\(\int \frac {(a^2+2 a b x^3+b^2 x^6)^{5/2}}{x^9} \, dx\) [72]

Optimal result

Integrand size = 26, antiderivative size = 247 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^9} \, dx=-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {5 a b^4 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {b^5 x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )} \]

[Out]

-1/8*a^5*((b*x^3+a)^2)^(1/2)/x^8/(b*x^3+a)-a^4*b*((b*x^3+a)^2)^(1/2)/x^5/(b*x^3+a)-5*a^3*b^2*((b*x^3+a)^2)^(1/
2)/x^2/(b*x^3+a)+10*a^2*b^3*x*((b*x^3+a)^2)^(1/2)/(b*x^3+a)+5/4*a*b^4*x^4*((b*x^3+a)^2)^(1/2)/(b*x^3+a)+1/7*b^
5*x^7*((b*x^3+a)^2)^(1/2)/(b*x^3+a)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 276} \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^9} \, dx=\frac {b^5 x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {5 a b^4 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )} \]

[In]

Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^9,x]

[Out]

-1/8*(a^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x^8*(a + b*x^3)) - (a^4*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x^5*(a
+ b*x^3)) - (5*a^3*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(x^2*(a + b*x^3)) + (10*a^2*b^3*x*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(a + b*x^3) + (5*a*b^4*x^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(4*(a + b*x^3)) + (b^5*x^7*Sqrt[a^2
+ 2*a*b*x^3 + b^2*x^6])/(7*(a + b*x^3))

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 1369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^5}{x^9} \, dx}{b^4 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (10 a^2 b^8+\frac {a^5 b^5}{x^9}+\frac {5 a^4 b^6}{x^6}+\frac {10 a^3 b^7}{x^3}+5 a b^9 x^3+b^{10} x^6\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )} \\ & = -\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^8 \left (a+b x^3\right )}-\frac {a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x^2 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {5 a b^4 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {b^5 x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.34 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^9} \, dx=\frac {\sqrt {\left (a+b x^3\right )^2} \left (-7 a^5-56 a^4 b x^3-280 a^3 b^2 x^6+560 a^2 b^3 x^9+70 a b^4 x^{12}+8 b^5 x^{15}\right )}{56 x^8 \left (a+b x^3\right )} \]

[In]

Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^9,x]

[Out]

(Sqrt[(a + b*x^3)^2]*(-7*a^5 - 56*a^4*b*x^3 - 280*a^3*b^2*x^6 + 560*a^2*b^3*x^9 + 70*a*b^4*x^12 + 8*b^5*x^15))
/(56*x^8*(a + b*x^3))

Maple [A] (verified)

Time = 8.88 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.32

method	result	size
gosper	\(-\frac {\left (-8 b^{5} x^{15}-70 a \,b^{4} x^{12}-560 a^{2} b^{3} x^{9}+280 a^{3} b^{2} x^{6}+56 a^{4} b \,x^{3}+7 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{56 \left (b \,x^{3}+a \right )^{5} x^{8}}\)	\(80\)
default	\(-\frac {\left (-8 b^{5} x^{15}-70 a \,b^{4} x^{12}-560 a^{2} b^{3} x^{9}+280 a^{3} b^{2} x^{6}+56 a^{4} b \,x^{3}+7 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{56 \left (b \,x^{3}+a \right )^{5} x^{8}}\)	\(80\)
risch	\(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, b^{3} \left (\frac {1}{7} b^{2} x^{7}+\frac {5}{4} a b \,x^{4}+10 a^{2} x \right )}{b \,x^{3}+a}+\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-5 a^{3} b^{2} x^{6}-a^{4} b \,x^{3}-\frac {1}{8} a^{5}\right )}{\left (b \,x^{3}+a \right ) x^{8}}\)	\(98\)

[In]

int((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/56*(-8*b^5*x^15-70*a*b^4*x^12-560*a^2*b^3*x^9+280*a^3*b^2*x^6+56*a^4*b*x^3+7*a^5)*((b*x^3+a)^2)^(5/2)/(b*x^
3+a)^5/x^8

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.24 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^9} \, dx=\frac {8 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 560 \, a^{2} b^{3} x^{9} - 280 \, a^{3} b^{2} x^{6} - 56 \, a^{4} b x^{3} - 7 \, a^{5}}{56 \, x^{8}} \]

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^9,x, algorithm="fricas")

[Out]

1/56*(8*b^5*x^15 + 70*a*b^4*x^12 + 560*a^2*b^3*x^9 - 280*a^3*b^2*x^6 - 56*a^4*b*x^3 - 7*a^5)/x^8

Sympy [F]

\[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^9} \, dx=\int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{9}}\, dx \]

[In]

integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x**9,x)

[Out]

Integral(((a + b*x**3)**2)**(5/2)/x**9, x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.24 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^9} \, dx=\frac {8 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 560 \, a^{2} b^{3} x^{9} - 280 \, a^{3} b^{2} x^{6} - 56 \, a^{4} b x^{3} - 7 \, a^{5}}{56 \, x^{8}} \]

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^9,x, algorithm="maxima")

[Out]

1/56*(8*b^5*x^15 + 70*a*b^4*x^12 + 560*a^2*b^3*x^9 - 280*a^3*b^2*x^6 - 56*a^4*b*x^3 - 7*a^5)/x^8

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.43 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^9} \, dx=\frac {1}{7} \, b^{5} x^{7} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{4} \, a b^{4} x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) + 10 \, a^{2} b^{3} x \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {40 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 8 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{8 \, x^{8}} \]

[In]

integrate((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^9,x, algorithm="giac")

[Out]

1/7*b^5*x^7*sgn(b*x^3 + a) + 5/4*a*b^4*x^4*sgn(b*x^3 + a) + 10*a^2*b^3*x*sgn(b*x^3 + a) - 1/8*(40*a^3*b^2*x^6*
sgn(b*x^3 + a) + 8*a^4*b*x^3*sgn(b*x^3 + a) + a^5*sgn(b*x^3 + a))/x^8

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^9} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}}{x^9} \,d x \]

[In]

int((a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)/x^9,x)

[Out]

int((a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)/x^9, x)