6*e^4 + 12/5*
a^2*b*d*x^5*e^3 + 9/2*a^2*b*d^2*x^4*e^2 + 4*a^2*b*d^3*x^3*e + 3/2*a^2*b*d^4*x^2 + 1/5*a^3*x^5*e^4 + a^3*d*x^4*
e^3 + 2*a^3*d^2*x^3*e^2 + 2*a^3*d^3*x^2*e + a^3*d^4*x