\(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{x^8} \, dx\) [176]

Optimal result

Integrand size = 24, antiderivative size = 76 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 a x^7}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{42 a^2 x^6} \]

[Out]

-1/7*(b*x+a)^5*((b*x+a)^2)^(1/2)/a/x^7+1/42*b*(b*x+a)^5*((b*x+a)^2)^(1/2)/a^2/x^6

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {660, 47, 37} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{42 a^2 x^6}-\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 a x^7} \]

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^8,x]

[Out]

-1/7*((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a*x^7) + (b*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(42*a
^2*x^6)

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]

Rule 47

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 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^8} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 a x^7}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^7} \, dx}{7 a b^3 \left (a b+b^2 x\right )} \\ & = -\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{7 a x^7}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{42 a^2 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.47 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {(6 a-b x) (a+b x)^5 \sqrt {(a+b x)^2}}{42 a^2 x^7} \]

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^8,x]

[Out]

-1/42*((6*a - b*x)*(a + b*x)^5*Sqrt[(a + b*x)^2])/(a^2*x^7)

Maple [A] (verified)

Time = 2.30 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {1}{2} b^{5} x^{5}-\frac {5}{3} a \,b^{4} x^{4}-\frac {5}{2} a^{2} b^{3} x^{3}-2 a^{3} b^{2} x^{2}-\frac {5}{6} a^{4} b x -\frac {1}{7} a^{5}\right )}{\left (b x +a \right ) x^{7}}\)	\(73\)
gosper	\(-\frac {\left (21 b^{5} x^{5}+70 a \,b^{4} x^{4}+105 a^{2} b^{3} x^{3}+84 a^{3} b^{2} x^{2}+35 a^{4} b x +6 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 x^{7} \left (b x +a \right )^{5}}\)	\(74\)
default	\(-\frac {\left (21 b^{5} x^{5}+70 a \,b^{4} x^{4}+105 a^{2} b^{3} x^{3}+84 a^{3} b^{2} x^{2}+35 a^{4} b x +6 a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 x^{7} \left (b x +a \right )^{5}}\)	\(74\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^8,x,method=_RETURNVERBOSE)

[Out]

((b*x+a)^2)^(1/2)/(b*x+a)/x^7*(-1/2*b^5*x^5-5/3*a*b^4*x^4-5/2*a^2*b^3*x^3-2*a^3*b^2*x^2-5/6*a^4*b*x-1/7*a^5)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {21 \, b^{5} x^{5} + 70 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} + 84 \, a^{3} b^{2} x^{2} + 35 \, a^{4} b x + 6 \, a^{5}}{42 \, x^{7}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^8,x, algorithm="fricas")

[Out]

-1/42*(21*b^5*x^5 + 70*a*b^4*x^4 + 105*a^2*b^3*x^3 + 84*a^3*b^2*x^2 + 35*a^4*b*x + 6*a^5)/x^7

Sympy [F]

\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{8}}\, dx \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/x**8,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**8, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (50) = 100\).

Time = 0.21 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.96 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{7}}{6 \, a^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{6}}{6 \, a^{6} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{5}}{6 \, a^{7} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{4}}{6 \, a^{6} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{3}}{6 \, a^{5} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{6 \, a^{4} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{6 \, a^{3} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{7 \, a^{2} x^{7}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^8,x, algorithm="maxima")

[Out]

-1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b^7/a^7 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b^6/(a^6*x) + 1/6*(b^2*x^2
+ 2*a*b*x + a^2)^(7/2)*b^5/(a^7*x^2) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^4/(a^6*x^3) + 1/6*(b^2*x^2 + 2*a*
b*x + a^2)^(7/2)*b^3/(a^5*x^4) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^2/(a^4*x^5) + 1/6*(b^2*x^2 + 2*a*b*x +
a^2)^(7/2)*b/(a^3*x^6) - 1/7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)/(a^2*x^7)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (50) = 100\).

Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=\frac {b^{7} \mathrm {sgn}\left (b x + a\right )}{42 \, a^{2}} - \frac {21 \, b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 70 \, a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 105 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 84 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{5} \mathrm {sgn}\left (b x + a\right )}{42 \, x^{7}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^8,x, algorithm="giac")

[Out]

1/42*b^7*sgn(b*x + a)/a^2 - 1/42*(21*b^5*x^5*sgn(b*x + a) + 70*a*b^4*x^4*sgn(b*x + a) + 105*a^2*b^3*x^3*sgn(b*
x + a) + 84*a^3*b^2*x^2*sgn(b*x + a) + 35*a^4*b*x*sgn(b*x + a) + 6*a^5*sgn(b*x + a))/x^7

Mupad [B] (verification not implemented)

Time = 10.36 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.72 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx=-\frac {a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^2\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^4\,\left (a+b\,x\right )}-\frac {2\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^3\,\left (a+b\,x\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\left (a+b\,x\right )} \]

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^(5/2)/x^8,x)

[Out]

- (a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(7*x^7*(a + b*x)) - (b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*x^2*(a +
b*x)) - (5*a^2*b^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*x^4*(a + b*x)) - (2*a^3*b^2*(a^2 + b^2*x^2 + 2*a*b*x)^(
1/2))/(x^5*(a + b*x)) - (5*a*b^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(3*x^3*(a + b*x)) - (5*a^4*b*(a^2 + b^2*x^2
+ 2*a*b*x)^(1/2))/(6*x^6*(a + b*x))