Integrand size = 28, antiderivative size = 428 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {e (e x)^{3/2} \left (a+b x^2\right )^2}{3 d \left (c+d x^2\right )^{3/2}}+\frac {(11 b c-3 a d) (b c-a d) e (e x)^{3/2}}{6 c d^3 \sqrt {c+d x^2}}+\frac {11 b^2 e (e x)^{3/2} \sqrt {c+d x^2}}{15 d^3}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{10 c d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{20 c^{3/4} d^{15/4} \sqrt {c+d x^2}} \] Output:
-1/3*e*(e*x)^(3/2)*(b*x^2+a)^2/d/(d*x^2+c)^(3/2)+1/6*(-3*a*d+11*b*c)*(-a*d +b*c)*e*(e*x)^(3/2)/c/d^3/(d*x^2+c)^(1/2)+11/15*b^2*e*(e*x)^(3/2)*(d*x^2+c )^(1/2)/d^3-1/10*(5*a^2*d^2-70*a*b*c*d+77*b^2*c^2)*e^2*(e*x)^(1/2)*(d*x^2+ c)^(1/2)/c/d^(7/2)/(c^(1/2)+d^(1/2)*x)+1/10*(5*a^2*d^2-70*a*b*c*d+77*b^2*c ^2)*e^(5/2)*(c^(1/2)+d^(1/2)*x)*((d*x^2+c)/(c^(1/2)+d^(1/2)*x)^2)^(1/2)*El lipticE(sin(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2))/c^ (3/4)/d^(15/4)/(d*x^2+c)^(1/2)-1/20*(5*a^2*d^2-70*a*b*c*d+77*b^2*c^2)*e^(5 /2)*(c^(1/2)+d^(1/2)*x)*((d*x^2+c)/(c^(1/2)+d^(1/2)*x)^2)^(1/2)*InverseJac obiAM(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)),1/2*2^(1/2))/c^(3/4)/d ^(15/4)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.14 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.36 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {e (e x)^{3/2} \left (5 a^2 d^2 \left (c+3 d x^2\right )-10 a b c d \left (7 c+9 d x^2\right )+b^2 c \left (77 c^2+99 c d x^2+12 d^2 x^4\right )-3 \left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \left (c+d x^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{30 c d^3 \left (c+d x^2\right )^{3/2}} \] Input:
Integrate[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]
Output:
(e*(e*x)^(3/2)*(5*a^2*d^2*(c + 3*d*x^2) - 10*a*b*c*d*(7*c + 9*d*x^2) + b^2 *c*(77*c^2 + 99*c*d*x^2 + 12*d^2*x^4) - 3*(77*b^2*c^2 - 70*a*b*c*d + 5*a^2 *d^2)*Sqrt[1 + c/(d*x^2)]*(c + d*x^2)*Hypergeometric2F1[-1/4, 1/2, 3/4, -( c/(d*x^2))]))/(30*c*d^3*(c + d*x^2)^(3/2))
Time = 0.50 (sec) , antiderivative size = 413, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {366, 27, 363, 252, 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.
| \(\displaystyle \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 366 |
| \(\displaystyle \frac {(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {\int -\frac {(e x)^{5/2} \left (6 a^2 d^2+6 b^2 c x^2 d-7 (b c-a d)^2\right )}{2 \left (d x^2+c\right )^{3/2}}dx}{3 c d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(e x)^{5/2} \left (6 a^2 d^2+6 b^2 c x^2 d-7 (b c-a d)^2\right )}{\left (d x^2+c\right )^{3/2}}dx}{6 c d^2}+\frac {(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {12 b^2 c (e x)^{7/2}}{5 e \sqrt {c+d x^2}}-\frac {1}{5} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \int \frac {(e x)^{5/2}}{\left (d x^2+c\right )^{3/2}}dx}{6 c d^2}+\frac {(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {12 b^2 c (e x)^{7/2}}{5 e \sqrt {c+d x^2}}-\frac {1}{5} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \left (\frac {3 e^2 \int \frac {\sqrt {e x}}{\sqrt {d x^2+c}}dx}{2 d}-\frac {e (e x)^{3/2}}{d \sqrt {c+d x^2}}\right )}{6 c d^2}+\frac {(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {12 b^2 c (e x)^{7/2}}{5 e \sqrt {c+d x^2}}-\frac {1}{5} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \left (\frac {3 e \int \frac {e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{d}-\frac {e (e x)^{3/2}}{d \sqrt {c+d x^2}}\right )}{6 c d^2}+\frac {(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {\frac {12 b^2 c (e x)^{7/2}}{5 e \sqrt {c+d x^2}}-\frac {1}{5} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \left (\frac {3 e \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\sqrt {c} e \int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {c} e \sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{d}-\frac {e (e x)^{3/2}}{d \sqrt {c+d x^2}}\right )}{6 c d^2}+\frac {(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {12 b^2 c (e x)^{7/2}}{5 e \sqrt {c+d x^2}}-\frac {1}{5} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \left (\frac {3 e \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{d}-\frac {e (e x)^{3/2}}{d \sqrt {c+d x^2}}\right )}{6 c d^2}+\frac {(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {12 b^2 c (e x)^{7/2}}{5 e \sqrt {c+d x^2}}-\frac {1}{5} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \left (\frac {3 e \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{d}-\frac {e (e x)^{3/2}}{d \sqrt {c+d x^2}}\right )}{6 c d^2}+\frac {(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {12 b^2 c (e x)^{7/2}}{5 e \sqrt {c+d x^2}}-\frac {1}{5} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \left (\frac {3 e \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^2}}-\frac {e^2 \sqrt {e x} \sqrt {c+d x^2}}{\sqrt {c} e+\sqrt {d} e x}}{\sqrt {d}}\right )}{d}-\frac {e (e x)^{3/2}}{d \sqrt {c+d x^2}}\right )}{6 c d^2}+\frac {(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}\) |
Input:
Int[((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x]
Output:
((b*c - a*d)^2*(e*x)^(7/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) + ((12*b^2*c*(e* x)^(7/2))/(5*e*Sqrt[c + d*x^2]) - ((77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*( -((e*(e*x)^(3/2))/(d*Sqrt[c + d*x^2])) + (3*e*(-((-((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)*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)*Sqrt[e])], 1/2])/(2*d^(3/4)*Sqrt[c + d*x^2])))/d))/5)/(6*c*d ^2)
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*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[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[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a* b^2*e*(p + 1))), x] + Simp[1/(2*a*b^2*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + 2*b^2*c^2*(p + 1) + 2*a*b*d^2*(p + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p , -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 = 2.98 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.90
| method | result | size |
| elliptic | \(\frac {\sqrt {e x \left (x^{2} d +c \right )}\, \sqrt {e x}\, \left (-\frac {e^{2} x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d^{5} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {e^{3} x^{2} \left (a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right )}{2 d^{3} c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} e^{2} x \sqrt {d e \,x^{3}+c e x}}{5 d^{3}}+\frac {\left (\frac {2 \left (a d -b c \right ) b \,e^{3}}{d^{3}}-\frac {e^{3} \left (a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right )}{4 c \,d^{3}}-\frac {3 b^{2} e^{3} c}{5 d^{3}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {x^{2} d +c}}\) | \(387\) |
| risch | \(\frac {2 b^{2} x^{2} \sqrt {x^{2} d +c}\, e^{3}}{5 d^{3} \sqrt {e x}}+\frac {\left (\frac {b \left (10 a d -13 b c \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}}+\left (5 a^{2} d^{2}-20 a b c d +15 b^{2} c^{2}\right ) \left (\frac {x^{2}}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}-\frac {\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 )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )-5 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {x \sqrt {d e \,x^{3}+c e x}}{3 c e \,d^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {x^{2}}{2 c^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}-\frac {\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 )}{4 c^{2} d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e^{3} \sqrt {e x \left (x^{2} d +c \right )}}{5 d^{3} \sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(691\) |
| default | \(\text {Expression too large to display}\) | \(1191\) |
Input:
int((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
(e*x*(d*x^2+c))^(1/2)/e/x*(e*x)^(1/2)/(d*x^2+c)^(1/2)*(-1/3*e^2/d^5*x*(a^2 *d^2-2*a*b*c*d+b^2*c^2)*(d*e*x^3+c*e*x)^(1/2)/(x^2+c/d)^2+1/2/d^3*e^3*x^2/ c*(a^2*d^2-6*a*b*c*d+5*b^2*c^2)/((x^2+c/d)*d*e*x)^(1/2)+2/5/d^3*b^2*e^2*x* (d*e*x^3+c*e*x)^(1/2)+(2*(a*d-b*c)*b*e^3/d^3-1/4/c/d^3*e^3*(a^2*d^2-6*a*b* c*d+5*b^2*c^2)-3/5/d^3*b^2*e^3*c)*(-c*d)^(1/2)/d*((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))))
Time = 0.12 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.61 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {3 \, {\left ({\left (77 \, b^{2} c^{2} d^{2} - 70 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} e^{2} x^{4} + 2 \, {\left (77 \, b^{2} c^{3} d - 70 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} e^{2} x^{2} + {\left (77 \, b^{2} c^{4} - 70 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} e^{2}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (12 \, b^{2} c d^{3} e^{2} x^{5} + 3 \, {\left (33 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} e^{2} x^{3} + {\left (77 \, b^{2} c^{3} d - 70 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} e^{2} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{30 \, {\left (c d^{6} x^{4} + 2 \, c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}} \] Input:
integrate((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="fricas")
Output:
1/30*(3*((77*b^2*c^2*d^2 - 70*a*b*c*d^3 + 5*a^2*d^4)*e^2*x^4 + 2*(77*b^2*c ^3*d - 70*a*b*c^2*d^2 + 5*a^2*c*d^3)*e^2*x^2 + (77*b^2*c^4 - 70*a*b*c^3*d + 5*a^2*c^2*d^2)*e^2)*sqrt(d*e)*weierstrassZeta(-4*c/d, 0, weierstrassPInv erse(-4*c/d, 0, x)) + (12*b^2*c*d^3*e^2*x^5 + 3*(33*b^2*c^2*d^2 - 30*a*b*c *d^3 + 5*a^2*d^4)*e^2*x^3 + (77*b^2*c^3*d - 70*a*b*c^2*d^2 + 5*a^2*c*d^3)* e^2*x)*sqrt(d*x^2 + c)*sqrt(e*x))/(c*d^6*x^4 + 2*c^2*d^5*x^2 + c^3*d^4)
Timed out. \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(5/2)*(b*x**2+a)**2/(d*x**2+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^2*(e*x)^(5/2)/(d*x^2 + c)^(5/2), x)
\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^2*(e*x)^(5/2)/(d*x^2 + c)^(5/2), x)
Timed out. \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:
int(((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2),x)
Output:
int(((e*x)^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^(5/2), x)
\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {e}\, e^{2} \left (-10 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, a^{2} d^{2} x +140 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, a b c d x +60 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, a b \,d^{2} x^{3}-154 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, b^{2} c^{2} x -66 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, b^{2} c d \,x^{3}+6 \sqrt {x}\, \sqrt {d \,x^{2}+c}\, b^{2} d^{2} x^{5}+15 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{6}+3 c \,d^{2} x^{4}+3 c^{2} d \,x^{2}+c^{3}}d x \right ) a^{2} c^{3} d^{2}+30 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{6}+3 c \,d^{2} x^{4}+3 c^{2} d \,x^{2}+c^{3}}d x \right ) a^{2} c^{2} d^{3} x^{2}+15 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{6}+3 c \,d^{2} x^{4}+3 c^{2} d \,x^{2}+c^{3}}d x \right ) a^{2} c \,d^{4} x^{4}-210 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{6}+3 c \,d^{2} x^{4}+3 c^{2} d \,x^{2}+c^{3}}d x \right ) a b \,c^{4} d -420 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{6}+3 c \,d^{2} x^{4}+3 c^{2} d \,x^{2}+c^{3}}d x \right ) a b \,c^{3} d^{2} x^{2}-210 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{6}+3 c \,d^{2} x^{4}+3 c^{2} d \,x^{2}+c^{3}}d x \right ) a b \,c^{2} d^{3} x^{4}+231 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{6}+3 c \,d^{2} x^{4}+3 c^{2} d \,x^{2}+c^{3}}d x \right ) b^{2} c^{5}+462 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{6}+3 c \,d^{2} x^{4}+3 c^{2} d \,x^{2}+c^{3}}d x \right ) b^{2} c^{4} d \,x^{2}+231 \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{6}+3 c \,d^{2} x^{4}+3 c^{2} d \,x^{2}+c^{3}}d x \right ) b^{2} c^{3} d^{2} x^{4}\right )}{15 d^{3} \left (d^{2} x^{4}+2 c d \,x^{2}+c^{2}\right )} \] Input:
int((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(5/2),x)
Output:
(sqrt(e)*e**2*( - 10*sqrt(x)*sqrt(c + d*x**2)*a**2*d**2*x + 140*sqrt(x)*sq rt(c + d*x**2)*a*b*c*d*x + 60*sqrt(x)*sqrt(c + d*x**2)*a*b*d**2*x**3 - 154 *sqrt(x)*sqrt(c + d*x**2)*b**2*c**2*x - 66*sqrt(x)*sqrt(c + d*x**2)*b**2*c *d*x**3 + 6*sqrt(x)*sqrt(c + d*x**2)*b**2*d**2*x**5 + 15*int((sqrt(x)*sqrt (c + d*x**2))/(c**3 + 3*c**2*d*x**2 + 3*c*d**2*x**4 + d**3*x**6),x)*a**2*c **3*d**2 + 30*int((sqrt(x)*sqrt(c + d*x**2))/(c**3 + 3*c**2*d*x**2 + 3*c*d **2*x**4 + d**3*x**6),x)*a**2*c**2*d**3*x**2 + 15*int((sqrt(x)*sqrt(c + d* x**2))/(c**3 + 3*c**2*d*x**2 + 3*c*d**2*x**4 + d**3*x**6),x)*a**2*c*d**4*x **4 - 210*int((sqrt(x)*sqrt(c + d*x**2))/(c**3 + 3*c**2*d*x**2 + 3*c*d**2* x**4 + d**3*x**6),x)*a*b*c**4*d - 420*int((sqrt(x)*sqrt(c + d*x**2))/(c**3 + 3*c**2*d*x**2 + 3*c*d**2*x**4 + d**3*x**6),x)*a*b*c**3*d**2*x**2 - 210* int((sqrt(x)*sqrt(c + d*x**2))/(c**3 + 3*c**2*d*x**2 + 3*c*d**2*x**4 + d** 3*x**6),x)*a*b*c**2*d**3*x**4 + 231*int((sqrt(x)*sqrt(c + d*x**2))/(c**3 + 3*c**2*d*x**2 + 3*c*d**2*x**4 + d**3*x**6),x)*b**2*c**5 + 462*int((sqrt(x )*sqrt(c + d*x**2))/(c**3 + 3*c**2*d*x**2 + 3*c*d**2*x**4 + d**3*x**6),x)* b**2*c**4*d*x**2 + 231*int((sqrt(x)*sqrt(c + d*x**2))/(c**3 + 3*c**2*d*x** 2 + 3*c*d**2*x**4 + d**3*x**6),x)*b**2*c**3*d**2*x**4))/(15*d**3*(c**2 + 2 *c*d*x**2 + d**2*x**4))