Integrand size = 24, antiderivative size = 592 \[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\frac {45}{7} x \left (a-b x^2\right )^{2/3}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}+\frac {324 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {108 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
45/7*x*(-b*x^2+a)^(2/3)+6*x*(b*x^2+3*a)/(-b*x^2+a)^(1/3)+324/7*a*x/(-(-b*x ^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))-108/7*3^(3/4)*a^(4/3)*(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))*2^(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) /b/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)+162/7*3^(1/4)*a^(4/3)*(a^(1/3)-(-b*x^2+a)^(1/3))*Elliptic E((-(-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/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)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 14.23 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.28 \[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\frac {x \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {Gamma}\left (\frac {1}{3}\right ) \left (63 a \left (45 a^2+10 a b x^2+b^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{2},\frac {b x^2}{a}\right )+32 b x^2 \left (18 a^2+9 a b x^2+b^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{3},\frac {9}{2},\frac {b x^2}{a}\right )+16 b \left (3 a x+b x^3\right )^2 \, _3F_2\left (\frac {3}{2},2,\frac {7}{3};1,\frac {9}{2};\frac {b x^2}{a}\right )\right )}{945 a^2 \sqrt [3]{a-b x^2} \operatorname {Gamma}\left (\frac {4}{3}\right )} \]
(x*(1 - (b*x^2)/a)^(1/3)*Gamma[1/3]*(63*a*(45*a^2 + 10*a*b*x^2 + b^2*x^4)* Hypergeometric2F1[1/2, 4/3, 7/2, (b*x^2)/a] + 32*b*x^2*(18*a^2 + 9*a*b*x^2 + b^2*x^4)*Hypergeometric2F1[3/2, 7/3, 9/2, (b*x^2)/a] + 16*b*(3*a*x + b* x^3)^2*HypergeometricPFQ[{3/2, 2, 7/3}, {1, 9/2}, (b*x^2)/a]))/(945*a^2*(a - b*x^2)^(1/3)*Gamma[4/3])
Time = 0.51 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {315, 27, 299, 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.
| \(\displaystyle \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-\frac {3 \int \frac {2 a b \left (5 b x^2+3 a\right )}{\sqrt [3]{a-b x^2}}dx}{2 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-3 \int \frac {5 b x^2+3 a}{\sqrt [3]{a-b x^2}}dx\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-3 \left (\frac {36}{7} a \int \frac {1}{\sqrt [3]{a-b x^2}}dx-\frac {15}{7} x \left (a-b x^2\right )^{2/3}\right )\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-3 \left (-\frac {54 a \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{7 b x}-\frac {15}{7} x \left (a-b x^2\right )^{2/3}\right )\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-3 \left (-\frac {54 a \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{7 b x}-\frac {15}{7} x \left (a-b x^2\right )^{2/3}\right )\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-3 \left (-\frac {54 a \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{7 b x}-\frac {15}{7} x \left (a-b x^2\right )^{2/3}\right )\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-3 \left (-\frac {54 a \sqrt {-b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {2 \sqrt {-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )}{7 b x}-\frac {15}{7} x \left (a-b x^2\right )^{2/3}\right )\) |
(6*x*(3*a + b*x^2))/(a - b*x^2)^(1/3) - 3*((-15*x*(a - b*x^2)^(2/3))/7 - ( 54*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 - S qrt[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*Sqrt[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 + 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]*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))
3.2.30.3.1 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_)^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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
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]
\[\int \frac {\left (b \,x^{2}+3 a \right )^{2}}{\left (-b \,x^{2}+a \right )^{\frac {4}{3}}}d x\]
\[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \]
\[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\int \frac {\left (3 a + b x^{2}\right )^{2}}{\left (a - b x^{2}\right )^{\frac {4}{3}}}\, dx \]
\[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \]
\[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \]
Timed out. \[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\int \frac {{\left (b\,x^2+3\,a\right )}^2}{{\left (a-b\,x^2\right )}^{4/3}} \,d x \]