3.3007 \(\int \frac{x^2}{a+b (c x^n)^{\frac{1}{n}}} \, dx\)

Optimal. Leaf size=77 \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^3}-\frac{a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac{x^3 \left (c x^n\right )^{-1/n}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0269364, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {368, 43} \[ \frac{a^2 x^3 \left (c x^n\right )^{-3/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^3}-\frac{a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac{x^3 \left (c x^n\right )^{-1/n}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*(c*x^n)^n^(-1)),x]

[Out]

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

Rule 368

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)]

Rule 43

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*} \int \frac{x^2}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx &=\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=-\frac{a x^3 \left (c x^n\right )^{-2/n}}{b^2}+\frac{x^3 \left (c x^n\right )^{-1/n}}{2 b}+\frac{a^2 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]  time = 0.0287248, size = 67, normalized size = 0.87 \[ \frac{x^3 \left (c x^n\right )^{-3/n} \left (2 a^2 \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )+b \left (c x^n\right )^{\frac{1}{n}} \left (b \left (c x^n\right )^{\frac{1}{n}}-2 a\right )\right )}{2 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.105, size = 438, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.30868, size = 113, normalized size = 1.47 \begin{align*} \frac{b^{2} c^{\frac{2}{n}} x^{2} - 2 \, a b c^{\left (\frac{1}{n}\right )} x + 2 \, a^{2} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{2 \, b^{3} c^{\frac{3}{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{a + b \left (c x^{n}\right )^{\frac{1}{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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