\(\int \frac {(a^2+2 a b x^2+b^2 x^4)^{5/2}}{x^{15}} \, dx\) [599]

Optimal result

Integrand size = 26, antiderivative size = 72 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 a x^{14}}+\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{84 a^2 x^{14}} \]

[Out]

-1/12*(b*x^2+a)*(b^2*x^4+2*a*b*x^2+a^2)^(5/2)/a/x^14+1/84*(b^2*x^4+2*a*b*x^2+a^2)^(7/2)/a^2/x^14

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1124} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{84 a^2 x^{14}}-\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 a x^{14}} \]

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^15,x]

[Out]

-1/12*((a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2))/(a*x^14) + (a^2 + 2*a*b*x^2 + b^2*x^4)^(7/2)/(84*a^2*x^1
4)

Rule 1124

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*x^2
+ c*x^4)^(p + 1)/(4*a*d*(p + 1)*(2*p + 1))), x] - Simp[(d*x)^(m + 1)*(2*a + b*x^2)*((a + b*x^2 + c*x^4)^p/(4*a
*d*(2*p + 1))), x] /; FreeQ[{a, b, c, d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && EqQ[m + 4*p + 5,
0] && NeQ[p, -2^(-1)]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{12 a x^{14}}+\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{7/2}}{84 a^2 x^{14}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {\sqrt {\left (a+b x^2\right )^2} \left (6 a^5+35 a^4 b x^2+84 a^3 b^2 x^4+105 a^2 b^3 x^6+70 a b^4 x^8+21 b^5 x^{10}\right )}{84 x^{14} \left (a+b x^2\right )} \]

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^15,x]

[Out]

-1/84*(Sqrt[(a + b*x^2)^2]*(6*a^5 + 35*a^4*b*x^2 + 84*a^3*b^2*x^4 + 105*a^2*b^3*x^6 + 70*a*b^4*x^8 + 21*b^5*x^
10))/(x^14*(a + b*x^2))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 2.

Time = 0.47 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(-\frac {\left (21 x^{10} b^{5}+70 a \,x^{8} b^{4}+105 a^{2} x^{6} b^{3}+84 a^{3} x^{4} b^{2}+35 x^{2} a^{4} b +6 a^{5}\right ) \operatorname {csgn}\left (b \,x^{2}+a \right )}{84 x^{14}}\)	\(68\)
risch	\(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {1}{14} a^{5}-\frac {5}{12} x^{2} a^{4} b -a^{3} x^{4} b^{2}-\frac {5}{4} a^{2} x^{6} b^{3}-\frac {5}{6} a \,x^{8} b^{4}-\frac {1}{4} x^{10} b^{5}\right )}{\left (b \,x^{2}+a \right ) x^{14}}\)	\(79\)
gosper	\(-\frac {\left (21 x^{10} b^{5}+70 a \,x^{8} b^{4}+105 a^{2} x^{6} b^{3}+84 a^{3} x^{4} b^{2}+35 x^{2} a^{4} b +6 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{84 x^{14} \left (b \,x^{2}+a \right )^{5}}\)	\(80\)
default	\(-\frac {\left (21 x^{10} b^{5}+70 a \,x^{8} b^{4}+105 a^{2} x^{6} b^{3}+84 a^{3} x^{4} b^{2}+35 x^{2} a^{4} b +6 a^{5}\right ) {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{84 x^{14} \left (b \,x^{2}+a \right )^{5}}\)	\(80\)

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x,method=_RETURNVERBOSE)

[Out]

-1/84*(21*b^5*x^10+70*a*b^4*x^8+105*a^2*b^3*x^6+84*a^3*b^2*x^4+35*a^4*b*x^2+6*a^5)*csgn(b*x^2+a)/x^14

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x, algorithm="fricas")

[Out]

-1/84*(21*b^5*x^10 + 70*a*b^4*x^8 + 105*a^2*b^3*x^6 + 84*a^3*b^2*x^4 + 35*a^4*b*x^2 + 6*a^5)/x^14

Sympy [F]

\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{x^{15}}\, dx \]

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**15,x)

[Out]

Integral(((a + b*x**2)**2)**(5/2)/x**15, x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {b^{5}}{4 \, x^{4}} - \frac {5 \, a b^{4}}{6 \, x^{6}} - \frac {5 \, a^{2} b^{3}}{4 \, x^{8}} - \frac {a^{3} b^{2}}{x^{10}} - \frac {5 \, a^{4} b}{12 \, x^{12}} - \frac {a^{5}}{14 \, x^{14}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x, algorithm="maxima")

[Out]

-1/4*b^5/x^4 - 5/6*a*b^4/x^6 - 5/4*a^2*b^3/x^8 - a^3*b^2/x^10 - 5/12*a^4*b/x^12 - 1/14*a^5/x^14

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {21 \, b^{5} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 70 \, a b^{4} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 105 \, a^{2} b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 84 \, a^{3} b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 35 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 6 \, a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{84 \, x^{14}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^15,x, algorithm="giac")

[Out]

-1/84*(21*b^5*x^10*sgn(b*x^2 + a) + 70*a*b^4*x^8*sgn(b*x^2 + a) + 105*a^2*b^3*x^6*sgn(b*x^2 + a) + 84*a^3*b^2*
x^4*sgn(b*x^2 + a) + 35*a^4*b*x^2*sgn(b*x^2 + a) + 6*a^5*sgn(b*x^2 + a))/x^14

Mupad [B] (verification not implemented)

Time = 13.24 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.21 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{15}} \, dx=-\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{14\,x^{14}\,\left (b\,x^2+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^4\,\left (b\,x^2+a\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{6\,x^6\,\left (b\,x^2+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{12\,x^{12}\,\left (b\,x^2+a\right )}-\frac {5\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{4\,x^8\,\left (b\,x^2+a\right )}-\frac {a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}}{x^{10}\,\left (b\,x^2+a\right )} \]

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2)/x^15,x)

[Out]

- (a^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(14*x^14*(a + b*x^2)) - (b^5*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(4*x
^4*(a + b*x^2)) - (5*a*b^4*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(6*x^6*(a + b*x^2)) - (5*a^4*b*(a^2 + b^2*x^4 +
2*a*b*x^2)^(1/2))/(12*x^12*(a + b*x^2)) - (5*a^2*b^3*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(4*x^8*(a + b*x^2)) -
(a^3*b^2*(a^2 + b^2*x^4 + 2*a*b*x^2)^(1/2))/(x^10*(a + b*x^2))