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

Optimal. Leaf size=52 \[ \frac{x^4}{60 a^3 (a+b x)^4}+\frac{x^4}{15 a^2 (a+b x)^5}+\frac{x^4}{6 a (a+b x)^6} \]

[Out]

x^4/(6*a*(a + b*x)^6) + x^4/(15*a^2*(a + b*x)^5) + x^4/(60*a^3*(a + b*x)^4)

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Rubi [A]  time = 0.0293148, antiderivative size = 64, normalized size of antiderivative = 1.23, 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} \[ \frac{a^3}{6 b^4 (a+b x)^6}-\frac{3 a^2}{5 b^4 (a+b x)^5}+\frac{3 a}{4 b^4 (a+b x)^4}-\frac{1}{3 b^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x)^7,x]

[Out]

a^3/(6*b^4*(a + b*x)^6) - (3*a^2)/(5*b^4*(a + b*x)^5) + (3*a)/(4*b^4*(a + b*x)^4) - 1/(3*b^4*(a + b*x)^3)

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^3}{(a+b x)^7} \, dx &=\int \left (-\frac{a^3}{b^3 (a+b x)^7}+\frac{3 a^2}{b^3 (a+b x)^6}-\frac{3 a}{b^3 (a+b x)^5}+\frac{1}{b^3 (a+b x)^4}\right ) \, dx\\ &=\frac{a^3}{6 b^4 (a+b x)^6}-\frac{3 a^2}{5 b^4 (a+b x)^5}+\frac{3 a}{4 b^4 (a+b x)^4}-\frac{1}{3 b^4 (a+b x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0182702, size = 42, normalized size = 0.81 \[ -\frac{6 a^2 b x+a^3+15 a b^2 x^2+20 b^3 x^3}{60 b^4 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x)^7,x]

[Out]

-(a^3 + 6*a^2*b*x + 15*a*b^2*x^2 + 20*b^3*x^3)/(60*b^4*(a + b*x)^6)

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Maple [A]  time = 0.005, size = 57, normalized size = 1.1 \begin{align*}{\frac{{a}^{3}}{6\,{b}^{4} \left ( bx+a \right ) ^{6}}}-{\frac{3\,{a}^{2}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}+{\frac{3\,a}{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{1}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a^3/b^4/(b*x+a)^6-3/5*a^2/b^4/(b*x+a)^5+3/4*a/b^4/(b*x+a)^4-1/3/b^4/(b*x+a)^3

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Maxima [B]  time = 1.04443, size = 132, normalized size = 2.54 \begin{align*} -\frac{20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \,{\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3
+ 15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^4)

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Fricas [B]  time = 1.47325, size = 207, normalized size = 3.98 \begin{align*} -\frac{20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \,{\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3
+ 15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^4)

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Sympy [B]  time = 0.873424, size = 104, normalized size = 2. \begin{align*} - \frac{a^{3} + 6 a^{2} b x + 15 a b^{2} x^{2} + 20 b^{3} x^{3}}{60 a^{6} b^{4} + 360 a^{5} b^{5} x + 900 a^{4} b^{6} x^{2} + 1200 a^{3} b^{7} x^{3} + 900 a^{2} b^{8} x^{4} + 360 a b^{9} x^{5} + 60 b^{10} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a**3 + 6*a**2*b*x + 15*a*b**2*x**2 + 20*b**3*x**3)/(60*a**6*b**4 + 360*a**5*b**5*x + 900*a**4*b**6*x**2 + 12
00*a**3*b**7*x**3 + 900*a**2*b**8*x**4 + 360*a*b**9*x**5 + 60*b**10*x**6)

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Giac [A]  time = 1.21901, size = 54, normalized size = 1.04 \begin{align*} -\frac{20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \,{\left (b x + a\right )}^{6} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)/((b*x + a)^6*b^4)