3.194 \(\int \frac{x^7}{(a+b x)^4} \, dx\)

Optimal. Leaf size=105 \[ \frac{5 a^2 x^2}{b^6}+\frac{a^7}{3 b^8 (a+b x)^3}-\frac{7 a^6}{2 b^8 (a+b x)^2}+\frac{21 a^5}{b^8 (a+b x)}-\frac{20 a^3 x}{b^7}+\frac{35 a^4 \log (a+b x)}{b^8}-\frac{4 a x^3}{3 b^5}+\frac{x^4}{4 b^4} \]

[Out]

(-20*a^3*x)/b^7 + (5*a^2*x^2)/b^6 - (4*a*x^3)/(3*b^5) + x^4/(4*b^4) + a^7/(3*b^8*(a + b*x)^3) - (7*a^6)/(2*b^8
*(a + b*x)^2) + (21*a^5)/(b^8*(a + b*x)) + (35*a^4*Log[a + b*x])/b^8

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Rubi [A]  time = 0.0701249, antiderivative size = 105, 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} \[ \frac{5 a^2 x^2}{b^6}+\frac{a^7}{3 b^8 (a+b x)^3}-\frac{7 a^6}{2 b^8 (a+b x)^2}+\frac{21 a^5}{b^8 (a+b x)}-\frac{20 a^3 x}{b^7}+\frac{35 a^4 \log (a+b x)}{b^8}-\frac{4 a x^3}{3 b^5}+\frac{x^4}{4 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x)^4,x]

[Out]

(-20*a^3*x)/b^7 + (5*a^2*x^2)/b^6 - (4*a*x^3)/(3*b^5) + x^4/(4*b^4) + a^7/(3*b^8*(a + b*x)^3) - (7*a^6)/(2*b^8
*(a + b*x)^2) + (21*a^5)/(b^8*(a + b*x)) + (35*a^4*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)^4} \, dx &=\int \left (-\frac{20 a^3}{b^7}+\frac{10 a^2 x}{b^6}-\frac{4 a x^2}{b^5}+\frac{x^3}{b^4}-\frac{a^7}{b^7 (a+b x)^4}+\frac{7 a^6}{b^7 (a+b x)^3}-\frac{21 a^5}{b^7 (a+b x)^2}+\frac{35 a^4}{b^7 (a+b x)}\right ) \, dx\\ &=-\frac{20 a^3 x}{b^7}+\frac{5 a^2 x^2}{b^6}-\frac{4 a x^3}{3 b^5}+\frac{x^4}{4 b^4}+\frac{a^7}{3 b^8 (a+b x)^3}-\frac{7 a^6}{2 b^8 (a+b x)^2}+\frac{21 a^5}{b^8 (a+b x)}+\frac{35 a^4 \log (a+b x)}{b^8}\\ \end{align*}

Mathematica [A]  time = 0.0513451, size = 90, normalized size = 0.86 \[ \frac{60 a^2 b^2 x^2+\frac{4 a^7}{(a+b x)^3}-\frac{42 a^6}{(a+b x)^2}+\frac{252 a^5}{a+b x}-240 a^3 b x+420 a^4 \log (a+b x)-16 a b^3 x^3+3 b^4 x^4}{12 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x)^4,x]

[Out]

(-240*a^3*b*x + 60*a^2*b^2*x^2 - 16*a*b^3*x^3 + 3*b^4*x^4 + (4*a^7)/(a + b*x)^3 - (42*a^6)/(a + b*x)^2 + (252*
a^5)/(a + b*x) + 420*a^4*Log[a + b*x])/(12*b^8)

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Maple [A]  time = 0.008, size = 98, normalized size = 0.9 \begin{align*} -20\,{\frac{{a}^{3}x}{{b}^{7}}}+5\,{\frac{{a}^{2}{x}^{2}}{{b}^{6}}}-{\frac{4\,a{x}^{3}}{3\,{b}^{5}}}+{\frac{{x}^{4}}{4\,{b}^{4}}}+{\frac{{a}^{7}}{3\,{b}^{8} \left ( bx+a \right ) ^{3}}}-{\frac{7\,{a}^{6}}{2\,{b}^{8} \left ( bx+a \right ) ^{2}}}+21\,{\frac{{a}^{5}}{{b}^{8} \left ( bx+a \right ) }}+35\,{\frac{{a}^{4}\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)^4,x)

[Out]

-20*a^3*x/b^7+5*a^2*x^2/b^6-4/3*a*x^3/b^5+1/4*x^4/b^4+1/3*a^7/b^8/(b*x+a)^3-7/2*a^6/b^8/(b*x+a)^2+21*a^5/b^8/(
b*x+a)+35*a^4*ln(b*x+a)/b^8

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Maxima [A]  time = 1.05678, size = 154, normalized size = 1.47 \begin{align*} \frac{126 \, a^{5} b^{2} x^{2} + 231 \, a^{6} b x + 107 \, a^{7}}{6 \,{\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} + \frac{35 \, a^{4} \log \left (b x + a\right )}{b^{8}} + \frac{3 \, b^{3} x^{4} - 16 \, a b^{2} x^{3} + 60 \, a^{2} b x^{2} - 240 \, a^{3} x}{12 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(126*a^5*b^2*x^2 + 231*a^6*b*x + 107*a^7)/(b^11*x^3 + 3*a*b^10*x^2 + 3*a^2*b^9*x + a^3*b^8) + 35*a^4*log(b
*x + a)/b^8 + 1/12*(3*b^3*x^4 - 16*a*b^2*x^3 + 60*a^2*b*x^2 - 240*a^3*x)/b^7

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Fricas [A]  time = 1.56855, size = 329, normalized size = 3.13 \begin{align*} \frac{3 \, b^{7} x^{7} - 7 \, a b^{6} x^{6} + 21 \, a^{2} b^{5} x^{5} - 105 \, a^{3} b^{4} x^{4} - 556 \, a^{4} b^{3} x^{3} - 408 \, a^{5} b^{2} x^{2} + 222 \, a^{6} b x + 214 \, a^{7} + 420 \,{\left (a^{4} b^{3} x^{3} + 3 \, a^{5} b^{2} x^{2} + 3 \, a^{6} b x + a^{7}\right )} \log \left (b x + a\right )}{12 \,{\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^7*x^7 - 7*a*b^6*x^6 + 21*a^2*b^5*x^5 - 105*a^3*b^4*x^4 - 556*a^4*b^3*x^3 - 408*a^5*b^2*x^2 + 222*a^6
*b*x + 214*a^7 + 420*(a^4*b^3*x^3 + 3*a^5*b^2*x^2 + 3*a^6*b*x + a^7)*log(b*x + a))/(b^11*x^3 + 3*a*b^10*x^2 +
3*a^2*b^9*x + a^3*b^8)

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Sympy [A]  time = 0.792181, size = 119, normalized size = 1.13 \begin{align*} \frac{35 a^{4} \log{\left (a + b x \right )}}{b^{8}} - \frac{20 a^{3} x}{b^{7}} + \frac{5 a^{2} x^{2}}{b^{6}} - \frac{4 a x^{3}}{3 b^{5}} + \frac{107 a^{7} + 231 a^{6} b x + 126 a^{5} b^{2} x^{2}}{6 a^{3} b^{8} + 18 a^{2} b^{9} x + 18 a b^{10} x^{2} + 6 b^{11} x^{3}} + \frac{x^{4}}{4 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35*a**4*log(a + b*x)/b**8 - 20*a**3*x/b**7 + 5*a**2*x**2/b**6 - 4*a*x**3/(3*b**5) + (107*a**7 + 231*a**6*b*x +
126*a**5*b**2*x**2)/(6*a**3*b**8 + 18*a**2*b**9*x + 18*a*b**10*x**2 + 6*b**11*x**3) + x**4/(4*b**4)

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Giac [A]  time = 1.19132, size = 128, normalized size = 1.22 \begin{align*} \frac{35 \, a^{4} \log \left ({\left | b x + a \right |}\right )}{b^{8}} + \frac{126 \, a^{5} b^{2} x^{2} + 231 \, a^{6} b x + 107 \, a^{7}}{6 \,{\left (b x + a\right )}^{3} b^{8}} + \frac{3 \, b^{12} x^{4} - 16 \, a b^{11} x^{3} + 60 \, a^{2} b^{10} x^{2} - 240 \, a^{3} b^{9} x}{12 \, b^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35*a^4*log(abs(b*x + a))/b^8 + 1/6*(126*a^5*b^2*x^2 + 231*a^6*b*x + 107*a^7)/((b*x + a)^3*b^8) + 1/12*(3*b^12*
x^4 - 16*a*b^11*x^3 + 60*a^2*b^10*x^2 - 240*a^3*b^9*x)/b^16