Integrand size = 19, antiderivative size = 418 \[ \int (c x)^{4/3} \sqrt [3]{a+b x^2} \, dx=\frac {2 a c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{9 b}+\frac {(c x)^{7/3} \sqrt [3]{a+b x^2}}{3 c}-\frac {a \sqrt [3]{c} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {c^{4/3}+\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{9 \sqrt [4]{3} b \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}} \]
2/9*a*c*(c*x)^(1/3)*(b*x^2+a)^(1/3)/b+1/3*(c*x)^(7/3)*(b*x^2+a)^(1/3)/c-1/ 27*a*c^(1/3)*(c*x)^(1/3)*(b*x^2+a)^(1/3)*(c^(2/3)-b^(1/3)*(c*x)^(2/3)/(b*x ^2+a)^(1/3))*((c^(2/3)-b^(1/3)*(c*x)^(2/3)*(1-3^(1/2))/(b*x^2+a)^(1/3))^2/ (c^(2/3)-b^(1/3)*(c*x)^(2/3)*(1+3^(1/2))/(b*x^2+a)^(1/3))^2)^(1/2)/(c^(2/3 )-b^(1/3)*(c*x)^(2/3)*(1-3^(1/2))/(b*x^2+a)^(1/3))*(c^(2/3)-b^(1/3)*(c*x)^ (2/3)*(1+3^(1/2))/(b*x^2+a)^(1/3))*EllipticF((1-(c^(2/3)-b^(1/3)*(c*x)^(2/ 3)*(1-3^(1/2))/(b*x^2+a)^(1/3))^2/(c^(2/3)-b^(1/3)*(c*x)^(2/3)*(1+3^(1/2)) /(b*x^2+a)^(1/3))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*((c^(4/3)+b^(2/3)*(c*x )^(4/3)/(b*x^2+a)^(2/3)+b^(1/3)*c^(2/3)*(c*x)^(2/3)/(b*x^2+a)^(1/3))/(c^(2 /3)-b^(1/3)*(c*x)^(2/3)*(1+3^(1/2))/(b*x^2+a)^(1/3))^2)^(1/2)*3^(3/4)/b/(- b^(1/3)*(c*x)^(2/3)*(c^(2/3)-b^(1/3)*(c*x)^(2/3)/(b*x^2+a)^(1/3))/(b*x^2+a )^(1/3)/(c^(2/3)-b^(1/3)*(c*x)^(2/3)*(1+3^(1/2))/(b*x^2+a)^(1/3))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.20 \[ \int (c x)^{4/3} \sqrt [3]{a+b x^2} \, dx=\frac {c \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (\left (a+b x^2\right ) \sqrt [3]{1+\frac {b x^2}{a}}-a \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{6},\frac {7}{6},-\frac {b x^2}{a}\right )\right )}{3 b \sqrt [3]{1+\frac {b x^2}{a}}} \]
(c*(c*x)^(1/3)*(a + b*x^2)^(1/3)*((a + b*x^2)*(1 + (b*x^2)/a)^(1/3) - a*Hy pergeometric2F1[-1/3, 1/6, 7/6, -((b*x^2)/a)]))/(3*b*(1 + (b*x^2)/a)^(1/3) )
Time = 0.35 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.86, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {248, 262, 266, 771, 766}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (c x)^{4/3} \sqrt [3]{a+b x^2} \, dx\) |
\(\Big \downarrow \) 248 |
\(\displaystyle \frac {2}{9} a \int \frac {(c x)^{4/3}}{\left (b x^2+a\right )^{2/3}}dx+\frac {(c x)^{7/3} \sqrt [3]{a+b x^2}}{3 c}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {2}{9} a \left (\frac {c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{b}-\frac {a c^2 \int \frac {1}{(c x)^{2/3} \left (b x^2+a\right )^{2/3}}dx}{3 b}\right )+\frac {(c x)^{7/3} \sqrt [3]{a+b x^2}}{3 c}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2}{9} a \left (\frac {c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{b}-\frac {a c \int \frac {1}{\left (b x^2+a\right )^{2/3}}d\sqrt [3]{c x}}{b}\right )+\frac {(c x)^{7/3} \sqrt [3]{a+b x^2}}{3 c}\) |
\(\Big \downarrow \) 771 |
\(\displaystyle \frac {2}{9} a \left (\frac {c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{b}-\frac {a c \int \frac {1}{\sqrt {1-b x^2}}d\frac {\sqrt [3]{c x}}{\sqrt [6]{b x^2+a}}}{b \sqrt {a+b x^2} \sqrt {\frac {a c^2}{a c^2+b c^2 x^2}}}\right )+\frac {(c x)^{7/3} \sqrt [3]{a+b x^2}}{3 c}\) |
\(\Big \downarrow \) 766 |
\(\displaystyle \frac {2}{9} a \left (\frac {c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{b}-\frac {a \sqrt [3]{c} \sqrt [3]{c x} \left (c^{2/3}-\sqrt [3]{b} (c x)^{2/3}\right ) \sqrt {\frac {b^{2/3} (c x)^{4/3}+\sqrt [3]{b} c^{2/3} (c x)^{2/3}+c^{4/3}}{\left (c^{2/3}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {c^{2/3}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{c^{2/3}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} b \sqrt {1-b x^2} \left (a+b x^2\right )^{2/3} \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\sqrt [3]{b} (c x)^{2/3}\right )}{\left (c^{2/3}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}\right )^2}} \sqrt {\frac {a c^2}{a c^2+b c^2 x^2}}}\right )+\frac {(c x)^{7/3} \sqrt [3]{a+b x^2}}{3 c}\) |
((c*x)^(7/3)*(a + b*x^2)^(1/3))/(3*c) + (2*a*((c*(c*x)^(1/3)*(a + b*x^2)^( 1/3))/b - (a*c^(1/3)*(c*x)^(1/3)*(c^(2/3) - b^(1/3)*(c*x)^(2/3))*Sqrt[(c^( 4/3) + b^(1/3)*c^(2/3)*(c*x)^(2/3) + b^(2/3)*(c*x)^(4/3))/(c^(2/3) - (1 + Sqrt[3])*b^(1/3)*(c*x)^(2/3))^2]*EllipticF[ArcCos[(c^(2/3) - (1 - Sqrt[3]) *b^(1/3)*(c*x)^(2/3))/(c^(2/3) - (1 + Sqrt[3])*b^(1/3)*(c*x)^(2/3))], (2 + Sqrt[3])/4])/(2*3^(1/4)*b*Sqrt[1 - b*x^2]*(a + b*x^2)^(2/3)*Sqrt[(a*c^2)/ (a*c^2 + b*c^2*x^2)]*Sqrt[-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - b^(1/3)*(c*x)^ (2/3)))/(c^(2/3) - (1 + Sqrt[3])*b^(1/3)*(c*x)^(2/3))^2)])))/9
3.8.47.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a/(a + b*x^n))^(p + 1 /n)*(a + b*x^n)^(p + 1/n) Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x /(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && LtQ[Denominator[p + 1/n], Denominator[p]]
\[\int \left (c x \right )^{\frac {4}{3}} \left (b \,x^{2}+a \right )^{\frac {1}{3}}d x\]
\[ \int (c x)^{4/3} \sqrt [3]{a+b x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {4}{3}} \,d x } \]
Result contains complex when optimal does not.
Time = 3.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.11 \[ \int (c x)^{4/3} \sqrt [3]{a+b x^2} \, dx=\frac {\sqrt [3]{a} c^{\frac {4}{3}} x^{\frac {7}{3}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{6}\right )} \]
a**(1/3)*c**(4/3)*x**(7/3)*gamma(7/6)*hyper((-1/3, 7/6), (13/6,), b*x**2*e xp_polar(I*pi)/a)/(2*gamma(13/6))
\[ \int (c x)^{4/3} \sqrt [3]{a+b x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {4}{3}} \,d x } \]
\[ \int (c x)^{4/3} \sqrt [3]{a+b x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {4}{3}} \,d x } \]
Timed out. \[ \int (c x)^{4/3} \sqrt [3]{a+b x^2} \, dx=\int {\left (c\,x\right )}^{4/3}\,{\left (b\,x^2+a\right )}^{1/3} \,d x \]