Integrand size = 28, antiderivative size = 482 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {4 c \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{3315 d^2 e}+\frac {8 c^2 \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{3315 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{1989 d^2 e}-\frac {2 b (7 b c-34 a d) (e x)^{3/2} \left (c+d x^2\right )^{5/2}}{221 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}-\frac {8 c^{9/4} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3315 d^{11/4} \sqrt {c+d x^2}}+\frac {4 c^{9/4} \left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{3315 d^{11/4} \sqrt {c+d x^2}} \]
2/1989*(221*a^2*d^2+3*b*c*(-34*a*d+7*b*c))*(e*x)^(3/2)*(d*x^2+c)^(3/2)/d^2 /e-2/221*b*(-34*a*d+7*b*c)*(e*x)^(3/2)*(d*x^2+c)^(5/2)/d^2/e+2/17*b^2*(e*x )^(7/2)*(d*x^2+c)^(5/2)/d/e^3+4/3315*c*(221*a^2*d^2+3*b*c*(-34*a*d+7*b*c)) *(e*x)^(3/2)*(d*x^2+c)^(1/2)/d^2/e+8/3315*c^2*(221*a^2*d^2+3*b*c*(-34*a*d+ 7*b*c))*(e*x)^(1/2)*(d*x^2+c)^(1/2)/d^(5/2)/(c^(1/2)+x*d^(1/2))-8/3315*c^( 9/4)*(221*a^2*d^2+3*b*c*(-34*a*d+7*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))*e^(1/2)*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/d ^(11/4)/(d*x^2+c)^(1/2)+4/3315*c^(9/4)*(221*a^2*d^2+3*b*c*(-34*a*d+7*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))*e^(1/2)*((d*x^2 +c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/d^(11/4)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.16 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.37 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {2 \sqrt {e x} \left (-x \left (c+d x^2\right ) \left (-221 a^2 d^2 \left (11 c+5 d x^2\right )-102 a b d \left (4 c^2+25 c d x^2+15 d^2 x^4\right )+b^2 \left (84 c^3-60 c^2 d x^2-855 c d^2 x^4-585 d^3 x^6\right )\right )+12 c^2 \left (21 b^2 c^2-102 a b c d+221 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{9945 d^2 \sqrt {c+d x^2}} \]
(2*Sqrt[e*x]*(-(x*(c + d*x^2)*(-221*a^2*d^2*(11*c + 5*d*x^2) - 102*a*b*d*( 4*c^2 + 25*c*d*x^2 + 15*d^2*x^4) + b^2*(84*c^3 - 60*c^2*d*x^2 - 855*c*d^2* x^4 - 585*d^3*x^6))) + 12*c^2*(21*b^2*c^2 - 102*a*b*c*d + 221*a^2*d^2)*Sqr t[1 + c/(d*x^2)]*x*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x^2))]))/(9945 *d^2*Sqrt[c + d*x^2])
Time = 0.53 (sec) , antiderivative size = 445, 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 = {367, 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.
| \(\displaystyle \int \sqrt {e x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 367 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {e x} \left (d x^2+c\right )^{3/2} \left (17 a^2 d-b (7 b c-34 a d) x^2\right )dx}{17 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {e x} \left (d x^2+c\right )^{3/2} \left (17 a^2 d-b (7 b c-34 a d) x^2\right )dx}{17 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {\left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \int \sqrt {e x} \left (d x^2+c\right )^{3/2}dx}{13 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{13 d e}}{17 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {\left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \left (\frac {2}{3} c \int \sqrt {e x} \sqrt {d x^2+c}dx+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{13 d e}}{17 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {\left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \left (\frac {2}{3} c \left (\frac {2}{5} c \int \frac {\sqrt {e x}}{\sqrt {d x^2+c}}dx+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{13 d e}}{17 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {\left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \left (\frac {2}{3} c \left (\frac {4 c \int \frac {e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{13 d e}}{17 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {\frac {\left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \left (\frac {2}{3} c \left (\frac {4 c \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\sqrt {c} e \int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {c} e \sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{13 d e}}{17 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \left (\frac {2}{3} c \left (\frac {4 c \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{13 d e}}{17 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \left (\frac {2}{3} c \left (\frac {4 c \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{13 d e}}{17 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {\left (221 a^2 d^2+3 b c (7 b c-34 a d)\right ) \left (\frac {2}{3} c \left (\frac {4 c \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^2}}-\frac {e^2 \sqrt {e x} \sqrt {c+d x^2}}{\sqrt {c} e+\sqrt {d} e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 e}\right )}{13 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{5/2} (7 b c-34 a d)}{13 d e}}{17 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{17 d e^3}\) |
(2*b^2*(e*x)^(7/2)*(c + d*x^2)^(5/2))/(17*d*e^3) + ((-2*b*(7*b*c - 34*a*d) *(e*x)^(3/2)*(c + d*x^2)^(5/2))/(13*d*e) + ((221*a^2*d^2 + 3*b*c*(7*b*c - 34*a*d))*((2*(e*x)^(3/2)*(c + d*x^2)^(3/2))/(9*e) + (2*c*((2*(e*x)^(3/2)*S qrt[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)*S qrt[e*x])/(c^(1/4)*Sqrt[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)*Sqr t[e])], 1/2])/(2*d^(3/4)*Sqrt[c + d*x^2])))/(5*e)))/3))/(13*d))/(17*d)
3.9.34.3.1 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[d^2*(e*x)^(m + 3)*((a + b*x^2)^(p + 1)/(b*e^3*(m + 2*p + 5))), x] + Simp[1/(b*(m + 2*p + 5)) Int[(e*x)^m*(a + b*x^2)^p*Simp[b*c^2* (m + 2*p + 5) - d*(a*d*(m + 3) - 2*b*c*(m + 2*p + 5))*x^2, x], x], x] /; Fr eeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + 2*p + 5, 0]
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 = 3.12 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(\frac {2 x^{2} \left (585 b^{2} d^{3} x^{6}+1530 a b \,d^{3} x^{4}+855 b^{2} c \,d^{2} x^{4}+1105 a^{2} d^{3} x^{2}+2550 a b c \,d^{2} x^{2}+60 b^{2} c^{2} d \,x^{2}+2431 c \,a^{2} d^{2}+408 a b \,c^{2} d -84 b^{2} c^{3}\right ) \sqrt {d \,x^{2}+c}\, e}{9945 d^{2} \sqrt {e x}}+\frac {4 c^{2} \left (221 a^{2} d^{2}-102 a b c d +21 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) e \sqrt {e x \left (d \,x^{2}+c \right )}}{3315 d^{3} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(331\) |
| elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {2 b^{2} d \,x^{7} \sqrt {d e \,x^{3}+c e x}}{17}+\frac {2 \left (2 b d \left (a d +b c \right ) e -\frac {15 b^{2} d c e}{17}\right ) x^{5} \sqrt {d e \,x^{3}+c e x}}{13 d e}+\frac {2 \left (\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) e -\frac {11 \left (2 b d \left (a d +b c \right ) e -\frac {15 b^{2} d c e}{17}\right ) c}{13 d}\right ) x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d e}+\frac {2 \left (2 a c \left (a d +b c \right ) e -\frac {7 \left (\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) e -\frac {11 \left (2 b d \left (a d +b c \right ) e -\frac {15 b^{2} d c e}{17}\right ) c}{13 d}\right ) c}{9 d}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (a^{2} c^{2} e -\frac {3 \left (2 a c \left (a d +b c \right ) e -\frac {7 \left (\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) e -\frac {11 \left (2 b d \left (a d +b c \right ) e -\frac {15 b^{2} d c e}{17}\right ) c}{13 d}\right ) c}{9 d}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(513\) |
| default | \(\frac {2 \sqrt {e x}\, \left (585 b^{2} d^{5} x^{10}+1530 a b \,d^{5} x^{8}+1440 b^{2} c \,d^{4} x^{8}+1105 a^{2} d^{5} x^{6}+4080 a b c \,d^{4} x^{6}+915 b^{2} c^{2} d^{3} x^{6}+2652 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{3} d^{2}-1224 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{4} d +252 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{5}-1326 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{3} d^{2}+612 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{4} d -126 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{5}+3536 a^{2} c \,d^{4} x^{4}+2958 a b \,c^{2} d^{3} x^{4}-24 b^{2} c^{3} d^{2} x^{4}+2431 a^{2} c^{2} d^{3} x^{2}+408 a b \,c^{3} d^{2} x^{2}-84 b^{2} c^{4} d \,x^{2}\right )}{9945 \sqrt {d \,x^{2}+c}\, d^{3} x}\) | \(699\) |
2/9945/d^2*x^2*(585*b^2*d^3*x^6+1530*a*b*d^3*x^4+855*b^2*c*d^2*x^4+1105*a^ 2*d^3*x^2+2550*a*b*c*d^2*x^2+60*b^2*c^2*d*x^2+2431*a^2*c*d^2+408*a*b*c^2*d -84*b^2*c^3)*(d*x^2+c)^(1/2)*e/(e*x)^(1/2)+4/3315*c^2/d^3*(221*a^2*d^2-102 *a*b*c*d+21*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*EllipticF(((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 = 173, normalized size of antiderivative = 0.36 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (12 \, {\left (21 \, b^{2} c^{4} - 102 \, a b c^{3} d + 221 \, a^{2} c^{2} d^{2}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (585 \, b^{2} d^{4} x^{7} + 45 \, {\left (19 \, b^{2} c d^{3} + 34 \, a b d^{4}\right )} x^{5} + 5 \, {\left (12 \, b^{2} c^{2} d^{2} + 510 \, a b c d^{3} + 221 \, a^{2} d^{4}\right )} x^{3} - {\left (84 \, b^{2} c^{3} d - 408 \, a b c^{2} d^{2} - 2431 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{9945 \, d^{3}} \]
-2/9945*(12*(21*b^2*c^4 - 102*a*b*c^3*d + 221*a^2*c^2*d^2)*sqrt(d*e)*weier strassZeta(-4*c/d, 0, weierstrassPInverse(-4*c/d, 0, x)) - (585*b^2*d^4*x^ 7 + 45*(19*b^2*c*d^3 + 34*a*b*d^4)*x^5 + 5*(12*b^2*c^2*d^2 + 510*a*b*c*d^3 + 221*a^2*d^4)*x^3 - (84*b^2*c^3*d - 408*a*b*c^2*d^2 - 2431*a^2*c*d^3)*x) *sqrt(d*x^2 + c)*sqrt(e*x))/d^3
Result contains complex when optimal does not.
Time = 10.48 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.63 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\frac {a^{2} c^{\frac {3}{2}} \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {a^{2} \sqrt {c} d \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {a b c^{\frac {3}{2}} \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\Gamma \left (\frac {11}{4}\right )} + \frac {a b \sqrt {c} d \sqrt {e} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\Gamma \left (\frac {15}{4}\right )} + \frac {b^{2} c^{\frac {3}{2}} \sqrt {e} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {15}{4}\right )} + \frac {b^{2} \sqrt {c} d \sqrt {e} x^{\frac {15}{2}} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {19}{4}\right )} \]
a**2*c**(3/2)*sqrt(e)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), d*x** 2*exp_polar(I*pi)/c)/(2*gamma(7/4)) + a**2*sqrt(c)*d*sqrt(e)*x**(7/2)*gamm a(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/(2*gamma(11/4 )) + a*b*c**(3/2)*sqrt(e)*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), d*x**2*exp_polar(I*pi)/c)/gamma(11/4) + a*b*sqrt(c)*d*sqrt(e)*x**(11/2)*ga mma(11/4)*hyper((-1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/gamma(15/ 4) + b**2*c**(3/2)*sqrt(e)*x**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4 ,), d*x**2*exp_polar(I*pi)/c)/(2*gamma(15/4)) + b**2*sqrt(c)*d*sqrt(e)*x** (15/2)*gamma(15/4)*hyper((-1/2, 15/4), (19/4,), d*x**2*exp_polar(I*pi)/c)/ (2*gamma(19/4))
\[ \int \sqrt {e x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\int { {\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {e x} \,d x } \]
\[ \int \sqrt {e x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\int { {\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {e x} \,d x } \]
Timed out. \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx=\int \sqrt {e\,x}\,{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2} \,d x \]