Integrand size = 22, antiderivative size = 588 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \, dx=\frac {18}{13} a x \left (a-b x^2\right )^{2/3}-\frac {3}{13} x \left (a-b x^2\right )^{5/3}-\frac {72 a^2 x}{13 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {36 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{13 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {24 \sqrt {2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{13 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \] Output:
18/13*a*x*(-b*x^2+a)^(2/3)-3/13*x*(-b*x^2+a)^(5/3)-72*a^2*x/(13*(1-3^(1/2) )*a^(1/3)-13*(-b*x^2+a)^(1/3))-36/13*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*a^( 7/3)*(a^(1/3)-(-b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b*x^2 +a)^(2/3))/((1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))^2)^(1/2)*EllipticE(((1+3 ^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3)),2 *I-I*3^(1/2))/b/x/(-a^(1/3)*(a^(1/3)-(-b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3 )-(-b*x^2+a)^(1/3))^2)^(1/2)+24/13*2^(1/2)*3^(3/4)*a^(7/3)*(a^(1/3)-(-b*x^ 2+a)^(1/3))*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b*x^2+a)^(2/3))/((1-3^(1/ 2))*a^(1/3)-(-b*x^2+a)^(1/3))^2)^(1/2)*EllipticF(((1+3^(1/2))*a^(1/3)-(-b* x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3)),2*I-I*3^(1/2))/b/x/(- a^(1/3)*(a^(1/3)-(-b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(-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 = 5.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.11 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \, dx=\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (-a+b x^2+\frac {14 a \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {3}{2},\frac {b x^2}{a}\right )}{\left (1-\frac {b x^2}{a}\right )^{2/3}}\right ) \] Input:
Integrate[(a - b*x^2)^(2/3)*(3*a + b*x^2),x]
Output:
(3*x*(a - b*x^2)^(2/3)*(-a + b*x^2 + (14*a*Hypergeometric2F1[-2/3, 1/2, 3/ 2, (b*x^2)/a])/(1 - (b*x^2)/a)^(2/3)))/13
Time = 0.48 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {299, 211, 233, 833, 760, 2418}
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.
| \(\displaystyle \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \, dx\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {42}{13} a \int \left (a-b x^2\right )^{2/3}dx-\frac {3}{13} x \left (a-b x^2\right )^{5/3}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {42}{13} a \left (\frac {4}{7} a \int \frac {1}{\sqrt [3]{a-b x^2}}dx+\frac {3}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {3}{13} x \left (a-b x^2\right )^{5/3}\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {42}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{7 b x}\right )-\frac {3}{13} x \left (a-b x^2\right )^{5/3}\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {42}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{7 b x}\right )-\frac {3}{13} x \left (a-b x^2\right )^{5/3}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {42}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{7 b x}\right )-\frac {3}{13} x \left (a-b x^2\right )^{5/3}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {42}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {2 \sqrt {-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )}{7 b x}\right )-\frac {3}{13} x \left (a-b x^2\right )^{5/3}\) |
Input:
Int[(a - b*x^2)^(2/3)*(3*a + b*x^2),x]
Output:
(-3*x*(a - b*x^2)^(5/3))/13 + (42*a*((3*x*(a - b*x^2)^(2/3))/7 - (6*a*Sqrt [-(b*x^2)]*((-2*Sqrt[-(b*x^2)])/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3) ) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(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]*EllipticE[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*Sqr t[3]])/(Sqrt[-(b*x^2)]*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)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + S qrt[3])*a^(1/3)*(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]])/(3^(1/4)*Sqrt[-( b*x^2)]*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)])))/(7*b*x)))/13
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b }, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
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]*Sqrt[(- 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]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && NegQ[a]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
\[\int \left (-b \,x^{2}+a \right )^{\frac {2}{3}} \left (b \,x^{2}+3 a \right )d x\]
Input:
int((-b*x^2+a)^(2/3)*(b*x^2+3*a),x)
Output:
int((-b*x^2+a)^(2/3)*(b*x^2+3*a),x)
\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \, dx=\int { {\left (b x^{2} + 3 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \] Input:
integrate((-b*x^2+a)^(2/3)*(b*x^2+3*a),x, algorithm="fricas")
Output:
integral((b*x^2 + 3*a)*(-b*x^2 + a)^(2/3), x)
Time = 1.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.11 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \, dx=3 a^{\frac {5}{3}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + \frac {a^{\frac {2}{3}} b x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{3} \] Input:
integrate((-b*x**2+a)**(2/3)*(b*x**2+3*a),x)
Output:
3*a**(5/3)*x*hyper((-2/3, 1/2), (3/2,), b*x**2*exp_polar(2*I*pi)/a) + a**( 2/3)*b*x**3*hyper((-2/3, 3/2), (5/2,), b*x**2*exp_polar(2*I*pi)/a)/3
\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \, dx=\int { {\left (b x^{2} + 3 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \] Input:
integrate((-b*x^2+a)^(2/3)*(b*x^2+3*a),x, algorithm="maxima")
Output:
integrate((b*x^2 + 3*a)*(-b*x^2 + a)^(2/3), x)
\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \, dx=\int { {\left (b x^{2} + 3 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \] Input:
integrate((-b*x^2+a)^(2/3)*(b*x^2+3*a),x, algorithm="giac")
Output:
integrate((b*x^2 + 3*a)*(-b*x^2 + a)^(2/3), x)
Timed out. \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \, dx=\int {\left (a-b\,x^2\right )}^{2/3}\,\left (b\,x^2+3\,a\right ) \,d x \] Input:
int((a - b*x^2)^(2/3)*(3*a + b*x^2),x)
Output:
int((a - b*x^2)^(2/3)*(3*a + b*x^2), x)
\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \, dx=\frac {15 \left (-b \,x^{2}+a \right )^{\frac {2}{3}} a x}{13}+\frac {3 \left (-b \,x^{2}+a \right )^{\frac {2}{3}} b \,x^{3}}{13}+\frac {24 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}}}d x \right ) a^{2}}{13} \] Input:
int((-b*x^2+a)^(2/3)*(b*x^2+3*a),x)
Output:
(3*(5*(a - b*x**2)**(2/3)*a*x + (a - b*x**2)**(2/3)*b*x**3 + 8*int((a - b* x**2)**(2/3)/(a - b*x**2),x)*a**2))/13