3.3.7 \(\int \frac {x^{10}}{(a+b x)^7} \, dx\) [207]

Optimal. Leaf size=150 \[ -\frac {84 a^3 x}{b^{10}}+\frac {14 a^2 x^2}{b^9}-\frac {7 a x^3}{3 b^8}+\frac {x^4}{4 b^7}-\frac {a^{10}}{6 b^{11} (a+b x)^6}+\frac {2 a^9}{b^{11} (a+b x)^5}-\frac {45 a^8}{4 b^{11} (a+b x)^4}+\frac {40 a^7}{b^{11} (a+b x)^3}-\frac {105 a^6}{b^{11} (a+b x)^2}+\frac {252 a^5}{b^{11} (a+b x)}+\frac {210 a^4 \log (a+b x)}{b^{11}} \]

[Out]

-84*a^3*x/b^10+14*a^2*x^2/b^9-7/3*a*x^3/b^8+1/4*x^4/b^7-1/6*a^10/b^11/(b*x+a)^6+2*a^9/b^11/(b*x+a)^5-45/4*a^8/
b^11/(b*x+a)^4+40*a^7/b^11/(b*x+a)^3-105*a^6/b^11/(b*x+a)^2+252*a^5/b^11/(b*x+a)+210*a^4*ln(b*x+a)/b^11

________________________________________________________________________________________

Rubi [A]
time = 0.10, antiderivative size = 150, 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^{10}}{6 b^{11} (a+b x)^6}+\frac {2 a^9}{b^{11} (a+b x)^5}-\frac {45 a^8}{4 b^{11} (a+b x)^4}+\frac {40 a^7}{b^{11} (a+b x)^3}-\frac {105 a^6}{b^{11} (a+b x)^2}+\frac {252 a^5}{b^{11} (a+b x)}+\frac {210 a^4 \log (a+b x)}{b^{11}}-\frac {84 a^3 x}{b^{10}}+\frac {14 a^2 x^2}{b^9}-\frac {7 a x^3}{3 b^8}+\frac {x^4}{4 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^10/(a + b*x)^7,x]

[Out]

(-84*a^3*x)/b^10 + (14*a^2*x^2)/b^9 - (7*a*x^3)/(3*b^8) + x^4/(4*b^7) - a^10/(6*b^11*(a + b*x)^6) + (2*a^9)/(b
^11*(a + b*x)^5) - (45*a^8)/(4*b^11*(a + b*x)^4) + (40*a^7)/(b^11*(a + b*x)^3) - (105*a^6)/(b^11*(a + b*x)^2)
+ (252*a^5)/(b^11*(a + b*x)) + (210*a^4*Log[a + b*x])/b^11

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 \frac {x^{10}}{(a+b x)^7} \, dx &=\int \left (-\frac {84 a^3}{b^{10}}+\frac {28 a^2 x}{b^9}-\frac {7 a x^2}{b^8}+\frac {x^3}{b^7}+\frac {a^{10}}{b^{10} (a+b x)^7}-\frac {10 a^9}{b^{10} (a+b x)^6}+\frac {45 a^8}{b^{10} (a+b x)^5}-\frac {120 a^7}{b^{10} (a+b x)^4}+\frac {210 a^6}{b^{10} (a+b x)^3}-\frac {252 a^5}{b^{10} (a+b x)^2}+\frac {210 a^4}{b^{10} (a+b x)}\right ) \, dx\\ &=-\frac {84 a^3 x}{b^{10}}+\frac {14 a^2 x^2}{b^9}-\frac {7 a x^3}{3 b^8}+\frac {x^4}{4 b^7}-\frac {a^{10}}{6 b^{11} (a+b x)^6}+\frac {2 a^9}{b^{11} (a+b x)^5}-\frac {45 a^8}{4 b^{11} (a+b x)^4}+\frac {40 a^7}{b^{11} (a+b x)^3}-\frac {105 a^6}{b^{11} (a+b x)^2}+\frac {252 a^5}{b^{11} (a+b x)}+\frac {210 a^4 \log (a+b x)}{b^{11}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 139, normalized size = 0.93 \begin {gather*} \frac {2131 a^{10}+10266 a^9 b x+18105 a^8 b^2 x^2+11540 a^7 b^3 x^3-3945 a^6 b^4 x^4-9138 a^5 b^5 x^5-4043 a^4 b^6 x^6-360 a^3 b^7 x^7+45 a^2 b^8 x^8-10 a b^9 x^9+3 b^{10} x^{10}+2520 a^4 (a+b x)^6 \log (a+b x)}{12 b^{11} (a+b x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^10/(a + b*x)^7,x]

[Out]

(2131*a^10 + 10266*a^9*b*x + 18105*a^8*b^2*x^2 + 11540*a^7*b^3*x^3 - 3945*a^6*b^4*x^4 - 9138*a^5*b^5*x^5 - 404
3*a^4*b^6*x^6 - 360*a^3*b^7*x^7 + 45*a^2*b^8*x^8 - 10*a*b^9*x^9 + 3*b^10*x^10 + 2520*a^4*(a + b*x)^6*Log[a + b
*x])/(12*b^11*(a + b*x)^6)

________________________________________________________________________________________

Mathics [A]
time = 3.73, size = 283, normalized size = 1.89 \begin {gather*} \frac {2131 a^{10}+2520 a^{10} \text {Log}\left [a+b x\right ]+10266 a^9 b x+15120 a^9 b x \text {Log}\left [a+b x\right ]+18105 a^8 b^2 x^2+37800 a^8 b^2 x^2 \text {Log}\left [a+b x\right ]+11540 a^7 b^3 x^3+50400 a^7 b^3 x^3 \text {Log}\left [a+b x\right ]-3945 a^6 b^4 x^4+37800 a^6 b^4 x^4 \text {Log}\left [a+b x\right ]-9138 a^5 b^5 x^5+15120 a^5 b^5 x^5 \text {Log}\left [a+b x\right ]-4043 a^4 b^6 x^6+2520 a^4 b^6 x^6 \text {Log}\left [a+b x\right ]-360 a^3 b^7 x^7+45 a^2 b^8 x^8-10 a b^9 x^9+3 b^{10} x^{10}}{12 b^{11} \left (a^6+6 a^5 b x+15 a^4 b^2 x^2+20 a^3 b^3 x^3+15 a^2 b^4 x^4+6 a b^5 x^5+b^6 x^6\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^10/(a + b*x)^7,x]')

[Out]

(2131 a ^ 10 + 2520 a ^ 10 Log[a + b x] + 10266 a ^ 9 b x + 15120 a ^ 9 b x Log[a + b x] + 18105 a ^ 8 b ^ 2 x
^ 2 + 37800 a ^ 8 b ^ 2 x ^ 2 Log[a + b x] + 11540 a ^ 7 b ^ 3 x ^ 3 + 50400 a ^ 7 b ^ 3 x ^ 3 Log[a + b x] -
3945 a ^ 6 b ^ 4 x ^ 4 + 37800 a ^ 6 b ^ 4 x ^ 4 Log[a + b x] - 9138 a ^ 5 b ^ 5 x ^ 5 + 15120 a ^ 5 b ^ 5 x
^ 5 Log[a + b x] - 4043 a ^ 4 b ^ 6 x ^ 6 + 2520 a ^ 4 b ^ 6 x ^ 6 Log[a + b x] - 360 a ^ 3 b ^ 7 x ^ 7 + 45 a
^ 2 b ^ 8 x ^ 8 - 10 a b ^ 9 x ^ 9 + 3 b ^ 10 x ^ 10) / (12 b ^ 11 (a ^ 6 + 6 a ^ 5 b x + 15 a ^ 4 b ^ 2 x ^
2 + 20 a ^ 3 b ^ 3 x ^ 3 + 15 a ^ 2 b ^ 4 x ^ 4 + 6 a b ^ 5 x ^ 5 + b ^ 6 x ^ 6))

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 144, normalized size = 0.96

method	result	size
risch	\(\frac {x^{4}}{4 b^{7}}-\frac {7 a \,x^{3}}{3 b^{8}}+\frac {14 a^{2} x^{2}}{b^{9}}-\frac {84 a^{3} x}{b^{10}}+\frac {252 a^{5} b^{4} x^{5}+1155 a^{6} b^{3} x^{4}+2140 a^{7} b^{2} x^{3}+\frac {7995 a^{8} b \,x^{2}}{4}+\frac {1879 a^{9} x}{2}+\frac {2131 a^{10}}{12 b}}{b^{10} \left (b x +a \right )^{6}}+\frac {210 a^{4} \ln \left (b x +a \right )}{b^{11}}\)	\(121\)
norman	\(\frac {\frac {x^{10}}{4 b}-\frac {5 a \,x^{9}}{6 b^{2}}-\frac {30 a^{3} x^{7}}{b^{4}}+\frac {1029 a^{10}}{2 b^{11}}+\frac {15 a^{2} x^{8}}{4 b^{3}}+\frac {1260 a^{5} x^{5}}{b^{6}}+\frac {4725 a^{6} x^{4}}{b^{7}}+\frac {7700 a^{7} x^{3}}{b^{8}}+\frac {13125 a^{8} x^{2}}{2 b^{9}}+\frac {2877 a^{9} x}{b^{10}}}{\left (b x +a \right )^{6}}+\frac {210 a^{4} \ln \left (b x +a \right )}{b^{11}}\)	\(125\)
default	\(-\frac {-\frac {1}{4} b^{3} x^{4}+\frac {7}{3} a \,b^{2} x^{3}-14 a^{2} b \,x^{2}+84 a^{3} x}{b^{10}}+\frac {252 a^{5}}{b^{11} \left (b x +a \right )}+\frac {2 a^{9}}{b^{11} \left (b x +a \right )^{5}}-\frac {45 a^{8}}{4 b^{11} \left (b x +a \right )^{4}}-\frac {105 a^{6}}{b^{11} \left (b x +a \right )^{2}}-\frac {a^{10}}{6 b^{11} \left (b x +a \right )^{6}}+\frac {210 a^{4} \ln \left (b x +a \right )}{b^{11}}+\frac {40 a^{7}}{b^{11} \left (b x +a \right )^{3}}\)	\(144\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

-1/b^10*(-1/4*b^3*x^4+7/3*a*b^2*x^3-14*a^2*b*x^2+84*a^3*x)+252*a^5/b^11/(b*x+a)+2*a^9/b^11/(b*x+a)^5-45/4*a^8/
b^11/(b*x+a)^4-105*a^6/b^11/(b*x+a)^2-1/6*a^10/b^11/(b*x+a)^6+210*a^4*ln(b*x+a)/b^11+40*a^7/b^11/(b*x+a)^3

________________________________________________________________________________________

Maxima [A]
time = 0.25, size = 180, normalized size = 1.20 \begin {gather*} \frac {3024 \, a^{5} b^{5} x^{5} + 13860 \, a^{6} b^{4} x^{4} + 25680 \, a^{7} b^{3} x^{3} + 23985 \, a^{8} b^{2} x^{2} + 11274 \, a^{9} b x + 2131 \, a^{10}}{12 \, {\left (b^{17} x^{6} + 6 \, a b^{16} x^{5} + 15 \, a^{2} b^{15} x^{4} + 20 \, a^{3} b^{14} x^{3} + 15 \, a^{4} b^{13} x^{2} + 6 \, a^{5} b^{12} x + a^{6} b^{11}\right )}} + \frac {210 \, a^{4} \log \left (b x + a\right )}{b^{11}} + \frac {3 \, b^{3} x^{4} - 28 \, a b^{2} x^{3} + 168 \, a^{2} b x^{2} - 1008 \, a^{3} x}{12 \, b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x+a)^7,x, algorithm="maxima")

[Out]

1/12*(3024*a^5*b^5*x^5 + 13860*a^6*b^4*x^4 + 25680*a^7*b^3*x^3 + 23985*a^8*b^2*x^2 + 11274*a^9*b*x + 2131*a^10
)/(b^17*x^6 + 6*a*b^16*x^5 + 15*a^2*b^15*x^4 + 20*a^3*b^14*x^3 + 15*a^4*b^13*x^2 + 6*a^5*b^12*x + a^6*b^11) +
210*a^4*log(b*x + a)/b^11 + 1/12*(3*b^3*x^4 - 28*a*b^2*x^3 + 168*a^2*b*x^2 - 1008*a^3*x)/b^10

________________________________________________________________________________________

Fricas [A]
time = 0.30, size = 250, normalized size = 1.67 \begin {gather*} \frac {3 \, b^{10} x^{10} - 10 \, a b^{9} x^{9} + 45 \, a^{2} b^{8} x^{8} - 360 \, a^{3} b^{7} x^{7} - 4043 \, a^{4} b^{6} x^{6} - 9138 \, a^{5} b^{5} x^{5} - 3945 \, a^{6} b^{4} x^{4} + 11540 \, a^{7} b^{3} x^{3} + 18105 \, a^{8} b^{2} x^{2} + 10266 \, a^{9} b x + 2131 \, a^{10} + 2520 \, {\left (a^{4} b^{6} x^{6} + 6 \, a^{5} b^{5} x^{5} + 15 \, a^{6} b^{4} x^{4} + 20 \, a^{7} b^{3} x^{3} + 15 \, a^{8} b^{2} x^{2} + 6 \, a^{9} b x + a^{10}\right )} \log \left (b x + a\right )}{12 \, {\left (b^{17} x^{6} + 6 \, a b^{16} x^{5} + 15 \, a^{2} b^{15} x^{4} + 20 \, a^{3} b^{14} x^{3} + 15 \, a^{4} b^{13} x^{2} + 6 \, a^{5} b^{12} x + a^{6} b^{11}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x+a)^7,x, algorithm="fricas")

[Out]

1/12*(3*b^10*x^10 - 10*a*b^9*x^9 + 45*a^2*b^8*x^8 - 360*a^3*b^7*x^7 - 4043*a^4*b^6*x^6 - 9138*a^5*b^5*x^5 - 39
45*a^6*b^4*x^4 + 11540*a^7*b^3*x^3 + 18105*a^8*b^2*x^2 + 10266*a^9*b*x + 2131*a^10 + 2520*(a^4*b^6*x^6 + 6*a^5
*b^5*x^5 + 15*a^6*b^4*x^4 + 20*a^7*b^3*x^3 + 15*a^8*b^2*x^2 + 6*a^9*b*x + a^10)*log(b*x + a))/(b^17*x^6 + 6*a*
b^16*x^5 + 15*a^2*b^15*x^4 + 20*a^3*b^14*x^3 + 15*a^4*b^13*x^2 + 6*a^5*b^12*x + a^6*b^11)

________________________________________________________________________________________

Sympy [A]
time = 0.46, size = 190, normalized size = 1.27 \begin {gather*} \frac {210 a^{4} \log {\left (a + b x \right )}}{b^{11}} - \frac {84 a^{3} x}{b^{10}} + \frac {14 a^{2} x^{2}}{b^{9}} - \frac {7 a x^{3}}{3 b^{8}} + \frac {2131 a^{10} + 11274 a^{9} b x + 23985 a^{8} b^{2} x^{2} + 25680 a^{7} b^{3} x^{3} + 13860 a^{6} b^{4} x^{4} + 3024 a^{5} b^{5} x^{5}}{12 a^{6} b^{11} + 72 a^{5} b^{12} x + 180 a^{4} b^{13} x^{2} + 240 a^{3} b^{14} x^{3} + 180 a^{2} b^{15} x^{4} + 72 a b^{16} x^{5} + 12 b^{17} x^{6}} + \frac {x^{4}}{4 b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10/(b*x+a)**7,x)

[Out]

210*a**4*log(a + b*x)/b**11 - 84*a**3*x/b**10 + 14*a**2*x**2/b**9 - 7*a*x**3/(3*b**8) + (2131*a**10 + 11274*a*
*9*b*x + 23985*a**8*b**2*x**2 + 25680*a**7*b**3*x**3 + 13860*a**6*b**4*x**4 + 3024*a**5*b**5*x**5)/(12*a**6*b*
*11 + 72*a**5*b**12*x + 180*a**4*b**13*x**2 + 240*a**3*b**14*x**3 + 180*a**2*b**15*x**4 + 72*a*b**16*x**5 + 12
*b**17*x**6) + x**4/(4*b**7)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 139, normalized size = 0.93 \begin {gather*} \frac {\frac {1}{4} x^{4} b^{21}-\frac {7}{3} x^{3} b^{20} a+14 x^{2} b^{19} a^{2}-84 x b^{18} a^{3}}{b^{28}}+\frac {\frac {1}{12} \left (3024 b^{5} a^{5} x^{5}+13860 b^{4} a^{6} x^{4}+25680 b^{3} a^{7} x^{3}+23985 b^{2} a^{8} x^{2}+11274 b a^{9} x+2131 a^{10}\right )}{b^{11} \left (x b+a\right )^{6}}+\frac {210 a^{4} \ln \left |x b+a\right |}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x+a)^7,x)

[Out]

210*a^4*log(abs(b*x + a))/b^11 + 1/12*(3024*a^5*b^5*x^5 + 13860*a^6*b^4*x^4 + 25680*a^7*b^3*x^3 + 23985*a^8*b^
2*x^2 + 11274*a^9*b*x + 2131*a^10)/((b*x + a)^6*b^11) + 1/12*(3*b^21*x^4 - 28*a*b^20*x^3 + 168*a^2*b^19*x^2 -
1008*a^3*b^18*x)/b^28

________________________________________________________________________________________

Mupad [B]
time = 1.09, size = 126, normalized size = 0.84 \begin {gather*} \frac {\frac {{\left (a+b\,x\right )}^4}{4}-\frac {10\,a\,{\left (a+b\,x\right )}^3}{3}+\frac {45\,a^2\,{\left (a+b\,x\right )}^2}{2}+\frac {252\,a^5}{a+b\,x}-\frac {105\,a^6}{{\left (a+b\,x\right )}^2}+\frac {40\,a^7}{{\left (a+b\,x\right )}^3}-\frac {45\,a^8}{4\,{\left (a+b\,x\right )}^4}+\frac {2\,a^9}{{\left (a+b\,x\right )}^5}-\frac {a^{10}}{6\,{\left (a+b\,x\right )}^6}+210\,a^4\,\ln \left (a+b\,x\right )-120\,a^3\,b\,x}{b^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(a + b*x)^7,x)

[Out]

((a + b*x)^4/4 - (10*a*(a + b*x)^3)/3 + (45*a^2*(a + b*x)^2)/2 + (252*a^5)/(a + b*x) - (105*a^6)/(a + b*x)^2 +
(40*a^7)/(a + b*x)^3 - (45*a^8)/(4*(a + b*x)^4) + (2*a^9)/(a + b*x)^5 - a^10/(6*(a + b*x)^6) + 210*a^4*log(a
+ b*x) - 120*a^3*b*x)/b^11

________________________________________________________________________________________