Integrand size = 30, antiderivative size = 414 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}+\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} (b c-a d) \sqrt {c-d x^2}}-\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{d^{3/4} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \] Output:
-e*(e*x)^(3/2)/(-a*d+b*c)/(-d*x^2+c)^(1/2)+c^(3/4)*e^(5/2)*(1-d*x^2/c)^(1/ 2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/d^(3/4)/(-a*d+b*c)/(-d *x^2+c)^(1/2)-c^(3/4)*e^(5/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1 /2)/c^(1/4)/e^(1/2),I)/d^(3/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)-a^(1/2)*c^(1/4) *e^(5/2)*(1-d*x^2/c)^(1/2)*EllipticPi(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2), -b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)/b^(1/2)/d^(1/4)/(-a*d+b*c)/(-d*x^2+c)^ (1/2)+a^(1/2)*c^(1/4)*e^(5/2)*(1-d*x^2/c)^(1/2)*EllipticPi(d^(1/4)*(e*x)^( 1/2)/c^(1/4)/e^(1/2),b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)/b^(1/2)/d^(1/4)/(- a*d+b*c)/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.32 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {e (e x)^{3/2} \left (7 a-7 a \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+b x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{7 a (b c-a d) \sqrt {c-d x^2}} \] Input:
Integrate[(e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]
Output:
-1/7*(e*(e*x)^(3/2)*(7*a - 7*a*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7 /4, (d*x^2)/c, (b*x^2)/a] + b*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1 , 11/4, (d*x^2)/c, (b*x^2)/a]))/(a*(b*c - a*d)*Sqrt[c - d*x^2])
Time = 0.70 (sec) , antiderivative size = 393, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {368, 27, 971, 1054, 2009}
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.
| \(\displaystyle \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^5 x^3}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \int \frac {e^3 x^3}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}\) |
| \(\Big \downarrow \) 971 |
| \(\displaystyle 2 e \left (\frac {\int \frac {e x \left (3 a e^2-b e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 (b c-a d)}-\frac {(e x)^{3/2}}{2 \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e \left (\frac {\int \left (\frac {2 a x e^3}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}+\frac {x e}{\sqrt {c-d x^2}}\right )d\sqrt {e x}}{2 (b c-a d)}-\frac {(e x)^{3/2}}{2 \sqrt {c-d x^2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e \left (\frac {-\frac {\sqrt {a} \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {c^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{d^{3/4} \sqrt {c-d x^2}}+\frac {c^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} \sqrt {c-d x^2}}}{2 (b c-a d)}-\frac {(e x)^{3/2}}{2 \sqrt {c-d x^2} (b c-a d)}\right )\) |
Input:
Int[(e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x]
Output:
2*e*(-1/2*(e*x)^(3/2)/((b*c - a*d)*Sqrt[c - d*x^2]) + ((c^(3/4)*e^(3/2)*Sq rt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(d^(3/4)*Sqrt[c - d*x^2]) - (c^(3/4)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*Ell ipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(d^(3/4)*Sqrt[c - d*x^2]) - (Sqrt[a]*c^(1/4)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sq rt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqr t[e])], -1])/(Sqrt[b]*d^(1/4)*Sqrt[c - d*x^2]) + (Sqrt[a]*c^(1/4)*e^(3/2)* Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin [(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(Sqrt[b]*d^(1/4)*Sqrt[c - d* x^2]))/(2*(b*c - a*d)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Simp[e^n/(n*(b*c - a*d) *(p + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e , q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Time = 1.33 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.39
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-x^{2} d +c \right ) e x}\, \left (\frac {e^{3} x^{2}}{\left (a d -b c \right ) \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}+\frac {e^{3} c \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{\left (a d -b c \right ) d \sqrt {-d e \,x^{3}+c e x}}-\frac {e^{3} c \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) d \sqrt {-d e \,x^{3}+c e x}}+\frac {e^{3} a \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {e^{3} a \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-x^{2} d +c}}\) | \(576\) |
| default | \(-\frac {\left (2 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d -2 \sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2}-\sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\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 +\sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\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}-\sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {a b}\, d +\sqrt {c d}\, b}, \frac {\sqrt {2}}{2}\right ) \sqrt {c d}\, a d +\sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \sqrt {a b}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) \sqrt {c d}\, a d +\sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {a b}\, d +\sqrt {c d}\, b}, \frac {\sqrt {2}}{2}\right ) a b c d +\sqrt {2}\, \sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-x d +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x d +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a b c d +2 x^{2} a b \,d^{2}-2 x^{2} b^{2} c d \right ) e^{2} \sqrt {e x}}{2 x \sqrt {-x^{2} d +c}\, \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {a b}\, d +\sqrt {c d}\, b \right ) \left (a d -b c \right )}\) | \(828\) |
Input:
int((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(e^3*x^2/(a*d-b* c)/(-(x^2-c/d)*d*e*x)^(1/2)+1/(a*d-b*c)*e^3/d*c*(1+x*d/(c*d)^(1/2))^(1/2)* (2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2 )*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/2/(a* d-b*c)*e^3/d*c*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d /(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))* d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/2/(a*d-b*c)*e^3*a/b/d*(c*d)^(1/2)*(1+x *d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2) /(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+ 1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1 /b*(a*b)^(1/2)),1/2*2^(1/2))+1/2/(a*d-b*c)*e^3*a/b/d*(c*d)^(1/2)*(1+x*d/(c *d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d* e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*( c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a *b)^(1/2)),1/2*2^(1/2)))
Timed out. \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=- \int \frac {\left (e x\right )^{\frac {5}{2}}}{- a c \sqrt {c - d x^{2}} + a d x^{2} \sqrt {c - d x^{2}} + b c x^{2} \sqrt {c - d x^{2}} - b d x^{4} \sqrt {c - d x^{2}}}\, dx \] Input:
integrate((e*x)**(5/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)
Output:
-Integral((e*x)**(5/2)/(-a*c*sqrt(c - d*x**2) + a*d*x**2*sqrt(c - d*x**2) + b*c*x**2*sqrt(c - d*x**2) - b*d*x**4*sqrt(c - d*x**2)), x)
\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {\left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x, algorithm="maxima")
Output:
-integrate((e*x)^(5/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)), x)
\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {\left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate(-(e*x)^(5/2)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)), x)
Timed out. \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}}{\left (a-b\,x^2\right )\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \] Input:
int((e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x)
Output:
int((e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(3/2)), x)
\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x^{2}}{-b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}-2 a c d \,x^{2}-b \,c^{2} x^{2}+a \,c^{2}}d x \right ) e^{2} \] Input:
int((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x)
Output:
sqrt(e)*int((sqrt(x)*sqrt(c - d*x**2)*x**2)/(a*c**2 - 2*a*c*d*x**2 + a*d** 2*x**4 - b*c**2*x**2 + 2*b*c*d*x**4 - b*d**2*x**6),x)*e**2