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

Optimal. Leaf size=99 \[ -\frac{5 a^3 x^2}{b^6}+\frac{2 a^2 x^3}{b^5}+\frac{a^7}{2 b^8 (a+b x)^2}-\frac{7 a^6}{b^8 (a+b x)}+\frac{15 a^4 x}{b^7}-\frac{21 a^5 \log (a+b x)}{b^8}-\frac{3 a x^4}{4 b^4}+\frac{x^5}{5 b^3} \]

[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

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Rubi [A]  time = 0.0703669, antiderivative size = 99, normalized size of antiderivative = 1., 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} \[ -\frac{5 a^3 x^2}{b^6}+\frac{2 a^2 x^3}{b^5}+\frac{a^7}{2 b^8 (a+b x)^2}-\frac{7 a^6}{b^8 (a+b x)}+\frac{15 a^4 x}{b^7}-\frac{21 a^5 \log (a+b x)}{b^8}-\frac{3 a x^4}{4 b^4}+\frac{x^5}{5 b^3} \]

Antiderivative was successfully verified.

[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 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 \frac{x^7}{(a+b x)^3} \, dx &=\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]  time = 0.0574518, size = 89, normalized size = 0.9 \[ \frac{-100 a^3 b^2 x^2+40 a^2 b^3 x^3+\frac{10 a^7}{(a+b x)^2}-\frac{140 a^6}{a+b x}+300 a^4 b x-420 a^5 \log (a+b x)-15 a b^4 x^4+4 b^5 x^5}{20 b^8} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.007, size = 94, normalized size = 1. \begin{align*} 15\,{\frac{{a}^{4}x}{{b}^{7}}}-5\,{\frac{{a}^{3}{x}^{2}}{{b}^{6}}}+2\,{\frac{{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} \left ( bx+a \right ) ^{2}}}-7\,{\frac{{a}^{6}}{{b}^{8} \left ( bx+a \right ) }}-21\,{\frac{{a}^{5}\ln \left ( bx+a \right ) }{{b}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 1.06132, size = 139, normalized size = 1.4 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.49915, size = 284, normalized size = 2.87 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.730562, size = 107, normalized size = 1.08 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.16591, size = 128, normalized size = 1.29 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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