\(\int \frac {x^2}{(a+b (c x^n)^{\frac {1}{n}})^2} \, dx\) [3014]

Optimal result

Integrand size = 19, antiderivative size = 90 \[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {x^3 \left (c x^n\right )^{-2/n}}{b^2}-\frac {a^2 x^3 \left (c x^n\right )^{-3/n}}{b^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {2 a x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3} \]

[Out]

x^3/b^2/((c*x^n)^(2/n))-a^2*x^3/b^3/((c*x^n)^(3/n))/(a+b*(c*x^n)^(1/n))-2*a*x^3*ln(a+b*(c*x^n)^(1/n))/b^3/((c*
x^n)^(3/n))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {375, 45} \[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=-\frac {a^2 x^3 \left (c x^n\right )^{-3/n}}{b^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {2 a x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3}+\frac {x^3 \left (c x^n\right )^{-2/n}}{b^2} \]

[In]

Int[x^2/(a + b*(c*x^n)^n^(-1))^2,x]

[Out]

x^3/(b^2*(c*x^n)^(2/n)) - (a^2*x^3)/(b^3*(c*x^n)^(3/n)*(a + b*(c*x^n)^n^(-1))) - (2*a*x^3*Log[a + b*(c*x^n)^n^
(-1)])/(b^3*(c*x^n)^(3/n))

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])

Rule 375

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*((c*x^q
)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q
}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rubi steps \begin{align*} \text {integral}& = \left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {x^2}{(a+b x)^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \left (\frac {1}{b^2}+\frac {a^2}{b^2 (a+b x)^2}-\frac {2 a}{b^2 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {x^3 \left (c x^n\right )^{-2/n}}{b^2}-\frac {a^2 x^3 \left (c x^n\right )^{-3/n}}{b^3 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}-\frac {2 a x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )}{b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {x^3 \left (c x^n\right )^{-3/n} \left (b \left (c x^n\right )^{\frac {1}{n}}-\frac {a^2}{a+b \left (c x^n\right )^{\frac {1}{n}}}-2 a \log \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )\right )}{b^3} \]

[In]

Integrate[x^2/(a + b*(c*x^n)^n^(-1))^2,x]

[Out]

(x^3*(b*(c*x^n)^n^(-1) - a^2/(a + b*(c*x^n)^n^(-1)) - 2*a*Log[a + b*(c*x^n)^n^(-1)]))/(b^3*(c*x^n)^(3/n))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.13 (sec) , antiderivative size = 386, normalized size of antiderivative = 4.29

method	result	size
risch	\(\frac {x^{3}}{a \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}-\frac {\left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} x^{3} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{a b}+\frac {2 \left (x^{n}\right )^{-\frac {2}{n}} c^{-\frac {2}{n}} x^{3} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{b^{2}}-\frac {2 a \ln \left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right ) \left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} c^{-\frac {2}{n}} \left (x^{n}\right )^{-\frac {2}{n}} x^{3} {\mathrm e}^{-\frac {3 i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{b^{3}}\)	\(386\)

[In]

int(x^2/(a+b*(c*x^n)^(1/n))^2,x,method=_RETURNVERBOSE)

[Out]

x^3/a/(b*(x^n)^(1/n)*c^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))
/n)+a)-1/a/((x^n)^(1/n))/(c^(1/n))*x^3*exp(-1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csg
n(I*c*x^n))/n)/b+2/((x^n)^(1/n))^2/(c^(1/n))^2*x^3*exp(-I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(
I*c)-csgn(I*c*x^n))/n)/b^2-2*a*ln(b*(x^n)^(1/n)*c^(1/n)*exp(1/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n)
)*(csgn(I*c)-csgn(I*c*x^n))/n)+a)/((x^n)^(1/n))/(c^(1/n))*c^(-2/n)*(x^n)^(-2/n)*x^3/b^3*exp(-3/2*I*Pi*csgn(I*c
*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\frac {b^{2} c^{\frac {2}{n}} x^{2} + a b c^{\left (\frac {1}{n}\right )} x - a^{2} - 2 \, {\left (a b c^{\left (\frac {1}{n}\right )} x + a^{2}\right )} \log \left (b c^{\left (\frac {1}{n}\right )} x + a\right )}{b^{4} c^{\frac {4}{n}} x + a b^{3} c^{\frac {3}{n}}} \]

[In]

integrate(x^2/(a+b*(c*x^n)^(1/n))^2,x, algorithm="fricas")

[Out]

(b^2*c^(2/n)*x^2 + a*b*c^(1/n)*x - a^2 - 2*(a*b*c^(1/n)*x + a^2)*log(b*c^(1/n)*x + a))/(b^4*c^(4/n)*x + a*b^3*
c^(3/n))

Sympy [F]

\[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {x^{2}}{\left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{2}}\, dx \]

[In]

integrate(x**2/(a+b*(c*x**n)**(1/n))**2,x)

[Out]

Integral(x**2/(a + b*(c*x**n)**(1/n))**2, x)

Maxima [F]

\[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {x^{2}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2/(a+b*(c*x^n)^(1/n))^2,x, algorithm="maxima")

[Out]

x^3/(a*b*c^(1/n)*(x^n)^(1/n) + a^2) - 2*integrate(x^2/(a*b*c^(1/n)*(x^n)^(1/n) + a^2), x)

Giac [F]

\[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int { \frac {x^{2}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2/(a+b*(c*x^n)^(1/n))^2,x, algorithm="giac")

[Out]

integrate(x^2/((c*x^n)^(1/n)*b + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \, dx=\int \frac {x^2}{{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^2} \,d x \]

[In]

int(x^2/(a + b*(c*x^n)^(1/n))^2,x)

[Out]

int(x^2/(a + b*(c*x^n)^(1/n))^2, x)