Integrand size = 28, antiderivative size = 276 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 \sqrt {e x}}{3 c^3 e^5 \left (c+d x^2\right )^{3/2}}+\frac {(5 b c-17 a d) (b c-a d) \sqrt {e x}}{6 c^4 e^5 \sqrt {c+d x^2}}-\frac {2 a^2 \sqrt {c+d x^2}}{7 c^3 e (e x)^{7/2}}-\frac {2 a (14 b c-19 a d) \sqrt {c+d x^2}}{21 c^4 e^3 (e x)^{3/2}}+\frac {5 \left (7 b^2 c^2-42 a b c d+39 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{84 c^{17/4} \sqrt [4]{d} e^{9/2} \sqrt {c+d x^2}} \] Output:
1/3*(-a*d+b*c)^2*(e*x)^(1/2)/c^3/e^5/(d*x^2+c)^(3/2)+1/6*(-17*a*d+5*b*c)*( -a*d+b*c)*(e*x)^(1/2)/c^4/e^5/(d*x^2+c)^(1/2)-2/7*a^2*(d*x^2+c)^(1/2)/c^3/ e/(e*x)^(7/2)-2/21*a*(-19*a*d+14*b*c)*(d*x^2+c)^(1/2)/c^4/e^3/(e*x)^(3/2)+ 5/84*(39*a^2*d^2-42*a*b*c*d+7*b^2*c^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))/c^(17/4)/d^(1/4)/e^(9/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.24 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {x^{9/2} \left (\frac {7 b^2 c^2 x^4 \left (7 c+5 d x^2\right )-14 a b c x^2 \left (4 c^2+21 c d x^2+15 d^2 x^4\right )+a^2 \left (-12 c^3+52 c^2 d x^2+273 c d^2 x^4+195 d^3 x^6\right )}{c^4 x^{7/2} \left (c+d x^2\right )}+\frac {5 i \left (7 b^2 c^2-42 a b c d+39 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{c^4 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{42 (e x)^{9/2} \sqrt {c+d x^2}} \] Input:
Integrate[(a + b*x^2)^2/((e*x)^(9/2)*(c + d*x^2)^(5/2)),x]
Output:
(x^(9/2)*((7*b^2*c^2*x^4*(7*c + 5*d*x^2) - 14*a*b*c*x^2*(4*c^2 + 21*c*d*x^ 2 + 15*d^2*x^4) + a^2*(-12*c^3 + 52*c^2*d*x^2 + 273*c*d^2*x^4 + 195*d^3*x^ 6))/(c^4*x^(7/2)*(c + d*x^2)) + ((5*I)*(7*b^2*c^2 - 42*a*b*c*d + 39*a^2*d^ 2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqr t[x]], -1])/(c^4*Sqrt[(I*Sqrt[c])/Sqrt[d]])))/(42*(e*x)^(9/2)*Sqrt[c + d*x ^2])
Time = 0.38 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {365, 27, 359, 253, 253, 266, 761}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {7 b^2 c x^2+a (14 b c-13 a d)}{2 (e x)^{5/2} \left (d x^2+c\right )^{5/2}}dx}{7 c e^2}-\frac {2 a^2}{7 c e (e x)^{7/2} \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {7 b^2 c x^2+a (14 b c-13 a d)}{(e x)^{5/2} \left (d x^2+c\right )^{5/2}}dx}{7 c e^2}-\frac {2 a^2}{7 c e (e x)^{7/2} \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {\frac {\left (7 b^2 c^2-3 a d (14 b c-13 a d)\right ) \int \frac {1}{\sqrt {e x} \left (d x^2+c\right )^{5/2}}dx}{c e^2}-\frac {2 a (14 b c-13 a d)}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}}{7 c e^2}-\frac {2 a^2}{7 c e (e x)^{7/2} \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {\left (7 b^2 c^2-3 a d (14 b c-13 a d)\right ) \left (\frac {5 \int \frac {1}{\sqrt {e x} \left (d x^2+c\right )^{3/2}}dx}{6 c}+\frac {\sqrt {e x}}{3 c e \left (c+d x^2\right )^{3/2}}\right )}{c e^2}-\frac {2 a (14 b c-13 a d)}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}}{7 c e^2}-\frac {2 a^2}{7 c e (e x)^{7/2} \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {\left (7 b^2 c^2-3 a d (14 b c-13 a d)\right ) \left (\frac {5 \left (\frac {\int \frac {1}{\sqrt {e x} \sqrt {d x^2+c}}dx}{2 c}+\frac {\sqrt {e x}}{c e \sqrt {c+d x^2}}\right )}{6 c}+\frac {\sqrt {e x}}{3 c e \left (c+d x^2\right )^{3/2}}\right )}{c e^2}-\frac {2 a (14 b c-13 a d)}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}}{7 c e^2}-\frac {2 a^2}{7 c e (e x)^{7/2} \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {\left (7 b^2 c^2-3 a d (14 b c-13 a d)\right ) \left (\frac {5 \left (\frac {\int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{c e}+\frac {\sqrt {e x}}{c e \sqrt {c+d x^2}}\right )}{6 c}+\frac {\sqrt {e x}}{3 c e \left (c+d x^2\right )^{3/2}}\right )}{c e^2}-\frac {2 a (14 b c-13 a d)}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}}{7 c e^2}-\frac {2 a^2}{7 c e (e x)^{7/2} \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\left (7 b^2 c^2-3 a d (14 b c-13 a d)\right ) \left (\frac {5 \left (\frac {\left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 c^{5/4} \sqrt [4]{d} e^{3/2} \sqrt {c+d x^2}}+\frac {\sqrt {e x}}{c e \sqrt {c+d x^2}}\right )}{6 c}+\frac {\sqrt {e x}}{3 c e \left (c+d x^2\right )^{3/2}}\right )}{c e^2}-\frac {2 a (14 b c-13 a d)}{3 c e (e x)^{3/2} \left (c+d x^2\right )^{3/2}}}{7 c e^2}-\frac {2 a^2}{7 c e (e x)^{7/2} \left (c+d x^2\right )^{3/2}}\) |
Input:
Int[(a + b*x^2)^2/((e*x)^(9/2)*(c + d*x^2)^(5/2)),x]
Output:
(-2*a^2)/(7*c*e*(e*x)^(7/2)*(c + d*x^2)^(3/2)) + ((-2*a*(14*b*c - 13*a*d)) /(3*c*e*(e*x)^(3/2)*(c + d*x^2)^(3/2)) + ((7*b^2*c^2 - 3*a*d*(14*b*c - 13* a*d))*(Sqrt[e*x]/(3*c*e*(c + d*x^2)^(3/2)) + (5*(Sqrt[e*x]/(c*e*Sqrt[c + d *x^2]) + ((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*c^(5/4)*d^(1/4)*e^(3/2)*Sqrt[c + d*x^2])))/(6*c)))/(c*e^2))/(7*c *e^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*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && 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[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
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]
Time = 4.22 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.28
| method | result | size |
| elliptic | \(\frac {\sqrt {e x \left (x^{2} d +c \right )}\, \left (-\frac {2 a^{2} \sqrt {d e \,x^{3}+c e x}}{7 e^{5} c^{3} x^{4}}+\frac {2 a \left (19 a d -14 b c \right ) \sqrt {d e \,x^{3}+c e x}}{21 e^{5} c^{4} x^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d e \,x^{3}+c e x}}{3 e^{5} c^{3} d^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {x \left (17 a^{2} d^{2}-22 a b c d +5 b^{2} c^{2}\right )}{6 e^{4} c^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\left (\frac {d a \left (19 a d -14 b c \right )}{21 c^{4} e^{4}}+\frac {17 a^{2} d^{2}-22 a b c d +5 b^{2} c^{2}}{12 c^{4} e^{4}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(353\) |
| risch | \(-\frac {2 \sqrt {x^{2} d +c}\, a \left (-19 a d \,x^{2}+14 x^{2} b c +3 a c \right )}{21 c^{4} x^{3} e^{4} \sqrt {e x}}+\frac {\left (\frac {19 a^{2} d \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}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}+21 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\sqrt {d e \,x^{3}+c e x}}{3 c e \,d^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {5 x}{6 c^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {5 \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}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{12 c^{2} d \sqrt {d e \,x^{3}+c e x}}\right )-\frac {14 a b c \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}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}+42 a c d \left (a d -b c \right ) \left (\frac {x}{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}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )\right ) \sqrt {e x \left (x^{2} d +c \right )}}{21 c^{4} e^{4} \sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(646\) |
| default | \(\frac {195 \sqrt {2}\, \sqrt {-c d}\, \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} d^{3} x^{5}-210 \sqrt {2}\, \sqrt {-c d}\, \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 \,d^{2} x^{5}+35 \sqrt {2}\, \sqrt {-c d}\, \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^{2} d \,x^{5}+195 \sqrt {2}\, \sqrt {-c d}\, \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 \,d^{2} x^{3}-210 \sqrt {2}\, \sqrt {-c d}\, \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^{2} d \,x^{3}+35 \sqrt {2}\, \sqrt {-c d}\, \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^{3} x^{3}+390 a^{2} d^{4} x^{6}-420 a b c \,d^{3} x^{6}+70 b^{2} c^{2} d^{2} x^{6}+546 a^{2} c \,d^{3} x^{4}-588 a b \,c^{2} d^{2} x^{4}+98 b^{2} c^{3} d \,x^{4}+104 a^{2} c^{2} d^{2} x^{2}-112 a b \,c^{3} d \,x^{2}-24 a^{2} c^{3} d}{84 x^{3} e^{4} \sqrt {e x}\, c^{4} d \left (x^{2} d +c \right )^{\frac {3}{2}}}\) | \(715\) |
Input:
int((b*x^2+a)^2/(e*x)^(9/2)/(d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
(e*x*(d*x^2+c))^(1/2)/(e*x)^(1/2)/(d*x^2+c)^(1/2)*(-2/7/e^5/c^3*a^2*(d*e*x ^3+c*e*x)^(1/2)/x^4+2/21/e^5/c^4*a*(19*a*d-14*b*c)*(d*e*x^3+c*e*x)^(1/2)/x ^2+1/3/e^5/c^3/d^2*(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+1/6/e^4*x/c^4*(17*a^2*d^2-22*a*b*c*d+5*b^2*c^2)/((x^2+c/d)*d*e*x)^( 1/2)+(1/21/c^4*d*a*(19*a*d-14*b*c)/e^4+1/12/c^4*(17*a^2*d^2-22*a*b*c*d+5*b ^2*c^2)/e^4)*(-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)*EllipticF(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2*2^( 1/2)))
Time = 0.10 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {5 \, {\left ({\left (7 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 39 \, a^{2} d^{4}\right )} x^{8} + 2 \, {\left (7 \, b^{2} c^{3} d - 42 \, a b c^{2} d^{2} + 39 \, a^{2} c d^{3}\right )} x^{6} + {\left (7 \, b^{2} c^{4} - 42 \, a b c^{3} d + 39 \, a^{2} c^{2} d^{2}\right )} x^{4}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (5 \, {\left (7 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 39 \, a^{2} d^{4}\right )} x^{6} - 12 \, a^{2} c^{3} d + 7 \, {\left (7 \, b^{2} c^{3} d - 42 \, a b c^{2} d^{2} + 39 \, a^{2} c d^{3}\right )} x^{4} - 4 \, {\left (14 \, a b c^{3} d - 13 \, a^{2} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{42 \, {\left (c^{4} d^{3} e^{5} x^{8} + 2 \, c^{5} d^{2} e^{5} x^{6} + c^{6} d e^{5} x^{4}\right )}} \] Input:
integrate((b*x^2+a)^2/(e*x)^(9/2)/(d*x^2+c)^(5/2),x, algorithm="fricas")
Output:
1/42*(5*((7*b^2*c^2*d^2 - 42*a*b*c*d^3 + 39*a^2*d^4)*x^8 + 2*(7*b^2*c^3*d - 42*a*b*c^2*d^2 + 39*a^2*c*d^3)*x^6 + (7*b^2*c^4 - 42*a*b*c^3*d + 39*a^2* c^2*d^2)*x^4)*sqrt(d*e)*weierstrassPInverse(-4*c/d, 0, x) + (5*(7*b^2*c^2* d^2 - 42*a*b*c*d^3 + 39*a^2*d^4)*x^6 - 12*a^2*c^3*d + 7*(7*b^2*c^3*d - 42* a*b*c^2*d^2 + 39*a^2*c*d^3)*x^4 - 4*(14*a*b*c^3*d - 13*a^2*c^2*d^2)*x^2)*s qrt(d*x^2 + c)*sqrt(e*x))/(c^4*d^3*e^5*x^8 + 2*c^5*d^2*e^5*x^6 + c^6*d*e^5 *x^4)
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**2/(e*x)**(9/2)/(d*x**2+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^2/(e*x)^(9/2)/(d*x^2+c)^(5/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*(e*x)^(9/2)), x)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^2/(e*x)^(9/2)/(d*x^2+c)^(5/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*(e*x)^(9/2)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{9/2}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:
int((a + b*x^2)^2/((e*x)^(9/2)*(c + d*x^2)^(5/2)),x)
Output:
int((a + b*x^2)^2/((e*x)^(9/2)*(c + d*x^2)^(5/2)), x)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {e}\, \left (-36 \sqrt {d \,x^{2}+c}\, a b d +6 \sqrt {d \,x^{2}+c}\, b^{2} c -26 \sqrt {d \,x^{2}+c}\, b^{2} d \,x^{2}+117 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{11}+3 c \,d^{2} x^{9}+3 c^{2} d \,x^{7}+c^{3} x^{5}}d x \right ) a^{2} c^{2} d^{2} x^{3}+234 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{11}+3 c \,d^{2} x^{9}+3 c^{2} d \,x^{7}+c^{3} x^{5}}d x \right ) a^{2} c \,d^{3} x^{5}+117 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{11}+3 c \,d^{2} x^{9}+3 c^{2} d \,x^{7}+c^{3} x^{5}}d x \right ) a^{2} d^{4} x^{7}-126 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{11}+3 c \,d^{2} x^{9}+3 c^{2} d \,x^{7}+c^{3} x^{5}}d x \right ) a b \,c^{3} d \,x^{3}-252 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{11}+3 c \,d^{2} x^{9}+3 c^{2} d \,x^{7}+c^{3} x^{5}}d x \right ) a b \,c^{2} d^{2} x^{5}-126 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{11}+3 c \,d^{2} x^{9}+3 c^{2} d \,x^{7}+c^{3} x^{5}}d x \right ) a b c \,d^{3} x^{7}+21 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{11}+3 c \,d^{2} x^{9}+3 c^{2} d \,x^{7}+c^{3} x^{5}}d x \right ) b^{2} c^{4} x^{3}+42 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{11}+3 c \,d^{2} x^{9}+3 c^{2} d \,x^{7}+c^{3} x^{5}}d x \right ) b^{2} c^{3} d \,x^{5}+21 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{3} x^{11}+3 c \,d^{2} x^{9}+3 c^{2} d \,x^{7}+c^{3} x^{5}}d x \right ) b^{2} c^{2} d^{2} x^{7}\right )}{117 \sqrt {x}\, d^{2} e^{5} x^{3} \left (d^{2} x^{4}+2 c d \,x^{2}+c^{2}\right )} \] Input:
int((b*x^2+a)^2/(e*x)^(9/2)/(d*x^2+c)^(5/2),x)
Output:
(sqrt(e)*( - 36*sqrt(c + d*x**2)*a*b*d + 6*sqrt(c + d*x**2)*b**2*c - 26*sq rt(c + d*x**2)*b**2*d*x**2 + 117*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c **3*x**5 + 3*c**2*d*x**7 + 3*c*d**2*x**9 + d**3*x**11),x)*a**2*c**2*d**2*x **3 + 234*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c**3*x**5 + 3*c**2*d*x** 7 + 3*c*d**2*x**9 + d**3*x**11),x)*a**2*c*d**3*x**5 + 117*sqrt(x)*int((sqr t(x)*sqrt(c + d*x**2))/(c**3*x**5 + 3*c**2*d*x**7 + 3*c*d**2*x**9 + d**3*x **11),x)*a**2*d**4*x**7 - 126*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c**3 *x**5 + 3*c**2*d*x**7 + 3*c*d**2*x**9 + d**3*x**11),x)*a*b*c**3*d*x**3 - 2 52*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c**3*x**5 + 3*c**2*d*x**7 + 3*c *d**2*x**9 + d**3*x**11),x)*a*b*c**2*d**2*x**5 - 126*sqrt(x)*int((sqrt(x)* sqrt(c + d*x**2))/(c**3*x**5 + 3*c**2*d*x**7 + 3*c*d**2*x**9 + d**3*x**11) ,x)*a*b*c*d**3*x**7 + 21*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c**3*x**5 + 3*c**2*d*x**7 + 3*c*d**2*x**9 + d**3*x**11),x)*b**2*c**4*x**3 + 42*sqrt (x)*int((sqrt(x)*sqrt(c + d*x**2))/(c**3*x**5 + 3*c**2*d*x**7 + 3*c*d**2*x **9 + d**3*x**11),x)*b**2*c**3*d*x**5 + 21*sqrt(x)*int((sqrt(x)*sqrt(c + d *x**2))/(c**3*x**5 + 3*c**2*d*x**7 + 3*c*d**2*x**9 + d**3*x**11),x)*b**2*c **2*d**2*x**7))/(117*sqrt(x)*d**2*e**5*x**3*(c**2 + 2*c*d*x**2 + d**2*x**4 ))