3.2.3 \(\int x^3 (a+b x)^7 \, dx\)

Optimal. Leaf size=64 \[ -\frac {a^3 (a+b x)^8}{8 b^4}+\frac {a^2 (a+b x)^9}{3 b^4}+\frac {(a+b x)^{11}}{11 b^4}-\frac {3 a (a+b x)^{10}}{10 b^4} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, 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 = {43} \begin {gather*} \frac {a^2 (a+b x)^9}{3 b^4}-\frac {a^3 (a+b x)^8}{8 b^4}+\frac {(a+b x)^{11}}{11 b^4}-\frac {3 a (a+b x)^{10}}{10 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^7,x]

[Out]

-(a^3*(a + b*x)^8)/(8*b^4) + (a^2*(a + b*x)^9)/(3*b^4) - (3*a*(a + b*x)^10)/(10*b^4) + (a + b*x)^11/(11*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)^7 \, dx &=\int \left (-\frac {a^3 (a+b x)^7}{b^3}+\frac {3 a^2 (a+b x)^8}{b^3}-\frac {3 a (a+b x)^9}{b^3}+\frac {(a+b x)^{10}}{b^3}\right ) \, dx\\ &=-\frac {a^3 (a+b x)^8}{8 b^4}+\frac {a^2 (a+b x)^9}{3 b^4}-\frac {3 a (a+b x)^{10}}{10 b^4}+\frac {(a+b x)^{11}}{11 b^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 93, normalized size = 1.45 \begin {gather*} \frac {a^7 x^4}{4}+\frac {7}{5} a^6 b x^5+\frac {7}{2} a^5 b^2 x^6+5 a^4 b^3 x^7+\frac {35}{8} a^3 b^4 x^8+\frac {7}{3} a^2 b^5 x^9+\frac {7}{10} a b^6 x^{10}+\frac {b^7 x^{11}}{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^7,x]

[Out]

(a^7*x^4)/4 + (7*a^6*b*x^5)/5 + (7*a^5*b^2*x^6)/2 + 5*a^4*b^3*x^7 + (35*a^3*b^4*x^8)/8 + (7*a^2*b^5*x^9)/3 + (
7*a*b^6*x^10)/10 + (b^7*x^11)/11

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 (a+b x)^7 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^3*(a + b*x)^7,x]

[Out]

IntegrateAlgebraic[x^3*(a + b*x)^7, x]

________________________________________________________________________________________

fricas [A]  time = 1.33, size = 79, normalized size = 1.23 \begin {gather*} \frac {1}{11} x^{11} b^{7} + \frac {7}{10} x^{10} b^{6} a + \frac {7}{3} x^{9} b^{5} a^{2} + \frac {35}{8} x^{8} b^{4} a^{3} + 5 x^{7} b^{3} a^{4} + \frac {7}{2} x^{6} b^{2} a^{5} + \frac {7}{5} x^{5} b a^{6} + \frac {1}{4} x^{4} a^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^7,x, algorithm="fricas")

[Out]

1/11*x^11*b^7 + 7/10*x^10*b^6*a + 7/3*x^9*b^5*a^2 + 35/8*x^8*b^4*a^3 + 5*x^7*b^3*a^4 + 7/2*x^6*b^2*a^5 + 7/5*x
^5*b*a^6 + 1/4*x^4*a^7

________________________________________________________________________________________

giac [A]  time = 1.22, size = 79, normalized size = 1.23 \begin {gather*} \frac {1}{11} \, b^{7} x^{11} + \frac {7}{10} \, a b^{6} x^{10} + \frac {7}{3} \, a^{2} b^{5} x^{9} + \frac {35}{8} \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{7} + \frac {7}{2} \, a^{5} b^{2} x^{6} + \frac {7}{5} \, a^{6} b x^{5} + \frac {1}{4} \, a^{7} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^7,x, algorithm="giac")

[Out]

1/11*b^7*x^11 + 7/10*a*b^6*x^10 + 7/3*a^2*b^5*x^9 + 35/8*a^3*b^4*x^8 + 5*a^4*b^3*x^7 + 7/2*a^5*b^2*x^6 + 7/5*a
^6*b*x^5 + 1/4*a^7*x^4

________________________________________________________________________________________

maple [A]  time = 0.00, size = 80, normalized size = 1.25 \begin {gather*} \frac {1}{11} b^{7} x^{11}+\frac {7}{10} a \,b^{6} x^{10}+\frac {7}{3} a^{2} b^{5} x^{9}+\frac {35}{8} a^{3} b^{4} x^{8}+5 a^{4} b^{3} x^{7}+\frac {7}{2} a^{5} b^{2} x^{6}+\frac {7}{5} a^{6} b \,x^{5}+\frac {1}{4} a^{7} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^7,x)

[Out]

1/11*b^7*x^11+7/10*a*b^6*x^10+7/3*a^2*b^5*x^9+35/8*a^3*b^4*x^8+5*a^4*b^3*x^7+7/2*a^5*b^2*x^6+7/5*a^6*b*x^5+1/4
*a^7*x^4

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 79, normalized size = 1.23 \begin {gather*} \frac {1}{11} \, b^{7} x^{11} + \frac {7}{10} \, a b^{6} x^{10} + \frac {7}{3} \, a^{2} b^{5} x^{9} + \frac {35}{8} \, a^{3} b^{4} x^{8} + 5 \, a^{4} b^{3} x^{7} + \frac {7}{2} \, a^{5} b^{2} x^{6} + \frac {7}{5} \, a^{6} b x^{5} + \frac {1}{4} \, a^{7} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^7,x, algorithm="maxima")

[Out]

1/11*b^7*x^11 + 7/10*a*b^6*x^10 + 7/3*a^2*b^5*x^9 + 35/8*a^3*b^4*x^8 + 5*a^4*b^3*x^7 + 7/2*a^5*b^2*x^6 + 7/5*a
^6*b*x^5 + 1/4*a^7*x^4

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 79, normalized size = 1.23 \begin {gather*} \frac {a^7\,x^4}{4}+\frac {7\,a^6\,b\,x^5}{5}+\frac {7\,a^5\,b^2\,x^6}{2}+5\,a^4\,b^3\,x^7+\frac {35\,a^3\,b^4\,x^8}{8}+\frac {7\,a^2\,b^5\,x^9}{3}+\frac {7\,a\,b^6\,x^{10}}{10}+\frac {b^7\,x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)^7,x)

[Out]

(a^7*x^4)/4 + (b^7*x^11)/11 + (7*a^6*b*x^5)/5 + (7*a*b^6*x^10)/10 + (7*a^5*b^2*x^6)/2 + 5*a^4*b^3*x^7 + (35*a^
3*b^4*x^8)/8 + (7*a^2*b^5*x^9)/3

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 92, normalized size = 1.44 \begin {gather*} \frac {a^{7} x^{4}}{4} + \frac {7 a^{6} b x^{5}}{5} + \frac {7 a^{5} b^{2} x^{6}}{2} + 5 a^{4} b^{3} x^{7} + \frac {35 a^{3} b^{4} x^{8}}{8} + \frac {7 a^{2} b^{5} x^{9}}{3} + \frac {7 a b^{6} x^{10}}{10} + \frac {b^{7} x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**7,x)

[Out]

a**7*x**4/4 + 7*a**6*b*x**5/5 + 7*a**5*b**2*x**6/2 + 5*a**4*b**3*x**7 + 35*a**3*b**4*x**8/8 + 7*a**2*b**5*x**9
/3 + 7*a*b**6*x**10/10 + b**7*x**11/11

________________________________________________________________________________________