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

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

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Rubi [A]  time = 0.07, antiderivative size = 105, 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 = {43} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 90, normalized size = 0.86 \begin {gather*} \frac {\frac {4 a^7}{(a+b x)^3}-\frac {42 a^6}{(a+b x)^2}+\frac {252 a^5}{a+b x}+420 a^4 \log (a+b x)-240 a^3 b x+60 a^2 b^2 x^2-16 a b^3 x^3+3 b^4 x^4}{12 b^8} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^7}{(a+b x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^7/(a + b*x)^4,x]

[Out]

IntegrateAlgebraic[x^7/(a + b*x)^4, x]

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fricas [A]  time = 1.04, size = 151, normalized size = 1.44 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.01, size = 95, normalized size = 0.90 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.01, size = 98, normalized size = 0.93 \begin {gather*} \frac {x^{4}}{4 b^{4}}+\frac {a^{7}}{3 \left (b x +a \right )^{3} b^{8}}-\frac {4 a \,x^{3}}{3 b^{5}}-\frac {7 a^{6}}{2 \left (b x +a \right )^{2} b^{8}}+\frac {5 a^{2} x^{2}}{b^{6}}+\frac {21 a^{5}}{\left (b x +a \right ) b^{8}}+\frac {35 a^{4} \ln \left (b x +a \right )}{b^{8}}-\frac {20 a^{3} x}{b^{7}} \end {gather*}

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 = 1.39, size = 114, normalized size = 1.09 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.22, size = 90, normalized size = 0.86 \begin {gather*} \frac {\frac {{\left (a+b\,x\right )}^4}{4}-\frac {7\,a\,{\left (a+b\,x\right )}^3}{3}+\frac {21\,a^2\,{\left (a+b\,x\right )}^2}{2}+\frac {21\,a^5}{a+b\,x}-\frac {7\,a^6}{2\,{\left (a+b\,x\right )}^2}+\frac {a^7}{3\,{\left (a+b\,x\right )}^3}+35\,a^4\,\ln \left (a+b\,x\right )-35\,a^3\,b\,x}{b^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x)^4,x)

[Out]

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

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sympy [A]  time = 0.48, size = 119, normalized size = 1.13 \begin {gather*} \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 {gather*}

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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