3.3071 \(\int (c x^n)^{\frac{1}{n}} (a+b (c x^n)^{\frac{1}{n}})^3 \, dx\)

Optimal. Leaf size=70 \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^5}{5 b^2}-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{4 b^2} \]

[Out]

-(a*x*(a + b*(c*x^n)^n^(-1))^4)/(4*b^2*(c*x^n)^n^(-1)) + (x*(a + b*(c*x^n)^n^(-1))^5)/(5*b^2*(c*x^n)^n^(-1))

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Rubi [A]  time = 0.0306298, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {15, 368, 43} \[ \frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^5}{5 b^2}-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^3,x]

[Out]

-(a*x*(a + b*(c*x^n)^n^(-1))^4)/(4*b^2*(c*x^n)^n^(-1)) + (x*(a + b*(c*x^n)^n^(-1))^5)/(5*b^2*(c*x^n)^n^(-1))

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

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 \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3 \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \int x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3 \, dx}{x}\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x (a+b x)^3 \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^3}{b}+\frac{(a+b x)^4}{b}\right ) \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )\\ &=-\frac{a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{4 b^2}+\frac{x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^5}{5 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0268751, size = 48, normalized size = 0.69 \[ -\frac{x \left (c x^n\right )^{-1/n} \left (a-4 b \left (c x^n\right )^{\frac{1}{n}}\right ) \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}{20 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^3,x]

[Out]

-(x*(a - 4*b*(c*x^n)^n^(-1))*(a + b*(c*x^n)^n^(-1))^4)/(20*b^2*(c*x^n)^n^(-1))

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Maple [F]  time = 0.247, size = 0, normalized size = 0. \begin{align*} \int \sqrt [n]{c{x}^{n}} \left ( a+b\sqrt [n]{c{x}^{n}} \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^3,x)

[Out]

int((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{3} \left (c x^{n}\right )^{\left (\frac{1}{n}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.62265, size = 117, normalized size = 1.67 \begin{align*} \frac{1}{5} \, b^{3} c^{\frac{4}{n}} x^{5} + \frac{3}{4} \, a b^{2} c^{\frac{3}{n}} x^{4} + a^{2} b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a^{3} c^{\left (\frac{1}{n}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^3,x, algorithm="fricas")

[Out]

1/5*b^3*c^(4/n)*x^5 + 3/4*a*b^2*c^(3/n)*x^4 + a^2*b*c^(2/n)*x^3 + 1/2*a^3*c^(1/n)*x^2

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Sympy [A]  time = 1.34525, size = 76, normalized size = 1.09 \begin{align*} \frac{a^{3} c^{\frac{1}{n}} x \left (x^{n}\right )^{\frac{1}{n}}}{2} + a^{2} b c^{\frac{2}{n}} x \left (x^{n}\right )^{\frac{2}{n}} + \frac{3 a b^{2} c^{\frac{3}{n}} x \left (x^{n}\right )^{\frac{3}{n}}}{4} + \frac{b^{3} c^{\frac{4}{n}} x \left (x^{n}\right )^{\frac{4}{n}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*c**(1/n)*x*(x**n)**(1/n)/2 + a**2*b*c**(2/n)*x*(x**n)**(2/n) + 3*a*b**2*c**(3/n)*x*(x**n)**(3/n)/4 + b**3
*c**(4/n)*x*(x**n)**(4/n)/5

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Giac [A]  time = 1.15871, size = 81, normalized size = 1.16 \begin{align*} \frac{1}{5} \, b^{3} c^{\frac{4}{n}} x^{5} + \frac{3}{4} \, a b^{2} c^{\frac{3}{n}} x^{4} + a^{2} b c^{\frac{2}{n}} x^{3} + \frac{1}{2} \, a^{3} c^{\left (\frac{1}{n}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^3,x, algorithm="giac")

[Out]

1/5*b^3*c^(4/n)*x^5 + 3/4*a*b^2*c^(3/n)*x^4 + a^2*b*c^(2/n)*x^3 + 1/2*a^3*c^(1/n)*x^2