3.7.72 \(\int x^4 \sqrt [3]{a+b x^2} \, dx\) [672]

Optimal. Leaf size=314 \[ -\frac {54 a^2 x \sqrt [3]{a+b x^2}}{935 b^2}+\frac {6 a x^3 \sqrt [3]{a+b x^2}}{187 b}+\frac {3}{17} x^5 \sqrt [3]{a+b x^2}-\frac {54\ 3^{3/4} \sqrt {2-\sqrt {3}} a^3 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{935 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \]

[Out]

-54/935*a^2*x*(b*x^2+a)^(1/3)/b^2+6/187*a*x^3*(b*x^2+a)^(1/3)/b+3/17*x^5*(b*x^2+a)^(1/3)-54/935*3^(3/4)*a^3*(a
^(1/3)-(b*x^2+a)^(1/3))*EllipticF((-(b*x^2+a)^(1/3)+a^(1/3)*(1+3^(1/2)))/(-(b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2))
),2*I-I*3^(1/2))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3))/(-(b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^
(1/2)*(1/2*6^(1/2)-1/2*2^(1/2))/b^3/x/(-a^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3))/(-(b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)
))^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.20, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {285, 327, 242, 225} \begin {gather*} -\frac {54 a^2 x \sqrt [3]{a+b x^2}}{935 b^2}-\frac {54\ 3^{3/4} \sqrt {2-\sqrt {3}} a^3 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\text {ArcSin}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{935 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {3}{17} x^5 \sqrt [3]{a+b x^2}+\frac {6 a x^3 \sqrt [3]{a+b x^2}}{187 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^2)^(1/3),x]

[Out]

(-54*a^2*x*(a + b*x^2)^(1/3))/(935*b^2) + (6*a*x^3*(a + b*x^2)^(1/3))/(187*b) + (3*x^5*(a + b*x^2)^(1/3))/17 -
(54*3^(3/4)*Sqrt[2 - Sqrt[3]]*a^3*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (
a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a
+ b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(935*b^3*x*Sqrt[-((a^(1/3)*(a^(
1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2)])

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r
*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 242

Int[((a_) + (b_.)*(x_)^2)^(-2/3), x_Symbol] :> Dist[3*(Sqrt[b*x^2]/(2*b*x)), Subst[Int[1/Sqrt[-a + x^3], x], x
, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]

Rule 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin {align*} \int x^4 \sqrt [3]{a+b x^2} \, dx &=\frac {3}{17} x^5 \sqrt [3]{a+b x^2}+\frac {1}{17} (2 a) \int \frac {x^4}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac {6 a x^3 \sqrt [3]{a+b x^2}}{187 b}+\frac {3}{17} x^5 \sqrt [3]{a+b x^2}-\frac {\left (18 a^2\right ) \int \frac {x^2}{\left (a+b x^2\right )^{2/3}} \, dx}{187 b}\\ &=-\frac {54 a^2 x \sqrt [3]{a+b x^2}}{935 b^2}+\frac {6 a x^3 \sqrt [3]{a+b x^2}}{187 b}+\frac {3}{17} x^5 \sqrt [3]{a+b x^2}+\frac {\left (54 a^3\right ) \int \frac {1}{\left (a+b x^2\right )^{2/3}} \, dx}{935 b^2}\\ &=-\frac {54 a^2 x \sqrt [3]{a+b x^2}}{935 b^2}+\frac {6 a x^3 \sqrt [3]{a+b x^2}}{187 b}+\frac {3}{17} x^5 \sqrt [3]{a+b x^2}+\frac {\left (81 a^3 \sqrt {b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a+b x^2}\right )}{935 b^3 x}\\ &=-\frac {54 a^2 x \sqrt [3]{a+b x^2}}{935 b^2}+\frac {6 a x^3 \sqrt [3]{a+b x^2}}{187 b}+\frac {3}{17} x^5 \sqrt [3]{a+b x^2}-\frac {54\ 3^{3/4} \sqrt {2-\sqrt {3}} a^3 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{935 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 5.86, size = 94, normalized size = 0.30 \begin {gather*} \frac {3 x \sqrt [3]{a+b x^2} \left (\sqrt [3]{1+\frac {b x^2}{a}} \left (-9 a^2+2 a b x^2+11 b^2 x^4\right )+9 a^2 \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )\right )}{187 b^2 \sqrt [3]{1+\frac {b x^2}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^2)^(1/3),x]

[Out]

(3*x*(a + b*x^2)^(1/3)*((1 + (b*x^2)/a)^(1/3)*(-9*a^2 + 2*a*b*x^2 + 11*b^2*x^4) + 9*a^2*Hypergeometric2F1[-1/3
, 1/2, 3/2, -((b*x^2)/a)]))/(187*b^2*(1 + (b*x^2)/a)^(1/3))

________________________________________________________________________________________

Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{4} \left (b \,x^{2}+a \right )^{\frac {1}{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^2+a)^(1/3),x)

[Out]

int(x^4*(b*x^2+a)^(1/3),x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^(1/3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/3)*x^4, x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^(1/3),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/3)*x^4, x)

________________________________________________________________________________________

Sympy [A]
time = 0.44, size = 29, normalized size = 0.09 \begin {gather*} \frac {\sqrt [3]{a} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a)**(1/3),x)

[Out]

a**(1/3)*x**5*hyper((-1/3, 5/2), (7/2,), b*x**2*exp_polar(I*pi)/a)/5

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^(1/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/3)*x^4, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\left (b\,x^2+a\right )}^{1/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^2)^(1/3),x)

[Out]

int(x^4*(a + b*x^2)^(1/3), x)

________________________________________________________________________________________