Integrand size = 27, antiderivative size = 313 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\frac {e \sqrt {e x} (c-d x)}{\left (b c^2-a d^2\right ) \sqrt {a-b x^2}}+\frac {a^{3/4} d e^{3/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} \left (b c^2-a d^2\right ) \sqrt {a-b x^2}}+\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{b^{3/4} \left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {a-b x^2}}-\frac {2 \sqrt [4]{a} c e^{3/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \left (b c^2-a d^2\right ) \sqrt {a-b x^2}} \] Output:
e*(e*x)^(1/2)*(-d*x+c)/(-a*d^2+b*c^2)/(-b*x^2+a)^(1/2)+a^(3/4)*d*e^(3/2)*( 1-b*x^2/a)^(1/2)*EllipticE(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I)/b^(3/4)/ (-a*d^2+b*c^2)/(-b*x^2+a)^(1/2)+a^(1/4)*e^(3/2)*(1-b*x^2/a)^(1/2)*Elliptic F(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I)/b^(3/4)/(b^(1/2)*c+a^(1/2)*d)/(-b *x^2+a)^(1/2)-2*a^(1/4)*c*e^(3/2)*(1-b*x^2/a)^(1/2)*EllipticPi(b^(1/4)*(e* x)^(1/2)/a^(1/4)/e^(1/2),-a^(1/2)*d/b^(1/2)/c,I)/b^(1/4)/(-a*d^2+b*c^2)/(- b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.22 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.98 \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\frac {e^2 \left (a \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} d-\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} b c x+i \sqrt {a} \sqrt {b} d \sqrt {1-\frac {a}{b x^2}} x^{3/2} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )-2 i b c \sqrt {1-\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} b \left (-b c^2+a d^2\right ) \sqrt {e x} \sqrt {a-b x^2}} \] Input:
Integrate[(e*x)^(3/2)/((c + d*x)*(a - b*x^2)^(3/2)),x]
Output:
(e^2*(a*Sqrt[-(Sqrt[a]/Sqrt[b])]*d - Sqrt[-(Sqrt[a]/Sqrt[b])]*b*c*x + I*Sq rt[a]*Sqrt[b]*d*Sqrt[1 - a/(b*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[-(Sqr t[a]/Sqrt[b])]/Sqrt[x]], -1] + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*Sqrt[1 - a/(b*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1] - (2*I)*b*c*Sqrt[1 - a/(b*x^2)]*x^(3/2)*EllipticPi[-((Sqrt[b]*c)/(Sqrt [a]*d)), I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1]))/(Sqrt[-(Sqrt[a ]/Sqrt[b])]*b*(-(b*c^2) + a*d^2)*Sqrt[e*x]*Sqrt[a - b*x^2])
Time = 0.82 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {616, 27, 1641, 25, 27, 1493, 25, 1513, 27, 765, 762, 1390, 1389, 327, 1543, 1542}
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.
| \(\displaystyle \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^{3/2} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 616 |
| \(\displaystyle \frac {2 \int \frac {e^3 x^2}{(c e+d x e) \left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {e^2 x^2}{(c e+d x e) \left (a-b x^2\right )^{3/2}}d\sqrt {e x}\) |
| \(\Big \downarrow \) 1641 |
| \(\displaystyle 2 \left (-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}-\frac {\int -\frac {a (c e-d e x)}{\left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\int \frac {a (c e-d e x)}{\left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{b c^2-a d^2}-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {a \int \frac {c e-d e x}{\left (a-b x^2\right )^{3/2}}d\sqrt {e x}}{b c^2-a d^2}-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 1493 |
| \(\displaystyle 2 \left (\frac {a \left (\frac {\sqrt {e x} (c e-d e x)}{2 a \sqrt {a-b x^2}}-\frac {\int -\frac {c e+d x e}{\sqrt {a-b x^2}}d\sqrt {e x}}{2 a}\right )}{b c^2-a d^2}-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {a \left (\frac {\int \frac {c e+d x e}{\sqrt {a-b x^2}}d\sqrt {e x}}{2 a}+\frac {\sqrt {e x} (c e-d e x)}{2 a \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle 2 \left (\frac {a \left (\frac {e \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}+\frac {\sqrt {a} d e \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e \sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}}{2 a}+\frac {\sqrt {e x} (c e-d e x)}{2 a \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {a \left (\frac {e \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}+\frac {d \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}}{2 a}+\frac {\sqrt {e x} (c e-d e x)}{2 a \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 \left (\frac {a \left (\frac {\frac {e \sqrt {1-\frac {b x^2}{a}} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {a-b x^2}}+\frac {d \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}}{2 a}+\frac {\sqrt {e x} (c e-d e x)}{2 a \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 \left (\frac {a \left (\frac {\frac {d \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}+\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}}{2 a}+\frac {\sqrt {e x} (c e-d e x)}{2 a \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle 2 \left (\frac {a \left (\frac {\frac {d \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {b} \sqrt {a-b x^2}}+\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}}{2 a}+\frac {\sqrt {e x} (c e-d e x)}{2 a \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle 2 \left (\frac {a \left (\frac {\frac {\sqrt {a} d e \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}d\sqrt {e x}}{\sqrt {b} \sqrt {a-b x^2}}+\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}}{2 a}+\frac {\sqrt {e x} (c e-d e x)}{2 a \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle 2 \left (\frac {a \left (\frac {\frac {a^{3/4} d e^{3/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^2}}+\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}}{2 a}+\frac {\sqrt {e x} (c e-d e x)}{2 a \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {c^2 e^2 \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{b c^2-a d^2}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 \left (\frac {a \left (\frac {\frac {a^{3/4} d e^{3/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^2}}+\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}}{2 a}+\frac {\sqrt {e x} (c e-d e x)}{2 a \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {c^2 e^2 \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{(c e+d x e) \sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 \left (\frac {a \left (\frac {\frac {a^{3/4} d e^{3/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^2}}+\frac {\sqrt [4]{a} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}}{2 a}+\frac {\sqrt {e x} (c e-d e x)}{2 a \sqrt {a-b x^2}}\right )}{b c^2-a d^2}-\frac {\sqrt [4]{a} c e^{3/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2} \left (b c^2-a d^2\right )}\right )\) |
Input:
Int[(e*x)^(3/2)/((c + d*x)*(a - b*x^2)^(3/2)),x]
Output:
2*((a*((Sqrt[e*x]*(c*e - d*e*x))/(2*a*Sqrt[a - b*x^2]) + ((a^(3/4)*d*e^(3/ 2)*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[ e])], -1])/(b^(3/4)*Sqrt[a - b*x^2]) + (a^(1/4)*(c - (Sqrt[a]*d)/Sqrt[b])* e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)* Sqrt[e])], -1])/(b^(1/4)*Sqrt[a - b*x^2]))/(2*a)))/(b*c^2 - a*d^2) - (a^(1 /4)*c*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticPi[-((Sqrt[a]*d)/(Sqrt[b]*c)), A rcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(1/4)*(b*c^2 - a*d^2 )*Sqrt[a - b*x^2]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[n, 0] && FractionQ[m]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x )*(d + e*x^2)*((a + c*x^4)^(p + 1)/(4*a*(p + 1))), x] + Simp[1/(4*a*(p + 1) ) Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && Integer Q[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Int[((x_)^(m_)*((a_) + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-d/e)^(m/2)*((c*d^2 + a*e^2)^(p + 1/2)/e^(2*p + 1)) Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] + Simp[(c*d^2 + a*e^2)^(p + 1/2) Int[(a + c*x^4)^p*ExpandToSum[((c*d^2 + a*e^2)^(-p - 1/2)*x^m - e^(-2*p - 1)*(-d/e )^(m/2)*(a + c*x^4)^(-p - 1/2))/(d + e*x^2), x], x], x] /; FreeQ[{a, c, d, e}, x] && ILtQ[p + 1/2, 0] && IGtQ[m/2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ [c/a]
Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(247)=494\).
Time = 1.39 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.69
| method | result | size |
| default | \(\frac {\left (\sqrt {2}\, \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2}-\sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2} \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}+2 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a b c d -2 \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a \,d^{2} \sqrt {-\frac {b x}{\sqrt {a b}}}\, \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}+2 \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{d \sqrt {a b}-b c}, \frac {\sqrt {2}}{2}\right ) b \,c^{2} \sqrt {-\frac {b x}{\sqrt {a b}}}\, \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}+2 b^{2} c \,x^{2} d -2 b \,d^{2} x^{2} \sqrt {a b}-2 x \,b^{2} c^{2}+2 b c d x \sqrt {a b}\right ) e \sqrt {e x}}{2 b x \sqrt {-b \,x^{2}+a}\, \left (b c -d \sqrt {a b}\right ) \left (a \,d^{2}-b \,c^{2}\right )}\) | \(528\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (\frac {2 b e x \left (\frac {d e x}{2 b \left (a \,d^{2}-b \,c^{2}\right )}-\frac {c e}{2 b \left (a \,d^{2}-b \,c^{2}\right )}\right )}{\sqrt {-\left (x^{2}-\frac {a}{b}\right ) b e x}}-\frac {e^{2} c \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{2 \left (a \,d^{2}-b \,c^{2}\right ) b \sqrt {-b e \,x^{3}+a e x}}-\frac {d \,e^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 \left (a \,d^{2}-b \,c^{2}\right ) b \sqrt {-b e \,x^{3}+a e x}}+\frac {e^{2} c^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{\left (a \,d^{2}-b \,c^{2}\right ) d b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-b \,x^{2}+a}}\) | \(562\) |
Input:
int((e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2/b*(2^(1/2)*(a*b)^(1/2)*((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*((-b*x+(a *b)^(1/2))/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)*EllipticF(((b*x+(a* b)^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*d^2-2^(1/2)*EllipticF(((b*x+(a *b)^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*b*c^2*(a*b)^(1/2)*((b*x+(a*b)^( 1/2))/(a*b)^(1/2))^(1/2)*((-b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*(-b*x/(a*b )^(1/2))^(1/2)+2*2^(1/2)*((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*((-b*x+(a*b )^(1/2))/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)*EllipticE(((b*x+(a*b) ^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c*d-2*2^(1/2)*EllipticE(((b*x+ (a*b)^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*d^2*(-b*x/(a*b)^(1/2))^(1/2 )*(a*b)^(1/2)*((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*((-b*x+(a*b)^(1/2))/(a *b)^(1/2))^(1/2)+2*2^(1/2)*EllipticPi(((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2 ),(a*b)^(1/2)*d/(d*(a*b)^(1/2)-b*c),1/2*2^(1/2))*b*c^2*(-b*x/(a*b)^(1/2))^ (1/2)*(a*b)^(1/2)*((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*((-b*x+(a*b)^(1/2) )/(a*b)^(1/2))^(1/2)+2*b^2*c*x^2*d-2*b*d^2*x^2*(a*b)^(1/2)-2*x*b^2*c^2+2*b *c*d*x*(a*b)^(1/2))*e*(e*x)^(1/2)/x/(-b*x^2+a)^(1/2)/(b*c-d*(a*b)^(1/2))/( a*d^2-b*c^2)
Timed out. \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(3/2)/(d*x+c)/(-b*x**2+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}} \,d x } \] Input:
integrate((e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((e*x)^(3/2)/((-b*x^2 + a)^(3/2)*(d*x + c)), x)
\[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}} \,d x } \] Input:
integrate((e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((e*x)^(3/2)/((-b*x^2 + a)^(3/2)*(d*x + c)), x)
Timed out. \[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}}{{\left (a-b\,x^2\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:
int((e*x)^(3/2)/((a - b*x^2)^(3/2)*(c + d*x)),x)
Output:
int((e*x)^(3/2)/((a - b*x^2)^(3/2)*(c + d*x)), x)
\[ \int \frac {(e x)^{3/2}}{(c+d x) \left (a-b x^2\right )^{3/2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}\, x}{b^{2} d \,x^{5}+b^{2} c \,x^{4}-2 a b d \,x^{3}-2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) e \] Input:
int((e*x)^(3/2)/(d*x+c)/(-b*x^2+a)^(3/2),x)
Output:
sqrt(e)*int((sqrt(x)*sqrt(a - b*x**2)*x)/(a**2*c + a**2*d*x - 2*a*b*c*x**2 - 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*e