\(\int \frac {x^7}{(a+b x)^3} \, dx\) [180]

Optimal result

Integrand size = 11, antiderivative size = 99 \[ \int \frac {x^7}{(a+b x)^3} \, dx=\frac {15 a^4 x}{b^7}-\frac {5 a^3 x^2}{b^6}+\frac {2 a^2 x^3}{b^5}-\frac {3 a x^4}{4 b^4}+\frac {x^5}{5 b^3}+\frac {a^7}{2 b^8 (a+b x)^2}-\frac {7 a^6}{b^8 (a+b x)}-\frac {21 a^5 \log (a+b x)}{b^8} \]

[Out]

15*a^4*x/b^7-5*a^3*x^2/b^6+2*a^2*x^3/b^5-3/4*a*x^4/b^4+1/5*x^5/b^3+1/2*a^7/b^8/(b*x+a)^2-7*a^6/b^8/(b*x+a)-21*
a^5*ln(b*x+a)/b^8

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^7}{(a+b x)^3} \, dx=\frac {a^7}{2 b^8 (a+b x)^2}-\frac {7 a^6}{b^8 (a+b x)}-\frac {21 a^5 \log (a+b x)}{b^8}+\frac {15 a^4 x}{b^7}-\frac {5 a^3 x^2}{b^6}+\frac {2 a^2 x^3}{b^5}-\frac {3 a x^4}{4 b^4}+\frac {x^5}{5 b^3} \]

[In]

Int[x^7/(a + b*x)^3,x]

[Out]

(15*a^4*x)/b^7 - (5*a^3*x^2)/b^6 + (2*a^2*x^3)/b^5 - (3*a*x^4)/(4*b^4) + x^5/(5*b^3) + a^7/(2*b^8*(a + b*x)^2)
- (7*a^6)/(b^8*(a + b*x)) - (21*a^5*Log[a + b*x])/b^8

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*} \text {integral}& = \int \left (\frac {15 a^4}{b^7}-\frac {10 a^3 x}{b^6}+\frac {6 a^2 x^2}{b^5}-\frac {3 a x^3}{b^4}+\frac {x^4}{b^3}-\frac {a^7}{b^7 (a+b x)^3}+\frac {7 a^6}{b^7 (a+b x)^2}-\frac {21 a^5}{b^7 (a+b x)}\right ) \, dx \\ & = \frac {15 a^4 x}{b^7}-\frac {5 a^3 x^2}{b^6}+\frac {2 a^2 x^3}{b^5}-\frac {3 a x^4}{4 b^4}+\frac {x^5}{5 b^3}+\frac {a^7}{2 b^8 (a+b x)^2}-\frac {7 a^6}{b^8 (a+b x)}-\frac {21 a^5 \log (a+b x)}{b^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.90 \[ \int \frac {x^7}{(a+b x)^3} \, dx=\frac {300 a^4 b x-100 a^3 b^2 x^2+40 a^2 b^3 x^3-15 a b^4 x^4+4 b^5 x^5+\frac {10 a^7}{(a+b x)^2}-\frac {140 a^6}{a+b x}-420 a^5 \log (a+b x)}{20 b^8} \]

[In]

Integrate[x^7/(a + b*x)^3,x]

[Out]

(300*a^4*b*x - 100*a^3*b^2*x^2 + 40*a^2*b^3*x^3 - 15*a*b^4*x^4 + 4*b^5*x^5 + (10*a^7)/(a + b*x)^2 - (140*a^6)/
(a + b*x) - 420*a^5*Log[a + b*x])/(20*b^8)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91

method	result	size
risch	\(\frac {x^{5}}{5 b^{3}}-\frac {3 a \,x^{4}}{4 b^{4}}+\frac {2 a^{2} x^{3}}{b^{5}}-\frac {5 a^{3} x^{2}}{b^{6}}+\frac {15 a^{4} x}{b^{7}}+\frac {-7 a^{6} x -\frac {13 a^{7}}{2 b}}{b^{7} \left (b x +a \right )^{2}}-\frac {21 a^{5} \ln \left (b x +a \right )}{b^{8}}\)	\(90\)
norman	\(\frac {\frac {x^{7}}{5 b}-\frac {7 a \,x^{6}}{20 b^{2}}+\frac {7 a^{2} x^{5}}{10 b^{3}}-\frac {63 a^{7}}{2 b^{8}}-\frac {7 a^{3} x^{4}}{4 b^{4}}+\frac {7 a^{4} x^{3}}{b^{5}}-\frac {42 a^{6} x}{b^{7}}}{\left (b x +a \right )^{2}}-\frac {21 a^{5} \ln \left (b x +a \right )}{b^{8}}\)	\(92\)
default	\(\frac {\frac {1}{5} b^{4} x^{5}-\frac {3}{4} a \,b^{3} x^{4}+2 a^{2} b^{2} x^{3}-5 a^{3} b \,x^{2}+15 a^{4} x}{b^{7}}-\frac {21 a^{5} \ln \left (b x +a \right )}{b^{8}}+\frac {a^{7}}{2 b^{8} \left (b x +a \right )^{2}}-\frac {7 a^{6}}{b^{8} \left (b x +a \right )}\)	\(94\)
parallelrisch	\(-\frac {-4 b^{7} x^{7}+7 a \,b^{6} x^{6}-14 a^{2} b^{5} x^{5}+35 a^{3} b^{4} x^{4}+420 \ln \left (b x +a \right ) x^{2} a^{5} b^{2}-140 a^{4} b^{3} x^{3}+840 \ln \left (b x +a \right ) x \,a^{6} b +420 \ln \left (b x +a \right ) a^{7}+840 a^{6} b x +630 a^{7}}{20 b^{8} \left (b x +a \right )^{2}}\)	\(117\)

[In]

int(x^7/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/5*x^5/b^3-3/4*a*x^4/b^4+2*a^2*x^3/b^5-5*a^3*x^2/b^6+15*a^4*x/b^7+(-7*a^6*x-13/2*a^7/b)/b^7/(b*x+a)^2-21*a^5*
ln(b*x+a)/b^8

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.30 \[ \int \frac {x^7}{(a+b x)^3} \, dx=\frac {4 \, b^{7} x^{7} - 7 \, a b^{6} x^{6} + 14 \, a^{2} b^{5} x^{5} - 35 \, a^{3} b^{4} x^{4} + 140 \, a^{4} b^{3} x^{3} + 500 \, a^{5} b^{2} x^{2} + 160 \, a^{6} b x - 130 \, a^{7} - 420 \, {\left (a^{5} b^{2} x^{2} + 2 \, a^{6} b x + a^{7}\right )} \log \left (b x + a\right )}{20 \, {\left (b^{10} x^{2} + 2 \, a b^{9} x + a^{2} b^{8}\right )}} \]

[In]

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

[Out]

1/20*(4*b^7*x^7 - 7*a*b^6*x^6 + 14*a^2*b^5*x^5 - 35*a^3*b^4*x^4 + 140*a^4*b^3*x^3 + 500*a^5*b^2*x^2 + 160*a^6*
b*x - 130*a^7 - 420*(a^5*b^2*x^2 + 2*a^6*b*x + a^7)*log(b*x + a))/(b^10*x^2 + 2*a*b^9*x + a^2*b^8)

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.10 \[ \int \frac {x^7}{(a+b x)^3} \, dx=- \frac {21 a^{5} \log {\left (a + b x \right )}}{b^{8}} + \frac {15 a^{4} x}{b^{7}} - \frac {5 a^{3} x^{2}}{b^{6}} + \frac {2 a^{2} x^{3}}{b^{5}} - \frac {3 a x^{4}}{4 b^{4}} + \frac {- 13 a^{7} - 14 a^{6} b x}{2 a^{2} b^{8} + 4 a b^{9} x + 2 b^{10} x^{2}} + \frac {x^{5}}{5 b^{3}} \]

[In]

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

[Out]

-21*a**5*log(a + b*x)/b**8 + 15*a**4*x/b**7 - 5*a**3*x**2/b**6 + 2*a**2*x**3/b**5 - 3*a*x**4/(4*b**4) + (-13*a
**7 - 14*a**6*b*x)/(2*a**2*b**8 + 4*a*b**9*x + 2*b**10*x**2) + x**5/(5*b**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.04 \[ \int \frac {x^7}{(a+b x)^3} \, dx=-\frac {14 \, a^{6} b x + 13 \, a^{7}}{2 \, {\left (b^{10} x^{2} + 2 \, a b^{9} x + a^{2} b^{8}\right )}} - \frac {21 \, a^{5} \log \left (b x + a\right )}{b^{8}} + \frac {4 \, b^{4} x^{5} - 15 \, a b^{3} x^{4} + 40 \, a^{2} b^{2} x^{3} - 100 \, a^{3} b x^{2} + 300 \, a^{4} x}{20 \, b^{7}} \]

[In]

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

[Out]

-1/2*(14*a^6*b*x + 13*a^7)/(b^10*x^2 + 2*a*b^9*x + a^2*b^8) - 21*a^5*log(b*x + a)/b^8 + 1/20*(4*b^4*x^5 - 15*a
*b^3*x^4 + 40*a^2*b^2*x^3 - 100*a^3*b*x^2 + 300*a^4*x)/b^7

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.96 \[ \int \frac {x^7}{(a+b x)^3} \, dx=-\frac {21 \, a^{5} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {14 \, a^{6} b x + 13 \, a^{7}}{2 \, {\left (b x + a\right )}^{2} b^{8}} + \frac {4 \, b^{12} x^{5} - 15 \, a b^{11} x^{4} + 40 \, a^{2} b^{10} x^{3} - 100 \, a^{3} b^{9} x^{2} + 300 \, a^{4} b^{8} x}{20 \, b^{15}} \]

[In]

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

[Out]

-21*a^5*log(abs(b*x + a))/b^8 - 1/2*(14*a^6*b*x + 13*a^7)/((b*x + a)^2*b^8) + 1/20*(4*b^12*x^5 - 15*a*b^11*x^4
+ 40*a^2*b^10*x^3 - 100*a^3*b^9*x^2 + 300*a^4*b^8*x)/b^15

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92 \[ \int \frac {x^7}{(a+b x)^3} \, dx=-\frac {\frac {7\,a\,{\left (a+b\,x\right )}^4}{4}-\frac {{\left (a+b\,x\right )}^5}{5}-7\,a^2\,{\left (a+b\,x\right )}^3+\frac {35\,a^3\,{\left (a+b\,x\right )}^2}{2}+\frac {7\,a^6}{a+b\,x}-\frac {a^7}{2\,{\left (a+b\,x\right )}^2}+21\,a^5\,\ln \left (a+b\,x\right )-35\,a^4\,b\,x}{b^8} \]

[In]

int(x^7/(a + b*x)^3,x)

[Out]

-((7*a*(a + b*x)^4)/4 - (a + b*x)^5/5 - 7*a^2*(a + b*x)^3 + (35*a^3*(a + b*x)^2)/2 + (7*a^6)/(a + b*x) - a^7/(
2*(a + b*x)^2) + 21*a^5*log(a + b*x) - 35*a^4*b*x)/b^8