Integrand size = 25, antiderivative size = 655 \[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\frac {b (c-d x)}{a \left (b c^2-a d^2\right ) (c+d x)^{3/2} \sqrt {a-b x^2}}+\frac {2 d^2 \left (3 b c^2+a d^2\right ) \sqrt {a-b x^2}}{3 a c \left (b c^2-a d^2\right )^2 (c+d x)^{3/2}}+\frac {d^2 \left (9 b^2 c^4+29 a b c^2 d^2-6 a^2 d^4\right ) \sqrt {a-b x^2}}{3 a c^2 \left (b c^2-a d^2\right )^3 \sqrt {c+d x}}-\frac {\sqrt {b} d \left (9 b^2 c^4+29 a b c^2 d^2-6 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {a} c^2 \left (b c^2-a d^2\right )^3 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {b} d \left (3 b c^2+a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {a} c \left (b c^2-a d^2\right )^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a c^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
b*(-d*x+c)/a/(-a*d^2+b*c^2)/(d*x+c)^(3/2)/(-b*x^2+a)^(1/2)+2/3*d^2*(a*d^2+ 3*b*c^2)*(-b*x^2+a)^(1/2)/a/c/(-a*d^2+b*c^2)^2/(d*x+c)^(3/2)+1/3*d^2*(-6*a ^2*d^4+29*a*b*c^2*d^2+9*b^2*c^4)*(-b*x^2+a)^(1/2)/a/c^2/(-a*d^2+b*c^2)^3/( d*x+c)^(1/2)-1/3*b^(1/2)*d*(-6*a^2*d^4+29*a*b*c^2*d^2+9*b^2*c^4)*(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^(1/2)/c^2/(-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)+2/3*b^(1 /2)*d*(a*d^2+3*b*c^2)*(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^(1/2)/c/(-a*d^2+b*c^2)^2/(d*x+c)^(1 /2)/(-b*x^2+a)^(1/2)-2*(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^2/(d*x+c)^(1/2)/(-b*x^2+a) ^(1/2)
Result contains complex when optimal does not.
Time = 26.50 (sec) , antiderivative size = 1696, normalized size of antiderivative = 2.59 \[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
Integrate[1/(x*(c + d*x)^(5/2)*(a - b*x^2)^(3/2)),x]
Output:
(Sqrt[a - b*x^2]*(3*(c + d*x)*((2*d^4)/(3*c*(b*c^2 - a*d^2)^2*(c + d*x)^2) + (2*d^4*(13*b*c^2 - 3*a*d^2))/(3*c^2*(b*c^2 - a*d^2)^3*(c + d*x)) + (b^2 *(-(b*c^2*(c - 3*d*x)) + a*d^2*(-3*c + d*x)))/(a*(-(b*c^2) + a*d^2)^3*(a - b*x^2))) - (9*b^3*c^7*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 20*a*b^2*c^5*d^2*S qrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 35*a^2*b*c^3*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqr t[b]] + 6*a^3*c*d^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 18*b^3*c^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 58*a*b^2*c^4*d^2*Sqrt[-c + (Sqrt[a]*d)/Sq rt[b]]*(c + d*x) + 12*a^2*b*c^2*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d* x) + 9*b^3*c^5*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + 29*a*b^2*c^3*d ^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 6*a^2*b*c*d^4*Sqrt[-c + (S qrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - I*Sqrt[b]*c*(9*b^(5/2)*c^5 - 9*Sqrt[a]*b^ 2*c^4*d + 29*a*b^(3/2)*c^3*d^2 - 29*a^(3/2)*b*c^2*d^3 - 6*a^2*Sqrt[b]*c*d^ 4 + 6*a^(5/2)*d^5)*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*ArcSinh[Sqrt [-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b ]*c - Sqrt[a]*d)] - I*(3*b^3*c^6 + 9*Sqrt[a]*b^(5/2)*c^5*d - 51*a*b^2*c^4* d^2 + 29*a^(3/2)*b^(3/2)*c^3*d^3 + 22*a^2*b*c^2*d^4 - 6*a^(5/2)*Sqrt[b]*c* d^5 - 6*a^3*d^6)*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)*EllipticF[I*ArcSinh[Sqrt[- c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[...
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.
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \int \left (-\frac {d}{c^2 \left (a-b x^2\right )^{3/2} (c+d x)^{3/2}}+\frac {1}{c^2 x \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}-\frac {d}{c \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{x \left (a-b x^2\right )^{3/2} (c+d x)^{5/2}}dx\) |
Input:
Int[1/(x*(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. \(1223\) vs. \(2(558)=1116\).
Time = 8.64 (sec) , antiderivative size = 1224, normalized size of antiderivative = 1.87
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1224\) |
| default | \(\text {Expression too large to display}\) | \(6064\) |
Input:
int(1/x/(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^2/(a*d^2- b*c^2)^2/c*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(x+c/d)^2+2/3*(-b*d*x^2+a*d) *d^3/(a*d^2-b*c^2)^3*(3*a*d^2-13*b*c^2)/c^2/((x+c/d)*(-b*d*x^2+a*d))^(1/2) -2*(-b*d*x-b*c)*(1/2*b*d*(a*d^2+3*b*c^2)/(a*d^2-b*c^2)^3/a*x-1/2*b*c*(3*a* d^2+b*c^2)/(a*d^2-b*c^2)^3/a)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2*(-1/3*b*d^3 /(a*d^2-b*c^2)^2/c+1/3*b/c*d^3*(3*a*d^2-13*b*c^2)/(a*d^2-b*c^2)^3-2/(a*d^2 -b*c^2)^2*b^2/a*c*d+1/2*b^2*c*d*(3*a*d^2+b*c^2)/a/(a*d^2-b*c^2)^3-b^2*c*d* (a*d^2+3*b*c^2)/(a*d^2-b*c^2)^3/a)*(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*(1/3*b*d^4*(3*a*d^2-13*b*c^2)/(a* d^2-b*c^2)^3/c^2-1/2*b^2/a*d^2*(a*d^2+3*b*c^2)/(a*d^2-b*c^2)^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)))^(1/2)))-2/a/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)...
Timed out. \[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{x \left (a - b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/x/(d*x+c)**(5/2)/(-b*x**2+a)**(3/2),x)
Output:
Integral(1/(x*(a - b*x**2)**(3/2)*(c + d*x)**(5/2)), x)
\[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}} x} \,d x } \] Input:
integrate(1/x/(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), x)
\[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {5}{2}} x} \,d x } \] Input:
integrate(1/x/(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), x)
Timed out. \[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (a-b\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int(1/(x*(a - b*x^2)^(3/2)*(c + d*x)^(5/2)),x)
Output:
int(1/(x*(a - b*x^2)^(3/2)*(c + d*x)^(5/2)), x)
\[ \int \frac {1}{x (c+d x)^{5/2} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{b^{2} d^{3} x^{8}+3 b^{2} c \,d^{2} x^{7}-2 a b \,d^{3} x^{6}+3 b^{2} c^{2} d \,x^{6}-6 a b c \,d^{2} x^{5}+b^{2} c^{3} x^{5}+a^{2} d^{3} x^{4}-6 a b \,c^{2} d \,x^{4}+3 a^{2} c \,d^{2} x^{3}-2 a b \,c^{3} x^{3}+3 a^{2} c^{2} d \,x^{2}+a^{2} c^{3} x}d x \] Input:
int(1/x/(d*x+c)^(5/2)/(-b*x^2+a)^(3/2),x)
Output:
int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a**2*c**3*x + 3*a**2*c**2*d*x**2 + 3 *a**2*c*d**2*x**3 + a**2*d**3*x**4 - 2*a*b*c**3*x**3 - 6*a*b*c**2*d*x**4 - 6*a*b*c*d**2*x**5 - 2*a*b*d**3*x**6 + b**2*c**3*x**5 + 3*b**2*c**2*d*x**6 + 3*b**2*c*d**2*x**7 + b**2*d**3*x**8),x)