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

Optimal. Leaf size=35 \[ \frac{x^5}{30 a^2 (a+b x)^5}+\frac{x^5}{6 a (a+b x)^6} \]

[Out]

x^5/(6*a*(a + b*x)^6) + x^5/(30*a^2*(a + b*x)^5)

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Rubi [A]  time = 0.0049189, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ \frac{x^5}{30 a^2 (a+b x)^5}+\frac{x^5}{6 a (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x)^7,x]

[Out]

x^5/(6*a*(a + b*x)^6) + x^5/(30*a^2*(a + b*x)^5)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{x^4}{(a+b x)^7} \, dx &=\frac{x^5}{6 a (a+b x)^6}+\frac{\int \frac{x^4}{(a+b x)^6} \, dx}{6 a}\\ &=\frac{x^5}{6 a (a+b x)^6}+\frac{x^5}{30 a^2 (a+b x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0155113, size = 53, normalized size = 1.51 \[ -\frac{15 a^2 b^2 x^2+6 a^3 b x+a^4+20 a b^3 x^3+15 b^4 x^4}{30 b^5 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x)^7,x]

[Out]

-(a^4 + 6*a^3*b*x + 15*a^2*b^2*x^2 + 20*a*b^3*x^3 + 15*b^4*x^4)/(30*b^5*(a + b*x)^6)

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Maple [B]  time = 0.006, size = 72, normalized size = 2.1 \begin{align*} -{\frac{3\,{a}^{2}}{2\,{b}^{5} \left ( bx+a \right ) ^{4}}}-{\frac{1}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{4}}{6\,{b}^{5} \left ( bx+a \right ) ^{6}}}+{\frac{4\,a}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}+{\frac{4\,{a}^{3}}{5\,{b}^{5} \left ( bx+a \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x+a)^7,x)

[Out]

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

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Maxima [B]  time = 1.07266, size = 147, normalized size = 4.2 \begin{align*} -\frac{15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \,{\left (b^{11} x^{6} + 6 \, a b^{10} x^{5} + 15 \, a^{2} b^{9} x^{4} + 20 \, a^{3} b^{8} x^{3} + 15 \, a^{4} b^{7} x^{2} + 6 \, a^{5} b^{6} x + a^{6} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(15*b^4*x^4 + 20*a*b^3*x^3 + 15*a^2*b^2*x^2 + 6*a^3*b*x + a^4)/(b^11*x^6 + 6*a*b^10*x^5 + 15*a^2*b^9*x^4
+ 20*a^3*b^8*x^3 + 15*a^4*b^7*x^2 + 6*a^5*b^6*x + a^6*b^5)

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Fricas [B]  time = 1.54071, size = 231, normalized size = 6.6 \begin{align*} -\frac{15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \,{\left (b^{11} x^{6} + 6 \, a b^{10} x^{5} + 15 \, a^{2} b^{9} x^{4} + 20 \, a^{3} b^{8} x^{3} + 15 \, a^{4} b^{7} x^{2} + 6 \, a^{5} b^{6} x + a^{6} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(15*b^4*x^4 + 20*a*b^3*x^3 + 15*a^2*b^2*x^2 + 6*a^3*b*x + a^4)/(b^11*x^6 + 6*a*b^10*x^5 + 15*a^2*b^9*x^4
+ 20*a^3*b^8*x^3 + 15*a^4*b^7*x^2 + 6*a^5*b^6*x + a^6*b^5)

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Sympy [B]  time = 0.899177, size = 116, normalized size = 3.31 \begin{align*} - \frac{a^{4} + 6 a^{3} b x + 15 a^{2} b^{2} x^{2} + 20 a b^{3} x^{3} + 15 b^{4} x^{4}}{30 a^{6} b^{5} + 180 a^{5} b^{6} x + 450 a^{4} b^{7} x^{2} + 600 a^{3} b^{8} x^{3} + 450 a^{2} b^{9} x^{4} + 180 a b^{10} x^{5} + 30 b^{11} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a**4 + 6*a**3*b*x + 15*a**2*b**2*x**2 + 20*a*b**3*x**3 + 15*b**4*x**4)/(30*a**6*b**5 + 180*a**5*b**6*x + 450
*a**4*b**7*x**2 + 600*a**3*b**8*x**3 + 450*a**2*b**9*x**4 + 180*a*b**10*x**5 + 30*b**11*x**6)

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Giac [A]  time = 1.15434, size = 69, normalized size = 1.97 \begin{align*} -\frac{15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \,{\left (b x + a\right )}^{6} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(15*b^4*x^4 + 20*a*b^3*x^3 + 15*a^2*b^2*x^2 + 6*a^3*b*x + a^4)/((b*x + a)^6*b^5)