\(\int \frac {x^2}{(a+b x)^7} \, dx\) [215]

Optimal result

Integrand size = 11, antiderivative size = 47 \[ \int \frac {x^2}{(a+b x)^7} \, dx=-\frac {a^2}{6 b^3 (a+b x)^6}+\frac {2 a}{5 b^3 (a+b x)^5}-\frac {1}{4 b^3 (a+b x)^4} \]

[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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^2}{(a+b x)^7} \, dx=-\frac {a^2}{6 b^3 (a+b x)^6}+\frac {2 a}{5 b^3 (a+b x)^5}-\frac {1}{4 b^3 (a+b x)^4} \]

[In]

Int[x^2/(a + b*x)^7,x]

[Out]

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

Rule 45

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int \frac {x^2}{(a+b x)^7} \, dx=-\frac {a^2+6 a b x+15 b^2 x^2}{60 b^3 (a+b x)^6} \]

[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)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.64

method	result	size
gosper	\(-\frac {15 b^{2} x^{2}+6 a b x +a^{2}}{60 \left (b x +a \right )^{6} b^{3}}\)	\(30\)
norman	\(\frac {-\frac {x^{2}}{4 b}-\frac {a x}{10 b^{2}}-\frac {a^{2}}{60 b^{3}}}{\left (b x +a \right )^{6}}\)	\(33\)
risch	\(\frac {-\frac {x^{2}}{4 b}-\frac {a x}{10 b^{2}}-\frac {a^{2}}{60 b^{3}}}{\left (b x +a \right )^{6}}\)	\(33\)
parallelrisch	\(\frac {-15 b^{5} x^{2}-6 a \,b^{4} x -a^{2} b^{3}}{60 b^{6} \left (b x +a \right )^{6}}\)	\(37\)
default	\(-\frac {a^{2}}{6 b^{3} \left (b x +a \right )^{6}}+\frac {2 a}{5 b^{3} \left (b x +a \right )^{5}}-\frac {1}{4 b^{3} \left (b x +a \right )^{4}}\)	\(42\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (41) = 82\).

Time = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.85 \[ \int \frac {x^2}{(a+b x)^7} \, dx=-\frac {15 \, b^{2} x^{2} + 6 \, a b x + a^{2}}{60 \, {\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (42) = 84\).

Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.96 \[ \int \frac {x^2}{(a+b x)^7} \, dx=\frac {- a^{2} - 6 a b x - 15 b^{2} x^{2}}{60 a^{6} b^{3} + 360 a^{5} b^{4} x + 900 a^{4} b^{5} x^{2} + 1200 a^{3} b^{6} x^{3} + 900 a^{2} b^{7} x^{4} + 360 a b^{8} x^{5} + 60 b^{9} x^{6}} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (41) = 82\).

Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.85 \[ \int \frac {x^2}{(a+b x)^7} \, dx=-\frac {15 \, b^{2} x^{2} + 6 \, a b x + a^{2}}{60 \, {\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.62 \[ \int \frac {x^2}{(a+b x)^7} \, dx=-\frac {15 \, b^{2} x^{2} + 6 \, a b x + a^{2}}{60 \, {\left (b x + a\right )}^{6} b^{3}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int \frac {x^2}{(a+b x)^7} \, dx=-\frac {8\,a^2+48\,a\,b\,x+120\,b^2\,x^2}{480\,b^3\,{\left (a+b\,x\right )}^6} \]

[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)