3.18.71 \(\int (a+b x) (a c+(b c+a d) x+b d x^2)^2 \, dx\) [1771]

Optimal. Leaf size=65 \[ \frac {(b c-a d)^2 (a+b x)^4}{4 b^3}+\frac {2 d (b c-a d) (a+b x)^5}{5 b^3}+\frac {d^2 (a+b x)^6}{6 b^3} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 = {640, 45} \begin {gather*} \frac {2 d (a+b x)^5 (b c-a d)}{5 b^3}+\frac {(a+b x)^4 (b c-a d)^2}{4 b^3}+\frac {d^2 (a+b x)^6}{6 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.01, size = 122, normalized size = 1.88 \begin {gather*} a^3 c^2 x+\frac {1}{2} a^2 c (3 b c+2 a d) x^2+\frac {1}{3} a \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^3+\frac {1}{4} b \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^4+\frac {1}{5} b^2 d (2 b c+3 a d) x^5+\frac {1}{6} b^3 d^2 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(146\) vs. \(2(59)=118\).
time = 0.68, size = 147, normalized size = 2.26

method	result	size
norman	\(\frac {b^{3} d^{2} x^{6}}{6}+\left (\frac {3}{5} a \,b^{2} d^{2}+\frac {2}{5} b^{3} d c \right ) x^{5}+\left (\frac {3}{4} a^{2} b \,d^{2}+\frac {3}{2} a \,b^{2} c d +\frac {1}{4} b^{3} c^{2}\right ) x^{4}+\left (\frac {1}{3} d^{2} a^{3}+2 b c d \,a^{2}+b^{2} c^{2} a \right ) x^{3}+\left (a^{3} c d +\frac {3}{2} a^{2} b \,c^{2}\right ) x^{2}+a^{3} c^{2} x\)	\(121\)
risch	\(\frac {1}{6} b^{3} d^{2} x^{6}+\frac {3}{5} x^{5} a \,b^{2} d^{2}+\frac {2}{5} x^{5} b^{3} d c +\frac {3}{4} a^{2} b \,d^{2} x^{4}+\frac {3}{2} a \,b^{2} c d \,x^{4}+\frac {1}{4} b^{3} x^{4} c^{2}+\frac {1}{3} a^{3} d^{2} x^{3}+2 a^{2} b c d \,x^{3}+a \,b^{2} c^{2} x^{3}+x^{2} a^{3} c d +\frac {3}{2} x^{2} a^{2} b \,c^{2}+a^{3} c^{2} x\)	\(131\)
gosper	\(\frac {x \left (10 b^{3} d^{2} x^{5}+36 x^{4} a \,b^{2} d^{2}+24 x^{4} b^{3} d c +45 a^{2} b \,d^{2} x^{3}+90 a \,b^{2} c d \,x^{3}+15 x^{3} b^{3} c^{2}+20 a^{3} d^{2} x^{2}+120 a^{2} b c d \,x^{2}+60 a \,b^{2} c^{2} x^{2}+60 x \,a^{3} c d +90 x \,a^{2} b \,c^{2}+60 c^{2} a^{3}\right )}{60}\)	\(132\)
default	\(\frac {b^{3} d^{2} x^{6}}{6}+\frac {\left (a \,b^{2} d^{2}+2 b^{2} d \left (a d +b c \right )\right ) x^{5}}{5}+\frac {\left (2 a b d \left (a d +b c \right )+b \left (\left (a d +b c \right )^{2}+2 a b c d \right )\right ) x^{4}}{4}+\frac {\left (a \left (\left (a d +b c \right )^{2}+2 a b c d \right )+2 b a c \left (a d +b c \right )\right ) x^{3}}{3}+\frac {\left (2 a^{2} c \left (a d +b c \right )+a^{2} b \,c^{2}\right ) x^{2}}{2}+a^{3} c^{2} x\)	\(147\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (59) = 118\).
time = 0.31, size = 124, normalized size = 1.91 \begin {gather*} \frac {1}{6} \, b^{3} d^{2} x^{6} + a^{3} c^{2} x + \frac {1}{5} \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{2} \end {gather*}

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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (59) = 118\).
time = 2.30, size = 124, normalized size = 1.91 \begin {gather*} \frac {1}{6} \, b^{3} d^{2} x^{6} + a^{3} c^{2} x + \frac {1}{5} \, {\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{2} \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (56) = 112\).
time = 0.03, size = 133, normalized size = 2.05 \begin {gather*} a^{3} c^{2} x + \frac {b^{3} d^{2} x^{6}}{6} + x^{5} \cdot \left (\frac {3 a b^{2} d^{2}}{5} + \frac {2 b^{3} c d}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} b d^{2}}{4} + \frac {3 a b^{2} c d}{2} + \frac {b^{3} c^{2}}{4}\right ) + x^{3} \left (\frac {a^{3} d^{2}}{3} + 2 a^{2} b c d + a b^{2} c^{2}\right ) + x^{2} \left (a^{3} c d + \frac {3 a^{2} b c^{2}}{2}\right ) \end {gather*}

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)**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (59) = 118\).
time = 0.83, size = 130, normalized size = 2.00 \begin {gather*} \frac {1}{6} \, b^{3} d^{2} x^{6} + \frac {2}{5} \, b^{3} c d x^{5} + \frac {3}{5} \, a b^{2} d^{2} x^{5} + \frac {1}{4} \, b^{3} c^{2} x^{4} + \frac {3}{2} \, a b^{2} c d x^{4} + \frac {3}{4} \, a^{2} b d^{2} x^{4} + a b^{2} c^{2} x^{3} + 2 \, a^{2} b c d x^{3} + \frac {1}{3} \, a^{3} d^{2} x^{3} + \frac {3}{2} \, a^{2} b c^{2} x^{2} + a^{3} c d x^{2} + a^{3} c^{2} x \end {gather*}

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)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.58, size = 115, normalized size = 1.77 \begin {gather*} x^3\,\left (\frac {a^3\,d^2}{3}+2\,a^2\,b\,c\,d+a\,b^2\,c^2\right )+x^4\,\left (\frac {3\,a^2\,b\,d^2}{4}+\frac {3\,a\,b^2\,c\,d}{2}+\frac {b^3\,c^2}{4}\right )+a^3\,c^2\,x+\frac {b^3\,d^2\,x^6}{6}+\frac {a^2\,c\,x^2\,\left (2\,a\,d+3\,b\,c\right )}{2}+\frac {b^2\,d\,x^5\,\left (3\,a\,d+2\,b\,c\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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