3.1258 \(\int (a+b x)^5 (c+d x)^3 \, dx\)

Optimal. Leaf size=92 \[ \frac {3 d^2 (a+b x)^8 (b c-a d)}{8 b^4}+\frac {3 d (a+b x)^7 (b c-a d)^2}{7 b^4}+\frac {(a+b x)^6 (b c-a d)^3}{6 b^4}+\frac {d^3 (a+b x)^9}{9 b^4} \]

[Out]

1/6*(-a*d+b*c)^3*(b*x+a)^6/b^4+3/7*d*(-a*d+b*c)^2*(b*x+a)^7/b^4+3/8*d^2*(-a*d+b*c)*(b*x+a)^8/b^4+1/9*d^3*(b*x+
a)^9/b^4

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Rubi [A]  time = 0.16, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ \frac {3 d^2 (a+b x)^8 (b c-a d)}{8 b^4}+\frac {3 d (a+b x)^7 (b c-a d)^2}{7 b^4}+\frac {(a+b x)^6 (b c-a d)^3}{6 b^4}+\frac {d^3 (a+b x)^9}{9 b^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5*(c + d*x)^3,x]

[Out]

((b*c - a*d)^3*(a + b*x)^6)/(6*b^4) + (3*d*(b*c - a*d)^2*(a + b*x)^7)/(7*b^4) + (3*d^2*(b*c - a*d)*(a + b*x)^8
)/(8*b^4) + (d^3*(a + b*x)^9)/(9*b^4)

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)^5 (c+d x)^3 \, dx &=\int \left (\frac {(b c-a d)^3 (a+b x)^5}{b^3}+\frac {3 d (b c-a d)^2 (a+b x)^6}{b^3}+\frac {3 d^2 (b c-a d) (a+b x)^7}{b^3}+\frac {d^3 (a+b x)^8}{b^3}\right ) \, dx\\ &=\frac {(b c-a d)^3 (a+b x)^6}{6 b^4}+\frac {3 d (b c-a d)^2 (a+b x)^7}{7 b^4}+\frac {3 d^2 (b c-a d) (a+b x)^8}{8 b^4}+\frac {d^3 (a+b x)^9}{9 b^4}\\ \end {align*}

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Mathematica [B]  time = 0.08, size = 235, normalized size = 2.55 \[ \frac {1}{504} x \left (126 a^5 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+126 a^4 b x \left (10 c^3+20 c^2 d x+15 c d^2 x^2+4 d^3 x^3\right )+84 a^3 b^2 x^2 \left (20 c^3+45 c^2 d x+36 c d^2 x^2+10 d^3 x^3\right )+36 a^2 b^3 x^3 \left (35 c^3+84 c^2 d x+70 c d^2 x^2+20 d^3 x^3\right )+9 a b^4 x^4 \left (56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right )+b^5 x^5 \left (84 c^3+216 c^2 d x+189 c d^2 x^2+56 d^3 x^3\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5*(c + d*x)^3,x]

[Out]

(x*(126*a^5*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + 126*a^4*b*x*(10*c^3 + 20*c^2*d*x + 15*c*d^2*x^2 + 4*
d^3*x^3) + 84*a^3*b^2*x^2*(20*c^3 + 45*c^2*d*x + 36*c*d^2*x^2 + 10*d^3*x^3) + 36*a^2*b^3*x^3*(35*c^3 + 84*c^2*
d*x + 70*c*d^2*x^2 + 20*d^3*x^3) + 9*a*b^4*x^4*(56*c^3 + 140*c^2*d*x + 120*c*d^2*x^2 + 35*d^3*x^3) + b^5*x^5*(
84*c^3 + 216*c^2*d*x + 189*c*d^2*x^2 + 56*d^3*x^3)))/504

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fricas [B]  time = 0.39, size = 303, normalized size = 3.29 \[ \frac {1}{9} x^{9} d^{3} b^{5} + \frac {3}{8} x^{8} d^{2} c b^{5} + \frac {5}{8} x^{8} d^{3} b^{4} a + \frac {3}{7} x^{7} d c^{2} b^{5} + \frac {15}{7} x^{7} d^{2} c b^{4} a + \frac {10}{7} x^{7} d^{3} b^{3} a^{2} + \frac {1}{6} x^{6} c^{3} b^{5} + \frac {5}{2} x^{6} d c^{2} b^{4} a + 5 x^{6} d^{2} c b^{3} a^{2} + \frac {5}{3} x^{6} d^{3} b^{2} a^{3} + x^{5} c^{3} b^{4} a + 6 x^{5} d c^{2} b^{3} a^{2} + 6 x^{5} d^{2} c b^{2} a^{3} + x^{5} d^{3} b a^{4} + \frac {5}{2} x^{4} c^{3} b^{3} a^{2} + \frac {15}{2} x^{4} d c^{2} b^{2} a^{3} + \frac {15}{4} x^{4} d^{2} c b a^{4} + \frac {1}{4} x^{4} d^{3} a^{5} + \frac {10}{3} x^{3} c^{3} b^{2} a^{3} + 5 x^{3} d c^{2} b a^{4} + x^{3} d^{2} c a^{5} + \frac {5}{2} x^{2} c^{3} b a^{4} + \frac {3}{2} x^{2} d c^{2} a^{5} + x c^{3} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(d*x+c)^3,x, algorithm="fricas")

[Out]

1/9*x^9*d^3*b^5 + 3/8*x^8*d^2*c*b^5 + 5/8*x^8*d^3*b^4*a + 3/7*x^7*d*c^2*b^5 + 15/7*x^7*d^2*c*b^4*a + 10/7*x^7*
d^3*b^3*a^2 + 1/6*x^6*c^3*b^5 + 5/2*x^6*d*c^2*b^4*a + 5*x^6*d^2*c*b^3*a^2 + 5/3*x^6*d^3*b^2*a^3 + x^5*c^3*b^4*
a + 6*x^5*d*c^2*b^3*a^2 + 6*x^5*d^2*c*b^2*a^3 + x^5*d^3*b*a^4 + 5/2*x^4*c^3*b^3*a^2 + 15/2*x^4*d*c^2*b^2*a^3 +
15/4*x^4*d^2*c*b*a^4 + 1/4*x^4*d^3*a^5 + 10/3*x^3*c^3*b^2*a^3 + 5*x^3*d*c^2*b*a^4 + x^3*d^2*c*a^5 + 5/2*x^2*c
^3*b*a^4 + 3/2*x^2*d*c^2*a^5 + x*c^3*a^5

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giac [B]  time = 1.01, size = 303, normalized size = 3.29 \[ \frac {1}{9} \, b^{5} d^{3} x^{9} + \frac {3}{8} \, b^{5} c d^{2} x^{8} + \frac {5}{8} \, a b^{4} d^{3} x^{8} + \frac {3}{7} \, b^{5} c^{2} d x^{7} + \frac {15}{7} \, a b^{4} c d^{2} x^{7} + \frac {10}{7} \, a^{2} b^{3} d^{3} x^{7} + \frac {1}{6} \, b^{5} c^{3} x^{6} + \frac {5}{2} \, a b^{4} c^{2} d x^{6} + 5 \, a^{2} b^{3} c d^{2} x^{6} + \frac {5}{3} \, a^{3} b^{2} d^{3} x^{6} + a b^{4} c^{3} x^{5} + 6 \, a^{2} b^{3} c^{2} d x^{5} + 6 \, a^{3} b^{2} c d^{2} x^{5} + a^{4} b d^{3} x^{5} + \frac {5}{2} \, a^{2} b^{3} c^{3} x^{4} + \frac {15}{2} \, a^{3} b^{2} c^{2} d x^{4} + \frac {15}{4} \, a^{4} b c d^{2} x^{4} + \frac {1}{4} \, a^{5} d^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} c^{3} x^{3} + 5 \, a^{4} b c^{2} d x^{3} + a^{5} c d^{2} x^{3} + \frac {5}{2} \, a^{4} b c^{3} x^{2} + \frac {3}{2} \, a^{5} c^{2} d x^{2} + a^{5} c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(d*x+c)^3,x, algorithm="giac")

[Out]

1/9*b^5*d^3*x^9 + 3/8*b^5*c*d^2*x^8 + 5/8*a*b^4*d^3*x^8 + 3/7*b^5*c^2*d*x^7 + 15/7*a*b^4*c*d^2*x^7 + 10/7*a^2*
b^3*d^3*x^7 + 1/6*b^5*c^3*x^6 + 5/2*a*b^4*c^2*d*x^6 + 5*a^2*b^3*c*d^2*x^6 + 5/3*a^3*b^2*d^3*x^6 + a*b^4*c^3*x^
5 + 6*a^2*b^3*c^2*d*x^5 + 6*a^3*b^2*c*d^2*x^5 + a^4*b*d^3*x^5 + 5/2*a^2*b^3*c^3*x^4 + 15/2*a^3*b^2*c^2*d*x^4 +
15/4*a^4*b*c*d^2*x^4 + 1/4*a^5*d^3*x^4 + 10/3*a^3*b^2*c^3*x^3 + 5*a^4*b*c^2*d*x^3 + a^5*c*d^2*x^3 + 5/2*a^4*b
*c^3*x^2 + 3/2*a^5*c^2*d*x^2 + a^5*c^3*x

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maple [B]  time = 0.00, size = 281, normalized size = 3.05 \[ \frac {b^{5} d^{3} x^{9}}{9}+a^{5} c^{3} x +\frac {\left (5 a \,b^{4} d^{3}+3 b^{5} c \,d^{2}\right ) x^{8}}{8}+\frac {\left (10 a^{2} b^{3} d^{3}+15 a \,b^{4} c \,d^{2}+3 b^{5} c^{2} d \right ) x^{7}}{7}+\frac {\left (10 a^{3} b^{2} d^{3}+30 a^{2} b^{3} c \,d^{2}+15 a \,b^{4} c^{2} d +b^{5} c^{3}\right ) x^{6}}{6}+\frac {\left (5 a^{4} b \,d^{3}+30 a^{3} b^{2} c \,d^{2}+30 a^{2} b^{3} c^{2} d +5 a \,b^{4} c^{3}\right ) x^{5}}{5}+\frac {\left (a^{5} d^{3}+15 a^{4} b c \,d^{2}+30 a^{3} b^{2} c^{2} d +10 a^{2} b^{3} c^{3}\right ) x^{4}}{4}+\frac {\left (3 a^{5} c \,d^{2}+15 a^{4} b \,c^{2} d +10 a^{3} b^{2} c^{3}\right ) x^{3}}{3}+\frac {\left (3 a^{5} c^{2} d +5 a^{4} b \,c^{3}\right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(d*x+c)^3,x)

[Out]

1/9*b^5*d^3*x^9+1/8*(5*a*b^4*d^3+3*b^5*c*d^2)*x^8+1/7*(10*a^2*b^3*d^3+15*a*b^4*c*d^2+3*b^5*c^2*d)*x^7+1/6*(10*
a^3*b^2*d^3+30*a^2*b^3*c*d^2+15*a*b^4*c^2*d+b^5*c^3)*x^6+1/5*(5*a^4*b*d^3+30*a^3*b^2*c*d^2+30*a^2*b^3*c^2*d+5*
a*b^4*c^3)*x^5+1/4*(a^5*d^3+15*a^4*b*c*d^2+30*a^3*b^2*c^2*d+10*a^2*b^3*c^3)*x^4+1/3*(3*a^5*c*d^2+15*a^4*b*c^2*
d+10*a^3*b^2*c^3)*x^3+1/2*(3*a^5*c^2*d+5*a^4*b*c^3)*x^2+a^5*c^3*x

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maxima [B]  time = 1.38, size = 277, normalized size = 3.01 \[ \frac {1}{9} \, b^{5} d^{3} x^{9} + a^{5} c^{3} x + \frac {1}{8} \, {\left (3 \, b^{5} c d^{2} + 5 \, a b^{4} d^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, b^{5} c^{2} d + 15 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{7} + \frac {1}{6} \, {\left (b^{5} c^{3} + 15 \, a b^{4} c^{2} d + 30 \, a^{2} b^{3} c d^{2} + 10 \, a^{3} b^{2} d^{3}\right )} x^{6} + {\left (a b^{4} c^{3} + 6 \, a^{2} b^{3} c^{2} d + 6 \, a^{3} b^{2} c d^{2} + a^{4} b d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, a^{2} b^{3} c^{3} + 30 \, a^{3} b^{2} c^{2} d + 15 \, a^{4} b c d^{2} + a^{5} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, a^{3} b^{2} c^{3} + 15 \, a^{4} b c^{2} d + 3 \, a^{5} c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b c^{3} + 3 \, a^{5} c^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(d*x+c)^3,x, algorithm="maxima")

[Out]

1/9*b^5*d^3*x^9 + a^5*c^3*x + 1/8*(3*b^5*c*d^2 + 5*a*b^4*d^3)*x^8 + 1/7*(3*b^5*c^2*d + 15*a*b^4*c*d^2 + 10*a^2
*b^3*d^3)*x^7 + 1/6*(b^5*c^3 + 15*a*b^4*c^2*d + 30*a^2*b^3*c*d^2 + 10*a^3*b^2*d^3)*x^6 + (a*b^4*c^3 + 6*a^2*b^
3*c^2*d + 6*a^3*b^2*c*d^2 + a^4*b*d^3)*x^5 + 1/4*(10*a^2*b^3*c^3 + 30*a^3*b^2*c^2*d + 15*a^4*b*c*d^2 + a^5*d^3
)*x^4 + 1/3*(10*a^3*b^2*c^3 + 15*a^4*b*c^2*d + 3*a^5*c*d^2)*x^3 + 1/2*(5*a^4*b*c^3 + 3*a^5*c^2*d)*x^2

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mupad [B]  time = 0.24, size = 261, normalized size = 2.84 \[ x^5\,\left (a^4\,b\,d^3+6\,a^3\,b^2\,c\,d^2+6\,a^2\,b^3\,c^2\,d+a\,b^4\,c^3\right )+x^4\,\left (\frac {a^5\,d^3}{4}+\frac {15\,a^4\,b\,c\,d^2}{4}+\frac {15\,a^3\,b^2\,c^2\,d}{2}+\frac {5\,a^2\,b^3\,c^3}{2}\right )+x^6\,\left (\frac {5\,a^3\,b^2\,d^3}{3}+5\,a^2\,b^3\,c\,d^2+\frac {5\,a\,b^4\,c^2\,d}{2}+\frac {b^5\,c^3}{6}\right )+a^5\,c^3\,x+\frac {b^5\,d^3\,x^9}{9}+\frac {a^4\,c^2\,x^2\,\left (3\,a\,d+5\,b\,c\right )}{2}+\frac {b^4\,d^2\,x^8\,\left (5\,a\,d+3\,b\,c\right )}{8}+\frac {a^3\,c\,x^3\,\left (3\,a^2\,d^2+15\,a\,b\,c\,d+10\,b^2\,c^2\right )}{3}+\frac {b^3\,d\,x^7\,\left (10\,a^2\,d^2+15\,a\,b\,c\,d+3\,b^2\,c^2\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5*(c + d*x)^3,x)

[Out]

x^5*(a*b^4*c^3 + a^4*b*d^3 + 6*a^2*b^3*c^2*d + 6*a^3*b^2*c*d^2) + x^4*((a^5*d^3)/4 + (5*a^2*b^3*c^3)/2 + (15*a
^3*b^2*c^2*d)/2 + (15*a^4*b*c*d^2)/4) + x^6*((b^5*c^3)/6 + (5*a^3*b^2*d^3)/3 + 5*a^2*b^3*c*d^2 + (5*a*b^4*c^2*
d)/2) + a^5*c^3*x + (b^5*d^3*x^9)/9 + (a^4*c^2*x^2*(3*a*d + 5*b*c))/2 + (b^4*d^2*x^8*(5*a*d + 3*b*c))/8 + (a^3
*c*x^3*(3*a^2*d^2 + 10*b^2*c^2 + 15*a*b*c*d))/3 + (b^3*d*x^7*(10*a^2*d^2 + 3*b^2*c^2 + 15*a*b*c*d))/7

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sympy [B]  time = 0.12, size = 308, normalized size = 3.35 \[ a^{5} c^{3} x + \frac {b^{5} d^{3} x^{9}}{9} + x^{8} \left (\frac {5 a b^{4} d^{3}}{8} + \frac {3 b^{5} c d^{2}}{8}\right ) + x^{7} \left (\frac {10 a^{2} b^{3} d^{3}}{7} + \frac {15 a b^{4} c d^{2}}{7} + \frac {3 b^{5} c^{2} d}{7}\right ) + x^{6} \left (\frac {5 a^{3} b^{2} d^{3}}{3} + 5 a^{2} b^{3} c d^{2} + \frac {5 a b^{4} c^{2} d}{2} + \frac {b^{5} c^{3}}{6}\right ) + x^{5} \left (a^{4} b d^{3} + 6 a^{3} b^{2} c d^{2} + 6 a^{2} b^{3} c^{2} d + a b^{4} c^{3}\right ) + x^{4} \left (\frac {a^{5} d^{3}}{4} + \frac {15 a^{4} b c d^{2}}{4} + \frac {15 a^{3} b^{2} c^{2} d}{2} + \frac {5 a^{2} b^{3} c^{3}}{2}\right ) + x^{3} \left (a^{5} c d^{2} + 5 a^{4} b c^{2} d + \frac {10 a^{3} b^{2} c^{3}}{3}\right ) + x^{2} \left (\frac {3 a^{5} c^{2} d}{2} + \frac {5 a^{4} b c^{3}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(d*x+c)**3,x)

[Out]

a**5*c**3*x + b**5*d**3*x**9/9 + x**8*(5*a*b**4*d**3/8 + 3*b**5*c*d**2/8) + x**7*(10*a**2*b**3*d**3/7 + 15*a*b
**4*c*d**2/7 + 3*b**5*c**2*d/7) + x**6*(5*a**3*b**2*d**3/3 + 5*a**2*b**3*c*d**2 + 5*a*b**4*c**2*d/2 + b**5*c**
3/6) + x**5*(a**4*b*d**3 + 6*a**3*b**2*c*d**2 + 6*a**2*b**3*c**2*d + a*b**4*c**3) + x**4*(a**5*d**3/4 + 15*a**
4*b*c*d**2/4 + 15*a**3*b**2*c**2*d/2 + 5*a**2*b**3*c**3/2) + x**3*(a**5*c*d**2 + 5*a**4*b*c**2*d + 10*a**3*b**
2*c**3/3) + x**2*(3*a**5*c**2*d/2 + 5*a**4*b*c**3/2)

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