3.18.86 \(\int (a c+(b c+a d) x+b d x^2)^3 \, dx\) [1786]

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

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {624, 45} \begin {gather*} -\frac {b^2 (c+d x)^6 (b c-a d)}{2 d^4}+\frac {3 b (c+d x)^5 (b c-a d)^2}{5 d^4}-\frac {(c+d x)^4 (b c-a d)^3}{4 d^4}+\frac {b^3 (c+d x)^7}{7 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 624

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[1/c^p, Int[Simp[
b/2 - q/2 + c*x, x]^p*Simp[b/2 + q/2 + c*x, x]^p, x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGt
Q[p, 0] && PerfectSquareQ[b^2 - 4*a*c]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 161, normalized size = 1.75 \begin {gather*} a^3 c^3 x+\frac {3}{2} a^2 c^2 (b c+a d) x^2+a c \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^3+\frac {1}{4} \left (b^3 c^3+9 a b^2 c^2 d+9 a^2 b c d^2+a^3 d^3\right ) x^4+\frac {3}{5} b d \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^5+\frac {1}{2} b^2 d^2 (b c+a d) x^6+\frac {1}{7} b^3 d^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(193\) vs. \(2(84)=168\).
time = 0.63, size = 194, normalized size = 2.11

method	result	size
norman	\(\frac {d^{3} x^{7} b^{3}}{7}+\left (\frac {1}{2} a \,b^{2} d^{3}+\frac {1}{2} b^{3} c \,d^{2}\right ) x^{6}+\left (\frac {3}{5} d^{3} a^{2} b +\frac {9}{5} a \,b^{2} c \,d^{2}+\frac {3}{5} c^{2} d \,b^{3}\right ) x^{5}+\left (\frac {1}{4} a^{3} d^{3}+\frac {9}{4} a^{2} b c \,d^{2}+\frac {9}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{4}+\left (a^{3} c \,d^{2}+3 a^{2} b \,c^{2} d +a \,b^{2} c^{3}\right ) x^{3}+\left (\frac {3}{2} a^{3} c^{2} d +\frac {3}{2} a^{2} b \,c^{3}\right ) x^{2}+a^{3} c^{3} x\)	\(172\)
risch	\(\frac {1}{7} d^{3} x^{7} b^{3}+\frac {1}{2} a \,b^{2} d^{3} x^{6}+\frac {1}{2} b^{3} c \,d^{2} x^{6}+\frac {3}{5} a^{2} b \,d^{3} x^{5}+\frac {9}{5} a \,b^{2} c \,d^{2} x^{5}+\frac {3}{5} b^{3} c^{2} d \,x^{5}+\frac {1}{4} a^{3} d^{3} x^{4}+\frac {9}{4} a^{2} b c \,d^{2} x^{4}+\frac {9}{4} a \,b^{2} c^{2} d \,x^{4}+\frac {1}{4} c^{3} b^{3} x^{4}+d^{2} a^{3} c \,x^{3}+3 a^{2} b \,c^{2} d \,x^{3}+a \,b^{2} c^{3} x^{3}+\frac {3}{2} d \,a^{3} c^{2} x^{2}+\frac {3}{2} a^{2} b \,c^{3} x^{2}+a^{3} c^{3} x\)	\(189\)
gosper	\(\frac {x \left (20 d^{3} x^{6} b^{3}+70 a \,b^{2} d^{3} x^{5}+70 b^{3} c \,d^{2} x^{5}+84 a^{2} b \,d^{3} x^{4}+252 a \,b^{2} c \,d^{2} x^{4}+84 b^{3} c^{2} d \,x^{4}+35 x^{3} a^{3} d^{3}+315 x^{3} a^{2} b c \,d^{2}+315 x^{3} a \,b^{2} c^{2} d +35 b^{3} x^{3} c^{3}+140 a^{3} c \,d^{2} x^{2}+420 a^{2} b \,c^{2} d \,x^{2}+140 a \,b^{2} c^{3} x^{2}+210 x \,a^{3} c^{2} d +210 a^{2} b \,c^{3} x +140 a^{3} c^{3}\right )}{140}\)	\(190\)
default	\(\frac {d^{3} x^{7} b^{3}}{7}+\frac {\left (a d +b c \right ) b^{2} d^{2} x^{6}}{2}+\frac {\left (a \,b^{2} c \,d^{2}+2 \left (a d +b c \right )^{2} b d +b d \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right ) x^{5}}{5}+\frac {\left (4 a c b d \left (a d +b c \right )+\left (a d +b c \right ) \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right ) x^{4}}{4}+\frac {\left (a c \left (\left (a d +b c \right )^{2}+2 a b c d \right )+2 \left (a d +b c \right )^{2} a c +a^{2} b \,c^{2} d \right ) x^{3}}{3}+\frac {3 a^{2} c^{2} \left (a d +b c \right ) x^{2}}{2}+a^{3} c^{3} x\)	\(194\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 140, normalized size = 1.52 \begin {gather*} \frac {1}{7} \, b^{3} d^{3} x^{7} + \frac {1}{2} \, {\left (b c + a d\right )} b^{2} d^{2} x^{6} + \frac {3}{5} \, {\left (b c + a d\right )}^{2} b d x^{5} + a^{3} c^{3} x + \frac {1}{4} \, {\left (b c + a d\right )}^{3} x^{4} + \frac {1}{2} \, {\left (2 \, b d x^{3} + 3 \, {\left (b c + a d\right )} x^{2}\right )} a^{2} c^{2} + \frac {1}{10} \, {\left (6 \, b^{2} d^{2} x^{5} + 15 \, {\left (b c + a d\right )} b d x^{4} + 10 \, {\left (b c + a d\right )}^{2} x^{3}\right )} a c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^3*d^3*x^7 + 1/2*(b*c + a*d)*b^2*d^2*x^6 + 3/5*(b*c + a*d)^2*b*d*x^5 + a^3*c^3*x + 1/4*(b*c + a*d)^3*x^4
+ 1/2*(2*b*d*x^3 + 3*(b*c + a*d)*x^2)*a^2*c^2 + 1/10*(6*b^2*d^2*x^5 + 15*(b*c + a*d)*b*d*x^4 + 10*(b*c + a*d)^
2*x^3)*a*c

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Fricas [A]
time = 3.22, size = 167, normalized size = 1.82 \begin {gather*} \frac {1}{7} \, b^{3} d^{3} x^{7} + a^{3} c^{3} x + \frac {1}{2} \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{6} + \frac {3}{5} \, {\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{4} + {\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (80) = 160\).
time = 0.03, size = 190, normalized size = 2.07 \begin {gather*} a^{3} c^{3} x + \frac {b^{3} d^{3} x^{7}}{7} + x^{6} \left (\frac {a b^{2} d^{3}}{2} + \frac {b^{3} c d^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{2} b d^{3}}{5} + \frac {9 a b^{2} c d^{2}}{5} + \frac {3 b^{3} c^{2} d}{5}\right ) + x^{4} \left (\frac {a^{3} d^{3}}{4} + \frac {9 a^{2} b c d^{2}}{4} + \frac {9 a b^{2} c^{2} d}{4} + \frac {b^{3} c^{3}}{4}\right ) + x^{3} \left (a^{3} c d^{2} + 3 a^{2} b c^{2} d + a b^{2} c^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{3} c^{2} d}{2} + \frac {3 a^{2} b c^{3}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (84) = 168\).
time = 2.12, size = 188, normalized size = 2.04 \begin {gather*} \frac {1}{7} \, b^{3} d^{3} x^{7} + \frac {1}{2} \, b^{3} c d^{2} x^{6} + \frac {1}{2} \, a b^{2} d^{3} x^{6} + \frac {3}{5} \, b^{3} c^{2} d x^{5} + \frac {9}{5} \, a b^{2} c d^{2} x^{5} + \frac {3}{5} \, a^{2} b d^{3} x^{5} + \frac {1}{4} \, b^{3} c^{3} x^{4} + \frac {9}{4} \, a b^{2} c^{2} d x^{4} + \frac {9}{4} \, a^{2} b c d^{2} x^{4} + \frac {1}{4} \, a^{3} d^{3} x^{4} + a b^{2} c^{3} x^{3} + 3 \, a^{2} b c^{2} d x^{3} + a^{3} c d^{2} x^{3} + \frac {3}{2} \, a^{2} b c^{3} x^{2} + \frac {3}{2} \, a^{3} c^{2} d x^{2} + a^{3} c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.60, size = 152, normalized size = 1.65 \begin {gather*} x^4\,\left (\frac {a^3\,d^3}{4}+\frac {9\,a^2\,b\,c\,d^2}{4}+\frac {9\,a\,b^2\,c^2\,d}{4}+\frac {b^3\,c^3}{4}\right )+a^3\,c^3\,x+\frac {b^3\,d^3\,x^7}{7}+a\,c\,x^3\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )+\frac {3\,b\,d\,x^5\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{5}+\frac {3\,a^2\,c^2\,x^2\,\left (a\,d+b\,c\right )}{2}+\frac {b^2\,d^2\,x^6\,\left (a\,d+b\,c\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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