3.131 \(\int x^3 (a+b x)^{10} \, dx\)

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

[Out]

-(a^3*(a + b*x)^11)/(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
)

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Rubi [A]  time = 0.035897, antiderivative size = 64, normalized size of antiderivative = 1., 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 = {43} \[ \frac{a^2 (a+b x)^{12}}{4 b^4}-\frac{a^3 (a+b x)^{11}}{11 b^4}+\frac{(a+b x)^{14}}{14 b^4}-\frac{3 a (a+b x)^{13}}{13 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^10,x]

[Out]

-(a^3*(a + b*x)^11)/(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 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 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*}

Mathematica [A]  time = 0.003124, size = 128, normalized size = 2. \[ \frac{15}{4} a^2 b^8 x^{12}+\frac{120}{11} a^3 b^7 x^{11}+21 a^4 b^6 x^{10}+28 a^5 b^5 x^9+\frac{105}{4} a^6 b^4 x^8+\frac{120}{7} a^7 b^3 x^7+\frac{15}{2} a^8 b^2 x^6+2 a^9 b x^5+\frac{a^{10} x^4}{4}+\frac{10}{13} a b^9 x^{13}+\frac{b^{10} x^{14}}{14} \]

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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Maple [A]  time = 0.001, size = 113, normalized size = 1.8 \begin{align*}{\frac{{b}^{10}{x}^{14}}{14}}+{\frac{10\,a{b}^{9}{x}^{13}}{13}}+{\frac{15\,{a}^{2}{b}^{8}{x}^{12}}{4}}+{\frac{120\,{a}^{3}{b}^{7}{x}^{11}}{11}}+21\,{a}^{4}{b}^{6}{x}^{10}+28\,{a}^{5}{b}^{5}{x}^{9}+{\frac{105\,{a}^{6}{b}^{4}{x}^{8}}{4}}+{\frac{120\,{a}^{7}{b}^{3}{x}^{7}}{7}}+{\frac{15\,{a}^{8}{b}^{2}{x}^{6}}{2}}+2\,{a}^{9}b{x}^{5}+{\frac{{a}^{10}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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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Maxima [A]  time = 1.0224, size = 151, normalized size = 2.36 \begin{align*} \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{align*}

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 = 1.53716, size = 270, normalized size = 4.22 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.105953, size = 129, normalized size = 2.02 \begin{align*} \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{align*}

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 = 1.16197, size = 151, normalized size = 2.36 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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