3.14.8 \(\int (a+b x)^3 (c+d x)^{10} \, dx\) [1308]

Optimal. Leaf size=92 \[ -\frac {(b c-a d)^3 (c+d x)^{11}}{11 d^4}+\frac {b (b c-a d)^2 (c+d x)^{12}}{4 d^4}-\frac {3 b^2 (b c-a d) (c+d x)^{13}}{13 d^4}+\frac {b^3 (c+d x)^{14}}{14 d^4} \]

[Out]

-1/11*(-a*d+b*c)^3*(d*x+c)^11/d^4+1/4*b*(-a*d+b*c)^2*(d*x+c)^12/d^4-3/13*b^2*(-a*d+b*c)*(d*x+c)^13/d^4+1/14*b^
3*(d*x+c)^14/d^4

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Rubi [A]
time = 0.25, 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 = {45} \begin {gather*} -\frac {3 b^2 (c+d x)^{13} (b c-a d)}{13 d^4}+\frac {b (c+d x)^{12} (b c-a d)^2}{4 d^4}-\frac {(c+d x)^{11} (b c-a d)^3}{11 d^4}+\frac {b^3 (c+d x)^{14}}{14 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(c + d*x)^10,x]

[Out]

-1/11*((b*c - a*d)^3*(c + d*x)^11)/d^4 + (b*(b*c - a*d)^2*(c + d*x)^12)/(4*d^4) - (3*b^2*(b*c - a*d)*(c + d*x)
^13)/(13*d^4) + (b^3*(c + d*x)^14)/(14*d^4)

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)^3 (c+d x)^{10} \, dx &=\int \left (\frac {(-b c+a d)^3 (c+d x)^{10}}{d^3}+\frac {3 b (b c-a d)^2 (c+d x)^{11}}{d^3}-\frac {3 b^2 (b c-a d) (c+d x)^{12}}{d^3}+\frac {b^3 (c+d x)^{13}}{d^3}\right ) \, dx\\ &=-\frac {(b c-a d)^3 (c+d x)^{11}}{11 d^4}+\frac {b (b c-a d)^2 (c+d x)^{12}}{4 d^4}-\frac {3 b^2 (b c-a d) (c+d x)^{13}}{13 d^4}+\frac {b^3 (c+d x)^{14}}{14 d^4}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(511\) vs. \(2(92)=184\).
time = 0.04, size = 511, normalized size = 5.55 \begin {gather*} a^3 c^{10} x+\frac {1}{2} a^2 c^9 (3 b c+10 a d) x^2+a c^8 \left (b^2 c^2+10 a b c d+15 a^2 d^2\right ) x^3+\frac {1}{4} c^7 \left (b^3 c^3+30 a b^2 c^2 d+135 a^2 b c d^2+120 a^3 d^3\right ) x^4+c^6 d \left (2 b^3 c^3+27 a b^2 c^2 d+72 a^2 b c d^2+42 a^3 d^3\right ) x^5+\frac {3}{2} c^5 d^2 \left (5 b^3 c^3+40 a b^2 c^2 d+70 a^2 b c d^2+28 a^3 d^3\right ) x^6+\frac {6}{7} c^4 d^3 \left (20 b^3 c^3+105 a b^2 c^2 d+126 a^2 b c d^2+35 a^3 d^3\right ) x^7+\frac {3}{4} c^3 d^4 \left (35 b^3 c^3+126 a b^2 c^2 d+105 a^2 b c d^2+20 a^3 d^3\right ) x^8+c^2 d^5 \left (28 b^3 c^3+70 a b^2 c^2 d+40 a^2 b c d^2+5 a^3 d^3\right ) x^9+\frac {1}{2} c d^6 \left (42 b^3 c^3+72 a b^2 c^2 d+27 a^2 b c d^2+2 a^3 d^3\right ) x^{10}+\frac {1}{11} d^7 \left (120 b^3 c^3+135 a b^2 c^2 d+30 a^2 b c d^2+a^3 d^3\right ) x^{11}+\frac {1}{4} b d^8 \left (15 b^2 c^2+10 a b c d+a^2 d^2\right ) x^{12}+\frac {1}{13} b^2 d^9 (10 b c+3 a d) x^{13}+\frac {1}{14} b^3 d^{10} x^{14} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(c + d*x)^10,x]

[Out]

a^3*c^10*x + (a^2*c^9*(3*b*c + 10*a*d)*x^2)/2 + a*c^8*(b^2*c^2 + 10*a*b*c*d + 15*a^2*d^2)*x^3 + (c^7*(b^3*c^3
+ 30*a*b^2*c^2*d + 135*a^2*b*c*d^2 + 120*a^3*d^3)*x^4)/4 + c^6*d*(2*b^3*c^3 + 27*a*b^2*c^2*d + 72*a^2*b*c*d^2
+ 42*a^3*d^3)*x^5 + (3*c^5*d^2*(5*b^3*c^3 + 40*a*b^2*c^2*d + 70*a^2*b*c*d^2 + 28*a^3*d^3)*x^6)/2 + (6*c^4*d^3*
(20*b^3*c^3 + 105*a*b^2*c^2*d + 126*a^2*b*c*d^2 + 35*a^3*d^3)*x^7)/7 + (3*c^3*d^4*(35*b^3*c^3 + 126*a*b^2*c^2*
d + 105*a^2*b*c*d^2 + 20*a^3*d^3)*x^8)/4 + c^2*d^5*(28*b^3*c^3 + 70*a*b^2*c^2*d + 40*a^2*b*c*d^2 + 5*a^3*d^3)*
x^9 + (c*d^6*(42*b^3*c^3 + 72*a*b^2*c^2*d + 27*a^2*b*c*d^2 + 2*a^3*d^3)*x^10)/2 + (d^7*(120*b^3*c^3 + 135*a*b^
2*c^2*d + 30*a^2*b*c*d^2 + a^3*d^3)*x^11)/11 + (b*d^8*(15*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*x^12)/4 + (b^2*d^9*(
10*b*c + 3*a*d)*x^13)/13 + (b^3*d^10*x^14)/14

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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(510\) vs. \(2(92)=184\).
time = 5.60, size = 490, normalized size = 5.33 \begin {gather*} x \left (a^3 c^{10}+\frac {a^2 c^9 x \left (10 a d+3 b c\right )}{2}+a c^8 x^2 \left (15 a^2 d^2+10 a b c d+b^2 c^2\right )+\frac {b d^8 x^{11} \left (a^2 d^2+10 a b c d+15 b^2 c^2\right )}{4}+\frac {b^2 d^9 x^{12} \left (3 a d+10 b c\right )}{13}+\frac {c^7 x^3 \left (120 a^3 d^3+135 a^2 b c d^2+30 a b^2 c^2 d+b^3 c^3\right )}{4}+\frac {6 c^4 d^3 x^6 \left (35 a^3 d^3+126 a^2 b c d^2+105 a b^2 c^2 d+20 b^3 c^3\right )}{7}+\frac {3 c^3 d^4 x^7 \left (20 a^3 d^3+105 a^2 b c d^2+126 a b^2 c^2 d+35 b^3 c^3\right )}{4}+c^2 d^5 x^8 \left (5 a^3 d^3+40 a^2 b c d^2+70 a b^2 c^2 d+28 b^3 c^3\right )+\frac {d^7 x^{10} \left (a^3 d^3+30 a^2 b c d^2+135 a b^2 c^2 d+120 b^3 c^3\right )}{11}+\frac {b^3 d^{10} x^{13}}{14}+c^6 d x^4 \left (42 a^3 d^3+72 a^2 b c d^2+27 a b^2 c^2 d+2 b^3 c^3\right )+\frac {3 c^5 d^2 x^5 \left (28 a^3 d^3+70 a^2 b c d^2+40 a b^2 c^2 d+5 b^3 c^3\right )}{2}+\frac {c d^6 x^9 \left (2 a^3 d^3+27 a^2 b c d^2+72 a b^2 c^2 d+42 b^3 c^3\right )}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^3*(c + d*x)^10,x]')

[Out]

x (a ^ 3 c ^ 10 + a ^ 2 c ^ 9 x (10 a d + 3 b c) / 2 + a c ^ 8 x ^ 2 (15 a ^ 2 d ^ 2 + 10 a b c d + b ^ 2 c ^
2) + b d ^ 8 x ^ 11 (a ^ 2 d ^ 2 + 10 a b c d + 15 b ^ 2 c ^ 2) / 4 + b ^ 2 d ^ 9 x ^ 12 (3 a d + 10 b c) / 13
+ c ^ 7 x ^ 3 (120 a ^ 3 d ^ 3 + 135 a ^ 2 b c d ^ 2 + 30 a b ^ 2 c ^ 2 d + b ^ 3 c ^ 3) / 4 + 6 c ^ 4 d ^ 3
x ^ 6 (35 a ^ 3 d ^ 3 + 126 a ^ 2 b c d ^ 2 + 105 a b ^ 2 c ^ 2 d + 20 b ^ 3 c ^ 3) / 7 + 3 c ^ 3 d ^ 4 x ^ 7
(20 a ^ 3 d ^ 3 + 105 a ^ 2 b c d ^ 2 + 126 a b ^ 2 c ^ 2 d + 35 b ^ 3 c ^ 3) / 4 + c ^ 2 d ^ 5 x ^ 8 (5 a ^ 3
d ^ 3 + 40 a ^ 2 b c d ^ 2 + 70 a b ^ 2 c ^ 2 d + 28 b ^ 3 c ^ 3) + d ^ 7 x ^ 10 (a ^ 3 d ^ 3 + 30 a ^ 2 b c
d ^ 2 + 135 a b ^ 2 c ^ 2 d + 120 b ^ 3 c ^ 3) / 11 + b ^ 3 d ^ 10 x ^ 13 / 14 + c ^ 6 d x ^ 4 (42 a ^ 3 d ^ 3
+ 72 a ^ 2 b c d ^ 2 + 27 a b ^ 2 c ^ 2 d + 2 b ^ 3 c ^ 3) + 3 c ^ 5 d ^ 2 x ^ 5 (28 a ^ 3 d ^ 3 + 70 a ^ 2 b
c d ^ 2 + 40 a b ^ 2 c ^ 2 d + 5 b ^ 3 c ^ 3) / 2 + c d ^ 6 x ^ 9 (2 a ^ 3 d ^ 3 + 27 a ^ 2 b c d ^ 2 + 72 a
b ^ 2 c ^ 2 d + 42 b ^ 3 c ^ 3) / 2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(540\) vs. \(2(84)=168\).
time = 0.23, size = 541, normalized size = 5.88

method	result	size
norman	\(a^{3} c^{10} x +\left (5 a^{3} c^{9} d +\frac {3}{2} a^{2} b \,c^{10}\right ) x^{2}+\left (15 a^{3} c^{8} d^{2}+10 a^{2} b \,c^{9} d +a \,b^{2} c^{10}\right ) x^{3}+\left (30 a^{3} c^{7} d^{3}+\frac {135}{4} a^{2} b \,c^{8} d^{2}+\frac {15}{2} a \,b^{2} c^{9} d +\frac {1}{4} b^{3} c^{10}\right ) x^{4}+\left (42 a^{3} c^{6} d^{4}+72 a^{2} b \,c^{7} d^{3}+27 a \,b^{2} c^{8} d^{2}+2 b^{3} c^{9} d \right ) x^{5}+\left (42 a^{3} c^{5} d^{5}+105 a^{2} b \,c^{6} d^{4}+60 a \,b^{2} c^{7} d^{3}+\frac {15}{2} b^{3} c^{8} d^{2}\right ) x^{6}+\left (30 a^{3} c^{4} d^{6}+108 a^{2} b \,c^{5} d^{5}+90 a \,b^{2} c^{6} d^{4}+\frac {120}{7} b^{3} c^{7} d^{3}\right ) x^{7}+\left (15 a^{3} c^{3} d^{7}+\frac {315}{4} a^{2} b \,c^{4} d^{6}+\frac {189}{2} a \,b^{2} c^{5} d^{5}+\frac {105}{4} b^{3} c^{6} d^{4}\right ) x^{8}+\left (5 a^{3} c^{2} d^{8}+40 a^{2} b \,c^{3} d^{7}+70 a \,b^{2} c^{4} d^{6}+28 b^{3} c^{5} d^{5}\right ) x^{9}+\left (a^{3} c \,d^{9}+\frac {27}{2} a^{2} b \,c^{2} d^{8}+36 a \,b^{2} c^{3} d^{7}+21 b^{3} c^{4} d^{6}\right ) x^{10}+\left (\frac {1}{11} a^{3} d^{10}+\frac {30}{11} a^{2} b c \,d^{9}+\frac {135}{11} a \,b^{2} c^{2} d^{8}+\frac {120}{11} b^{3} c^{3} d^{7}\right ) x^{11}+\left (\frac {1}{4} a^{2} b \,d^{10}+\frac {5}{2} a \,b^{2} c \,d^{9}+\frac {15}{4} b^{3} c^{2} d^{8}\right ) x^{12}+\left (\frac {3}{13} a \,b^{2} d^{10}+\frac {10}{13} b^{3} c \,d^{9}\right ) x^{13}+\frac {b^{3} d^{10} x^{14}}{14}\)	\(529\)
default	\(\frac {b^{3} d^{10} x^{14}}{14}+\frac {\left (3 a \,b^{2} d^{10}+10 b^{3} c \,d^{9}\right ) x^{13}}{13}+\frac {\left (3 a^{2} b \,d^{10}+30 a \,b^{2} c \,d^{9}+45 b^{3} c^{2} d^{8}\right ) x^{12}}{12}+\frac {\left (a^{3} d^{10}+30 a^{2} b c \,d^{9}+135 a \,b^{2} c^{2} d^{8}+120 b^{3} c^{3} d^{7}\right ) x^{11}}{11}+\frac {\left (10 a^{3} c \,d^{9}+135 a^{2} b \,c^{2} d^{8}+360 a \,b^{2} c^{3} d^{7}+210 b^{3} c^{4} d^{6}\right ) x^{10}}{10}+\frac {\left (45 a^{3} c^{2} d^{8}+360 a^{2} b \,c^{3} d^{7}+630 a \,b^{2} c^{4} d^{6}+252 b^{3} c^{5} d^{5}\right ) x^{9}}{9}+\frac {\left (120 a^{3} c^{3} d^{7}+630 a^{2} b \,c^{4} d^{6}+756 a \,b^{2} c^{5} d^{5}+210 b^{3} c^{6} d^{4}\right ) x^{8}}{8}+\frac {\left (210 a^{3} c^{4} d^{6}+756 a^{2} b \,c^{5} d^{5}+630 a \,b^{2} c^{6} d^{4}+120 b^{3} c^{7} d^{3}\right ) x^{7}}{7}+\frac {\left (252 a^{3} c^{5} d^{5}+630 a^{2} b \,c^{6} d^{4}+360 a \,b^{2} c^{7} d^{3}+45 b^{3} c^{8} d^{2}\right ) x^{6}}{6}+\frac {\left (210 a^{3} c^{6} d^{4}+360 a^{2} b \,c^{7} d^{3}+135 a \,b^{2} c^{8} d^{2}+10 b^{3} c^{9} d \right ) x^{5}}{5}+\frac {\left (120 a^{3} c^{7} d^{3}+135 a^{2} b \,c^{8} d^{2}+30 a \,b^{2} c^{9} d +b^{3} c^{10}\right ) x^{4}}{4}+\frac {\left (45 a^{3} c^{8} d^{2}+30 a^{2} b \,c^{9} d +3 a \,b^{2} c^{10}\right ) x^{3}}{3}+\frac {\left (10 a^{3} c^{9} d +3 a^{2} b \,c^{10}\right ) x^{2}}{2}+a^{3} c^{10} x\)	\(541\)
gosper	\(5 x^{2} a^{3} c^{9} d +\frac {3}{2} x^{2} a^{2} b \,c^{10}+30 x^{4} a^{3} c^{7} d^{3}+42 x^{6} a^{3} c^{5} d^{5}+\frac {15}{2} x^{6} b^{3} c^{8} d^{2}+30 x^{7} a^{3} c^{4} d^{6}+\frac {120}{7} x^{7} b^{3} c^{7} d^{3}+\frac {3}{13} x^{13} a \,b^{2} d^{10}+\frac {10}{13} x^{13} b^{3} c \,d^{9}+15 a^{3} c^{8} d^{2} x^{3}+a \,b^{2} c^{10} x^{3}+42 a^{3} c^{6} d^{4} x^{5}+2 b^{3} c^{9} d \,x^{5}+5 a^{3} c^{2} d^{8} x^{9}+28 b^{3} c^{5} d^{5} x^{9}+15 x^{8} a^{3} c^{3} d^{7}+\frac {105}{4} x^{8} b^{3} c^{6} d^{4}+x^{10} a^{3} c \,d^{9}+21 x^{10} b^{3} c^{4} d^{6}+\frac {120}{11} x^{11} b^{3} c^{3} d^{7}+\frac {1}{4} x^{12} a^{2} b \,d^{10}+\frac {15}{4} x^{12} b^{3} c^{2} d^{8}+\frac {1}{4} x^{4} b^{3} c^{10}+\frac {1}{11} x^{11} a^{3} d^{10}+\frac {1}{14} b^{3} d^{10} x^{14}+a^{3} c^{10} x +40 a^{2} b \,c^{3} d^{7} x^{9}+70 a \,b^{2} c^{4} d^{6} x^{9}+\frac {27}{2} x^{10} a^{2} b \,c^{2} d^{8}+36 x^{10} a \,b^{2} c^{3} d^{7}+\frac {30}{11} x^{11} a^{2} b c \,d^{9}+\frac {135}{11} x^{11} a \,b^{2} c^{2} d^{8}+\frac {5}{2} x^{12} a \,b^{2} c \,d^{9}+10 a^{2} b \,c^{9} d \,x^{3}+72 a^{2} b \,c^{7} d^{3} x^{5}+27 a \,b^{2} c^{8} d^{2} x^{5}+\frac {135}{4} x^{4} a^{2} b \,c^{8} d^{2}+\frac {15}{2} x^{4} a \,b^{2} c^{9} d +105 x^{6} a^{2} b \,c^{6} d^{4}+60 x^{6} a \,b^{2} c^{7} d^{3}+108 x^{7} a^{2} b \,c^{5} d^{5}+90 x^{7} a \,b^{2} c^{6} d^{4}+\frac {315}{4} x^{8} a^{2} b \,c^{4} d^{6}+\frac {189}{2} x^{8} a \,b^{2} c^{5} d^{5}\)	\(595\)
risch	\(5 x^{2} a^{3} c^{9} d +\frac {3}{2} x^{2} a^{2} b \,c^{10}+30 x^{4} a^{3} c^{7} d^{3}+42 x^{6} a^{3} c^{5} d^{5}+\frac {15}{2} x^{6} b^{3} c^{8} d^{2}+30 x^{7} a^{3} c^{4} d^{6}+\frac {120}{7} x^{7} b^{3} c^{7} d^{3}+\frac {3}{13} x^{13} a \,b^{2} d^{10}+\frac {10}{13} x^{13} b^{3} c \,d^{9}+15 a^{3} c^{8} d^{2} x^{3}+a \,b^{2} c^{10} x^{3}+42 a^{3} c^{6} d^{4} x^{5}+2 b^{3} c^{9} d \,x^{5}+5 a^{3} c^{2} d^{8} x^{9}+28 b^{3} c^{5} d^{5} x^{9}+15 x^{8} a^{3} c^{3} d^{7}+\frac {105}{4} x^{8} b^{3} c^{6} d^{4}+x^{10} a^{3} c \,d^{9}+21 x^{10} b^{3} c^{4} d^{6}+\frac {120}{11} x^{11} b^{3} c^{3} d^{7}+\frac {1}{4} x^{12} a^{2} b \,d^{10}+\frac {15}{4} x^{12} b^{3} c^{2} d^{8}+\frac {1}{4} x^{4} b^{3} c^{10}+\frac {1}{11} x^{11} a^{3} d^{10}+\frac {1}{14} b^{3} d^{10} x^{14}+a^{3} c^{10} x +40 a^{2} b \,c^{3} d^{7} x^{9}+70 a \,b^{2} c^{4} d^{6} x^{9}+\frac {27}{2} x^{10} a^{2} b \,c^{2} d^{8}+36 x^{10} a \,b^{2} c^{3} d^{7}+\frac {30}{11} x^{11} a^{2} b c \,d^{9}+\frac {135}{11} x^{11} a \,b^{2} c^{2} d^{8}+\frac {5}{2} x^{12} a \,b^{2} c \,d^{9}+10 a^{2} b \,c^{9} d \,x^{3}+72 a^{2} b \,c^{7} d^{3} x^{5}+27 a \,b^{2} c^{8} d^{2} x^{5}+\frac {135}{4} x^{4} a^{2} b \,c^{8} d^{2}+\frac {15}{2} x^{4} a \,b^{2} c^{9} d +105 x^{6} a^{2} b \,c^{6} d^{4}+60 x^{6} a \,b^{2} c^{7} d^{3}+108 x^{7} a^{2} b \,c^{5} d^{5}+90 x^{7} a \,b^{2} c^{6} d^{4}+\frac {315}{4} x^{8} a^{2} b \,c^{4} d^{6}+\frac {189}{2} x^{8} a \,b^{2} c^{5} d^{5}\)	\(595\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(d*x+c)^10,x,method=_RETURNVERBOSE)

[Out]

1/14*b^3*d^10*x^14+1/13*(3*a*b^2*d^10+10*b^3*c*d^9)*x^13+1/12*(3*a^2*b*d^10+30*a*b^2*c*d^9+45*b^3*c^2*d^8)*x^1
2+1/11*(a^3*d^10+30*a^2*b*c*d^9+135*a*b^2*c^2*d^8+120*b^3*c^3*d^7)*x^11+1/10*(10*a^3*c*d^9+135*a^2*b*c^2*d^8+3
60*a*b^2*c^3*d^7+210*b^3*c^4*d^6)*x^10+1/9*(45*a^3*c^2*d^8+360*a^2*b*c^3*d^7+630*a*b^2*c^4*d^6+252*b^3*c^5*d^5
)*x^9+1/8*(120*a^3*c^3*d^7+630*a^2*b*c^4*d^6+756*a*b^2*c^5*d^5+210*b^3*c^6*d^4)*x^8+1/7*(210*a^3*c^4*d^6+756*a
^2*b*c^5*d^5+630*a*b^2*c^6*d^4+120*b^3*c^7*d^3)*x^7+1/6*(252*a^3*c^5*d^5+630*a^2*b*c^6*d^4+360*a*b^2*c^7*d^3+4
5*b^3*c^8*d^2)*x^6+1/5*(210*a^3*c^6*d^4+360*a^2*b*c^7*d^3+135*a*b^2*c^8*d^2+10*b^3*c^9*d)*x^5+1/4*(120*a^3*c^7
*d^3+135*a^2*b*c^8*d^2+30*a*b^2*c^9*d+b^3*c^10)*x^4+1/3*(45*a^3*c^8*d^2+30*a^2*b*c^9*d+3*a*b^2*c^10)*x^3+1/2*(
10*a^3*c^9*d+3*a^2*b*c^10)*x^2+a^3*c^10*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (84) = 168\).
time = 0.27, size = 535, normalized size = 5.82 \begin {gather*} \frac {1}{14} \, b^{3} d^{10} x^{14} + a^{3} c^{10} x + \frac {1}{13} \, {\left (10 \, b^{3} c d^{9} + 3 \, a b^{2} d^{10}\right )} x^{13} + \frac {1}{4} \, {\left (15 \, b^{3} c^{2} d^{8} + 10 \, a b^{2} c d^{9} + a^{2} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (120 \, b^{3} c^{3} d^{7} + 135 \, a b^{2} c^{2} d^{8} + 30 \, a^{2} b c d^{9} + a^{3} d^{10}\right )} x^{11} + \frac {1}{2} \, {\left (42 \, b^{3} c^{4} d^{6} + 72 \, a b^{2} c^{3} d^{7} + 27 \, a^{2} b c^{2} d^{8} + 2 \, a^{3} c d^{9}\right )} x^{10} + {\left (28 \, b^{3} c^{5} d^{5} + 70 \, a b^{2} c^{4} d^{6} + 40 \, a^{2} b c^{3} d^{7} + 5 \, a^{3} c^{2} d^{8}\right )} x^{9} + \frac {3}{4} \, {\left (35 \, b^{3} c^{6} d^{4} + 126 \, a b^{2} c^{5} d^{5} + 105 \, a^{2} b c^{4} d^{6} + 20 \, a^{3} c^{3} d^{7}\right )} x^{8} + \frac {6}{7} \, {\left (20 \, b^{3} c^{7} d^{3} + 105 \, a b^{2} c^{6} d^{4} + 126 \, a^{2} b c^{5} d^{5} + 35 \, a^{3} c^{4} d^{6}\right )} x^{7} + \frac {3}{2} \, {\left (5 \, b^{3} c^{8} d^{2} + 40 \, a b^{2} c^{7} d^{3} + 70 \, a^{2} b c^{6} d^{4} + 28 \, a^{3} c^{5} d^{5}\right )} x^{6} + {\left (2 \, b^{3} c^{9} d + 27 \, a b^{2} c^{8} d^{2} + 72 \, a^{2} b c^{7} d^{3} + 42 \, a^{3} c^{6} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{10} + 30 \, a b^{2} c^{9} d + 135 \, a^{2} b c^{8} d^{2} + 120 \, a^{3} c^{7} d^{3}\right )} x^{4} + {\left (a b^{2} c^{10} + 10 \, a^{2} b c^{9} d + 15 \, a^{3} c^{8} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{10} + 10 \, a^{3} c^{9} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(d*x+c)^10,x, algorithm="maxima")

[Out]

1/14*b^3*d^10*x^14 + a^3*c^10*x + 1/13*(10*b^3*c*d^9 + 3*a*b^2*d^10)*x^13 + 1/4*(15*b^3*c^2*d^8 + 10*a*b^2*c*d
^9 + a^2*b*d^10)*x^12 + 1/11*(120*b^3*c^3*d^7 + 135*a*b^2*c^2*d^8 + 30*a^2*b*c*d^9 + a^3*d^10)*x^11 + 1/2*(42*
b^3*c^4*d^6 + 72*a*b^2*c^3*d^7 + 27*a^2*b*c^2*d^8 + 2*a^3*c*d^9)*x^10 + (28*b^3*c^5*d^5 + 70*a*b^2*c^4*d^6 + 4
0*a^2*b*c^3*d^7 + 5*a^3*c^2*d^8)*x^9 + 3/4*(35*b^3*c^6*d^4 + 126*a*b^2*c^5*d^5 + 105*a^2*b*c^4*d^6 + 20*a^3*c^
3*d^7)*x^8 + 6/7*(20*b^3*c^7*d^3 + 105*a*b^2*c^6*d^4 + 126*a^2*b*c^5*d^5 + 35*a^3*c^4*d^6)*x^7 + 3/2*(5*b^3*c^
8*d^2 + 40*a*b^2*c^7*d^3 + 70*a^2*b*c^6*d^4 + 28*a^3*c^5*d^5)*x^6 + (2*b^3*c^9*d + 27*a*b^2*c^8*d^2 + 72*a^2*b
*c^7*d^3 + 42*a^3*c^6*d^4)*x^5 + 1/4*(b^3*c^10 + 30*a*b^2*c^9*d + 135*a^2*b*c^8*d^2 + 120*a^3*c^7*d^3)*x^4 + (
a*b^2*c^10 + 10*a^2*b*c^9*d + 15*a^3*c^8*d^2)*x^3 + 1/2*(3*a^2*b*c^10 + 10*a^3*c^9*d)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 535 vs. \(2 (84) = 168\).
time = 0.29, size = 535, normalized size = 5.82 \begin {gather*} \frac {1}{14} \, b^{3} d^{10} x^{14} + a^{3} c^{10} x + \frac {1}{13} \, {\left (10 \, b^{3} c d^{9} + 3 \, a b^{2} d^{10}\right )} x^{13} + \frac {1}{4} \, {\left (15 \, b^{3} c^{2} d^{8} + 10 \, a b^{2} c d^{9} + a^{2} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (120 \, b^{3} c^{3} d^{7} + 135 \, a b^{2} c^{2} d^{8} + 30 \, a^{2} b c d^{9} + a^{3} d^{10}\right )} x^{11} + \frac {1}{2} \, {\left (42 \, b^{3} c^{4} d^{6} + 72 \, a b^{2} c^{3} d^{7} + 27 \, a^{2} b c^{2} d^{8} + 2 \, a^{3} c d^{9}\right )} x^{10} + {\left (28 \, b^{3} c^{5} d^{5} + 70 \, a b^{2} c^{4} d^{6} + 40 \, a^{2} b c^{3} d^{7} + 5 \, a^{3} c^{2} d^{8}\right )} x^{9} + \frac {3}{4} \, {\left (35 \, b^{3} c^{6} d^{4} + 126 \, a b^{2} c^{5} d^{5} + 105 \, a^{2} b c^{4} d^{6} + 20 \, a^{3} c^{3} d^{7}\right )} x^{8} + \frac {6}{7} \, {\left (20 \, b^{3} c^{7} d^{3} + 105 \, a b^{2} c^{6} d^{4} + 126 \, a^{2} b c^{5} d^{5} + 35 \, a^{3} c^{4} d^{6}\right )} x^{7} + \frac {3}{2} \, {\left (5 \, b^{3} c^{8} d^{2} + 40 \, a b^{2} c^{7} d^{3} + 70 \, a^{2} b c^{6} d^{4} + 28 \, a^{3} c^{5} d^{5}\right )} x^{6} + {\left (2 \, b^{3} c^{9} d + 27 \, a b^{2} c^{8} d^{2} + 72 \, a^{2} b c^{7} d^{3} + 42 \, a^{3} c^{6} d^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} c^{10} + 30 \, a b^{2} c^{9} d + 135 \, a^{2} b c^{8} d^{2} + 120 \, a^{3} c^{7} d^{3}\right )} x^{4} + {\left (a b^{2} c^{10} + 10 \, a^{2} b c^{9} d + 15 \, a^{3} c^{8} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{10} + 10 \, a^{3} c^{9} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(d*x+c)^10,x, algorithm="fricas")

[Out]

1/14*b^3*d^10*x^14 + a^3*c^10*x + 1/13*(10*b^3*c*d^9 + 3*a*b^2*d^10)*x^13 + 1/4*(15*b^3*c^2*d^8 + 10*a*b^2*c*d
^9 + a^2*b*d^10)*x^12 + 1/11*(120*b^3*c^3*d^7 + 135*a*b^2*c^2*d^8 + 30*a^2*b*c*d^9 + a^3*d^10)*x^11 + 1/2*(42*
b^3*c^4*d^6 + 72*a*b^2*c^3*d^7 + 27*a^2*b*c^2*d^8 + 2*a^3*c*d^9)*x^10 + (28*b^3*c^5*d^5 + 70*a*b^2*c^4*d^6 + 4
0*a^2*b*c^3*d^7 + 5*a^3*c^2*d^8)*x^9 + 3/4*(35*b^3*c^6*d^4 + 126*a*b^2*c^5*d^5 + 105*a^2*b*c^4*d^6 + 20*a^3*c^
3*d^7)*x^8 + 6/7*(20*b^3*c^7*d^3 + 105*a*b^2*c^6*d^4 + 126*a^2*b*c^5*d^5 + 35*a^3*c^4*d^6)*x^7 + 3/2*(5*b^3*c^
8*d^2 + 40*a*b^2*c^7*d^3 + 70*a^2*b*c^6*d^4 + 28*a^3*c^5*d^5)*x^6 + (2*b^3*c^9*d + 27*a*b^2*c^8*d^2 + 72*a^2*b
*c^7*d^3 + 42*a^3*c^6*d^4)*x^5 + 1/4*(b^3*c^10 + 30*a*b^2*c^9*d + 135*a^2*b*c^8*d^2 + 120*a^3*c^7*d^3)*x^4 + (
a*b^2*c^10 + 10*a^2*b*c^9*d + 15*a^3*c^8*d^2)*x^3 + 1/2*(3*a^2*b*c^10 + 10*a^3*c^9*d)*x^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (80) = 160\).
time = 0.08, size = 586, normalized size = 6.37 \begin {gather*} a^{3} c^{10} x + \frac {b^{3} d^{10} x^{14}}{14} + x^{13} \cdot \left (\frac {3 a b^{2} d^{10}}{13} + \frac {10 b^{3} c d^{9}}{13}\right ) + x^{12} \left (\frac {a^{2} b d^{10}}{4} + \frac {5 a b^{2} c d^{9}}{2} + \frac {15 b^{3} c^{2} d^{8}}{4}\right ) + x^{11} \left (\frac {a^{3} d^{10}}{11} + \frac {30 a^{2} b c d^{9}}{11} + \frac {135 a b^{2} c^{2} d^{8}}{11} + \frac {120 b^{3} c^{3} d^{7}}{11}\right ) + x^{10} \left (a^{3} c d^{9} + \frac {27 a^{2} b c^{2} d^{8}}{2} + 36 a b^{2} c^{3} d^{7} + 21 b^{3} c^{4} d^{6}\right ) + x^{9} \cdot \left (5 a^{3} c^{2} d^{8} + 40 a^{2} b c^{3} d^{7} + 70 a b^{2} c^{4} d^{6} + 28 b^{3} c^{5} d^{5}\right ) + x^{8} \cdot \left (15 a^{3} c^{3} d^{7} + \frac {315 a^{2} b c^{4} d^{6}}{4} + \frac {189 a b^{2} c^{5} d^{5}}{2} + \frac {105 b^{3} c^{6} d^{4}}{4}\right ) + x^{7} \cdot \left (30 a^{3} c^{4} d^{6} + 108 a^{2} b c^{5} d^{5} + 90 a b^{2} c^{6} d^{4} + \frac {120 b^{3} c^{7} d^{3}}{7}\right ) + x^{6} \cdot \left (42 a^{3} c^{5} d^{5} + 105 a^{2} b c^{6} d^{4} + 60 a b^{2} c^{7} d^{3} + \frac {15 b^{3} c^{8} d^{2}}{2}\right ) + x^{5} \cdot \left (42 a^{3} c^{6} d^{4} + 72 a^{2} b c^{7} d^{3} + 27 a b^{2} c^{8} d^{2} + 2 b^{3} c^{9} d\right ) + x^{4} \cdot \left (30 a^{3} c^{7} d^{3} + \frac {135 a^{2} b c^{8} d^{2}}{4} + \frac {15 a b^{2} c^{9} d}{2} + \frac {b^{3} c^{10}}{4}\right ) + x^{3} \cdot \left (15 a^{3} c^{8} d^{2} + 10 a^{2} b c^{9} d + a b^{2} c^{10}\right ) + x^{2} \cdot \left (5 a^{3} c^{9} d + \frac {3 a^{2} b c^{10}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(d*x+c)**10,x)

[Out]

a**3*c**10*x + b**3*d**10*x**14/14 + x**13*(3*a*b**2*d**10/13 + 10*b**3*c*d**9/13) + x**12*(a**2*b*d**10/4 + 5
*a*b**2*c*d**9/2 + 15*b**3*c**2*d**8/4) + x**11*(a**3*d**10/11 + 30*a**2*b*c*d**9/11 + 135*a*b**2*c**2*d**8/11
+ 120*b**3*c**3*d**7/11) + x**10*(a**3*c*d**9 + 27*a**2*b*c**2*d**8/2 + 36*a*b**2*c**3*d**7 + 21*b**3*c**4*d*
*6) + x**9*(5*a**3*c**2*d**8 + 40*a**2*b*c**3*d**7 + 70*a*b**2*c**4*d**6 + 28*b**3*c**5*d**5) + x**8*(15*a**3*
c**3*d**7 + 315*a**2*b*c**4*d**6/4 + 189*a*b**2*c**5*d**5/2 + 105*b**3*c**6*d**4/4) + x**7*(30*a**3*c**4*d**6
+ 108*a**2*b*c**5*d**5 + 90*a*b**2*c**6*d**4 + 120*b**3*c**7*d**3/7) + x**6*(42*a**3*c**5*d**5 + 105*a**2*b*c*
*6*d**4 + 60*a*b**2*c**7*d**3 + 15*b**3*c**8*d**2/2) + x**5*(42*a**3*c**6*d**4 + 72*a**2*b*c**7*d**3 + 27*a*b*
*2*c**8*d**2 + 2*b**3*c**9*d) + x**4*(30*a**3*c**7*d**3 + 135*a**2*b*c**8*d**2/4 + 15*a*b**2*c**9*d/2 + b**3*c
**10/4) + x**3*(15*a**3*c**8*d**2 + 10*a**2*b*c**9*d + a*b**2*c**10) + x**2*(5*a**3*c**9*d + 3*a**2*b*c**10/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (84) = 168\).
time = 0.00, size = 634, normalized size = 6.89 \begin {gather*} \frac {1}{14} x^{14} b^{3} d^{10}+\frac {10}{13} x^{13} b^{3} d^{9} c+\frac {3}{13} x^{13} b^{2} a d^{10}+\frac {15}{4} x^{12} b^{3} d^{8} c^{2}+\frac {5}{2} x^{12} b^{2} a d^{9} c+\frac {1}{4} x^{12} b a^{2} d^{10}+\frac {120}{11} x^{11} b^{3} d^{7} c^{3}+\frac {135}{11} x^{11} b^{2} a d^{8} c^{2}+\frac {30}{11} x^{11} b a^{2} d^{9} c+\frac {1}{11} x^{11} a^{3} d^{10}+21 x^{10} b^{3} d^{6} c^{4}+36 x^{10} b^{2} a d^{7} c^{3}+\frac {27}{2} x^{10} b a^{2} d^{8} c^{2}+x^{10} a^{3} d^{9} c+28 x^{9} b^{3} d^{5} c^{5}+70 x^{9} b^{2} a d^{6} c^{4}+40 x^{9} b a^{2} d^{7} c^{3}+5 x^{9} a^{3} d^{8} c^{2}+\frac {105}{4} x^{8} b^{3} d^{4} c^{6}+\frac {189}{2} x^{8} b^{2} a d^{5} c^{5}+\frac {315}{4} x^{8} b a^{2} d^{6} c^{4}+15 x^{8} a^{3} d^{7} c^{3}+\frac {120}{7} x^{7} b^{3} d^{3} c^{7}+90 x^{7} b^{2} a d^{4} c^{6}+108 x^{7} b a^{2} d^{5} c^{5}+30 x^{7} a^{3} d^{6} c^{4}+\frac {15}{2} x^{6} b^{3} d^{2} c^{8}+60 x^{6} b^{2} a d^{3} c^{7}+105 x^{6} b a^{2} d^{4} c^{6}+42 x^{6} a^{3} d^{5} c^{5}+2 x^{5} b^{3} d c^{9}+27 x^{5} b^{2} a d^{2} c^{8}+72 x^{5} b a^{2} d^{3} c^{7}+42 x^{5} a^{3} d^{4} c^{6}+\frac {1}{4} x^{4} b^{3} c^{10}+\frac {15}{2} x^{4} b^{2} a d c^{9}+\frac {135}{4} x^{4} b a^{2} d^{2} c^{8}+30 x^{4} a^{3} d^{3} c^{7}+x^{3} b^{2} a c^{10}+10 x^{3} b a^{2} d c^{9}+15 x^{3} a^{3} d^{2} c^{8}+\frac {3}{2} x^{2} b a^{2} c^{10}+5 x^{2} a^{3} d c^{9}+x a^{3} c^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(d*x+c)^10,x)

[Out]

1/14*b^3*d^10*x^14 + 10/13*b^3*c*d^9*x^13 + 3/13*a*b^2*d^10*x^13 + 15/4*b^3*c^2*d^8*x^12 + 5/2*a*b^2*c*d^9*x^1
2 + 1/4*a^2*b*d^10*x^12 + 120/11*b^3*c^3*d^7*x^11 + 135/11*a*b^2*c^2*d^8*x^11 + 30/11*a^2*b*c*d^9*x^11 + 1/11*
a^3*d^10*x^11 + 21*b^3*c^4*d^6*x^10 + 36*a*b^2*c^3*d^7*x^10 + 27/2*a^2*b*c^2*d^8*x^10 + a^3*c*d^9*x^10 + 28*b^
3*c^5*d^5*x^9 + 70*a*b^2*c^4*d^6*x^9 + 40*a^2*b*c^3*d^7*x^9 + 5*a^3*c^2*d^8*x^9 + 105/4*b^3*c^6*d^4*x^8 + 189/
2*a*b^2*c^5*d^5*x^8 + 315/4*a^2*b*c^4*d^6*x^8 + 15*a^3*c^3*d^7*x^8 + 120/7*b^3*c^7*d^3*x^7 + 90*a*b^2*c^6*d^4*
x^7 + 108*a^2*b*c^5*d^5*x^7 + 30*a^3*c^4*d^6*x^7 + 15/2*b^3*c^8*d^2*x^6 + 60*a*b^2*c^7*d^3*x^6 + 105*a^2*b*c^6
*d^4*x^6 + 42*a^3*c^5*d^5*x^6 + 2*b^3*c^9*d*x^5 + 27*a*b^2*c^8*d^2*x^5 + 72*a^2*b*c^7*d^3*x^5 + 42*a^3*c^6*d^4
*x^5 + 1/4*b^3*c^10*x^4 + 15/2*a*b^2*c^9*d*x^4 + 135/4*a^2*b*c^8*d^2*x^4 + 30*a^3*c^7*d^3*x^4 + a*b^2*c^10*x^3
+ 10*a^2*b*c^9*d*x^3 + 15*a^3*c^8*d^2*x^3 + 3/2*a^2*b*c^10*x^2 + 5*a^3*c^9*d*x^2 + a^3*c^10*x

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Mupad [B]
time = 0.23, size = 495, normalized size = 5.38 \begin {gather*} x^4\,\left (30\,a^3\,c^7\,d^3+\frac {135\,a^2\,b\,c^8\,d^2}{4}+\frac {15\,a\,b^2\,c^9\,d}{2}+\frac {b^3\,c^{10}}{4}\right )+x^{11}\,\left (\frac {a^3\,d^{10}}{11}+\frac {30\,a^2\,b\,c\,d^9}{11}+\frac {135\,a\,b^2\,c^2\,d^8}{11}+\frac {120\,b^3\,c^3\,d^7}{11}\right )+a^3\,c^{10}\,x+\frac {b^3\,d^{10}\,x^{14}}{14}+\frac {3\,c^5\,d^2\,x^6\,\left (28\,a^3\,d^3+70\,a^2\,b\,c\,d^2+40\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}{2}+c^2\,d^5\,x^9\,\left (5\,a^3\,d^3+40\,a^2\,b\,c\,d^2+70\,a\,b^2\,c^2\,d+28\,b^3\,c^3\right )+\frac {6\,c^4\,d^3\,x^7\,\left (35\,a^3\,d^3+126\,a^2\,b\,c\,d^2+105\,a\,b^2\,c^2\,d+20\,b^3\,c^3\right )}{7}+\frac {3\,c^3\,d^4\,x^8\,\left (20\,a^3\,d^3+105\,a^2\,b\,c\,d^2+126\,a\,b^2\,c^2\,d+35\,b^3\,c^3\right )}{4}+\frac {a^2\,c^9\,x^2\,\left (10\,a\,d+3\,b\,c\right )}{2}+\frac {b^2\,d^9\,x^{13}\,\left (3\,a\,d+10\,b\,c\right )}{13}+a\,c^8\,x^3\,\left (15\,a^2\,d^2+10\,a\,b\,c\,d+b^2\,c^2\right )+\frac {b\,d^8\,x^{12}\,\left (a^2\,d^2+10\,a\,b\,c\,d+15\,b^2\,c^2\right )}{4}+c^6\,d\,x^5\,\left (42\,a^3\,d^3+72\,a^2\,b\,c\,d^2+27\,a\,b^2\,c^2\,d+2\,b^3\,c^3\right )+\frac {c\,d^6\,x^{10}\,\left (2\,a^3\,d^3+27\,a^2\,b\,c\,d^2+72\,a\,b^2\,c^2\,d+42\,b^3\,c^3\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3*(c + d*x)^10,x)

[Out]

x^4*((b^3*c^10)/4 + 30*a^3*c^7*d^3 + (135*a^2*b*c^8*d^2)/4 + (15*a*b^2*c^9*d)/2) + x^11*((a^3*d^10)/11 + (120*
b^3*c^3*d^7)/11 + (135*a*b^2*c^2*d^8)/11 + (30*a^2*b*c*d^9)/11) + a^3*c^10*x + (b^3*d^10*x^14)/14 + (3*c^5*d^2
*x^6*(28*a^3*d^3 + 5*b^3*c^3 + 40*a*b^2*c^2*d + 70*a^2*b*c*d^2))/2 + c^2*d^5*x^9*(5*a^3*d^3 + 28*b^3*c^3 + 70*
a*b^2*c^2*d + 40*a^2*b*c*d^2) + (6*c^4*d^3*x^7*(35*a^3*d^3 + 20*b^3*c^3 + 105*a*b^2*c^2*d + 126*a^2*b*c*d^2))/
7 + (3*c^3*d^4*x^8*(20*a^3*d^3 + 35*b^3*c^3 + 126*a*b^2*c^2*d + 105*a^2*b*c*d^2))/4 + (a^2*c^9*x^2*(10*a*d + 3
*b*c))/2 + (b^2*d^9*x^13*(3*a*d + 10*b*c))/13 + a*c^8*x^3*(15*a^2*d^2 + b^2*c^2 + 10*a*b*c*d) + (b*d^8*x^12*(a
^2*d^2 + 15*b^2*c^2 + 10*a*b*c*d))/4 + c^6*d*x^5*(42*a^3*d^3 + 2*b^3*c^3 + 27*a*b^2*c^2*d + 72*a^2*b*c*d^2) +
(c*d^6*x^10*(2*a^3*d^3 + 42*b^3*c^3 + 72*a*b^2*c^2*d + 27*a^2*b*c*d^2))/2

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