3.14.10 \(\int (a+b x) (c+d x)^{10} \, dx\) [1310]

Optimal. Leaf size=38 \[ -\frac {(b c-a d) (c+d x)^{11}}{11 d^2}+\frac {b (c+d x)^{12}}{12 d^2} \]

[Out]

-1/11*(-a*d+b*c)*(d*x+c)^11/d^2+1/12*b*(d*x+c)^12/d^2

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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \begin {gather*} \frac {b (c+d x)^{12}}{12 d^2}-\frac {(c+d x)^{11} (b c-a d)}{11 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(c + d*x)^10,x]

[Out]

-1/11*((b*c - a*d)*(c + d*x)^11)/d^2 + (b*(c + d*x)^12)/(12*d^2)

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])

Rubi steps

\begin {align*} \int (a+b x) (c+d x)^{10} \, dx &=\int \left (\frac {(-b c+a d) (c+d x)^{10}}{d}+\frac {b (c+d x)^{11}}{d}\right ) \, dx\\ &=-\frac {(b c-a d) (c+d x)^{11}}{11 d^2}+\frac {b (c+d x)^{12}}{12 d^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(220\) vs. \(2(38)=76\).
time = 0.02, size = 220, normalized size = 5.79 \begin {gather*} a c^{10} x+\frac {1}{2} c^9 (b c+10 a d) x^2+\frac {5}{3} c^8 d (2 b c+9 a d) x^3+\frac {15}{4} c^7 d^2 (3 b c+8 a d) x^4+6 c^6 d^3 (4 b c+7 a d) x^5+7 c^5 d^4 (5 b c+6 a d) x^6+6 c^4 d^5 (6 b c+5 a d) x^7+\frac {15}{4} c^3 d^6 (7 b c+4 a d) x^8+\frac {5}{3} c^2 d^7 (8 b c+3 a d) x^9+\frac {1}{2} c d^8 (9 b c+2 a d) x^{10}+\frac {1}{11} d^9 (10 b c+a d) x^{11}+\frac {1}{12} b d^{10} x^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(c + d*x)^10,x]

[Out]

a*c^10*x + (c^9*(b*c + 10*a*d)*x^2)/2 + (5*c^8*d*(2*b*c + 9*a*d)*x^3)/3 + (15*c^7*d^2*(3*b*c + 8*a*d)*x^4)/4 +
6*c^6*d^3*(4*b*c + 7*a*d)*x^5 + 7*c^5*d^4*(5*b*c + 6*a*d)*x^6 + 6*c^4*d^5*(6*b*c + 5*a*d)*x^7 + (15*c^3*d^6*(
7*b*c + 4*a*d)*x^8)/4 + (5*c^2*d^7*(8*b*c + 3*a*d)*x^9)/3 + (c*d^8*(9*b*c + 2*a*d)*x^10)/2 + (d^9*(10*b*c + a*
d)*x^11)/11 + (b*d^10*x^12)/12

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(201\) vs. \(2(38)=76\).
time = 3.23, size = 199, normalized size = 5.24 \begin {gather*} \frac {x \left (132 a c^{10}+d^9 x^{10} \left (12 a d+120 b c\right )+11 b d^{10} x^{11}+66 c^9 x \left (10 a d+b c\right )+220 c^8 d x^2 \left (9 a d+2 b c\right )+c^7 d^2 x^3 \left (3960 a d+1485 b c\right )+c^6 d^3 x^4 \left (5544 a d+3168 b c\right )+c^5 d^4 x^5 \left (5544 a d+4620 b c\right )+c^4 d^5 x^6 \left (3960 a d+4752 b c\right )+c^3 d^6 x^7 \left (1980 a d+3465 b c\right )+c^2 d^7 x^8 \left (660 a d+1760 b c\right )+66 c d^8 x^9 \left (2 a d+9 b c\right )\right )}{132} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^1*(c + d*x)^10,x]')

[Out]

x (132 a c ^ 10 + d ^ 9 x ^ 10 (12 a d + 120 b c) + 11 b d ^ 10 x ^ 11 + 66 c ^ 9 x (10 a d + b c) + 220 c ^ 8
d x ^ 2 (9 a d + 2 b c) + c ^ 7 d ^ 2 x ^ 3 (3960 a d + 1485 b c) + c ^ 6 d ^ 3 x ^ 4 (5544 a d + 3168 b c) +
c ^ 5 d ^ 4 x ^ 5 (5544 a d + 4620 b c) + c ^ 4 d ^ 5 x ^ 6 (3960 a d + 4752 b c) + c ^ 3 d ^ 6 x ^ 7 (1980 a
d + 3465 b c) + c ^ 2 d ^ 7 x ^ 8 (660 a d + 1760 b c) + 66 c d ^ 8 x ^ 9 (2 a d + 9 b c)) / 132

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(240\) vs. \(2(34)=68\).
time = 0.12, size = 241, normalized size = 6.34

method	result	size
norman	\(\frac {b \,d^{10} x^{12}}{12}+\left (\frac {1}{11} a \,d^{10}+\frac {10}{11} b c \,d^{9}\right ) x^{11}+\left (a c \,d^{9}+\frac {9}{2} b \,c^{2} d^{8}\right ) x^{10}+\left (5 a \,c^{2} d^{8}+\frac {40}{3} b \,c^{3} d^{7}\right ) x^{9}+\left (15 a \,c^{3} d^{7}+\frac {105}{4} b \,c^{4} d^{6}\right ) x^{8}+\left (30 a \,c^{4} d^{6}+36 b \,c^{5} d^{5}\right ) x^{7}+\left (42 a \,c^{5} d^{5}+35 b \,c^{6} d^{4}\right ) x^{6}+\left (42 a \,c^{6} d^{4}+24 b \,c^{7} d^{3}\right ) x^{5}+\left (30 a \,c^{7} d^{3}+\frac {45}{4} b \,c^{8} d^{2}\right ) x^{4}+\left (15 a \,c^{8} d^{2}+\frac {10}{3} b \,c^{9} d \right ) x^{3}+\left (5 a \,c^{9} d +\frac {1}{2} b \,c^{10}\right ) x^{2}+a \,c^{10} x\)	\(232\)
default	\(\frac {b \,d^{10} x^{12}}{12}+\frac {\left (a \,d^{10}+10 b c \,d^{9}\right ) x^{11}}{11}+\frac {\left (10 a c \,d^{9}+45 b \,c^{2} d^{8}\right ) x^{10}}{10}+\frac {\left (45 a \,c^{2} d^{8}+120 b \,c^{3} d^{7}\right ) x^{9}}{9}+\frac {\left (120 a \,c^{3} d^{7}+210 b \,c^{4} d^{6}\right ) x^{8}}{8}+\frac {\left (210 a \,c^{4} d^{6}+252 b \,c^{5} d^{5}\right ) x^{7}}{7}+\frac {\left (252 a \,c^{5} d^{5}+210 b \,c^{6} d^{4}\right ) x^{6}}{6}+\frac {\left (210 a \,c^{6} d^{4}+120 b \,c^{7} d^{3}\right ) x^{5}}{5}+\frac {\left (120 a \,c^{7} d^{3}+45 b \,c^{8} d^{2}\right ) x^{4}}{4}+\frac {\left (45 a \,c^{8} d^{2}+10 b \,c^{9} d \right ) x^{3}}{3}+\frac {\left (10 a \,c^{9} d +b \,c^{10}\right ) x^{2}}{2}+a \,c^{10} x\)	\(241\)
gosper	\(\frac {1}{12} b \,d^{10} x^{12}+\frac {1}{11} x^{11} a \,d^{10}+\frac {10}{11} x^{11} b c \,d^{9}+x^{10} a c \,d^{9}+\frac {9}{2} x^{10} b \,c^{2} d^{8}+5 x^{9} a \,c^{2} d^{8}+\frac {40}{3} x^{9} b \,c^{3} d^{7}+15 x^{8} a \,c^{3} d^{7}+\frac {105}{4} x^{8} b \,c^{4} d^{6}+30 a \,c^{4} d^{6} x^{7}+36 b \,c^{5} d^{5} x^{7}+42 a \,c^{5} d^{5} x^{6}+35 b \,c^{6} d^{4} x^{6}+42 a \,c^{6} d^{4} x^{5}+24 b \,c^{7} d^{3} x^{5}+30 x^{4} a \,c^{7} d^{3}+\frac {45}{4} x^{4} b \,c^{8} d^{2}+15 x^{3} a \,c^{8} d^{2}+\frac {10}{3} x^{3} b \,c^{9} d +5 x^{2} a \,c^{9} d +\frac {1}{2} x^{2} b \,c^{10}+a \,c^{10} x\)	\(242\)
risch	\(\frac {1}{12} b \,d^{10} x^{12}+\frac {1}{11} x^{11} a \,d^{10}+\frac {10}{11} x^{11} b c \,d^{9}+x^{10} a c \,d^{9}+\frac {9}{2} x^{10} b \,c^{2} d^{8}+5 x^{9} a \,c^{2} d^{8}+\frac {40}{3} x^{9} b \,c^{3} d^{7}+15 x^{8} a \,c^{3} d^{7}+\frac {105}{4} x^{8} b \,c^{4} d^{6}+30 a \,c^{4} d^{6} x^{7}+36 b \,c^{5} d^{5} x^{7}+42 a \,c^{5} d^{5} x^{6}+35 b \,c^{6} d^{4} x^{6}+42 a \,c^{6} d^{4} x^{5}+24 b \,c^{7} d^{3} x^{5}+30 x^{4} a \,c^{7} d^{3}+\frac {45}{4} x^{4} b \,c^{8} d^{2}+15 x^{3} a \,c^{8} d^{2}+\frac {10}{3} x^{3} b \,c^{9} d +5 x^{2} a \,c^{9} d +\frac {1}{2} x^{2} b \,c^{10}+a \,c^{10} x\)	\(242\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(d*x+c)^10,x,method=_RETURNVERBOSE)

[Out]

1/12*b*d^10*x^12+1/11*(a*d^10+10*b*c*d^9)*x^11+1/10*(10*a*c*d^9+45*b*c^2*d^8)*x^10+1/9*(45*a*c^2*d^8+120*b*c^3
*d^7)*x^9+1/8*(120*a*c^3*d^7+210*b*c^4*d^6)*x^8+1/7*(210*a*c^4*d^6+252*b*c^5*d^5)*x^7+1/6*(252*a*c^5*d^5+210*b
*c^6*d^4)*x^6+1/5*(210*a*c^6*d^4+120*b*c^7*d^3)*x^5+1/4*(120*a*c^7*d^3+45*b*c^8*d^2)*x^4+1/3*(45*a*c^8*d^2+10*
b*c^9*d)*x^3+1/2*(10*a*c^9*d+b*c^10)*x^2+a*c^10*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (34) = 68\).
time = 0.28, size = 240, normalized size = 6.32 \begin {gather*} \frac {1}{12} \, b d^{10} x^{12} + a c^{10} x + \frac {1}{11} \, {\left (10 \, b c d^{9} + a d^{10}\right )} x^{11} + \frac {1}{2} \, {\left (9 \, b c^{2} d^{8} + 2 \, a c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (8 \, b c^{3} d^{7} + 3 \, a c^{2} d^{8}\right )} x^{9} + \frac {15}{4} \, {\left (7 \, b c^{4} d^{6} + 4 \, a c^{3} d^{7}\right )} x^{8} + 6 \, {\left (6 \, b c^{5} d^{5} + 5 \, a c^{4} d^{6}\right )} x^{7} + 7 \, {\left (5 \, b c^{6} d^{4} + 6 \, a c^{5} d^{5}\right )} x^{6} + 6 \, {\left (4 \, b c^{7} d^{3} + 7 \, a c^{6} d^{4}\right )} x^{5} + \frac {15}{4} \, {\left (3 \, b c^{8} d^{2} + 8 \, a c^{7} d^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, b c^{9} d + 9 \, a c^{8} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{10} + 10 \, a c^{9} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^10,x, algorithm="maxima")

[Out]

1/12*b*d^10*x^12 + a*c^10*x + 1/11*(10*b*c*d^9 + a*d^10)*x^11 + 1/2*(9*b*c^2*d^8 + 2*a*c*d^9)*x^10 + 5/3*(8*b*
c^3*d^7 + 3*a*c^2*d^8)*x^9 + 15/4*(7*b*c^4*d^6 + 4*a*c^3*d^7)*x^8 + 6*(6*b*c^5*d^5 + 5*a*c^4*d^6)*x^7 + 7*(5*b
*c^6*d^4 + 6*a*c^5*d^5)*x^6 + 6*(4*b*c^7*d^3 + 7*a*c^6*d^4)*x^5 + 15/4*(3*b*c^8*d^2 + 8*a*c^7*d^3)*x^4 + 5/3*(
2*b*c^9*d + 9*a*c^8*d^2)*x^3 + 1/2*(b*c^10 + 10*a*c^9*d)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (34) = 68\).
time = 0.29, size = 240, normalized size = 6.32 \begin {gather*} \frac {1}{12} \, b d^{10} x^{12} + a c^{10} x + \frac {1}{11} \, {\left (10 \, b c d^{9} + a d^{10}\right )} x^{11} + \frac {1}{2} \, {\left (9 \, b c^{2} d^{8} + 2 \, a c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (8 \, b c^{3} d^{7} + 3 \, a c^{2} d^{8}\right )} x^{9} + \frac {15}{4} \, {\left (7 \, b c^{4} d^{6} + 4 \, a c^{3} d^{7}\right )} x^{8} + 6 \, {\left (6 \, b c^{5} d^{5} + 5 \, a c^{4} d^{6}\right )} x^{7} + 7 \, {\left (5 \, b c^{6} d^{4} + 6 \, a c^{5} d^{5}\right )} x^{6} + 6 \, {\left (4 \, b c^{7} d^{3} + 7 \, a c^{6} d^{4}\right )} x^{5} + \frac {15}{4} \, {\left (3 \, b c^{8} d^{2} + 8 \, a c^{7} d^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, b c^{9} d + 9 \, a c^{8} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{10} + 10 \, a c^{9} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^10,x, algorithm="fricas")

[Out]

1/12*b*d^10*x^12 + a*c^10*x + 1/11*(10*b*c*d^9 + a*d^10)*x^11 + 1/2*(9*b*c^2*d^8 + 2*a*c*d^9)*x^10 + 5/3*(8*b*
c^3*d^7 + 3*a*c^2*d^8)*x^9 + 15/4*(7*b*c^4*d^6 + 4*a*c^3*d^7)*x^8 + 6*(6*b*c^5*d^5 + 5*a*c^4*d^6)*x^7 + 7*(5*b
*c^6*d^4 + 6*a*c^5*d^5)*x^6 + 6*(4*b*c^7*d^3 + 7*a*c^6*d^4)*x^5 + 15/4*(3*b*c^8*d^2 + 8*a*c^7*d^3)*x^4 + 5/3*(
2*b*c^9*d + 9*a*c^8*d^2)*x^3 + 1/2*(b*c^10 + 10*a*c^9*d)*x^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (32) = 64\).
time = 0.06, size = 248, normalized size = 6.53 \begin {gather*} a c^{10} x + \frac {b d^{10} x^{12}}{12} + x^{11} \left (\frac {a d^{10}}{11} + \frac {10 b c d^{9}}{11}\right ) + x^{10} \left (a c d^{9} + \frac {9 b c^{2} d^{8}}{2}\right ) + x^{9} \cdot \left (5 a c^{2} d^{8} + \frac {40 b c^{3} d^{7}}{3}\right ) + x^{8} \cdot \left (15 a c^{3} d^{7} + \frac {105 b c^{4} d^{6}}{4}\right ) + x^{7} \cdot \left (30 a c^{4} d^{6} + 36 b c^{5} d^{5}\right ) + x^{6} \cdot \left (42 a c^{5} d^{5} + 35 b c^{6} d^{4}\right ) + x^{5} \cdot \left (42 a c^{6} d^{4} + 24 b c^{7} d^{3}\right ) + x^{4} \cdot \left (30 a c^{7} d^{3} + \frac {45 b c^{8} d^{2}}{4}\right ) + x^{3} \cdot \left (15 a c^{8} d^{2} + \frac {10 b c^{9} d}{3}\right ) + x^{2} \cdot \left (5 a c^{9} d + \frac {b c^{10}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)**10,x)

[Out]

a*c**10*x + b*d**10*x**12/12 + x**11*(a*d**10/11 + 10*b*c*d**9/11) + x**10*(a*c*d**9 + 9*b*c**2*d**8/2) + x**9
*(5*a*c**2*d**8 + 40*b*c**3*d**7/3) + x**8*(15*a*c**3*d**7 + 105*b*c**4*d**6/4) + x**7*(30*a*c**4*d**6 + 36*b*
c**5*d**5) + x**6*(42*a*c**5*d**5 + 35*b*c**6*d**4) + x**5*(42*a*c**6*d**4 + 24*b*c**7*d**3) + x**4*(30*a*c**7
*d**3 + 45*b*c**8*d**2/4) + x**3*(15*a*c**8*d**2 + 10*b*c**9*d/3) + x**2*(5*a*c**9*d + b*c**10/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (34) = 68\).
time = 0.00, size = 259, normalized size = 6.82 \begin {gather*} \frac {1}{12} x^{12} b d^{10}+\frac {10}{11} x^{11} b d^{9} c+\frac {1}{11} x^{11} a d^{10}+\frac {9}{2} x^{10} b d^{8} c^{2}+x^{10} a d^{9} c+\frac {40}{3} x^{9} b d^{7} c^{3}+5 x^{9} a d^{8} c^{2}+\frac {105}{4} x^{8} b d^{6} c^{4}+15 x^{8} a d^{7} c^{3}+36 x^{7} b d^{5} c^{5}+30 x^{7} a d^{6} c^{4}+35 x^{6} b d^{4} c^{6}+42 x^{6} a d^{5} c^{5}+24 x^{5} b d^{3} c^{7}+42 x^{5} a d^{4} c^{6}+\frac {45}{4} x^{4} b d^{2} c^{8}+30 x^{4} a d^{3} c^{7}+\frac {10}{3} x^{3} b d c^{9}+15 x^{3} a d^{2} c^{8}+\frac {1}{2} x^{2} b c^{10}+5 x^{2} a d c^{9}+x a c^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^10,x)

[Out]

1/12*b*d^10*x^12 + 10/11*b*c*d^9*x^11 + 1/11*a*d^10*x^11 + 9/2*b*c^2*d^8*x^10 + a*c*d^9*x^10 + 40/3*b*c^3*d^7*
x^9 + 5*a*c^2*d^8*x^9 + 105/4*b*c^4*d^6*x^8 + 15*a*c^3*d^7*x^8 + 36*b*c^5*d^5*x^7 + 30*a*c^4*d^6*x^7 + 35*b*c^
6*d^4*x^6 + 42*a*c^5*d^5*x^6 + 24*b*c^7*d^3*x^5 + 42*a*c^6*d^4*x^5 + 45/4*b*c^8*d^2*x^4 + 30*a*c^7*d^3*x^4 + 1
0/3*b*c^9*d*x^3 + 15*a*c^8*d^2*x^3 + 1/2*b*c^10*x^2 + 5*a*c^9*d*x^2 + a*c^10*x

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Mupad [B]
time = 0.13, size = 208, normalized size = 5.47 \begin {gather*} x^2\,\left (\frac {b\,c^{10}}{2}+5\,a\,d\,c^9\right )+x^{11}\,\left (\frac {a\,d^{10}}{11}+\frac {10\,b\,c\,d^9}{11}\right )+\frac {b\,d^{10}\,x^{12}}{12}+a\,c^{10}\,x+\frac {5\,c^8\,d\,x^3\,\left (9\,a\,d+2\,b\,c\right )}{3}+\frac {c\,d^8\,x^{10}\,\left (2\,a\,d+9\,b\,c\right )}{2}+\frac {15\,c^7\,d^2\,x^4\,\left (8\,a\,d+3\,b\,c\right )}{4}+6\,c^6\,d^3\,x^5\,\left (7\,a\,d+4\,b\,c\right )+7\,c^5\,d^4\,x^6\,\left (6\,a\,d+5\,b\,c\right )+6\,c^4\,d^5\,x^7\,\left (5\,a\,d+6\,b\,c\right )+\frac {15\,c^3\,d^6\,x^8\,\left (4\,a\,d+7\,b\,c\right )}{4}+\frac {5\,c^2\,d^7\,x^9\,\left (3\,a\,d+8\,b\,c\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(c + d*x)^10,x)

[Out]

x^2*((b*c^10)/2 + 5*a*c^9*d) + x^11*((a*d^10)/11 + (10*b*c*d^9)/11) + (b*d^10*x^12)/12 + a*c^10*x + (5*c^8*d*x
^3*(9*a*d + 2*b*c))/3 + (c*d^8*x^10*(2*a*d + 9*b*c))/2 + (15*c^7*d^2*x^4*(8*a*d + 3*b*c))/4 + 6*c^6*d^3*x^5*(7
*a*d + 4*b*c) + 7*c^5*d^4*x^6*(6*a*d + 5*b*c) + 6*c^4*d^5*x^7*(5*a*d + 6*b*c) + (15*c^3*d^6*x^8*(4*a*d + 7*b*c
))/4 + (5*c^2*d^7*x^9*(3*a*d + 8*b*c))/3

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