Integrand size = 28, antiderivative size = 464 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2}{c d^2 e (e x)^{5/2} \sqrt {c+d x^2}}-\frac {\left (5 b^2 c^2-10 a b c d+7 a^2 d^2\right ) \sqrt {c+d x^2}}{5 c^2 d^2 e (e x)^{5/2}}+\frac {\left (5 b^2 c^2-30 a b c d+21 a^2 d^2\right ) \sqrt {c+d x^2}}{5 c^3 d e^3 \sqrt {e x}}-\frac {\left (5 b^2 c^2-30 a b c d+21 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{5 c^3 \sqrt {d} e^4 \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (5 b^2 c^2-30 a b c d+21 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 c^{11/4} d^{3/4} e^{7/2} \sqrt {c+d x^2}}-\frac {\left (5 b^2 c^2-30 a b c d+21 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{10 c^{11/4} d^{3/4} e^{7/2} \sqrt {c+d x^2}} \] Output:
(-a*d+b*c)^2/c/d^2/e/(e*x)^(5/2)/(d*x^2+c)^(1/2)-1/5*(7*a^2*d^2-10*a*b*c*d +5*b^2*c^2)*(d*x^2+c)^(1/2)/c^2/d^2/e/(e*x)^(5/2)+1/5*(21*a^2*d^2-30*a*b*c *d+5*b^2*c^2)*(d*x^2+c)^(1/2)/c^3/d/e^3/(e*x)^(1/2)-1/5*(21*a^2*d^2-30*a*b *c*d+5*b^2*c^2)*(e*x)^(1/2)*(d*x^2+c)^(1/2)/c^3/d^(1/2)/e^4/(c^(1/2)+d^(1/ 2)*x)+1/5*(21*a^2*d^2-30*a*b*c*d+5*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)*EllipticE(sin(2*arctan(d^(1/4)*(e*x)^(1/2)/c ^(1/4)/e^(1/2))),1/2*2^(1/2))/c^(11/4)/d^(3/4)/e^(7/2)/(d*x^2+c)^(1/2)-1/1 0*(21*a^2*d^2-30*a*b*c*d+5*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^(11/4)/d^(3/4)/e^(7/2)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.30 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {x \left (15 b^2 c^2 x^4-30 a b c x^2 \left (2 c+3 d x^2\right )+a^2 \left (-6 c^2+42 c d x^2+63 d^2 x^4\right )+\left (-5 b^2 c^2+30 a b c d-21 a^2 d^2\right ) x^4 \sqrt {1+\frac {d x^2}{c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {d x^2}{c}\right )\right )}{15 c^3 (e x)^{7/2} \sqrt {c+d x^2}} \] Input:
Integrate[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)),x]
Output:
(x*(15*b^2*c^2*x^4 - 30*a*b*c*x^2*(2*c + 3*d*x^2) + a^2*(-6*c^2 + 42*c*d*x ^2 + 63*d^2*x^4) + (-5*b^2*c^2 + 30*a*b*c*d - 21*a^2*d^2)*x^4*Sqrt[1 + (d* x^2)/c]*Hypergeometric2F1[1/2, 3/4, 7/4, -((d*x^2)/c)]))/(15*c^3*(e*x)^(7/ 2)*Sqrt[c + d*x^2])
Time = 0.51 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.89, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {365, 27, 359, 253, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {5 b^2 c x^2+a (10 b c-7 a d)}{2 (e x)^{3/2} \left (d x^2+c\right )^{3/2}}dx}{5 c e^2}-\frac {2 a^2}{5 c e (e x)^{5/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 b^2 c x^2+a (10 b c-7 a d)}{(e x)^{3/2} \left (d x^2+c\right )^{3/2}}dx}{5 c e^2}-\frac {2 a^2}{5 c e (e x)^{5/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {\frac {\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \int \frac {\sqrt {e x}}{\left (d x^2+c\right )^{3/2}}dx}{c e^2}-\frac {2 a (10 b c-7 a d)}{c e \sqrt {e x} \sqrt {c+d x^2}}}{5 c e^2}-\frac {2 a^2}{5 c e (e x)^{5/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \left (\frac {(e x)^{3/2}}{c e \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {e x}}{\sqrt {d x^2+c}}dx}{2 c}\right )}{c e^2}-\frac {2 a (10 b c-7 a d)}{c e \sqrt {e x} \sqrt {c+d x^2}}}{5 c e^2}-\frac {2 a^2}{5 c e (e x)^{5/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \left (\frac {(e x)^{3/2}}{c e \sqrt {c+d x^2}}-\frac {\int \frac {e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{c e}\right )}{c e^2}-\frac {2 a (10 b c-7 a d)}{c e \sqrt {e x} \sqrt {c+d x^2}}}{5 c e^2}-\frac {2 a^2}{5 c e (e x)^{5/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {\frac {\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \left (\frac {(e x)^{3/2}}{c e \sqrt {c+d x^2}}-\frac {\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\sqrt {c} e \int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {c} e \sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}}{c e}\right )}{c e^2}-\frac {2 a (10 b c-7 a d)}{c e \sqrt {e x} \sqrt {c+d x^2}}}{5 c e^2}-\frac {2 a^2}{5 c e (e x)^{5/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \left (\frac {(e x)^{3/2}}{c e \sqrt {c+d x^2}}-\frac {\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}}{c e}\right )}{c e^2}-\frac {2 a (10 b c-7 a d)}{c e \sqrt {e x} \sqrt {c+d x^2}}}{5 c e^2}-\frac {2 a^2}{5 c e (e x)^{5/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \left (\frac {(e x)^{3/2}}{c e \sqrt {c+d x^2}}-\frac {\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}}{c e}\right )}{c e^2}-\frac {2 a (10 b c-7 a d)}{c e \sqrt {e x} \sqrt {c+d x^2}}}{5 c e^2}-\frac {2 a^2}{5 c e (e x)^{5/2} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {\left (5 b^2 c^2-3 a d (10 b c-7 a d)\right ) \left (\frac {(e x)^{3/2}}{c e \sqrt {c+d x^2}}-\frac {\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^2}}-\frac {e^2 \sqrt {e x} \sqrt {c+d x^2}}{\sqrt {c} e+\sqrt {d} e x}}{\sqrt {d}}}{c e}\right )}{c e^2}-\frac {2 a (10 b c-7 a d)}{c e \sqrt {e x} \sqrt {c+d x^2}}}{5 c e^2}-\frac {2 a^2}{5 c e (e x)^{5/2} \sqrt {c+d x^2}}\) |
Input:
Int[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)),x]
Output:
(-2*a^2)/(5*c*e*(e*x)^(5/2)*Sqrt[c + d*x^2]) + ((-2*a*(10*b*c - 7*a*d))/(c *e*Sqrt[e*x]*Sqrt[c + d*x^2]) + ((5*b^2*c^2 - 3*a*d*(10*b*c - 7*a*d))*((e* x)^(3/2)/(c*e*Sqrt[c + d*x^2]) - (-((-((e^2*Sqrt[e*x]*Sqrt[c + d*x^2])/(Sq rt[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)/(S qrt[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]))/(c*e)))/(c*e^2))/(5*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]
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 = 2.40 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.76
| method | result | size |
| elliptic | \(\frac {\sqrt {e x \left (x^{2} d +c \right )}\, \left (-\frac {2 a^{2} \sqrt {d e \,x^{3}+c e x}}{5 e^{4} c^{2} x^{3}}+\frac {4 \left (d e \,x^{2}+c e \right ) a \left (4 a d -5 b c \right )}{5 e^{4} c^{3} \sqrt {x \left (d e \,x^{2}+c e \right )}}+\frac {x^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{e^{3} c^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\left (-\frac {2 d a \left (4 a d -5 b c \right )}{5 c^{3} e^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 c^{3} e^{3}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(352\) |
| risch | \(-\frac {2 \sqrt {x^{2} d +c}\, a \left (-8 a d \,x^{2}+10 x^{2} b c +a c \right )}{5 c^{3} x^{2} e^{3} \sqrt {e x}}-\frac {\left (\frac {2 a \left (4 a d -5 b c \right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{\sqrt {d e \,x^{3}+c e x}}-5 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {x^{2}}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}-\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )\right ) \sqrt {e x \left (x^{2} d +c \right )}}{5 c^{3} e^{3} \sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(458\) |
| default | \(-\frac {42 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2} x^{2}-60 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d \,x^{2}+10 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3} x^{2}-21 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2} x^{2}+30 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d \,x^{2}-5 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3} x^{2}-42 a^{2} d^{3} x^{4}+60 a b c \,d^{2} x^{4}-10 b^{2} c^{2} d \,x^{4}-28 a^{2} c \,d^{2} x^{2}+40 a b \,c^{2} d \,x^{2}+4 a^{2} c^{2} d}{10 x^{2} \sqrt {x^{2} d +c}\, d \,e^{3} \sqrt {e x}\, c^{3}}\) | \(638\) |
Input:
int((b*x^2+a)^2/(e*x)^(7/2)/(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)*(-2/5/e^4/c^2*a^2*(d*e*x ^3+c*e*x)^(1/2)/x^3+4/5*(d*e*x^2+c*e)/e^4/c^3*a*(4*a*d-5*b*c)/(x*(d*e*x^2+ c*e))^(1/2)+1/e^3*x^2/c^3*(a^2*d^2-2*a*b*c*d+b^2*c^2)/((x^2+c/d)*d*e*x)^(1 /2)+(-2/5/c^3*d*a*(4*a*d-5*b*c)/e^3-1/2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/c^3/e^ 3)*(-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 = 0.09 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (5 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} x^{5} + {\left (5 \, b^{2} c^{3} - 30 \, a b c^{2} d + 21 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (2 \, a^{2} c^{2} d - {\left (5 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} x^{4} + 2 \, {\left (10 \, a b c^{2} d - 7 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{5 \, {\left (c^{3} d^{2} e^{4} x^{5} + c^{4} d e^{4} x^{3}\right )}} \] Input:
integrate((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
1/5*(((5*b^2*c^2*d - 30*a*b*c*d^2 + 21*a^2*d^3)*x^5 + (5*b^2*c^3 - 30*a*b* c^2*d + 21*a^2*c*d^2)*x^3)*sqrt(d*e)*weierstrassZeta(-4*c/d, 0, weierstras sPInverse(-4*c/d, 0, x)) - (2*a^2*c^2*d - (5*b^2*c^2*d - 30*a*b*c*d^2 + 21 *a^2*d^3)*x^4 + 2*(10*a*b*c^2*d - 7*a^2*c*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(e *x))/(c^3*d^2*e^4*x^5 + c^4*d*e^4*x^3)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{\left (e x\right )^{\frac {7}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x**2+a)**2/(e*x)**(7/2)/(d*x**2+c)**(3/2),x)
Output:
Integral((a + b*x**2)**2/((e*x)**(7/2)*(c + d*x**2)**(3/2)), x)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(3/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)), x)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(7/2)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{7/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int((a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)),x)
Output:
int((a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(3/2)), x)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (-12 \sqrt {d \,x^{2}+c}\, a b d +2 \sqrt {d \,x^{2}+c}\, b^{2} c -14 \sqrt {d \,x^{2}+c}\, b^{2} d \,x^{2}+21 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{8}+2 c d \,x^{6}+c^{2} x^{4}}d x \right ) a^{2} c \,d^{2} x^{2}+21 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{8}+2 c d \,x^{6}+c^{2} x^{4}}d x \right ) a^{2} d^{3} x^{4}-30 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{8}+2 c d \,x^{6}+c^{2} x^{4}}d x \right ) a b \,c^{2} d \,x^{2}-30 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{8}+2 c d \,x^{6}+c^{2} x^{4}}d x \right ) a b c \,d^{2} x^{4}+5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{8}+2 c d \,x^{6}+c^{2} x^{4}}d x \right ) b^{2} c^{3} x^{2}+5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d^{2} x^{8}+2 c d \,x^{6}+c^{2} x^{4}}d x \right ) b^{2} c^{2} d \,x^{4}\right )}{21 \sqrt {x}\, d^{2} e^{4} x^{2} \left (d \,x^{2}+c \right )} \] Input:
int((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(3/2),x)
Output:
(sqrt(e)*( - 12*sqrt(c + d*x**2)*a*b*d + 2*sqrt(c + d*x**2)*b**2*c - 14*sq rt(c + d*x**2)*b**2*d*x**2 + 21*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c* *2*x**4 + 2*c*d*x**6 + d**2*x**8),x)*a**2*c*d**2*x**2 + 21*sqrt(x)*int((sq rt(x)*sqrt(c + d*x**2))/(c**2*x**4 + 2*c*d*x**6 + d**2*x**8),x)*a**2*d**3* x**4 - 30*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c**2*x**4 + 2*c*d*x**6 + d**2*x**8),x)*a*b*c**2*d*x**2 - 30*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2)) /(c**2*x**4 + 2*c*d*x**6 + d**2*x**8),x)*a*b*c*d**2*x**4 + 5*sqrt(x)*int(( sqrt(x)*sqrt(c + d*x**2))/(c**2*x**4 + 2*c*d*x**6 + d**2*x**8),x)*b**2*c** 3*x**2 + 5*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c**2*x**4 + 2*c*d*x**6 + d**2*x**8),x)*b**2*c**2*d*x**4))/(21*sqrt(x)*d**2*e**4*x**2*(c + d*x**2) )