3.2.1 \(\int x^5 (a+b x)^7 \, dx\) [101]

Optimal. Leaf size=96 \[ -\frac {a^5 (a+b x)^8}{8 b^6}+\frac {5 a^4 (a+b x)^9}{9 b^6}-\frac {a^3 (a+b x)^{10}}{b^6}+\frac {10 a^2 (a+b x)^{11}}{11 b^6}-\frac {5 a (a+b x)^{12}}{12 b^6}+\frac {(a+b x)^{13}}{13 b^6} \]

[Out]

-1/8*a^5*(b*x+a)^8/b^6+5/9*a^4*(b*x+a)^9/b^6-a^3*(b*x+a)^10/b^6+10/11*a^2*(b*x+a)^11/b^6-5/12*a*(b*x+a)^12/b^6
+1/13*(b*x+a)^13/b^6

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Rubi [A]
time = 0.03, antiderivative size = 96, 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^5 (a+b x)^8}{8 b^6}+\frac {5 a^4 (a+b x)^9}{9 b^6}-\frac {a^3 (a+b x)^{10}}{b^6}+\frac {10 a^2 (a+b x)^{11}}{11 b^6}+\frac {(a+b x)^{13}}{13 b^6}-\frac {5 a (a+b x)^{12}}{12 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x)^7,x]

[Out]

-1/8*(a^5*(a + b*x)^8)/b^6 + (5*a^4*(a + b*x)^9)/(9*b^6) - (a^3*(a + b*x)^10)/b^6 + (10*a^2*(a + b*x)^11)/(11*
b^6) - (5*a*(a + b*x)^12)/(12*b^6) + (a + b*x)^13/(13*b^6)

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^5 (a+b x)^7 \, dx &=\int \left (-\frac {a^5 (a+b x)^7}{b^5}+\frac {5 a^4 (a+b x)^8}{b^5}-\frac {10 a^3 (a+b x)^9}{b^5}+\frac {10 a^2 (a+b x)^{10}}{b^5}-\frac {5 a (a+b x)^{11}}{b^5}+\frac {(a+b x)^{12}}{b^5}\right ) \, dx\\ &=-\frac {a^5 (a+b x)^8}{8 b^6}+\frac {5 a^4 (a+b x)^9}{9 b^6}-\frac {a^3 (a+b x)^{10}}{b^6}+\frac {10 a^2 (a+b x)^{11}}{11 b^6}-\frac {5 a (a+b x)^{12}}{12 b^6}+\frac {(a+b x)^{13}}{13 b^6}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 92, normalized size = 0.96 \begin {gather*} \frac {a^7 x^6}{6}+a^6 b x^7+\frac {21}{8} a^5 b^2 x^8+\frac {35}{9} a^4 b^3 x^9+\frac {7}{2} a^3 b^4 x^{10}+\frac {21}{11} a^2 b^5 x^{11}+\frac {7}{12} a b^6 x^{12}+\frac {b^7 x^{13}}{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x)^7,x]

[Out]

(a^7*x^6)/6 + a^6*b*x^7 + (21*a^5*b^2*x^8)/8 + (35*a^4*b^3*x^9)/9 + (7*a^3*b^4*x^10)/2 + (21*a^2*b^5*x^11)/11
+ (7*a*b^6*x^12)/12 + (b^7*x^13)/13

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Maple [A]
time = 0.07, size = 79, normalized size = 0.82

method	result	size
gosper	\(\frac {1}{13} b^{7} x^{13}+\frac {7}{12} a \,b^{6} x^{12}+\frac {21}{11} a^{2} b^{5} x^{11}+\frac {7}{2} a^{3} b^{4} x^{10}+\frac {35}{9} a^{4} b^{3} x^{9}+\frac {21}{8} a^{5} b^{2} x^{8}+a^{6} b \,x^{7}+\frac {1}{6} a^{7} x^{6}\)	\(79\)
default	\(\frac {1}{13} b^{7} x^{13}+\frac {7}{12} a \,b^{6} x^{12}+\frac {21}{11} a^{2} b^{5} x^{11}+\frac {7}{2} a^{3} b^{4} x^{10}+\frac {35}{9} a^{4} b^{3} x^{9}+\frac {21}{8} a^{5} b^{2} x^{8}+a^{6} b \,x^{7}+\frac {1}{6} a^{7} x^{6}\)	\(79\)
norman	\(\frac {1}{13} b^{7} x^{13}+\frac {7}{12} a \,b^{6} x^{12}+\frac {21}{11} a^{2} b^{5} x^{11}+\frac {7}{2} a^{3} b^{4} x^{10}+\frac {35}{9} a^{4} b^{3} x^{9}+\frac {21}{8} a^{5} b^{2} x^{8}+a^{6} b \,x^{7}+\frac {1}{6} a^{7} x^{6}\)	\(79\)
risch	\(\frac {1}{13} b^{7} x^{13}+\frac {7}{12} a \,b^{6} x^{12}+\frac {21}{11} a^{2} b^{5} x^{11}+\frac {7}{2} a^{3} b^{4} x^{10}+\frac {35}{9} a^{4} b^{3} x^{9}+\frac {21}{8} a^{5} b^{2} x^{8}+a^{6} b \,x^{7}+\frac {1}{6} a^{7} x^{6}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

1/13*b^7*x^13+7/12*a*b^6*x^12+21/11*a^2*b^5*x^11+7/2*a^3*b^4*x^10+35/9*a^4*b^3*x^9+21/8*a^5*b^2*x^8+a^6*b*x^7+
1/6*a^7*x^6

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Maxima [A]
time = 0.27, size = 78, normalized size = 0.81 \begin {gather*} \frac {1}{13} \, b^{7} x^{13} + \frac {7}{12} \, a b^{6} x^{12} + \frac {21}{11} \, a^{2} b^{5} x^{11} + \frac {7}{2} \, a^{3} b^{4} x^{10} + \frac {35}{9} \, a^{4} b^{3} x^{9} + \frac {21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac {1}{6} \, a^{7} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^7,x, algorithm="maxima")

[Out]

1/13*b^7*x^13 + 7/12*a*b^6*x^12 + 21/11*a^2*b^5*x^11 + 7/2*a^3*b^4*x^10 + 35/9*a^4*b^3*x^9 + 21/8*a^5*b^2*x^8
+ a^6*b*x^7 + 1/6*a^7*x^6

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Fricas [A]
time = 0.96, size = 78, normalized size = 0.81 \begin {gather*} \frac {1}{13} \, b^{7} x^{13} + \frac {7}{12} \, a b^{6} x^{12} + \frac {21}{11} \, a^{2} b^{5} x^{11} + \frac {7}{2} \, a^{3} b^{4} x^{10} + \frac {35}{9} \, a^{4} b^{3} x^{9} + \frac {21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac {1}{6} \, a^{7} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^7,x, algorithm="fricas")

[Out]

1/13*b^7*x^13 + 7/12*a*b^6*x^12 + 21/11*a^2*b^5*x^11 + 7/2*a^3*b^4*x^10 + 35/9*a^4*b^3*x^9 + 21/8*a^5*b^2*x^8
+ a^6*b*x^7 + 1/6*a^7*x^6

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Sympy [A]
time = 0.01, size = 90, normalized size = 0.94 \begin {gather*} \frac {a^{7} x^{6}}{6} + a^{6} b x^{7} + \frac {21 a^{5} b^{2} x^{8}}{8} + \frac {35 a^{4} b^{3} x^{9}}{9} + \frac {7 a^{3} b^{4} x^{10}}{2} + \frac {21 a^{2} b^{5} x^{11}}{11} + \frac {7 a b^{6} x^{12}}{12} + \frac {b^{7} x^{13}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x+a)**7,x)

[Out]

a**7*x**6/6 + a**6*b*x**7 + 21*a**5*b**2*x**8/8 + 35*a**4*b**3*x**9/9 + 7*a**3*b**4*x**10/2 + 21*a**2*b**5*x**
11/11 + 7*a*b**6*x**12/12 + b**7*x**13/13

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Giac [A]
time = 1.12, size = 78, normalized size = 0.81 \begin {gather*} \frac {1}{13} \, b^{7} x^{13} + \frac {7}{12} \, a b^{6} x^{12} + \frac {21}{11} \, a^{2} b^{5} x^{11} + \frac {7}{2} \, a^{3} b^{4} x^{10} + \frac {35}{9} \, a^{4} b^{3} x^{9} + \frac {21}{8} \, a^{5} b^{2} x^{8} + a^{6} b x^{7} + \frac {1}{6} \, a^{7} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^7,x, algorithm="giac")

[Out]

1/13*b^7*x^13 + 7/12*a*b^6*x^12 + 21/11*a^2*b^5*x^11 + 7/2*a^3*b^4*x^10 + 35/9*a^4*b^3*x^9 + 21/8*a^5*b^2*x^8
+ a^6*b*x^7 + 1/6*a^7*x^6

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Mupad [B]
time = 0.06, size = 78, normalized size = 0.81 \begin {gather*} \frac {a^7\,x^6}{6}+a^6\,b\,x^7+\frac {21\,a^5\,b^2\,x^8}{8}+\frac {35\,a^4\,b^3\,x^9}{9}+\frac {7\,a^3\,b^4\,x^{10}}{2}+\frac {21\,a^2\,b^5\,x^{11}}{11}+\frac {7\,a\,b^6\,x^{12}}{12}+\frac {b^7\,x^{13}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x)^7,x)

[Out]

(a^7*x^6)/6 + (b^7*x^13)/13 + a^6*b*x^7 + (7*a*b^6*x^12)/12 + (21*a^5*b^2*x^8)/8 + (35*a^4*b^3*x^9)/9 + (7*a^3
*b^4*x^10)/2 + (21*a^2*b^5*x^11)/11

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