Integrand size = 28, antiderivative size = 476 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=-\frac {4 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 d e^3}-\frac {8 c \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{195 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e^3}+\frac {8 c^{5/4} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {4 c^{5/4} \left (3 b^2 c^2-13 a d (2 b c+9 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{195 d^{7/4} e^{3/2} \sqrt {c+d x^2}} \]
-2/117*(3*b^2*c^2-13*a*d*(9*a*d+2*b*c))*(e*x)^(3/2)*(d*x^2+c)^(3/2)/c/d/e^ 3+2/13*b^2*(e*x)^(3/2)*(d*x^2+c)^(5/2)/d/e^3-2*a^2*(d*x^2+c)^(5/2)/c/e/(e* x)^(1/2)-4/195*(3*b^2*c^2-13*a*d*(9*a*d+2*b*c))*(e*x)^(3/2)*(d*x^2+c)^(1/2 )/d/e^3-8/195*c*(3*b^2*c^2-13*a*d*(9*a*d+2*b*c))*(e*x)^(1/2)*(d*x^2+c)^(1/ 2)/d^(3/2)/e^2/(c^(1/2)+x*d^(1/2))+8/195*c^(5/4)*(3*b^2*c^2-13*a*d*(9*a*d+ 2*b*c))*(cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))^2)^(1/2)/cos(2 *arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))*EllipticE(sin(2*arctan(d^(1/ 4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2))*(c^(1/2)+x*d^(1/2))*((d*x^2+ c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/d^(7/4)/e^(3/2)/(d*x^2+c)^(1/2)-4/195*c^(5 /4)*(3*b^2*c^2-13*a*d*(9*a*d+2*b*c))*(cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^( 1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))) *EllipticF(sin(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2)) *(c^(1/2)+x*d^(1/2))*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/d^(7/4)/e^(3/ 2)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.12 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.34 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\frac {x \left (2 \left (c+d x^2\right ) \left (117 a^2 d \left (-5 c+d x^2\right )+26 a b d x^2 \left (11 c+5 d x^2\right )+3 b^2 x^2 \left (4 c^2+25 c d x^2+15 d^2 x^4\right )\right )+24 c \left (-3 b^2 c^2+26 a b c d+117 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{585 d (e x)^{3/2} \sqrt {c+d x^2}} \]
(x*(2*(c + d*x^2)*(117*a^2*d*(-5*c + d*x^2) + 26*a*b*d*x^2*(11*c + 5*d*x^2 ) + 3*b^2*x^2*(4*c^2 + 25*c*d*x^2 + 15*d^2*x^4)) + 24*c*(-3*b^2*c^2 + 26*a *b*c*d + 117*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(585*d*(e*x)^(3/2)*Sqrt[c + d*x^2])
Time = 0.51 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.92, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {365, 27, 363, 248, 248, 266, 834, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {e x} \left (b^2 c x^2+a (2 b c+9 a d)\right ) \left (d x^2+c\right )^{3/2}dx}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {e x} \left (b^2 c x^2+a (2 b c+9 a d)\right ) \left (d x^2+c\right )^{3/2}dx}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {2 b^2 c (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e}-\frac {\left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) \int \sqrt {e x} \left (d x^2+c\right )^{3/2}dx}{13 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {2 b^2 c (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e}-\frac {\left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) \left (\frac {2}{3} c \int \sqrt {e x} \sqrt {d x^2+c}dx+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {2 b^2 c (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e}-\frac {\left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) \left (\frac {2}{3} c \left (\frac {2}{5} c \int \frac {\sqrt {e x}}{\sqrt {d x^2+c}}dx+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {2 b^2 c (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e}-\frac {\left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) \left (\frac {2}{3} c \left (\frac {4 c \int \frac {e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {\frac {2 b^2 c (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e}-\frac {\left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) \left (\frac {2}{3} c \left (\frac {4 c \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\sqrt {c} e \int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {c} e \sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 b^2 c (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e}-\frac {\left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) \left (\frac {2}{3} c \left (\frac {4 c \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {2 b^2 c (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e}-\frac {\left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) \left (\frac {2}{3} c \left (\frac {4 c \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {2 b^2 c (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{13 d e}-\frac {\left (3 b^2 c^2-13 a d (9 a d+2 b c)\right ) \left (\frac {2}{3} c \left (\frac {4 c \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^2}}-\frac {e^2 \sqrt {e x} \sqrt {c+d x^2}}{\sqrt {c} e+\sqrt {d} e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}}{c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{c e \sqrt {e x}}\) |
(-2*a^2*(c + d*x^2)^(5/2))/(c*e*Sqrt[e*x]) + ((2*b^2*c*(e*x)^(3/2)*(c + d* x^2)^(5/2))/(13*d*e) - ((3*b^2*c^2 - 13*a*d*(2*b*c + 9*a*d))*((2*(e*x)^(3/ 2)*(c + d*x^2)^(3/2))/(9*e) + (2*c*((2*(e*x)^(3/2)*Sqrt[c + d*x^2])/(5*e) + (4*c*(-((-((e^2*Sqrt[e*x]*Sqrt[c + d*x^2])/(Sqrt[c]*e + Sqrt[d]*e*x)) + (c^(1/4)*Sqrt[e]*(Sqrt[c]*e + Sqrt[d]*e*x)*Sqrt[(c*e^2 + d*e^2*x^2)/(Sqrt[ c]*e + Sqrt[d]*e*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqr t[e])], 1/2])/(d^(1/4)*Sqrt[c + d*x^2]))/Sqrt[d]) + (c^(1/4)*Sqrt[e]*(Sqrt [c]*e + Sqrt[d]*e*x)*Sqrt[(c*e^2 + d*e^2*x^2)/(Sqrt[c]*e + Sqrt[d]*e*x)^2] *EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(2*d^(3/ 4)*Sqrt[c + d*x^2])))/(5*e)))/3))/(13*d))/(c*e^2)
3.9.36.3.1 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[((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] :> 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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Time = 3.18 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(-\frac {2 \sqrt {d \,x^{2}+c}\, \left (-45 b^{2} d^{2} x^{6}-130 a b \,d^{2} x^{4}-75 b^{2} c d \,x^{4}-117 a^{2} d^{2} x^{2}-286 a b c d \,x^{2}-12 b^{2} c^{2} x^{2}+585 a^{2} c d \right )}{585 d e \sqrt {e x}}+\frac {4 c \left (117 a^{2} d^{2}+26 a b c d -3 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{195 d^{2} \sqrt {d e \,x^{3}+c e x}\, e \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(307\) |
| elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 \left (d e \,x^{2}+c e \right ) c \,a^{2}}{e^{2} \sqrt {x \left (d e \,x^{2}+c e \right )}}+\frac {2 b^{2} d \,x^{5} \sqrt {d e \,x^{3}+c e x}}{13 e^{2}}+\frac {2 \left (\frac {2 b d \left (a d +b c \right )}{e}-\frac {11 b^{2} d c}{13 e}\right ) x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d e}+\frac {2 \left (\frac {a^{2} d^{2}+4 a b c d +b^{2} c^{2}}{e}-\frac {7 \left (\frac {2 b d \left (a d +b c \right )}{e}-\frac {11 b^{2} d c}{13 e}\right ) c}{9 d}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (\frac {2 a c \left (a d +b c \right )}{e}+\frac {d c \,a^{2}}{e}-\frac {3 \left (\frac {a^{2} d^{2}+4 a b c d +b^{2} c^{2}}{e}-\frac {7 \left (\frac {2 b d \left (a d +b c \right )}{e}-\frac {11 b^{2} d c}{13 e}\right ) c}{9 d}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(461\) |
| default | \(\frac {\frac {2 b^{2} d^{4} x^{8}}{13}+\frac {4 a b \,d^{4} x^{6}}{9}+\frac {16 b^{2} c \,d^{3} x^{6}}{39}+\frac {24 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}}{5}+\frac {16 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d}{15}-\frac {8 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}}{65}-\frac {12 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}}{5}-\frac {8 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d}{15}+\frac {4 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}}{65}+\frac {2 a^{2} d^{4} x^{4}}{5}+\frac {64 c a b \,x^{4} d^{3}}{45}+\frac {58 b^{2} c^{2} d^{2} x^{4}}{195}-\frac {8 a^{2} c \,d^{3} x^{2}}{5}+\frac {44 a b \,c^{2} d^{2} x^{2}}{45}+\frac {8 b^{2} c^{3} d \,x^{2}}{195}-2 a^{2} c^{2} d^{2}}{\sqrt {d \,x^{2}+c}\, d^{2} e \sqrt {e x}}\) | \(669\) |
-2/585*(d*x^2+c)^(1/2)*(-45*b^2*d^2*x^6-130*a*b*d^2*x^4-75*b^2*c*d*x^4-117 *a^2*d^2*x^2-286*a*b*c*d*x^2-12*b^2*c^2*x^2+585*a^2*c*d)/d/e/(e*x)^(1/2)+4 /195*c/d^2*(117*a^2*d^2+26*a*b*c*d-3*b^2*c^2)*(-c*d)^(1/2)*((x+(-c*d)^(1/2 )/d)/(-c*d)^(1/2)*d)^(1/2)*(-2*(x-(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*(- x/(-c*d)^(1/2)*d)^(1/2)/(d*e*x^3+c*e*x)^(1/2)*(-2*(-c*d)^(1/2)/d*EllipticE (((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2*2^(1/2))+(-c*d)^(1/2)/d*Ell ipticF(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2*2^(1/2)))/e*(e*x*(d*x ^2+c))^(1/2)/(e*x)^(1/2)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.32 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\frac {2 \, {\left (12 \, {\left (3 \, b^{2} c^{3} - 26 \, a b c^{2} d - 117 \, a^{2} c d^{2}\right )} \sqrt {d e} x {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (45 \, b^{2} d^{3} x^{6} - 585 \, a^{2} c d^{2} + 5 \, {\left (15 \, b^{2} c d^{2} + 26 \, a b d^{3}\right )} x^{4} + {\left (12 \, b^{2} c^{2} d + 286 \, a b c d^{2} + 117 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{585 \, d^{2} e^{2} x} \]
2/585*(12*(3*b^2*c^3 - 26*a*b*c^2*d - 117*a^2*c*d^2)*sqrt(d*e)*x*weierstra ssZeta(-4*c/d, 0, weierstrassPInverse(-4*c/d, 0, x)) + (45*b^2*d^3*x^6 - 5 85*a^2*c*d^2 + 5*(15*b^2*c*d^2 + 26*a*b*d^3)*x^4 + (12*b^2*c^2*d + 286*a*b *c*d^2 + 117*a^2*d^3)*x^2)*sqrt(d*x^2 + c)*sqrt(e*x))/(d^2*e^2*x)
Result contains complex when optimal does not.
Time = 13.68 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.65 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\frac {a^{2} c^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {a^{2} \sqrt {c} d x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {a b c^{\frac {3}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {a b \sqrt {c} d x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} c^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} \sqrt {c} d x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {15}{4}\right )} \]
a**2*c**(3/2)*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), d*x**2*exp_polar(I*p i)/c)/(2*e**(3/2)*sqrt(x)*gamma(3/4)) + a**2*sqrt(c)*d*x**(3/2)*gamma(3/4) *hyper((-1/2, 3/4), (7/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(3/2)*gamma(7/ 4)) + a*b*c**(3/2)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), d*x**2*e xp_polar(I*pi)/c)/(e**(3/2)*gamma(7/4)) + a*b*sqrt(c)*d*x**(7/2)*gamma(7/4 )*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(e**(3/2)*gamma(11 /4)) + b**2*c**(3/2)*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x** 2*exp_polar(I*pi)/c)/(2*e**(3/2)*gamma(11/4)) + b**2*sqrt(c)*d*x**(11/2)*g amma(11/4)*hyper((-1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(3 /2)*gamma(15/4))
\[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}}{{\left (e\,x\right )}^{3/2}} \,d x \]