Integrand size = 28, antiderivative size = 422 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {2 \left (39 a^2+\frac {b c (7 b c-26 a d)}{d^2}\right ) (e x)^{3/2} \sqrt {c+d x^2}}{195 e}+\frac {4 c \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{195 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 b (7 b c-26 a d) (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{117 d^2 e}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}-\frac {4 c^{5/4} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{11/4} \sqrt {c+d x^2}}+\frac {2 c^{5/4} \left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{195 d^{11/4} \sqrt {c+d x^2}} \] Output:
2/195*(39*a^2+b*c*(-26*a*d+7*b*c)/d^2)*(e*x)^(3/2)*(d*x^2+c)^(1/2)/e+4/195 *c*(39*a^2*d^2+b*c*(-26*a*d+7*b*c))*(e*x)^(1/2)*(d*x^2+c)^(1/2)/d^(5/2)/(c ^(1/2)+d^(1/2)*x)-2/117*b*(-26*a*d+7*b*c)*(e*x)^(3/2)*(d*x^2+c)^(3/2)/d^2/ e+2/13*b^2*(e*x)^(7/2)*(d*x^2+c)^(3/2)/d/e^3-4/195*c^(5/4)*(39*a^2*d^2+b*c *(-26*a*d+7*b*c))*e^(1/2)*(c^(1/2)+d^(1/2)*x)*((d*x^2+c)/(c^(1/2)+d^(1/2)* x)^2)^(1/2)*EllipticE(sin(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1 /2*2^(1/2))/d^(11/4)/(d*x^2+c)^(1/2)+2/195*c^(5/4)*(39*a^2*d^2+b*c*(-26*a* d+7*b*c))*e^(1/2)*(c^(1/2)+d^(1/2)*x)*((d*x^2+c)/(c^(1/2)+d^(1/2)*x)^2)^(1 /2)*InverseJacobiAM(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)),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 = 21.13 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.34 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {2 \sqrt {e x} \left (-x \left (c+d x^2\right ) \left (-117 a^2 d^2-26 a b d \left (2 c+5 d x^2\right )+b^2 \left (14 c^2-10 c d x^2-45 d^2 x^4\right )\right )+6 c \left (7 b^2 c^2-26 a b c d+39 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{585 d^2 \sqrt {c+d x^2}} \] Input:
Integrate[Sqrt[e*x]*(a + b*x^2)^2*Sqrt[c + d*x^2],x]
Output:
(2*Sqrt[e*x]*(-(x*(c + d*x^2)*(-117*a^2*d^2 - 26*a*b*d*(2*c + 5*d*x^2) + b ^2*(14*c^2 - 10*c*d*x^2 - 45*d^2*x^4))) + 6*c*(7*b^2*c^2 - 26*a*b*c*d + 39 *a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c/(d*x ^2))]))/(585*d^2*Sqrt[c + d*x^2])
Time = 0.51 (sec) , antiderivative size = 413, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {367, 27, 363, 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.
| \(\displaystyle \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx\) |
| \(\Big \downarrow \) 367 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {e x} \sqrt {d x^2+c} \left (13 a^2 d-b (7 b c-26 a d) x^2\right )dx}{13 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {e x} \sqrt {d x^2+c} \left (13 a^2 d-b (7 b c-26 a d) x^2\right )dx}{13 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {\left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \int \sqrt {e x} \sqrt {d x^2+c}dx}{3 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{9 d e}}{13 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {\left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \left (\frac {2}{5} c \int \frac {\sqrt {e x}}{\sqrt {d x^2+c}}dx+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )}{3 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{9 d e}}{13 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {\left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \left (\frac {4 c \int \frac {e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )}{3 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{9 d e}}{13 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {\frac {\left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \left (\frac {4 c \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\sqrt {c} e \int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {c} e \sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )}{3 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{9 d e}}{13 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \left (\frac {4 c \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )}{3 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{9 d e}}{13 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \left (\frac {4 c \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )}{3 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{9 d e}}{13 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {\left (39 a^2 d^2+b c (7 b c-26 a d)\right ) \left (\frac {4 c \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^2}}-\frac {e^2 \sqrt {e x} \sqrt {c+d x^2}}{\sqrt {c} e+\sqrt {d} e x}}{\sqrt {d}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {c+d x^2}}{5 e}\right )}{3 d}-\frac {2 b (e x)^{3/2} \left (c+d x^2\right )^{3/2} (7 b c-26 a d)}{9 d e}}{13 d}+\frac {2 b^2 (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{13 d e^3}\) |
Input:
Int[Sqrt[e*x]*(a + b*x^2)^2*Sqrt[c + d*x^2],x]
Output:
(2*b^2*(e*x)^(7/2)*(c + d*x^2)^(3/2))/(13*d*e^3) + ((-2*b*(7*b*c - 26*a*d) *(e*x)^(3/2)*(c + d*x^2)^(3/2))/(9*d*e) + ((39*a^2*d^2 + b*c*(7*b*c - 26*a *d))*((2*(e*x)^(3/2)*Sqrt[c + d*x^2])/(5*e) + (4*c*(-((-((e^2*Sqrt[e*x]*Sq rt[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]*Ellipti cE[2*ArcTan[(d^(1/4)*Sqrt[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)*S qrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(2*d^(3/4)*Sqrt[c + d*x^2])))/(5*e)))/ (3*d))/(13*d)
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 = 0.91 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(\frac {2 x^{2} \left (45 b^{2} d^{2} x^{4}+130 x^{2} a b \,d^{2}+10 x^{2} b^{2} c d +117 a^{2} d^{2}+52 a b c d -14 b^{2} c^{2}\right ) \sqrt {x^{2} d +c}\, e}{585 d^{2} \sqrt {e x}}+\frac {2 c \left (39 a^{2} d^{2}-26 a b c d +7 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) e \sqrt {e x \left (x^{2} d +c \right )}}{195 d^{3} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(290\) |
| elliptic | \(\frac {\sqrt {e x \left (x^{2} d +c \right )}\, \sqrt {e x}\, \left (\frac {2 b^{2} x^{5} \sqrt {d e \,x^{3}+c e x}}{13}+\frac {2 \left (b \left (2 a d +b c \right ) e -\frac {11 c e \,b^{2}}{13}\right ) x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d e}+\frac {2 \left (a \left (a d +2 b c \right ) e -\frac {7 \left (b \left (2 a d +b c \right ) e -\frac {11 c e \,b^{2}}{13}\right ) c}{9 d}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (a^{2} c e -\frac {3 \left (a \left (a d +2 b c \right ) e -\frac {7 \left (b \left (2 a d +b c \right ) e -\frac {11 c e \,b^{2}}{13}\right ) c}{9 d}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {x^{2} d +c}}\) | \(367\) |
| default | \(\frac {2 \sqrt {e x}\, \left (45 b^{2} x^{8} d^{4}+130 a b \,d^{4} x^{6}+55 c \,x^{6} d^{3} b^{2}+234 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}-156 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d +42 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}-117 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}+78 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d -21 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}+117 a^{2} d^{4} x^{4}+182 a b c \,d^{3} x^{4}-4 b^{2} c^{2} d^{2} x^{4}+117 a^{2} c \,d^{3} x^{2}+52 c^{2} x^{2} d^{2} a b -14 b^{2} c^{3} d \,x^{2}\right )}{585 \sqrt {x^{2} d +c}\, d^{3} x}\) | \(658\) |
Input:
int((e*x)^(1/2)*(b*x^2+a)^2*(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/585*x^2*(45*b^2*d^2*x^4+130*a*b*d^2*x^2+10*b^2*c*d*x^2+117*a^2*d^2+52*a* b*c*d-14*b^2*c^2)*(d*x^2+c)^(1/2)/d^2*e/(e*x)^(1/2)+2/195*c*(39*a^2*d^2-26 *a*b*c*d+7*b^2*c^2)/d^3*(-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)*(-d/(-c*d)^(1/2)*x)^(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)
Time = 0.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.32 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=-\frac {2 \, {\left (6 \, {\left (7 \, b^{2} c^{3} - 26 \, a b c^{2} d + 39 \, a^{2} c d^{2}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (45 \, b^{2} d^{3} x^{5} + 10 \, {\left (b^{2} c d^{2} + 13 \, a b d^{3}\right )} x^{3} - {\left (14 \, b^{2} c^{2} d - 52 \, a b c d^{2} - 117 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{585 \, d^{3}} \] Input:
integrate((e*x)^(1/2)*(b*x^2+a)^2*(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
-2/585*(6*(7*b^2*c^3 - 26*a*b*c^2*d + 39*a^2*c*d^2)*sqrt(d*e)*weierstrassZ eta(-4*c/d, 0, weierstrassPInverse(-4*c/d, 0, x)) - (45*b^2*d^3*x^5 + 10*( b^2*c*d^2 + 13*a*b*d^3)*x^3 - (14*b^2*c^2*d - 52*a*b*c*d^2 - 117*a^2*d^3)* x)*sqrt(d*x^2 + c)*sqrt(e*x))/d^3
Result contains complex when optimal does not.
Time = 3.75 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.36 \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {a^{2} \sqrt {c} \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {a b \sqrt {c} \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} \sqrt {c} \sqrt {e} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {15}{4}\right )} \] Input:
integrate((e*x)**(1/2)*(b*x**2+a)**2*(d*x**2+c)**(1/2),x)
Output:
a**2*sqrt(c)*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*b*sqrt(c)*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) + b** 2*sqrt(c)*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))
\[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c} \sqrt {e x} \,d x } \] Input:
integrate((e*x)^(1/2)*(b*x^2+a)^2*(d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)*sqrt(e*x), x)
\[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c} \sqrt {e x} \,d x } \] Input:
integrate((e*x)^(1/2)*(b*x^2+a)^2*(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)*sqrt(e*x), x)
Timed out. \[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\int \sqrt {e\,x}\,{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c} \,d x \] Input:
int((e*x)^(1/2)*(a + b*x^2)^2*(c + d*x^2)^(1/2),x)
Output:
int((e*x)^(1/2)*(a + b*x^2)^2*(c + d*x^2)^(1/2), x)
\[ \int \sqrt {e x} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {2 \sqrt {e}\, \left (117 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, a^{2} d^{2} x +52 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, a b c d x +130 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, a b \,d^{2} x^{3}-14 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, b^{2} c^{2} x +10 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, b^{2} c d \,x^{3}+45 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, b^{2} d^{2} x^{5}+117 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{2}+c}d x \right ) a^{2} c \,d^{2}-78 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{2}+c}d x \right ) a b \,c^{2} d +21 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{2}+c}d x \right ) b^{2} c^{3}\right )}{585 d^{2}} \] Input:
int((e*x)^(1/2)*(b*x^2+a)^2*(d*x^2+c)^(1/2),x)
Output:
(2*sqrt(e)*(117*sqrt(x)*sqrt(c + d*x**2)*a**2*d**2*x + 52*sqrt(x)*sqrt(c + d*x**2)*a*b*c*d*x + 130*sqrt(x)*sqrt(c + d*x**2)*a*b*d**2*x**3 - 14*sqrt( x)*sqrt(c + d*x**2)*b**2*c**2*x + 10*sqrt(x)*sqrt(c + d*x**2)*b**2*c*d*x** 3 + 45*sqrt(x)*sqrt(c + d*x**2)*b**2*d**2*x**5 + 117*int((sqrt(x)*sqrt(c + d*x**2))/(c + d*x**2),x)*a**2*c*d**2 - 78*int((sqrt(x)*sqrt(c + d*x**2))/ (c + d*x**2),x)*a*b*c**2*d + 21*int((sqrt(x)*sqrt(c + d*x**2))/(c + d*x**2 ),x)*b**2*c**3))/(585*d**2)