3.102 \(\int x^4 (a+b x)^7 \, dx\)

Optimal. Leaf size=81 \[ \frac {a^4 (a+b x)^8}{8 b^5}-\frac {4 a^3 (a+b x)^9}{9 b^5}+\frac {3 a^2 (a+b x)^{10}}{5 b^5}+\frac {(a+b x)^{12}}{12 b^5}-\frac {4 a (a+b x)^{11}}{11 b^5} \]

[Out]

1/8*a^4*(b*x+a)^8/b^5-4/9*a^3*(b*x+a)^9/b^5+3/5*a^2*(b*x+a)^10/b^5-4/11*a*(b*x+a)^11/b^5+1/12*(b*x+a)^12/b^5

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Rubi [A]  time = 0.04, antiderivative size = 81, 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} \[ \frac {3 a^2 (a+b x)^{10}}{5 b^5}-\frac {4 a^3 (a+b x)^9}{9 b^5}+\frac {a^4 (a+b x)^8}{8 b^5}+\frac {(a+b x)^{12}}{12 b^5}-\frac {4 a (a+b x)^{11}}{11 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x)^7,x]

[Out]

(a^4*(a + b*x)^8)/(8*b^5) - (4*a^3*(a + b*x)^9)/(9*b^5) + (3*a^2*(a + b*x)^10)/(5*b^5) - (4*a*(a + b*x)^11)/(1
1*b^5) + (a + b*x)^12/(12*b^5)

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^4 (a+b x)^7 \, dx &=\int \left (\frac {a^4 (a+b x)^7}{b^4}-\frac {4 a^3 (a+b x)^8}{b^4}+\frac {6 a^2 (a+b x)^9}{b^4}-\frac {4 a (a+b x)^{10}}{b^4}+\frac {(a+b x)^{11}}{b^4}\right ) \, dx\\ &=\frac {a^4 (a+b x)^8}{8 b^5}-\frac {4 a^3 (a+b x)^9}{9 b^5}+\frac {3 a^2 (a+b x)^{10}}{5 b^5}-\frac {4 a (a+b x)^{11}}{11 b^5}+\frac {(a+b x)^{12}}{12 b^5}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 93, normalized size = 1.15 \[ \frac {a^7 x^5}{5}+\frac {7}{6} a^6 b x^6+3 a^5 b^2 x^7+\frac {35}{8} a^4 b^3 x^8+\frac {35}{9} a^3 b^4 x^9+\frac {21}{10} a^2 b^5 x^{10}+\frac {7}{11} a b^6 x^{11}+\frac {b^7 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x)^7,x]

[Out]

(a^7*x^5)/5 + (7*a^6*b*x^6)/6 + 3*a^5*b^2*x^7 + (35*a^4*b^3*x^8)/8 + (35*a^3*b^4*x^9)/9 + (21*a^2*b^5*x^10)/10
+ (7*a*b^6*x^11)/11 + (b^7*x^12)/12

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fricas [A]  time = 0.36, size = 79, normalized size = 0.98 \[ \frac {1}{12} x^{12} b^{7} + \frac {7}{11} x^{11} b^{6} a + \frac {21}{10} x^{10} b^{5} a^{2} + \frac {35}{9} x^{9} b^{4} a^{3} + \frac {35}{8} x^{8} b^{3} a^{4} + 3 x^{7} b^{2} a^{5} + \frac {7}{6} x^{6} b a^{6} + \frac {1}{5} x^{5} a^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^7,x, algorithm="fricas")

[Out]

1/12*x^12*b^7 + 7/11*x^11*b^6*a + 21/10*x^10*b^5*a^2 + 35/9*x^9*b^4*a^3 + 35/8*x^8*b^3*a^4 + 3*x^7*b^2*a^5 + 7
/6*x^6*b*a^6 + 1/5*x^5*a^7

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giac [A]  time = 1.13, size = 79, normalized size = 0.98 \[ \frac {1}{12} \, b^{7} x^{12} + \frac {7}{11} \, a b^{6} x^{11} + \frac {21}{10} \, a^{2} b^{5} x^{10} + \frac {35}{9} \, a^{3} b^{4} x^{9} + \frac {35}{8} \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{7} + \frac {7}{6} \, a^{6} b x^{6} + \frac {1}{5} \, a^{7} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^7,x, algorithm="giac")

[Out]

1/12*b^7*x^12 + 7/11*a*b^6*x^11 + 21/10*a^2*b^5*x^10 + 35/9*a^3*b^4*x^9 + 35/8*a^4*b^3*x^8 + 3*a^5*b^2*x^7 + 7
/6*a^6*b*x^6 + 1/5*a^7*x^5

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maple [A]  time = 0.00, size = 80, normalized size = 0.99 \[ \frac {1}{12} b^{7} x^{12}+\frac {7}{11} a \,b^{6} x^{11}+\frac {21}{10} a^{2} b^{5} x^{10}+\frac {35}{9} a^{3} b^{4} x^{9}+\frac {35}{8} a^{4} b^{3} x^{8}+3 a^{5} b^{2} x^{7}+\frac {7}{6} a^{6} b \,x^{6}+\frac {1}{5} a^{7} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x+a)^7,x)

[Out]

1/12*b^7*x^12+7/11*a*b^6*x^11+21/10*a^2*b^5*x^10+35/9*a^3*b^4*x^9+35/8*a^4*b^3*x^8+3*a^5*b^2*x^7+7/6*a^6*b*x^6
+1/5*a^7*x^5

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maxima [A]  time = 1.35, size = 79, normalized size = 0.98 \[ \frac {1}{12} \, b^{7} x^{12} + \frac {7}{11} \, a b^{6} x^{11} + \frac {21}{10} \, a^{2} b^{5} x^{10} + \frac {35}{9} \, a^{3} b^{4} x^{9} + \frac {35}{8} \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{7} + \frac {7}{6} \, a^{6} b x^{6} + \frac {1}{5} \, a^{7} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x+a)^7,x, algorithm="maxima")

[Out]

1/12*b^7*x^12 + 7/11*a*b^6*x^11 + 21/10*a^2*b^5*x^10 + 35/9*a^3*b^4*x^9 + 35/8*a^4*b^3*x^8 + 3*a^5*b^2*x^7 + 7
/6*a^6*b*x^6 + 1/5*a^7*x^5

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mupad [B]  time = 0.06, size = 79, normalized size = 0.98 \[ \frac {a^7\,x^5}{5}+\frac {7\,a^6\,b\,x^6}{6}+3\,a^5\,b^2\,x^7+\frac {35\,a^4\,b^3\,x^8}{8}+\frac {35\,a^3\,b^4\,x^9}{9}+\frac {21\,a^2\,b^5\,x^{10}}{10}+\frac {7\,a\,b^6\,x^{11}}{11}+\frac {b^7\,x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x)^7,x)

[Out]

(a^7*x^5)/5 + (b^7*x^12)/12 + (7*a^6*b*x^6)/6 + (7*a*b^6*x^11)/11 + 3*a^5*b^2*x^7 + (35*a^4*b^3*x^8)/8 + (35*a
^3*b^4*x^9)/9 + (21*a^2*b^5*x^10)/10

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sympy [A]  time = 0.10, size = 92, normalized size = 1.14 \[ \frac {a^{7} x^{5}}{5} + \frac {7 a^{6} b x^{6}}{6} + 3 a^{5} b^{2} x^{7} + \frac {35 a^{4} b^{3} x^{8}}{8} + \frac {35 a^{3} b^{4} x^{9}}{9} + \frac {21 a^{2} b^{5} x^{10}}{10} + \frac {7 a b^{6} x^{11}}{11} + \frac {b^{7} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x+a)**7,x)

[Out]

a**7*x**5/5 + 7*a**6*b*x**6/6 + 3*a**5*b**2*x**7 + 35*a**4*b**3*x**8/8 + 35*a**3*b**4*x**9/9 + 21*a**2*b**5*x*
*10/10 + 7*a*b**6*x**11/11 + b**7*x**12/12

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