3.2.31 \(\int x^3 (a+b x)^{10} \, dx\) [131]

Optimal. Leaf size=64 \[ -\frac {a^3 (a+b x)^{11}}{11 b^4}+\frac {a^2 (a+b x)^{12}}{4 b^4}-\frac {3 a (a+b x)^{13}}{13 b^4}+\frac {(a+b x)^{14}}{14 b^4} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \begin {gather*} -\frac {a^3 (a+b x)^{11}}{11 b^4}+\frac {a^2 (a+b x)^{12}}{4 b^4}+\frac {(a+b x)^{14}}{14 b^4}-\frac {3 a (a+b x)^{13}}{13 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^10,x]

[Out]

-1/11*(a^3*(a + b*x)^11)/b^4 + (a^2*(a + b*x)^12)/(4*b^4) - (3*a*(a + b*x)^13)/(13*b^4) + (a + b*x)^14/(14*b^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 x^3 (a+b x)^{10} \, dx &=\int \left (-\frac {a^3 (a+b x)^{10}}{b^3}+\frac {3 a^2 (a+b x)^{11}}{b^3}-\frac {3 a (a+b x)^{12}}{b^3}+\frac {(a+b x)^{13}}{b^3}\right ) \, dx\\ &=-\frac {a^3 (a+b x)^{11}}{11 b^4}+\frac {a^2 (a+b x)^{12}}{4 b^4}-\frac {3 a (a+b x)^{13}}{13 b^4}+\frac {(a+b x)^{14}}{14 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 128, normalized size = 2.00 \begin {gather*} \frac {a^{10} x^4}{4}+2 a^9 b x^5+\frac {15}{2} a^8 b^2 x^6+\frac {120}{7} a^7 b^3 x^7+\frac {105}{4} a^6 b^4 x^8+28 a^5 b^5 x^9+21 a^4 b^6 x^{10}+\frac {120}{11} a^3 b^7 x^{11}+\frac {15}{4} a^2 b^8 x^{12}+\frac {10}{13} a b^9 x^{13}+\frac {b^{10} x^{14}}{14} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^10,x]

[Out]

(a^10*x^4)/4 + 2*a^9*b*x^5 + (15*a^8*b^2*x^6)/2 + (120*a^7*b^3*x^7)/7 + (105*a^6*b^4*x^8)/4 + 28*a^5*b^5*x^9 +
21*a^4*b^6*x^10 + (120*a^3*b^7*x^11)/11 + (15*a^2*b^8*x^12)/4 + (10*a*b^9*x^13)/13 + (b^10*x^14)/14

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Mathics [A]
time = 2.18, size = 112, normalized size = 1.75 \begin {gather*} \frac {x^4 \left (1001 a^{10}+8008 a^9 b x+30030 a^8 b^2 x^2+68640 a^7 b^3 x^3+105105 a^6 b^4 x^4+112112 a^5 b^5 x^5+84084 a^4 b^6 x^6+43680 a^3 b^7 x^7+15015 a^2 b^8 x^8+3080 a b^9 x^9+286 b^{10} x^{10}\right )}{4004} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x ^ 4 (1001 a ^ 10 + 8008 a ^ 9 b x + 30030 a ^ 8 b ^ 2 x ^ 2 + 68640 a ^ 7 b ^ 3 x ^ 3 + 105105 a ^ 6 b ^ 4 x
^ 4 + 112112 a ^ 5 b ^ 5 x ^ 5 + 84084 a ^ 4 b ^ 6 x ^ 6 + 43680 a ^ 3 b ^ 7 x ^ 7 + 15015 a ^ 2 b ^ 8 x ^ 8
+ 3080 a b ^ 9 x ^ 9 + 286 b ^ 10 x ^ 10) / 4004

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Maple [A]
time = 0.08, size = 113, normalized size = 1.77

method	result	size
gosper	\(\frac {1}{4} a^{10} x^{4}+2 a^{9} b \,x^{5}+\frac {15}{2} a^{8} b^{2} x^{6}+\frac {120}{7} a^{7} b^{3} x^{7}+\frac {105}{4} a^{6} b^{4} x^{8}+28 a^{5} b^{5} x^{9}+21 a^{4} b^{6} x^{10}+\frac {120}{11} a^{3} b^{7} x^{11}+\frac {15}{4} a^{2} b^{8} x^{12}+\frac {10}{13} a \,b^{9} x^{13}+\frac {1}{14} b^{10} x^{14}\)	\(113\)
default	\(\frac {1}{4} a^{10} x^{4}+2 a^{9} b \,x^{5}+\frac {15}{2} a^{8} b^{2} x^{6}+\frac {120}{7} a^{7} b^{3} x^{7}+\frac {105}{4} a^{6} b^{4} x^{8}+28 a^{5} b^{5} x^{9}+21 a^{4} b^{6} x^{10}+\frac {120}{11} a^{3} b^{7} x^{11}+\frac {15}{4} a^{2} b^{8} x^{12}+\frac {10}{13} a \,b^{9} x^{13}+\frac {1}{14} b^{10} x^{14}\)	\(113\)
norman	\(\frac {1}{4} a^{10} x^{4}+2 a^{9} b \,x^{5}+\frac {15}{2} a^{8} b^{2} x^{6}+\frac {120}{7} a^{7} b^{3} x^{7}+\frac {105}{4} a^{6} b^{4} x^{8}+28 a^{5} b^{5} x^{9}+21 a^{4} b^{6} x^{10}+\frac {120}{11} a^{3} b^{7} x^{11}+\frac {15}{4} a^{2} b^{8} x^{12}+\frac {10}{13} a \,b^{9} x^{13}+\frac {1}{14} b^{10} x^{14}\)	\(113\)
risch	\(\frac {1}{4} a^{10} x^{4}+2 a^{9} b \,x^{5}+\frac {15}{2} a^{8} b^{2} x^{6}+\frac {120}{7} a^{7} b^{3} x^{7}+\frac {105}{4} a^{6} b^{4} x^{8}+28 a^{5} b^{5} x^{9}+21 a^{4} b^{6} x^{10}+\frac {120}{11} a^{3} b^{7} x^{11}+\frac {15}{4} a^{2} b^{8} x^{12}+\frac {10}{13} a \,b^{9} x^{13}+\frac {1}{14} b^{10} x^{14}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^10*x^4+2*a^9*b*x^5+15/2*a^8*b^2*x^6+120/7*a^7*b^3*x^7+105/4*a^6*b^4*x^8+28*a^5*b^5*x^9+21*a^4*b^6*x^10+1
20/11*a^3*b^7*x^11+15/4*a^2*b^8*x^12+10/13*a*b^9*x^13+1/14*b^10*x^14

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Maxima [A]
time = 0.25, size = 112, normalized size = 1.75 \begin {gather*} \frac {1}{14} \, b^{10} x^{14} + \frac {10}{13} \, a b^{9} x^{13} + \frac {15}{4} \, a^{2} b^{8} x^{12} + \frac {120}{11} \, a^{3} b^{7} x^{11} + 21 \, a^{4} b^{6} x^{10} + 28 \, a^{5} b^{5} x^{9} + \frac {105}{4} \, a^{6} b^{4} x^{8} + \frac {120}{7} \, a^{7} b^{3} x^{7} + \frac {15}{2} \, a^{8} b^{2} x^{6} + 2 \, a^{9} b x^{5} + \frac {1}{4} \, a^{10} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^10*x^14 + 10/13*a*b^9*x^13 + 15/4*a^2*b^8*x^12 + 120/11*a^3*b^7*x^11 + 21*a^4*b^6*x^10 + 28*a^5*b^5*x^9
+ 105/4*a^6*b^4*x^8 + 120/7*a^7*b^3*x^7 + 15/2*a^8*b^2*x^6 + 2*a^9*b*x^5 + 1/4*a^10*x^4

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Fricas [A]
time = 0.31, size = 112, normalized size = 1.75 \begin {gather*} \frac {1}{14} \, b^{10} x^{14} + \frac {10}{13} \, a b^{9} x^{13} + \frac {15}{4} \, a^{2} b^{8} x^{12} + \frac {120}{11} \, a^{3} b^{7} x^{11} + 21 \, a^{4} b^{6} x^{10} + 28 \, a^{5} b^{5} x^{9} + \frac {105}{4} \, a^{6} b^{4} x^{8} + \frac {120}{7} \, a^{7} b^{3} x^{7} + \frac {15}{2} \, a^{8} b^{2} x^{6} + 2 \, a^{9} b x^{5} + \frac {1}{4} \, a^{10} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^10*x^14 + 10/13*a*b^9*x^13 + 15/4*a^2*b^8*x^12 + 120/11*a^3*b^7*x^11 + 21*a^4*b^6*x^10 + 28*a^5*b^5*x^9
+ 105/4*a^6*b^4*x^8 + 120/7*a^7*b^3*x^7 + 15/2*a^8*b^2*x^6 + 2*a^9*b*x^5 + 1/4*a^10*x^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (56) = 112\).
time = 0.05, size = 129, normalized size = 2.02 \begin {gather*} \frac {a^{10} x^{4}}{4} + 2 a^{9} b x^{5} + \frac {15 a^{8} b^{2} x^{6}}{2} + \frac {120 a^{7} b^{3} x^{7}}{7} + \frac {105 a^{6} b^{4} x^{8}}{4} + 28 a^{5} b^{5} x^{9} + 21 a^{4} b^{6} x^{10} + \frac {120 a^{3} b^{7} x^{11}}{11} + \frac {15 a^{2} b^{8} x^{12}}{4} + \frac {10 a b^{9} x^{13}}{13} + \frac {b^{10} x^{14}}{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**10*x**4/4 + 2*a**9*b*x**5 + 15*a**8*b**2*x**6/2 + 120*a**7*b**3*x**7/7 + 105*a**6*b**4*x**8/4 + 28*a**5*b**
5*x**9 + 21*a**4*b**6*x**10 + 120*a**3*b**7*x**11/11 + 15*a**2*b**8*x**12/4 + 10*a*b**9*x**13/13 + b**10*x**14
/14

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Giac [A]
time = 0.00, size = 128, normalized size = 2.00 \begin {gather*} \frac {1}{14} x^{14} b^{10}+\frac {10}{13} x^{13} b^{9} a+\frac {15}{4} x^{12} b^{8} a^{2}+\frac {120}{11} x^{11} b^{7} a^{3}+21 x^{10} b^{6} a^{4}+28 x^{9} b^{5} a^{5}+\frac {105}{4} x^{8} b^{4} a^{6}+\frac {120}{7} x^{7} b^{3} a^{7}+\frac {15}{2} x^{6} b^{2} a^{8}+2 x^{5} b a^{9}+\frac {1}{4} x^{4} a^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^10*x^14 + 10/13*a*b^9*x^13 + 15/4*a^2*b^8*x^12 + 120/11*a^3*b^7*x^11 + 21*a^4*b^6*x^10 + 28*a^5*b^5*x^9
+ 105/4*a^6*b^4*x^8 + 120/7*a^7*b^3*x^7 + 15/2*a^8*b^2*x^6 + 2*a^9*b*x^5 + 1/4*a^10*x^4

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Mupad [B]
time = 0.12, size = 112, normalized size = 1.75 \begin {gather*} \frac {a^{10}\,x^4}{4}+2\,a^9\,b\,x^5+\frac {15\,a^8\,b^2\,x^6}{2}+\frac {120\,a^7\,b^3\,x^7}{7}+\frac {105\,a^6\,b^4\,x^8}{4}+28\,a^5\,b^5\,x^9+21\,a^4\,b^6\,x^{10}+\frac {120\,a^3\,b^7\,x^{11}}{11}+\frac {15\,a^2\,b^8\,x^{12}}{4}+\frac {10\,a\,b^9\,x^{13}}{13}+\frac {b^{10}\,x^{14}}{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)^10,x)

[Out]

(a^10*x^4)/4 + (b^10*x^14)/14 + 2*a^9*b*x^5 + (10*a*b^9*x^13)/13 + (15*a^8*b^2*x^6)/2 + (120*a^7*b^3*x^7)/7 +
(105*a^6*b^4*x^8)/4 + 28*a^5*b^5*x^9 + 21*a^4*b^6*x^10 + (120*a^3*b^7*x^11)/11 + (15*a^2*b^8*x^12)/4

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