Integrand size = 28, antiderivative size = 281 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {d (b c-a d)^2 \sqrt {e x}}{c^4 e^7 \sqrt {c+d x^2}}-\frac {2 a^2 \sqrt {c+d x^2}}{11 c^2 e (e x)^{11/2}}-\frac {4 a (11 b c-10 a d) \sqrt {c+d x^2}}{77 c^3 e^3 (e x)^{7/2}}-\frac {2 \left (77 b^2 c^2-264 a b c d+177 a^2 d^2\right ) \sqrt {c+d x^2}}{231 c^4 e^5 (e x)^{3/2}}-\frac {5 d^{3/4} \left (77 b^2 c^2-198 a b c d+117 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{462 c^{17/4} e^{13/2} \sqrt {c+d x^2}} \] Output:
-d*(-a*d+b*c)^2*(e*x)^(1/2)/c^4/e^7/(d*x^2+c)^(1/2)-2/11*a^2*(d*x^2+c)^(1/ 2)/c^2/e/(e*x)^(11/2)-4/77*a*(-10*a*d+11*b*c)*(d*x^2+c)^(1/2)/c^3/e^3/(e*x )^(7/2)-2/231*(177*a^2*d^2-264*a*b*c*d+77*b^2*c^2)*(d*x^2+c)^(1/2)/c^4/e^5 /(e*x)^(3/2)-5/462*d^(3/4)*(117*a^2*d^2-198*a*b*c*d+77*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)/e^(13/2)/(d*x^ 2+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.23 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {e x} \left (-\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} \left (77 b^2 c^2 x^4 \left (2 c+5 d x^2\right )+66 a b c x^2 \left (2 c^2-6 c d x^2-15 d^2 x^4\right )+3 a^2 \left (14 c^3-26 c^2 d x^2+78 c d^2 x^4+195 d^3 x^6\right )\right )-5 i d \left (77 b^2 c^2-198 a b c d+117 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x^{13/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )\right )}{231 c^4 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} e^7 x^6 \sqrt {c+d x^2}} \] Input:
Integrate[(a + b*x^2)^2/((e*x)^(13/2)*(c + d*x^2)^(3/2)),x]
Output:
(Sqrt[e*x]*(-(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(77*b^2*c^2*x^4*(2*c + 5*d*x^2) + 66*a*b*c*x^2*(2*c^2 - 6*c*d*x^2 - 15*d^2*x^4) + 3*a^2*(14*c^3 - 26*c^2*d*x ^2 + 78*c*d^2*x^4 + 195*d^3*x^6))) - (5*I)*d*(77*b^2*c^2 - 198*a*b*c*d + 1 17*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^(13/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[ c])/Sqrt[d]]/Sqrt[x]], -1]))/(231*c^4*Sqrt[(I*Sqrt[c])/Sqrt[d]]*e^7*x^6*Sq rt[c + d*x^2])
Time = 0.38 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.02, 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, 264, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {11 b^2 c x^2+a (22 b c-13 a d)}{2 (e x)^{9/2} \left (d x^2+c\right )^{3/2}}dx}{11 c e^2}-\frac {2 a^2}{11 c e (e x)^{11/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {11 b^2 c x^2+a (22 b c-13 a d)}{(e x)^{9/2} \left (d x^2+c\right )^{3/2}}dx}{11 c e^2}-\frac {2 a^2}{11 c e (e x)^{11/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {\frac {\left (77 b^2 c^2-9 a d (22 b c-13 a d)\right ) \int \frac {1}{(e x)^{5/2} \left (d x^2+c\right )^{3/2}}dx}{7 c e^2}-\frac {2 a (22 b c-13 a d)}{7 c e (e x)^{7/2} \sqrt {c+d x^2}}}{11 c e^2}-\frac {2 a^2}{11 c e (e x)^{11/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {\left (77 b^2 c^2-9 a d (22 b c-13 a d)\right ) \left (\frac {5 \int \frac {1}{(e x)^{5/2} \sqrt {d x^2+c}}dx}{2 c}+\frac {1}{c e (e x)^{3/2} \sqrt {c+d x^2}}\right )}{7 c e^2}-\frac {2 a (22 b c-13 a d)}{7 c e (e x)^{7/2} \sqrt {c+d x^2}}}{11 c e^2}-\frac {2 a^2}{11 c e (e x)^{11/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\frac {\left (77 b^2 c^2-9 a d (22 b c-13 a d)\right ) \left (\frac {5 \left (-\frac {d \int \frac {1}{\sqrt {e x} \sqrt {d x^2+c}}dx}{3 c e^2}-\frac {2 \sqrt {c+d x^2}}{3 c e (e x)^{3/2}}\right )}{2 c}+\frac {1}{c e (e x)^{3/2} \sqrt {c+d x^2}}\right )}{7 c e^2}-\frac {2 a (22 b c-13 a d)}{7 c e (e x)^{7/2} \sqrt {c+d x^2}}}{11 c e^2}-\frac {2 a^2}{11 c e (e x)^{11/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {\left (77 b^2 c^2-9 a d (22 b c-13 a d)\right ) \left (\frac {5 \left (-\frac {2 d \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{3 c e^3}-\frac {2 \sqrt {c+d x^2}}{3 c e (e x)^{3/2}}\right )}{2 c}+\frac {1}{c e (e x)^{3/2} \sqrt {c+d x^2}}\right )}{7 c e^2}-\frac {2 a (22 b c-13 a d)}{7 c e (e x)^{7/2} \sqrt {c+d x^2}}}{11 c e^2}-\frac {2 a^2}{11 c e (e x)^{11/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\left (77 b^2 c^2-9 a d (22 b c-13 a d)\right ) \left (\frac {5 \left (-\frac {d^{3/4} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 c^{5/4} e^{7/2} \sqrt {c+d x^2}}-\frac {2 \sqrt {c+d x^2}}{3 c e (e x)^{3/2}}\right )}{2 c}+\frac {1}{c e (e x)^{3/2} \sqrt {c+d x^2}}\right )}{7 c e^2}-\frac {2 a (22 b c-13 a d)}{7 c e (e x)^{7/2} \sqrt {c+d x^2}}}{11 c e^2}-\frac {2 a^2}{11 c e (e x)^{11/2} \sqrt {c+d x^2}}\) |
Input:
Int[(a + b*x^2)^2/((e*x)^(13/2)*(c + d*x^2)^(3/2)),x]
Output:
(-2*a^2)/(11*c*e*(e*x)^(11/2)*Sqrt[c + d*x^2]) + ((-2*a*(22*b*c - 13*a*d)) /(7*c*e*(e*x)^(7/2)*Sqrt[c + d*x^2]) + ((77*b^2*c^2 - 9*a*d*(22*b*c - 13*a *d))*(1/(c*e*(e*x)^(3/2)*Sqrt[c + d*x^2]) + (5*((-2*Sqrt[c + d*x^2])/(3*c* e*(e*x)^(3/2)) - (d^(3/4)*(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])/(3*c^(5/4)*e^(7/2)*Sqrt[c + d*x^2])))/(2*c)))/(7*c* e^2))/(11*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] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -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 = 3.35 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.26
| method | result | size |
| elliptic | \(\frac {\sqrt {e x \left (x^{2} d +c \right )}\, \left (-\frac {2 a^{2} \sqrt {d e \,x^{3}+c e x}}{11 e^{7} c^{2} x^{6}}+\frac {4 a \left (10 a d -11 b c \right ) \sqrt {d e \,x^{3}+c e x}}{77 e^{7} c^{3} x^{4}}-\frac {2 \left (177 a^{2} d^{2}-264 a b c d +77 b^{2} c^{2}\right ) \sqrt {d e \,x^{3}+c e x}}{231 e^{7} c^{4} x^{2}}-\frac {d x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{e^{6} c^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\left (-\frac {d \left (177 a^{2} d^{2}-264 a b c d +77 b^{2} c^{2}\right )}{231 c^{4} e^{6}}-\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 c^{4} e^{6}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(355\) |
| default | \(-\frac {585 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2} x^{5}-990 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d \,x^{5}+385 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2} x^{5}+1170 a^{2} d^{3} x^{6}-1980 a b c \,d^{2} x^{6}+770 b^{2} c^{2} d \,x^{6}+468 a^{2} c \,d^{2} x^{4}-792 a b \,c^{2} d \,x^{4}+308 b^{2} c^{3} x^{4}-156 a^{2} c^{2} d \,x^{2}+264 a b \,c^{3} x^{2}+84 a^{2} c^{3}}{462 x^{5} \sqrt {x^{2} d +c}\, c^{4} e^{6} \sqrt {e x}}\) | \(411\) |
| risch | \(-\frac {2 \sqrt {x^{2} d +c}\, \left (177 a^{2} d^{2} x^{4}-264 a b c d \,x^{4}+77 b^{2} c^{2} x^{4}-60 a^{2} c d \,x^{2}+66 a b \,c^{2} x^{2}+21 a^{2} c^{2}\right )}{231 c^{4} x^{5} e^{6} \sqrt {e x}}-\frac {d \left (\frac {177 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}}+\frac {77 b^{2} c^{2} \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}}+231 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\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 )-\frac {264 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}}\right ) \sqrt {e x \left (x^{2} d +c \right )}}{231 c^{4} e^{6} \sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(618\) |
Input:
int((b*x^2+a)^2/(e*x)^(13/2)/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
(e*x*(d*x^2+c))^(1/2)/(e*x)^(1/2)/(d*x^2+c)^(1/2)*(-2/11/e^7/c^2*a^2*(d*e* x^3+c*e*x)^(1/2)/x^6+4/77/e^7/c^3*a*(10*a*d-11*b*c)*(d*e*x^3+c*e*x)^(1/2)/ x^4-2/231/e^7/c^4*(177*a^2*d^2-264*a*b*c*d+77*b^2*c^2)*(d*e*x^3+c*e*x)^(1/ 2)/x^2-d/e^6*x/c^4*(a^2*d^2-2*a*b*c*d+b^2*c^2)/((x^2+c/d)*d*e*x)^(1/2)+(-1 /231/c^4*d*(177*a^2*d^2-264*a*b*c*d+77*b^2*c^2)/e^6-1/2/c^4*d/e^6*(a^2*d^2 -2*a*b*c*d+b^2*c^2))*(-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 = 212, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {5 \, {\left ({\left (77 \, b^{2} c^{2} d - 198 \, a b c d^{2} + 117 \, a^{2} d^{3}\right )} x^{8} + {\left (77 \, b^{2} c^{3} - 198 \, a b c^{2} d + 117 \, a^{2} c d^{2}\right )} x^{6}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (5 \, {\left (77 \, b^{2} c^{2} d - 198 \, a b c d^{2} + 117 \, a^{2} d^{3}\right )} x^{6} + 42 \, a^{2} c^{3} + 2 \, {\left (77 \, b^{2} c^{3} - 198 \, a b c^{2} d + 117 \, a^{2} c d^{2}\right )} x^{4} + 6 \, {\left (22 \, a b c^{3} - 13 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{231 \, {\left (c^{4} d e^{7} x^{8} + c^{5} e^{7} x^{6}\right )}} \] Input:
integrate((b*x^2+a)^2/(e*x)^(13/2)/(d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
-1/231*(5*((77*b^2*c^2*d - 198*a*b*c*d^2 + 117*a^2*d^3)*x^8 + (77*b^2*c^3 - 198*a*b*c^2*d + 117*a^2*c*d^2)*x^6)*sqrt(d*e)*weierstrassPInverse(-4*c/d , 0, x) + (5*(77*b^2*c^2*d - 198*a*b*c*d^2 + 117*a^2*d^3)*x^6 + 42*a^2*c^3 + 2*(77*b^2*c^3 - 198*a*b*c^2*d + 117*a^2*c*d^2)*x^4 + 6*(22*a*b*c^3 - 13 *a^2*c^2*d)*x^2)*sqrt(d*x^2 + c)*sqrt(e*x))/(c^4*d*e^7*x^8 + c^5*e^7*x^6)
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \left (c+d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**2/(e*x)**(13/2)/(d*x**2+c)**(3/2),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^2/(e*x)^(13/2)/(d*x^2+c)^(3/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(13/2)), x)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^2/(e*x)^(13/2)/(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(13/2)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{13/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int((a + b*x^2)^2/((e*x)^(13/2)*(c + d*x^2)^(3/2)),x)
Output:
int((a + b*x^2)^2/((e*x)^(13/2)*(c + d*x^2)^(3/2)), x)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{13/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (-36 \sqrt {d \,x^{2}+c}\, a b d +14 \sqrt {d \,x^{2}+c}\, b^{2} c -26 \sqrt {d \,x^{2}+c}\, b^{2} d \,x^{2}+117 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{11}+2 c d \,x^{9}+c^{2} x^{7}}d x \right ) a^{2} c \,d^{2} x^{5}+117 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{11}+2 c d \,x^{9}+c^{2} x^{7}}d x \right ) a^{2} d^{3} x^{7}-198 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{11}+2 c d \,x^{9}+c^{2} x^{7}}d x \right ) a b \,c^{2} d \,x^{5}-198 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{11}+2 c d \,x^{9}+c^{2} x^{7}}d x \right ) a b c \,d^{2} x^{7}+77 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{11}+2 c d \,x^{9}+c^{2} x^{7}}d x \right ) b^{2} c^{3} x^{5}+77 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{11}+2 c d \,x^{9}+c^{2} x^{7}}d x \right ) b^{2} c^{2} d \,x^{7}\right )}{117 \sqrt {x}\, d^{2} e^{7} x^{5} \left (d \,x^{2}+c \right )} \] Input:
int((b*x^2+a)^2/(e*x)^(13/2)/(d*x^2+c)^(3/2),x)
Output:
(sqrt(e)*( - 36*sqrt(c + d*x**2)*a*b*d + 14*sqrt(c + d*x**2)*b**2*c - 26*s qrt(c + d*x**2)*b**2*d*x**2 + 117*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/( c**2*x**7 + 2*c*d*x**9 + d**2*x**11),x)*a**2*c*d**2*x**5 + 117*sqrt(x)*int ((sqrt(x)*sqrt(c + d*x**2))/(c**2*x**7 + 2*c*d*x**9 + d**2*x**11),x)*a**2* d**3*x**7 - 198*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c**2*x**7 + 2*c*d* x**9 + d**2*x**11),x)*a*b*c**2*d*x**5 - 198*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c**2*x**7 + 2*c*d*x**9 + d**2*x**11),x)*a*b*c*d**2*x**7 + 77*sqr t(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c**2*x**7 + 2*c*d*x**9 + d**2*x**11), x)*b**2*c**3*x**5 + 77*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c**2*x**7 + 2*c*d*x**9 + d**2*x**11),x)*b**2*c**2*d*x**7))/(117*sqrt(x)*d**2*e**7*x** 5*(c + d*x**2))