\(\int (a+b x)^2 (a c+(b c+a d) x+b d x^2)^2 \, dx\) [1770]

Optimal result

Integrand size = 29, antiderivative size = 65 \[ \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {(b c-a d)^2 (a+b x)^5}{5 b^3}+\frac {d (b c-a d) (a+b x)^6}{3 b^3}+\frac {d^2 (a+b x)^7}{7 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 45} \[ \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {d (a+b x)^6 (b c-a d)}{3 b^3}+\frac {(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac {d^2 (a+b x)^7}{7 b^3} \]

[In]

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

[Out]

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

Rule 45

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])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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]

Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^4 (c+d x)^2 \, dx \\ & = \int \left (\frac {(b c-a d)^2 (a+b x)^4}{b^2}+\frac {2 d (b c-a d) (a+b x)^5}{b^2}+\frac {d^2 (a+b x)^6}{b^2}\right ) \, dx \\ & = \frac {(b c-a d)^2 (a+b x)^5}{5 b^3}+\frac {d (b c-a d) (a+b x)^6}{3 b^3}+\frac {d^2 (a+b x)^7}{7 b^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(148\) vs. \(2(65)=130\).

Time = 0.01 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.28 \[ \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=a^4 c^2 x+a^3 c (2 b c+a d) x^2+\frac {1}{3} a^2 \left (6 b^2 c^2+8 a b c d+a^2 d^2\right ) x^3+a b \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^4+\frac {1}{5} b^2 \left (b^2 c^2+8 a b c d+6 a^2 d^2\right ) x^5+\frac {1}{3} b^3 d (b c+2 a d) x^6+\frac {1}{7} b^4 d^2 x^7 \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(59)=118\).

Time = 2.36 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.42

method	result	size
norman	\(\frac {b^{4} d^{2} x^{7}}{7}+\left (\frac {2}{3} a \,b^{3} d^{2}+\frac {1}{3} b^{4} c d \right ) x^{6}+\left (\frac {6}{5} b^{2} d^{2} a^{2}+\frac {8}{5} a \,b^{3} c d +\frac {1}{5} b^{4} c^{2}\right ) x^{5}+\left (a^{3} b \,d^{2}+3 a^{2} b^{2} c d +a \,b^{3} c^{2}\right ) x^{4}+\left (\frac {1}{3} a^{4} d^{2}+\frac {8}{3} a^{3} b c d +2 a^{2} b^{2} c^{2}\right ) x^{3}+\left (a^{4} c d +2 a^{3} b \,c^{2}\right ) x^{2}+a^{4} c^{2} x\)	\(157\)
risch	\(\frac {1}{7} b^{4} d^{2} x^{7}+\frac {2}{3} x^{6} a \,b^{3} d^{2}+\frac {1}{3} x^{6} b^{4} c d +\frac {6}{5} x^{5} b^{2} d^{2} a^{2}+\frac {8}{5} x^{5} b^{3} d c a +\frac {1}{5} b^{4} c^{2} x^{5}+a^{3} b \,d^{2} x^{4}+3 a^{2} b^{2} c d \,x^{4}+a \,b^{3} c^{2} x^{4}+\frac {1}{3} x^{3} a^{4} d^{2}+\frac {8}{3} x^{3} a^{3} b c d +2 x^{3} a^{2} b^{2} c^{2}+a^{4} c d \,x^{2}+2 a^{3} b \,c^{2} x^{2}+a^{4} c^{2} x\)	\(171\)
parallelrisch	\(\frac {1}{7} b^{4} d^{2} x^{7}+\frac {2}{3} x^{6} a \,b^{3} d^{2}+\frac {1}{3} x^{6} b^{4} c d +\frac {6}{5} x^{5} b^{2} d^{2} a^{2}+\frac {8}{5} x^{5} b^{3} d c a +\frac {1}{5} b^{4} c^{2} x^{5}+a^{3} b \,d^{2} x^{4}+3 a^{2} b^{2} c d \,x^{4}+a \,b^{3} c^{2} x^{4}+\frac {1}{3} x^{3} a^{4} d^{2}+\frac {8}{3} x^{3} a^{3} b c d +2 x^{3} a^{2} b^{2} c^{2}+a^{4} c d \,x^{2}+2 a^{3} b \,c^{2} x^{2}+a^{4} c^{2} x\)	\(171\)
gosper	\(\frac {x \left (15 b^{4} d^{2} x^{6}+70 x^{5} a \,b^{3} d^{2}+35 x^{5} b^{4} c d +126 x^{4} b^{2} d^{2} a^{2}+168 x^{4} a \,b^{3} c d +21 c^{2} x^{4} b^{4}+105 b \,d^{2} x^{3} a^{3}+315 a^{2} b^{2} c d \,x^{3}+105 a \,b^{3} c^{2} x^{3}+35 x^{2} a^{4} d^{2}+280 x^{2} a^{3} b c d +210 x^{2} a^{2} b^{2} c^{2}+105 a^{4} c d x +210 a^{3} b \,c^{2} x +105 a^{4} c^{2}\right )}{105}\)	\(173\)
default	\(\frac {b^{4} d^{2} x^{7}}{7}+\frac {\left (2 a \,b^{3} d^{2}+2 b^{3} d \left (a d +b c \right )\right ) x^{6}}{6}+\frac {\left (b^{2} d^{2} a^{2}+4 a \,b^{2} d \left (a d +b c \right )+b^{2} \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right ) x^{5}}{5}+\frac {\left (2 a^{2} b d \left (a d +b c \right )+2 a b \left (\left (a d +b c \right )^{2}+2 a b c d \right )+2 b^{2} a c \left (a d +b c \right )\right ) x^{4}}{4}+\frac {\left (a^{2} \left (\left (a d +b c \right )^{2}+2 a b c d \right )+4 a^{2} b c \left (a d +b c \right )+a^{2} b^{2} c^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{3} c \left (a d +b c \right )+2 a^{3} b \,c^{2}\right ) x^{2}}{2}+a^{4} c^{2} x\)	\(231\)

[In]

int((b*x+a)^2*(b*d*x^2+(a*d+b*c)*x+a*c)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (59) = 118\).

Time = 0.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.40 \[ \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {1}{7} \, b^{4} d^{2} x^{7} + a^{4} c^{2} x + \frac {1}{3} \, {\left (b^{4} c d + 2 \, a b^{3} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{2} + 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x^{5} + {\left (a b^{3} c^{2} + 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{2} + a^{4} c d\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (54) = 108\).

Time = 0.03 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.58 \[ \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=a^{4} c^{2} x + \frac {b^{4} d^{2} x^{7}}{7} + x^{6} \cdot \left (\frac {2 a b^{3} d^{2}}{3} + \frac {b^{4} c d}{3}\right ) + x^{5} \cdot \left (\frac {6 a^{2} b^{2} d^{2}}{5} + \frac {8 a b^{3} c d}{5} + \frac {b^{4} c^{2}}{5}\right ) + x^{4} \left (a^{3} b d^{2} + 3 a^{2} b^{2} c d + a b^{3} c^{2}\right ) + x^{3} \left (\frac {a^{4} d^{2}}{3} + \frac {8 a^{3} b c d}{3} + 2 a^{2} b^{2} c^{2}\right ) + x^{2} \left (a^{4} c d + 2 a^{3} b c^{2}\right ) \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (59) = 118\).

Time = 0.21 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.40 \[ \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {1}{7} \, b^{4} d^{2} x^{7} + a^{4} c^{2} x + \frac {1}{3} \, {\left (b^{4} c d + 2 \, a b^{3} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{2} + 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x^{5} + {\left (a b^{3} c^{2} + 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{2} + a^{4} c d\right )} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (59) = 118\).

Time = 0.26 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.62 \[ \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=\frac {1}{7} \, b^{4} d^{2} x^{7} + \frac {1}{3} \, b^{4} c d x^{6} + \frac {2}{3} \, a b^{3} d^{2} x^{6} + \frac {1}{5} \, b^{4} c^{2} x^{5} + \frac {8}{5} \, a b^{3} c d x^{5} + \frac {6}{5} \, a^{2} b^{2} d^{2} x^{5} + a b^{3} c^{2} x^{4} + 3 \, a^{2} b^{2} c d x^{4} + a^{3} b d^{2} x^{4} + 2 \, a^{2} b^{2} c^{2} x^{3} + \frac {8}{3} \, a^{3} b c d x^{3} + \frac {1}{3} \, a^{4} d^{2} x^{3} + 2 \, a^{3} b c^{2} x^{2} + a^{4} c d x^{2} + a^{4} c^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.22 \[ \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right )^2 \, dx=x^3\,\left (\frac {a^4\,d^2}{3}+\frac {8\,a^3\,b\,c\,d}{3}+2\,a^2\,b^2\,c^2\right )+x^5\,\left (\frac {6\,a^2\,b^2\,d^2}{5}+\frac {8\,a\,b^3\,c\,d}{5}+\frac {b^4\,c^2}{5}\right )+a^4\,c^2\,x+\frac {b^4\,d^2\,x^7}{7}+a^3\,c\,x^2\,\left (a\,d+2\,b\,c\right )+\frac {b^3\,d\,x^6\,\left (2\,a\,d+b\,c\right )}{3}+a\,b\,x^4\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right ) \]

[In]

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

[Out]

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