3.1280 \(\int (a+b x)^2 (c+d x)^7 \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 b (c+d x)^9 (b c-a d)}{9 d^3}+\frac{(c+d x)^8 (b c-a d)^2}{8 d^3}+\frac{b^2 (c+d x)^{10}}{10 d^3} \]

[Out]

((b*c - a*d)^2*(c + d*x)^8)/(8*d^3) - (2*b*(b*c - a*d)*(c + d*x)^9)/(9*d^3) + (b^2*(c + d*x)^10)/(10*d^3)

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Rubi [A]  time = 0.159342, antiderivative size = 65, 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} \[ -\frac{2 b (c+d x)^9 (b c-a d)}{9 d^3}+\frac{(c+d x)^8 (b c-a d)^2}{8 d^3}+\frac{b^2 (c+d x)^{10}}{10 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(c + d*x)^7,x]

[Out]

((b*c - a*d)^2*(c + d*x)^8)/(8*d^3) - (2*b*(b*c - a*d)*(c + d*x)^9)/(9*d^3) + (b^2*(c + d*x)^10)/(10*d^3)

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)^2 (c+d x)^7 \, dx &=\int \left (\frac{(-b c+a d)^2 (c+d x)^7}{d^2}-\frac{2 b (b c-a d) (c+d x)^8}{d^2}+\frac{b^2 (c+d x)^9}{d^2}\right ) \, dx\\ &=\frac{(b c-a d)^2 (c+d x)^8}{8 d^3}-\frac{2 b (b c-a d) (c+d x)^9}{9 d^3}+\frac{b^2 (c+d x)^{10}}{10 d^3}\\ \end{align*}

Mathematica [B]  time = 0.0270711, size = 261, normalized size = 4.02 \[ \frac{1}{8} d^5 x^8 \left (a^2 d^2+14 a b c d+21 b^2 c^2\right )+c d^4 x^7 \left (a^2 d^2+6 a b c d+5 b^2 c^2\right )+\frac{7}{6} c^2 d^3 x^6 \left (3 a^2 d^2+10 a b c d+5 b^2 c^2\right )+\frac{7}{5} c^3 d^2 x^5 \left (5 a^2 d^2+10 a b c d+3 b^2 c^2\right )+\frac{7}{4} c^4 d x^4 \left (5 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{3} c^5 x^3 \left (21 a^2 d^2+14 a b c d+b^2 c^2\right )+a^2 c^7 x+\frac{1}{2} a c^6 x^2 (7 a d+2 b c)+\frac{1}{9} b d^6 x^9 (2 a d+7 b c)+\frac{1}{10} b^2 d^7 x^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2*(c + d*x)^7,x]

[Out]

a^2*c^7*x + (a*c^6*(2*b*c + 7*a*d)*x^2)/2 + (c^5*(b^2*c^2 + 14*a*b*c*d + 21*a^2*d^2)*x^3)/3 + (7*c^4*d*(b^2*c^
2 + 6*a*b*c*d + 5*a^2*d^2)*x^4)/4 + (7*c^3*d^2*(3*b^2*c^2 + 10*a*b*c*d + 5*a^2*d^2)*x^5)/5 + (7*c^2*d^3*(5*b^2
*c^2 + 10*a*b*c*d + 3*a^2*d^2)*x^6)/6 + c*d^4*(5*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^7 + (d^5*(21*b^2*c^2 + 14*a*
b*c*d + a^2*d^2)*x^8)/8 + (b*d^6*(7*b*c + 2*a*d)*x^9)/9 + (b^2*d^7*x^10)/10

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Maple [B]  time = 0.002, size = 277, normalized size = 4.3 \begin{align*}{\frac{{b}^{2}{d}^{7}{x}^{10}}{10}}+{\frac{ \left ( 2\,ab{d}^{7}+7\,{b}^{2}c{d}^{6} \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{2}{d}^{7}+14\,abc{d}^{6}+21\,{b}^{2}{c}^{2}{d}^{5} \right ){x}^{8}}{8}}+{\frac{ \left ( 7\,{a}^{2}c{d}^{6}+42\,ab{c}^{2}{d}^{5}+35\,{b}^{2}{c}^{3}{d}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( 21\,{a}^{2}{c}^{2}{d}^{5}+70\,ab{c}^{3}{d}^{4}+35\,{b}^{2}{c}^{4}{d}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 35\,{a}^{2}{c}^{3}{d}^{4}+70\,ab{c}^{4}{d}^{3}+21\,{b}^{2}{c}^{5}{d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 35\,{a}^{2}{c}^{4}{d}^{3}+42\,ab{c}^{5}{d}^{2}+7\,{b}^{2}{c}^{6}d \right ){x}^{4}}{4}}+{\frac{ \left ( 21\,{a}^{2}{c}^{5}{d}^{2}+14\,ab{c}^{6}d+{b}^{2}{c}^{7} \right ){x}^{3}}{3}}+{\frac{ \left ( 7\,{a}^{2}{c}^{6}d+2\,ab{c}^{7} \right ){x}^{2}}{2}}+{a}^{2}{c}^{7}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(d*x+c)^7,x)

[Out]

1/10*b^2*d^7*x^10+1/9*(2*a*b*d^7+7*b^2*c*d^6)*x^9+1/8*(a^2*d^7+14*a*b*c*d^6+21*b^2*c^2*d^5)*x^8+1/7*(7*a^2*c*d
^6+42*a*b*c^2*d^5+35*b^2*c^3*d^4)*x^7+1/6*(21*a^2*c^2*d^5+70*a*b*c^3*d^4+35*b^2*c^4*d^3)*x^6+1/5*(35*a^2*c^3*d
^4+70*a*b*c^4*d^3+21*b^2*c^5*d^2)*x^5+1/4*(35*a^2*c^4*d^3+42*a*b*c^5*d^2+7*b^2*c^6*d)*x^4+1/3*(21*a^2*c^5*d^2+
14*a*b*c^6*d+b^2*c^7)*x^3+1/2*(7*a^2*c^6*d+2*a*b*c^7)*x^2+a^2*c^7*x

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Maxima [B]  time = 0.986425, size = 369, normalized size = 5.68 \begin{align*} \frac{1}{10} \, b^{2} d^{7} x^{10} + a^{2} c^{7} x + \frac{1}{9} \,{\left (7 \, b^{2} c d^{6} + 2 \, a b d^{7}\right )} x^{9} + \frac{1}{8} \,{\left (21 \, b^{2} c^{2} d^{5} + 14 \, a b c d^{6} + a^{2} d^{7}\right )} x^{8} +{\left (5 \, b^{2} c^{3} d^{4} + 6 \, a b c^{2} d^{5} + a^{2} c d^{6}\right )} x^{7} + \frac{7}{6} \,{\left (5 \, b^{2} c^{4} d^{3} + 10 \, a b c^{3} d^{4} + 3 \, a^{2} c^{2} d^{5}\right )} x^{6} + \frac{7}{5} \,{\left (3 \, b^{2} c^{5} d^{2} + 10 \, a b c^{4} d^{3} + 5 \, a^{2} c^{3} d^{4}\right )} x^{5} + \frac{7}{4} \,{\left (b^{2} c^{6} d + 6 \, a b c^{5} d^{2} + 5 \, a^{2} c^{4} d^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} c^{7} + 14 \, a b c^{6} d + 21 \, a^{2} c^{5} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b c^{7} + 7 \, a^{2} c^{6} d\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^7,x, algorithm="maxima")

[Out]

1/10*b^2*d^7*x^10 + a^2*c^7*x + 1/9*(7*b^2*c*d^6 + 2*a*b*d^7)*x^9 + 1/8*(21*b^2*c^2*d^5 + 14*a*b*c*d^6 + a^2*d
^7)*x^8 + (5*b^2*c^3*d^4 + 6*a*b*c^2*d^5 + a^2*c*d^6)*x^7 + 7/6*(5*b^2*c^4*d^3 + 10*a*b*c^3*d^4 + 3*a^2*c^2*d^
5)*x^6 + 7/5*(3*b^2*c^5*d^2 + 10*a*b*c^4*d^3 + 5*a^2*c^3*d^4)*x^5 + 7/4*(b^2*c^6*d + 6*a*b*c^5*d^2 + 5*a^2*c^4
*d^3)*x^4 + 1/3*(b^2*c^7 + 14*a*b*c^6*d + 21*a^2*c^5*d^2)*x^3 + 1/2*(2*a*b*c^7 + 7*a^2*c^6*d)*x^2

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Fricas [B]  time = 1.94814, size = 644, normalized size = 9.91 \begin{align*} \frac{1}{10} x^{10} d^{7} b^{2} + \frac{7}{9} x^{9} d^{6} c b^{2} + \frac{2}{9} x^{9} d^{7} b a + \frac{21}{8} x^{8} d^{5} c^{2} b^{2} + \frac{7}{4} x^{8} d^{6} c b a + \frac{1}{8} x^{8} d^{7} a^{2} + 5 x^{7} d^{4} c^{3} b^{2} + 6 x^{7} d^{5} c^{2} b a + x^{7} d^{6} c a^{2} + \frac{35}{6} x^{6} d^{3} c^{4} b^{2} + \frac{35}{3} x^{6} d^{4} c^{3} b a + \frac{7}{2} x^{6} d^{5} c^{2} a^{2} + \frac{21}{5} x^{5} d^{2} c^{5} b^{2} + 14 x^{5} d^{3} c^{4} b a + 7 x^{5} d^{4} c^{3} a^{2} + \frac{7}{4} x^{4} d c^{6} b^{2} + \frac{21}{2} x^{4} d^{2} c^{5} b a + \frac{35}{4} x^{4} d^{3} c^{4} a^{2} + \frac{1}{3} x^{3} c^{7} b^{2} + \frac{14}{3} x^{3} d c^{6} b a + 7 x^{3} d^{2} c^{5} a^{2} + x^{2} c^{7} b a + \frac{7}{2} x^{2} d c^{6} a^{2} + x c^{7} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^7,x, algorithm="fricas")

[Out]

1/10*x^10*d^7*b^2 + 7/9*x^9*d^6*c*b^2 + 2/9*x^9*d^7*b*a + 21/8*x^8*d^5*c^2*b^2 + 7/4*x^8*d^6*c*b*a + 1/8*x^8*d
^7*a^2 + 5*x^7*d^4*c^3*b^2 + 6*x^7*d^5*c^2*b*a + x^7*d^6*c*a^2 + 35/6*x^6*d^3*c^4*b^2 + 35/3*x^6*d^4*c^3*b*a +
7/2*x^6*d^5*c^2*a^2 + 21/5*x^5*d^2*c^5*b^2 + 14*x^5*d^3*c^4*b*a + 7*x^5*d^4*c^3*a^2 + 7/4*x^4*d*c^6*b^2 + 21/
2*x^4*d^2*c^5*b*a + 35/4*x^4*d^3*c^4*a^2 + 1/3*x^3*c^7*b^2 + 14/3*x^3*d*c^6*b*a + 7*x^3*d^2*c^5*a^2 + x^2*c^7*
b*a + 7/2*x^2*d*c^6*a^2 + x*c^7*a^2

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Sympy [B]  time = 0.105788, size = 303, normalized size = 4.66 \begin{align*} a^{2} c^{7} x + \frac{b^{2} d^{7} x^{10}}{10} + x^{9} \left (\frac{2 a b d^{7}}{9} + \frac{7 b^{2} c d^{6}}{9}\right ) + x^{8} \left (\frac{a^{2} d^{7}}{8} + \frac{7 a b c d^{6}}{4} + \frac{21 b^{2} c^{2} d^{5}}{8}\right ) + x^{7} \left (a^{2} c d^{6} + 6 a b c^{2} d^{5} + 5 b^{2} c^{3} d^{4}\right ) + x^{6} \left (\frac{7 a^{2} c^{2} d^{5}}{2} + \frac{35 a b c^{3} d^{4}}{3} + \frac{35 b^{2} c^{4} d^{3}}{6}\right ) + x^{5} \left (7 a^{2} c^{3} d^{4} + 14 a b c^{4} d^{3} + \frac{21 b^{2} c^{5} d^{2}}{5}\right ) + x^{4} \left (\frac{35 a^{2} c^{4} d^{3}}{4} + \frac{21 a b c^{5} d^{2}}{2} + \frac{7 b^{2} c^{6} d}{4}\right ) + x^{3} \left (7 a^{2} c^{5} d^{2} + \frac{14 a b c^{6} d}{3} + \frac{b^{2} c^{7}}{3}\right ) + x^{2} \left (\frac{7 a^{2} c^{6} d}{2} + a b c^{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(d*x+c)**7,x)

[Out]

a**2*c**7*x + b**2*d**7*x**10/10 + x**9*(2*a*b*d**7/9 + 7*b**2*c*d**6/9) + x**8*(a**2*d**7/8 + 7*a*b*c*d**6/4
+ 21*b**2*c**2*d**5/8) + x**7*(a**2*c*d**6 + 6*a*b*c**2*d**5 + 5*b**2*c**3*d**4) + x**6*(7*a**2*c**2*d**5/2 +
35*a*b*c**3*d**4/3 + 35*b**2*c**4*d**3/6) + x**5*(7*a**2*c**3*d**4 + 14*a*b*c**4*d**3 + 21*b**2*c**5*d**2/5) +
x**4*(35*a**2*c**4*d**3/4 + 21*a*b*c**5*d**2/2 + 7*b**2*c**6*d/4) + x**3*(7*a**2*c**5*d**2 + 14*a*b*c**6*d/3
+ b**2*c**7/3) + x**2*(7*a**2*c**6*d/2 + a*b*c**7)

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Giac [B]  time = 1.04615, size = 397, normalized size = 6.11 \begin{align*} \frac{1}{10} \, b^{2} d^{7} x^{10} + \frac{7}{9} \, b^{2} c d^{6} x^{9} + \frac{2}{9} \, a b d^{7} x^{9} + \frac{21}{8} \, b^{2} c^{2} d^{5} x^{8} + \frac{7}{4} \, a b c d^{6} x^{8} + \frac{1}{8} \, a^{2} d^{7} x^{8} + 5 \, b^{2} c^{3} d^{4} x^{7} + 6 \, a b c^{2} d^{5} x^{7} + a^{2} c d^{6} x^{7} + \frac{35}{6} \, b^{2} c^{4} d^{3} x^{6} + \frac{35}{3} \, a b c^{3} d^{4} x^{6} + \frac{7}{2} \, a^{2} c^{2} d^{5} x^{6} + \frac{21}{5} \, b^{2} c^{5} d^{2} x^{5} + 14 \, a b c^{4} d^{3} x^{5} + 7 \, a^{2} c^{3} d^{4} x^{5} + \frac{7}{4} \, b^{2} c^{6} d x^{4} + \frac{21}{2} \, a b c^{5} d^{2} x^{4} + \frac{35}{4} \, a^{2} c^{4} d^{3} x^{4} + \frac{1}{3} \, b^{2} c^{7} x^{3} + \frac{14}{3} \, a b c^{6} d x^{3} + 7 \, a^{2} c^{5} d^{2} x^{3} + a b c^{7} x^{2} + \frac{7}{2} \, a^{2} c^{6} d x^{2} + a^{2} c^{7} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^7,x, algorithm="giac")

[Out]

1/10*b^2*d^7*x^10 + 7/9*b^2*c*d^6*x^9 + 2/9*a*b*d^7*x^9 + 21/8*b^2*c^2*d^5*x^8 + 7/4*a*b*c*d^6*x^8 + 1/8*a^2*d
^7*x^8 + 5*b^2*c^3*d^4*x^7 + 6*a*b*c^2*d^5*x^7 + a^2*c*d^6*x^7 + 35/6*b^2*c^4*d^3*x^6 + 35/3*a*b*c^3*d^4*x^6 +
7/2*a^2*c^2*d^5*x^6 + 21/5*b^2*c^5*d^2*x^5 + 14*a*b*c^4*d^3*x^5 + 7*a^2*c^3*d^4*x^5 + 7/4*b^2*c^6*d*x^4 + 21/
2*a*b*c^5*d^2*x^4 + 35/4*a^2*c^4*d^3*x^4 + 1/3*b^2*c^7*x^3 + 14/3*a*b*c^6*d*x^3 + 7*a^2*c^5*d^2*x^3 + a*b*c^7*
x^2 + 7/2*a^2*c^6*d*x^2 + a^2*c^7*x