3.3.15 \(\int \frac {x^2}{(a+b x)^7} \, dx\)

Optimal. Leaf size=47 \[ -\frac {a^2}{6 b^3 (a+b x)^6}+\frac {2 a}{5 b^3 (a+b x)^5}-\frac {1}{4 b^3 (a+b x)^4} \]

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Rubi [A]  time = 0.02, antiderivative size = 47, 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 {a^2}{6 b^3 (a+b x)^6}+\frac {2 a}{5 b^3 (a+b x)^5}-\frac {1}{4 b^3 (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^7,x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 31, normalized size = 0.66 \begin {gather*} -\frac {a^2+6 a b x+15 b^2 x^2}{60 b^3 (a+b x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^7,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{(a+b x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^2/(a + b*x)^7,x]

[Out]

IntegrateAlgebraic[x^2/(a + b*x)^7, x]

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fricas [B]  time = 0.88, size = 87, normalized size = 1.85 \begin {gather*} -\frac {15 \, b^{2} x^{2} + 6 \, a b x + a^{2}}{60 \, {\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^7,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.00, size = 29, normalized size = 0.62 \begin {gather*} -\frac {15 \, b^{2} x^{2} + 6 \, a b x + a^{2}}{60 \, {\left (b x + a\right )}^{6} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^7,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 42, normalized size = 0.89 \begin {gather*} -\frac {a^{2}}{6 \left (b x +a \right )^{6} b^{3}}+\frac {2 a}{5 \left (b x +a \right )^{5} b^{3}}-\frac {1}{4 \left (b x +a \right )^{4} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x+a)^7,x)

[Out]

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

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maxima [B]  time = 1.45, size = 87, normalized size = 1.85 \begin {gather*} -\frac {15 \, b^{2} x^{2} + 6 \, a b x + a^{2}}{60 \, {\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^7,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.08, size = 31, normalized size = 0.66 \begin {gather*} -\frac {8\,a^2+48\,a\,b\,x+120\,b^2\,x^2}{480\,b^3\,{\left (a+b\,x\right )}^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x)^7,x)

[Out]

-(8*a^2 + 120*b^2*x^2 + 48*a*b*x)/(480*b^3*(a + b*x)^6)

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sympy [B]  time = 0.55, size = 92, normalized size = 1.96 \begin {gather*} \frac {- a^{2} - 6 a b x - 15 b^{2} x^{2}}{60 a^{6} b^{3} + 360 a^{5} b^{4} x + 900 a^{4} b^{5} x^{2} + 1200 a^{3} b^{6} x^{3} + 900 a^{2} b^{7} x^{4} + 360 a b^{8} x^{5} + 60 b^{9} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x+a)**7,x)

[Out]

(-a**2 - 6*a*b*x - 15*b**2*x**2)/(60*a**6*b**3 + 360*a**5*b**4*x + 900*a**4*b**5*x**2 + 1200*a**3*b**6*x**3 +
900*a**2*b**7*x**4 + 360*a*b**8*x**5 + 60*b**9*x**6)

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