Integrand size = 25, antiderivative size = 733 \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {b (c-d x)}{a \left (b c^2-a d^2\right ) x (c+d x)^{3/2} \sqrt {a-b x^2}}-\frac {2 d^3 \left (3 b c^2+a d^2\right ) \sqrt {a-b x^2}}{3 a c^2 \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}-\frac {d^3 \left (15 b^2 c^4+29 a b c^2 d^2-12 a^2 d^4\right ) \sqrt {a-b x^2}}{3 a c^3 \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}-\frac {\left (2 b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{a^2 c^3 \left (b c^2-a d^2\right ) x}+\frac {\sqrt {b} \left (6 b^3 c^6+41 a^2 b c^2 d^4-15 a^3 d^6\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 a^{3/2} c^3 \left (b c^2-a d^2\right )^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \left (6 b^2 c^4-3 a b c^2 d^2+5 a^2 d^4\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 a^{3/2} c^2 \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {5 d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a c^3 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
b*(-d*x+c)/a/(-a*d^2+b*c^2)/x/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2)-2/3*d^3*(a*d^ 2+3*b*c^2)*(-b*x^2+a)^(1/2)/a/c^2/(-a*d^2+b*c^2)^2/(d*x+c)^(3/2)-1/3*d^3*( -12*a^2*d^4+29*a*b*c^2*d^2+15*b^2*c^4)*(-b*x^2+a)^(1/2)/a/c^3/(-a*d^2+b*c^ 2)^3/(d*x+c)^(1/2)-(-a*d^2+2*b*c^2)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^2/c^3 /(-a*d^2+b*c^2)/x+1/3*b^(1/2)*(-15*a^3*d^6+41*a^2*b*c^2*d^4+6*b^3*c^6)*(d* x+c)^(1/2)*(1-b*x^2/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^( 1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(3/2)/c^3/(-a*d^2+ b*c^2)^3/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)/(-b*x^2+a)^(1/2)-1/ 3*b^(1/2)*(5*a^2*d^4-3*a*b*c^2*d^2+6*b^2*c^4)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+ a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1 /2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(3/2)/c^2/( -a*d^2+b*c^2)^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)+5*d*(b^(1/2)*(d*x+c)/(b^(1/ 2)*c+a^(1/2)*d))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/2)*x/a^ (1/2))^(1/2)*2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a/ c^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 27.10 (sec) , antiderivative size = 1769, normalized size of antiderivative = 2.41 \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
Integrate[1/(x^2*(c + d*x)^(5/2)*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(3*(c + d*x)*(-(1/(a^2*c^3*x)) - (2*d^5)/(3*c^2*(b*c^2 - a*d^2)^2*(c + d*x)^2) + (4*d^5*(-8*b*c^2 + 3*a*d^2))/(3*(b*c^3 - a*c*d^2)^ 3*(c + d*x)) + (b^2*(a^2*d^3 - b^2*c^3*x + 3*a*b*c*d*(c - d*x)))/(a^2*(-(b *c^2) + a*d^2)^3*(a - b*x^2))) + (6*b^4*c^9*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 6*a*b^3*c^7*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 41*a^2*b^2*c^5*d^4*Sqr t[-c + (Sqrt[a]*d)/Sqrt[b]] - 56*a^3*b*c^3*d^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[ b]] + 15*a^4*c*d^8*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 12*b^4*c^8*Sqrt[-c + ( Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 82*a^2*b^2*c^4*d^4*Sqrt[-c + (Sqrt[a]*d)/S qrt[b]]*(c + d*x) + 30*a^3*b*c^2*d^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d *x) + 6*b^4*c^7*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + 41*a^2*b^2*c^ 3*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 15*a^3*b*c*d^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - I*Sqrt[b]*c*(6*b^(7/2)*c^7 - 6*Sqrt[a ]*b^3*c^6*d + 41*a^2*b^(3/2)*c^3*d^4 - 41*a^(5/2)*b*c^2*d^5 - 15*a^3*Sqrt[ b]*c*d^6 + 15*a^(7/2)*d^7)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[ -(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcS inh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d) /(Sqrt[b]*c - Sqrt[a]*d)] - I*Sqrt[a]*d*(6*b^(7/2)*c^7 + 6*Sqrt[a]*b^3*c^6 *d - 78*a^(3/2)*b^2*c^4*d^3 + 41*a^2*b^(3/2)*c^3*d^4 + 55*a^(5/2)*b*c^2*d^ 5 - 15*a^3*Sqrt[b]*c*d^6 - 15*a^(7/2)*d^7)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/ (c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/...
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.
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (\frac {2 d^2}{c^3 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}-\frac {2 d}{c^3 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}+\frac {d^2}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}+\frac {1}{c^2 x^2 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
Input:
Int[1/(x^2*(c + d*x)^(5/2)*(a - b*x^2)^(3/2)),x]
Output:
$Aborted
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p/Sqrt[c + d*x], x^m*(c + d*x)^(n + 1 /2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p + 1/2] && IntegerQ[n + 1/2] && IntegerQ[m]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(1280\) vs. \(2(632)=1264\).
Time = 15.18 (sec) , antiderivative size = 1281, normalized size of antiderivative = 1.75
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1281\) |
| risch | \(\text {Expression too large to display}\) | \(2207\) |
| default | \(\text {Expression too large to display}\) | \(6863\) |
Input:
int(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*(-2/3*d^3/(a*d^2 -b*c^2)^2/c^2*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2-4/3*(-b*d*x^2+a *d)*d^4/(a*d^2-b*c^2)^3*(3*a*d^2-8*b*c^2)/c^3/((x+c/d)*(-b*d*x^2+a*d))^(1/ 2)-2*(-b*d*x-b*c)*(-1/2*b^2*c*(3*a*d^2+b*c^2)/(a*d^2-b*c^2)^3/a^2*x+1/2*b* d*(a*d^2+3*b*c^2)/(a*d^2-b*c^2)^3/a)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)-1/a^2/ c^3*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x+2*(1/3*b*d^4/(a*d^2-b*c^2)^2/c^2- 2/3*b/c^2*d^4*(3*a*d^2-8*b*c^2)/(a*d^2-b*c^2)^3+b^2/(a*d^2-b*c^2)^2*(a*d^2 +b*c^2)/a^2-1/2*b^2/a*d^2*(a*d^2+3*b*c^2)/(a*d^2-b*c^2)^3+b^3*c^2*(3*a*d^2 +b*c^2)/(a*d^2-b*c^2)^3/a^2)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b) ^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b* (a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1 /2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2) )/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(-2/3*d^5*b*(3*a*d^2-8*b*c^2)/(a*d^2-b* c^2)^3/c^3+1/2*b^3*c*d*(3*a*d^2+b*c^2)/(a*d^2-b*c^2)^3/a^2-1/2/a^2*d*b/c^3 )*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b )^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b )^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d-1/b*(a*b)^(1/2)) *EllipticE(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/( -c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b* (a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/...
Timed out. \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x**2/(d*x+c)**(5/2)/(-b*x**2+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*(d*x + c)^(5/2)*x^2), x)
\[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((-b*x^2 + a)^(3/2)*(d*x + c)^(5/2)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (a-b\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int(1/(x^2*(a - b*x^2)^(3/2)*(c + d*x)^(5/2)),x)
Output:
int(1/(x^2*(a - b*x^2)^(3/2)*(c + d*x)^(5/2)), x)
\[ \int \frac {1}{x^2 (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (d x +c \right )^{\frac {5}{2}} \left (-b \,x^{2}+a \right )^{\frac {3}{2}}}d x \] Input:
int(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x)
Output:
int(1/x^2/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x)