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]

((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)

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Rubi [A]  time = 0.157189, antiderivative size = 92, normalized size of antiderivative = 1., 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*}

Mathematica [B]  time = 0.0739038, size = 235, normalized size = 2.55 \[ \frac{1}{504} x \left (84 a^3 b^2 x^2 \left (45 c^2 d x+20 c^3+36 c d^2 x^2+10 d^3 x^3\right )+36 a^2 b^3 x^3 \left (84 c^2 d x+35 c^3+70 c d^2 x^2+20 d^3 x^3\right )+126 a^4 b x \left (20 c^2 d x+10 c^3+15 c d^2 x^2+4 d^3 x^3\right )+126 a^5 \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )+9 a b^4 x^4 \left (140 c^2 d x+56 c^3+120 c d^2 x^2+35 d^3 x^3\right )+b^5 x^5 \left (216 c^2 d x+84 c^3+189 c d^2 x^2+56 d^3 x^3\right )\right ) \]

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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Maple [B]  time = 0.001, size = 281, normalized size = 3.1 \begin{align*}{\frac{{b}^{5}{d}^{3}{x}^{9}}{9}}+{\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}bc{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}}+{a}^{5}{c}^{3}x \end{align*}

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 = 0.947845, size = 374, normalized size = 4.07 \begin{align*} \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} \end{align*}

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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Fricas [B]  time = 1.72465, size = 651, normalized size = 7.08 \begin{align*} \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} \end{align*}

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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Sympy [B]  time = 0.101289, size = 308, normalized size = 3.35 \begin{align*} 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 ) \end{align*}

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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Giac [B]  time = 1.05207, size = 409, normalized size = 4.45 \begin{align*} \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 \end{align*}

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