Integrand size = 22, antiderivative size = 94 \[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{4/3} \, dx=-\frac {9}{44} a x \left (a+b x^3\right )^{7/3}-\frac {1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3}+\frac {57 a^3 x \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{44 \sqrt [3]{1+\frac {b x^3}{a}}} \]
-9/44*a*x*(b*x^3+a)^(7/3)-1/11*x*(-b*x^3+a)*(b*x^3+a)^(7/3)+57/44*a^3*x*(b *x^3+a)^(1/3)*hypergeom([-4/3, 1/3],[4/3],-b*x^3/a)/(1+b*x^3/a)^(1/3)
Time = 8.40 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.03 \[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{4/3} \, dx=\frac {x \left (106 a^4+53 a^3 b x^3-78 a^2 b^2 x^6-5 a b^3 x^9+20 b^4 x^{12}+114 a^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},-\frac {b x^3}{a}\right )\right )}{220 \left (a+b x^3\right )^{2/3}} \]
(x*(106*a^4 + 53*a^3*b*x^3 - 78*a^2*b^2*x^6 - 5*a*b^3*x^9 + 20*b^4*x^12 + 114*a^4*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^3)/a )]))/(220*(a + b*x^3)^(2/3))
Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {933, 27, 913, 779, 778}
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.
\(\displaystyle \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{4/3} \, dx\) |
\(\Big \downarrow \) 933 |
\(\displaystyle \frac {\int 6 a b \left (2 a-3 b x^3\right ) \left (b x^3+a\right )^{4/3}dx}{11 b}-\frac {1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {6}{11} a \int \left (2 a-3 b x^3\right ) \left (b x^3+a\right )^{4/3}dx-\frac {1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3}\) |
\(\Big \downarrow \) 913 |
\(\displaystyle \frac {6}{11} a \left (\frac {19}{8} a \int \left (b x^3+a\right )^{4/3}dx-\frac {3}{8} x \left (a+b x^3\right )^{7/3}\right )-\frac {1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3}\) |
\(\Big \downarrow \) 779 |
\(\displaystyle \frac {6}{11} a \left (\frac {19 a^2 \sqrt [3]{a+b x^3} \int \left (\frac {b x^3}{a}+1\right )^{4/3}dx}{8 \sqrt [3]{\frac {b x^3}{a}+1}}-\frac {3}{8} x \left (a+b x^3\right )^{7/3}\right )-\frac {1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3}\) |
\(\Big \downarrow \) 778 |
\(\displaystyle \frac {6}{11} a \left (\frac {19 a^2 x \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{8 \sqrt [3]{\frac {b x^3}{a}+1}}-\frac {3}{8} x \left (a+b x^3\right )^{7/3}\right )-\frac {1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3}\) |
-1/11*(x*(a - b*x^3)*(a + b*x^3)^(7/3)) + (6*a*((-3*x*(a + b*x^3)^(7/3))/8 + (19*a^2*x*(a + b*x^3)^(1/3)*Hypergeometric2F1[-4/3, 1/3, 4/3, -((b*x^3) /a)])/(8*(1 + (b*x^3)/a)^(1/3))))/11
3.1.48.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p , 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x ^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p, x], x ] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p, 0] && !IntegerQ[1/n] && !ILtQ[Si mplify[1/n + p], 0] && !(IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Simp[1/(b*(n*(p + q) + 1)) Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d , 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[ a, b, c, d, n, p, q, x]
\[\int \left (-b \,x^{3}+a \right )^{2} \left (b \,x^{3}+a \right )^{\frac {4}{3}}d x\]
\[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{4/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (b x^{3} - a\right )}^{2} \,d x } \]
Result contains complex when optimal does not.
Time = 2.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.79 \[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{4/3} \, dx=\frac {a^{\frac {10}{3}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - \frac {a^{\frac {7}{3}} b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} - \frac {a^{\frac {4}{3}} b^{2} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {\sqrt [3]{a} b^{3} x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} \]
a**(10/3)*x*gamma(1/3)*hyper((-1/3, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a )/(3*gamma(4/3)) - a**(7/3)*b*x**4*gamma(4/3)*hyper((-1/3, 4/3), (7/3,), b *x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) - a**(4/3)*b**2*x**7*gamma(7/3)*hy per((-1/3, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + a**( 1/3)*b**3*x**10*gamma(10/3)*hyper((-1/3, 10/3), (13/3,), b*x**3*exp_polar( I*pi)/a)/(3*gamma(13/3))
\[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{4/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (b x^{3} - a\right )}^{2} \,d x } \]
\[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{4/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (b x^{3} - a\right )}^{2} \,d x } \]
Timed out. \[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{4/3} \, dx=\int {\left (b\,x^3+a\right )}^{4/3}\,{\left (a-b\,x^3\right )}^2 \,d x \]