Integrand size = 28, antiderivative size = 438 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{11/2} \sqrt {c+d x^2}} \, dx=-\frac {2 a^2 \sqrt {c+d x^2}}{9 c e (e x)^{9/2}}-\frac {2 a (18 b c-7 a d) \sqrt {c+d x^2}}{45 c^2 e^3 (e x)^{5/2}}-\frac {2 \left (15 b^2 c^2-18 a b c d+7 a^2 d^2\right ) \sqrt {c+d x^2}}{15 c^3 e^5 \sqrt {e x}}+\frac {2 \sqrt {d} \left (15 b^2 c^2-18 a b c d+7 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{15 c^3 e^6 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \sqrt [4]{d} \left (15 b^2 c^2-18 a b c d+7 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt {c+d x^2}}+\frac {\sqrt [4]{d} \left (15 b^2 c^2-18 a b c d+7 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{15 c^{11/4} e^{11/2} \sqrt {c+d x^2}} \] Output:
-2/9*a^2*(d*x^2+c)^(1/2)/c/e/(e*x)^(9/2)-2/45*a*(-7*a*d+18*b*c)*(d*x^2+c)^ (1/2)/c^2/e^3/(e*x)^(5/2)-2/15*(7*a^2*d^2-18*a*b*c*d+15*b^2*c^2)*(d*x^2+c) ^(1/2)/c^3/e^5/(e*x)^(1/2)+2/15*d^(1/2)*(7*a^2*d^2-18*a*b*c*d+15*b^2*c^2)* (e*x)^(1/2)*(d*x^2+c)^(1/2)/c^3/e^6/(c^(1/2)+d^(1/2)*x)-2/15*d^(1/4)*(7*a^ 2*d^2-18*a*b*c*d+15*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)/e^(11/2)/(d*x^2+c)^(1/2)+1/15*d^(1/4)*(7*a^2*d^2-1 8*a*b*c*d+15*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)/e^(11/2)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.35 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{11/2} \sqrt {c+d x^2}} \, dx=\frac {2 \sqrt {e x} \left (-\left (\left (c+d x^2\right ) \left (45 b^2 c^2 x^4+18 a b c x^2 \left (c-3 d x^2\right )+a^2 \left (5 c^2-7 c d x^2+21 d^2 x^4\right )\right )\right )+d \left (15 b^2 c^2-18 a b c d+7 a^2 d^2\right ) x^6 \sqrt {1+\frac {d x^2}{c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {d x^2}{c}\right )\right )}{45 c^3 e^6 x^5 \sqrt {c+d x^2}} \] Input:
Integrate[(a + b*x^2)^2/((e*x)^(11/2)*Sqrt[c + d*x^2]),x]
Output:
(2*Sqrt[e*x]*(-((c + d*x^2)*(45*b^2*c^2*x^4 + 18*a*b*c*x^2*(c - 3*d*x^2) + a^2*(5*c^2 - 7*c*d*x^2 + 21*d^2*x^4))) + d*(15*b^2*c^2 - 18*a*b*c*d + 7*a ^2*d^2)*x^6*Sqrt[1 + (d*x^2)/c]*Hypergeometric2F1[1/2, 3/4, 7/4, -((d*x^2) /c)]))/(45*c^3*e^6*x^5*Sqrt[c + d*x^2])
Time = 0.51 (sec) , antiderivative size = 422, 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 = {365, 27, 359, 264, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{(e x)^{11/2} \sqrt {c+d x^2}} \, dx\) |
\(\Big \downarrow \) 365 |
\(\displaystyle \frac {2 \int \frac {9 b^2 c x^2+a (18 b c-7 a d)}{2 (e x)^{7/2} \sqrt {d x^2+c}}dx}{9 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {9 b^2 c x^2+a (18 b c-7 a d)}{(e x)^{7/2} \sqrt {d x^2+c}}dx}{9 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 359 |
\(\displaystyle \frac {\frac {3 \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \int \frac {1}{(e x)^{3/2} \sqrt {d x^2+c}}dx}{5 c e^2}-\frac {2 a \sqrt {c+d x^2} (18 b c-7 a d)}{5 c e (e x)^{5/2}}}{9 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {\frac {3 \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \left (\frac {d \int \frac {\sqrt {e x}}{\sqrt {d x^2+c}}dx}{c e^2}-\frac {2 \sqrt {c+d x^2}}{c e \sqrt {e x}}\right )}{5 c e^2}-\frac {2 a \sqrt {c+d x^2} (18 b c-7 a d)}{5 c e (e x)^{5/2}}}{9 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\frac {3 \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \left (\frac {2 d \int \frac {e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{c e^3}-\frac {2 \sqrt {c+d x^2}}{c e \sqrt {e x}}\right )}{5 c e^2}-\frac {2 a \sqrt {c+d x^2} (18 b c-7 a d)}{5 c e (e x)^{5/2}}}{9 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 834 |
\(\displaystyle \frac {\frac {3 \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \left (\frac {2 d \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\sqrt {c} e \int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {c} e \sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{c e^3}-\frac {2 \sqrt {c+d x^2}}{c e \sqrt {e x}}\right )}{5 c e^2}-\frac {2 a \sqrt {c+d x^2} (18 b c-7 a d)}{5 c e (e x)^{5/2}}}{9 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {3 \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \left (\frac {2 d \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{c e^3}-\frac {2 \sqrt {c+d x^2}}{c e \sqrt {e x}}\right )}{5 c e^2}-\frac {2 a \sqrt {c+d x^2} (18 b c-7 a d)}{5 c e (e x)^{5/2}}}{9 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {\frac {3 \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \left (\frac {2 d \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{c e^3}-\frac {2 \sqrt {c+d x^2}}{c e \sqrt {e x}}\right )}{5 c e^2}-\frac {2 a \sqrt {c+d x^2} (18 b c-7 a d)}{5 c e (e x)^{5/2}}}{9 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{9 c e (e x)^{9/2}}\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle \frac {\frac {3 \left (15 b^2 c^2-a d (18 b c-7 a d)\right ) \left (\frac {2 d \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^2}}-\frac {e^2 \sqrt {e x} \sqrt {c+d x^2}}{\sqrt {c} e+\sqrt {d} e x}}{\sqrt {d}}\right )}{c e^3}-\frac {2 \sqrt {c+d x^2}}{c e \sqrt {e x}}\right )}{5 c e^2}-\frac {2 a \sqrt {c+d x^2} (18 b c-7 a d)}{5 c e (e x)^{5/2}}}{9 c e^2}-\frac {2 a^2 \sqrt {c+d x^2}}{9 c e (e x)^{9/2}}\) |
Input:
Int[(a + b*x^2)^2/((e*x)^(11/2)*Sqrt[c + d*x^2]),x]
Output:
(-2*a^2*Sqrt[c + d*x^2])/(9*c*e*(e*x)^(9/2)) + ((-2*a*(18*b*c - 7*a*d)*Sqr t[c + d*x^2])/(5*c*e*(e*x)^(5/2)) + (3*(15*b^2*c^2 - a*d*(18*b*c - 7*a*d)) *((-2*Sqrt[c + d*x^2])/(c*e*Sqrt[e*x]) + (2*d*(-((-((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])))/(c*e^3)))/(5* c*e^2))/(9*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)/(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]
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 = 1.72 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {2 \sqrt {x^{2} d +c}\, \left (21 a^{2} d^{2} x^{4}-54 a b c d \,x^{4}+45 b^{2} c^{2} x^{4}-7 a^{2} c d \,x^{2}+18 a b \,c^{2} x^{2}+5 a^{2} c^{2}\right )}{45 c^{3} x^{4} e^{5} \sqrt {e x}}+\frac {\left (7 a^{2} d^{2}-18 a b c d +15 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) \sqrt {e x \left (x^{2} d +c \right )}}{15 c^{3} \sqrt {d e \,x^{3}+c e x}\, e^{5} \sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(299\) |
elliptic | \(\frac {\sqrt {e x \left (x^{2} d +c \right )}\, \left (-\frac {2 a^{2} \sqrt {d e \,x^{3}+c e x}}{9 e^{6} c \,x^{5}}+\frac {2 a \left (7 a d -18 b c \right ) \sqrt {d e \,x^{3}+c e x}}{45 e^{6} c^{2} x^{3}}-\frac {2 \left (d e \,x^{2}+c e \right ) \left (7 a^{2} d^{2}-18 a b c d +15 b^{2} c^{2}\right )}{15 e^{6} c^{3} \sqrt {x \left (d e \,x^{2}+c e \right )}}+\frac {\left (7 a^{2} d^{2}-18 a b c d +15 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{15 c^{3} e^{5} \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {x^{2} d +c}}\) | \(331\) |
default | \(\frac {42 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2} x^{4}-108 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d \,x^{4}+90 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3} x^{4}-21 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2} x^{4}+54 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d \,x^{4}-45 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-x d +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x d +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3} x^{4}-42 a^{2} d^{3} x^{6}+108 a b c \,d^{2} x^{6}-90 b^{2} c^{2} d \,x^{6}-28 a^{2} c \,d^{2} x^{4}+72 a b \,c^{2} d \,x^{4}-90 b^{2} c^{3} x^{4}+4 a^{2} c^{2} d \,x^{2}-36 a b \,c^{3} x^{2}-10 a^{2} c^{3}}{45 x^{4} \sqrt {x^{2} d +c}\, e^{5} \sqrt {e x}\, c^{3}}\) | \(667\) |
Input:
int((b*x^2+a)^2/(e*x)^(11/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/45*(d*x^2+c)^(1/2)*(21*a^2*d^2*x^4-54*a*b*c*d*x^4+45*b^2*c^2*x^4-7*a^2* c*d*x^2+18*a*b*c^2*x^2+5*a^2*c^2)/c^3/x^4/e^5/(e*x)^(1/2)+1/15*(7*a^2*d^2- 18*a*b*c*d+15*b^2*c^2)/c^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^5*(e*x*(d*x^2+c))^(1/2)/(e*x )^(1/2)/(d*x^2+c)^(1/2)
Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.30 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{11/2} \sqrt {c+d x^2}} \, dx=-\frac {2 \, {\left (3 \, {\left (15 \, b^{2} c^{2} - 18 \, a b c d + 7 \, a^{2} d^{2}\right )} \sqrt {d e} x^{5} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (3 \, {\left (15 \, b^{2} c^{2} - 18 \, a b c d + 7 \, a^{2} d^{2}\right )} x^{4} + 5 \, a^{2} c^{2} + {\left (18 \, a b c^{2} - 7 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{45 \, c^{3} e^{6} x^{5}} \] Input:
integrate((b*x^2+a)^2/(e*x)^(11/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Output:
-2/45*(3*(15*b^2*c^2 - 18*a*b*c*d + 7*a^2*d^2)*sqrt(d*e)*x^5*weierstrassZe ta(-4*c/d, 0, weierstrassPInverse(-4*c/d, 0, x)) + (3*(15*b^2*c^2 - 18*a*b *c*d + 7*a^2*d^2)*x^4 + 5*a^2*c^2 + (18*a*b*c^2 - 7*a^2*c*d)*x^2)*sqrt(d*x ^2 + c)*sqrt(e*x))/(c^3*e^6*x^5)
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{11/2} \sqrt {c+d x^2}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**2/(e*x)**(11/2)/(d*x**2+c)**(1/2),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{11/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^2/(e*x)^(11/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(11/2)), x)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{11/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \left (e x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^2/(e*x)^(11/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^2/(sqrt(d*x^2 + c)*(e*x)^(11/2)), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{11/2} \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{11/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int((a + b*x^2)^2/((e*x)^(11/2)*(c + d*x^2)^(1/2)),x)
Output:
int((a + b*x^2)^2/((e*x)^(11/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{11/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {e}\, \left (-12 \sqrt {d \,x^{2}+c}\, a b d +10 \sqrt {d \,x^{2}+c}\, b^{2} c -14 \sqrt {d \,x^{2}+c}\, b^{2} d \,x^{2}+21 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{8}+c \,x^{6}}d x \right ) a^{2} d^{2} x^{4}-54 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{8}+c \,x^{6}}d x \right ) a b c d \,x^{4}+45 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {d \,x^{2}+c}}{d \,x^{8}+c \,x^{6}}d x \right ) b^{2} c^{2} x^{4}\right )}{21 \sqrt {x}\, d^{2} e^{6} x^{4}} \] Input:
int((b*x^2+a)^2/(e*x)^(11/2)/(d*x^2+c)^(1/2),x)
Output:
(sqrt(e)*( - 12*sqrt(c + d*x**2)*a*b*d + 10*sqrt(c + d*x**2)*b**2*c - 14*s qrt(c + d*x**2)*b**2*d*x**2 + 21*sqrt(x)*int((sqrt(x)*sqrt(c + d*x**2))/(c *x**6 + d*x**8),x)*a**2*d**2*x**4 - 54*sqrt(x)*int((sqrt(x)*sqrt(c + d*x** 2))/(c*x**6 + d*x**8),x)*a*b*c*d*x**4 + 45*sqrt(x)*int((sqrt(x)*sqrt(c + d *x**2))/(c*x**6 + d*x**8),x)*b**2*c**2*x**4))/(21*sqrt(x)*d**2*e**6*x**4)