### 3.1785 $$\int (a+b x) (a c+(b c+a d) x+b d x^2)^3 \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.120538, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.074, Rules used = {626, 43} $\frac{3 d^2 (a+b x)^7 (b c-a d)}{7 b^4}+\frac{d (a+b x)^6 (b c-a d)^2}{2 b^4}+\frac{(a+b x)^5 (b c-a d)^3}{5 b^4}+\frac{d^3 (a+b x)^8}{8 b^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 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) \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx &=\int (a+b x)^4 (c+d x)^3 \, dx\\ &=\int \left (\frac{(b c-a d)^3 (a+b x)^4}{b^3}+\frac{3 d (b c-a d)^2 (a+b x)^5}{b^3}+\frac{3 d^2 (b c-a d) (a+b x)^6}{b^3}+\frac{d^3 (a+b x)^7}{b^3}\right ) \, dx\\ &=\frac{(b c-a d)^3 (a+b x)^5}{5 b^4}+\frac{d (b c-a d)^2 (a+b x)^6}{2 b^4}+\frac{3 d^2 (b c-a d) (a+b x)^7}{7 b^4}+\frac{d^3 (a+b x)^8}{8 b^4}\\ \end{align*}

Mathematica [B]  time = 0.0247937, size = 217, normalized size = 2.36 $\frac{1}{2} b^2 d x^6 \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{1}{5} b x^5 \left (18 a^2 b c d^2+4 a^3 d^3+12 a b^2 c^2 d+b^3 c^3\right )+\frac{1}{4} a x^4 \left (12 a^2 b c d^2+a^3 d^3+18 a b^2 c^2 d+4 b^3 c^3\right )+a^2 c x^3 \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )+\frac{1}{2} a^3 c^2 x^2 (3 a d+4 b c)+a^4 c^3 x+\frac{1}{7} b^3 d^2 x^7 (4 a d+3 b c)+\frac{1}{8} b^4 d^3 x^8$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.04555, size = 304, normalized size = 3.3 \begin{align*} \frac{1}{8} \, b^{4} d^{3} x^{8} + a^{4} c^{3} x + \frac{1}{7} \,{\left (3 \, b^{4} c d^{2} + 4 \, a b^{3} d^{3}\right )} x^{7} + \frac{1}{2} \,{\left (b^{4} c^{2} d + 4 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} c^{3} + 12 \, a b^{3} c^{2} d + 18 \, a^{2} b^{2} c d^{2} + 4 \, a^{3} b d^{3}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, a b^{3} c^{3} + 18 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{4} +{\left (2 \, a^{2} b^{2} c^{3} + 4 \, a^{3} b c^{2} d + a^{4} c d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b c^{3} + 3 \, a^{4} c^{2} d\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.39332, size = 520, normalized size = 5.65 \begin{align*} \frac{1}{8} x^{8} d^{3} b^{4} + \frac{3}{7} x^{7} d^{2} c b^{4} + \frac{4}{7} x^{7} d^{3} b^{3} a + \frac{1}{2} x^{6} d c^{2} b^{4} + 2 x^{6} d^{2} c b^{3} a + x^{6} d^{3} b^{2} a^{2} + \frac{1}{5} x^{5} c^{3} b^{4} + \frac{12}{5} x^{5} d c^{2} b^{3} a + \frac{18}{5} x^{5} d^{2} c b^{2} a^{2} + \frac{4}{5} x^{5} d^{3} b a^{3} + x^{4} c^{3} b^{3} a + \frac{9}{2} x^{4} d c^{2} b^{2} a^{2} + 3 x^{4} d^{2} c b a^{3} + \frac{1}{4} x^{4} d^{3} a^{4} + 2 x^{3} c^{3} b^{2} a^{2} + 4 x^{3} d c^{2} b a^{3} + x^{3} d^{2} c a^{4} + 2 x^{2} c^{3} b a^{3} + \frac{3}{2} x^{2} d c^{2} a^{4} + x c^{3} a^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.279936, size = 243, normalized size = 2.64 \begin{align*} a^{4} c^{3} x + \frac{b^{4} d^{3} x^{8}}{8} + x^{7} \left (\frac{4 a b^{3} d^{3}}{7} + \frac{3 b^{4} c d^{2}}{7}\right ) + x^{6} \left (a^{2} b^{2} d^{3} + 2 a b^{3} c d^{2} + \frac{b^{4} c^{2} d}{2}\right ) + x^{5} \left (\frac{4 a^{3} b d^{3}}{5} + \frac{18 a^{2} b^{2} c d^{2}}{5} + \frac{12 a b^{3} c^{2} d}{5} + \frac{b^{4} c^{3}}{5}\right ) + x^{4} \left (\frac{a^{4} d^{3}}{4} + 3 a^{3} b c d^{2} + \frac{9 a^{2} b^{2} c^{2} d}{2} + a b^{3} c^{3}\right ) + x^{3} \left (a^{4} c d^{2} + 4 a^{3} b c^{2} d + 2 a^{2} b^{2} c^{3}\right ) + x^{2} \left (\frac{3 a^{4} c^{2} d}{2} + 2 a^{3} b c^{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.18625, size = 331, normalized size = 3.6 \begin{align*} \frac{1}{8} \, b^{4} d^{3} x^{8} + \frac{3}{7} \, b^{4} c d^{2} x^{7} + \frac{4}{7} \, a b^{3} d^{3} x^{7} + \frac{1}{2} \, b^{4} c^{2} d x^{6} + 2 \, a b^{3} c d^{2} x^{6} + a^{2} b^{2} d^{3} x^{6} + \frac{1}{5} \, b^{4} c^{3} x^{5} + \frac{12}{5} \, a b^{3} c^{2} d x^{5} + \frac{18}{5} \, a^{2} b^{2} c d^{2} x^{5} + \frac{4}{5} \, a^{3} b d^{3} x^{5} + a b^{3} c^{3} x^{4} + \frac{9}{2} \, a^{2} b^{2} c^{2} d x^{4} + 3 \, a^{3} b c d^{2} x^{4} + \frac{1}{4} \, a^{4} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{3} x^{3} + 4 \, a^{3} b c^{2} d x^{3} + a^{4} c d^{2} x^{3} + 2 \, a^{3} b c^{3} x^{2} + \frac{3}{2} \, a^{4} c^{2} d x^{2} + a^{4} c^{3} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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