\(\int \frac {1}{x^3 \sqrt {c+d x} (a-b x^2)^{3/2}} \, dx\) [1561]

Optimal result

Integrand size = 25, antiderivative size = 617 \[ \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\frac {b (c-d x) \sqrt {c+d x}}{a \left (b c^2-a d^2\right ) x^2 \sqrt {a-b x^2}}-\frac {\left (3 b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{2 a^2 c \left (b c^2-a d^2\right ) x^2}+\frac {d \left (7 b-\frac {3 a d^2}{c^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{4 a^2 \left (b c^2-a d^2\right ) x}-\frac {\sqrt {b} d \left (7 b c^2-3 a d^2\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 a^{3/2} c^2 \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {\sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 a^{3/2} c \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {3 \left (4 b c^2+a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 a^2 c^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

b*(-d*x+c)*(d*x+c)^(1/2)/a/(-a*d^2+b*c^2)/x^2/(-b*x^2+a)^(1/2)-1/2*(-a*d^2 
+3*b*c^2)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^2/c/(-a*d^2+b*c^2)/x^2+1/4*d*(7 
*b-3*a*d^2/c^2)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a^2/(-a*d^2+b*c^2)/x-1/4*b^ 
(1/2)*d*(-3*a*d^2+7*b*c^2)*(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^2/(-a*d^2+b*c^2)/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d 
))^(1/2)/(-b*x^2+a)^(1/2)+1/4*b^(1/2)*d*(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/(d*x+c)^( 
1/2)/(-b*x^2+a)^(1/2)-3/4*(a*d^2+4*b*c^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^2/c^2/(d* 
x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.81 (sec) , antiderivative size = 1268, normalized size of antiderivative = 2.06 \[ \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

Integrate[1/(x^3*Sqrt[c + d*x]*(a - b*x^2)^(3/2)),x]
 

Output:

(Sqrt[a - b*x^2]*(((c + d*x)*(-2/(c*x^2) + (3*d)/(c^2*x) - (4*b^2*(c - d*x 
))/((b*c^2 - a*d^2)*(-a + b*x^2))))/a^2 - (7*b^2*c^5*Sqrt[-c + (Sqrt[a]*d) 
/Sqrt[b]] - 10*a*b*c^3*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 3*a^2*c*d^4*Sq 
rt[-c + (Sqrt[a]*d)/Sqrt[b]] - 14*b^2*c^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*( 
c + d*x) + 6*a*b*c^2*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 7*b^2* 
c^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 3*a*b*c*d^2*Sqrt[-c + (Sq 
rt[a]*d)/Sqrt[b]]*(c + d*x)^2 - I*Sqrt[b]*c*(7*b^(3/2)*c^3 - 7*Sqrt[a]*b*c 
^2*d - 3*a*Sqrt[b]*c*d^2 + 3*a^(3/2)*d^3)*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)*E 
llipticE[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*(6*b^2*c^4 + 7*Sqrt[a]*b^(3/2 
)*c^3*d - 7*a*b*c^2*d^2 - 3*a^(3/2)*Sqrt[b]*c*d^3 - 3*a^2*d^4)*Sqrt[(d*(Sq 
rt[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]]/Sqr 
t[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + (12*I)*b^2 
*c^4*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)*EllipticPi[(Sqrt[b]*c)/(Sqrt[b]*c - Sq 
rt[a]*d), 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)] - (9*I)*a*b*c^2*d^2*Sqrt[(d*(Sqr 
t[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d...
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 638

\(\displaystyle \int \frac {1}{x^3 \left (a-b x^2\right )^{3/2} \sqrt {c+d x}}dx\)

Input:

Int[1/(x^3*Sqrt[c + d*x]*(a - b*x^2)^(3/2)),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 638

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Unintegrable[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x] /; FreeQ 
[{a, b, c, d, e, m, n, p}, x]
 

Maple [A] (verified)

Time = 8.57 (sec) , antiderivative size = 1004, normalized size of antiderivative = 1.63

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1004\)
risch	\(\text {Expression too large to display}\)	\(1179\)
default	\(\text {Expression too large to display}\)	\(2370\)

Input:

int(1/x^3/(d*x+c)^(1/2)/(-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)*(-1/2/a^2/c/x^2* 
(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+3/4*d/a^2/c^2*(-b*d*x^3-b*c*x^2+a*d*x+a 
*c)^(1/2)/x-2*(-b*d*x-b*c)*(1/2*d/(a*d^2-b*c^2)/a^2*b*x-1/2*b/a^2*c/(a*d^2 
-b*c^2))/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2*(1/4/a^2*d*b/c-1/2*c*d*b^2/a^2/( 
a*d^2-b*c^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*(3/8/c^2/a^2*b*d^2-1/2/a^2*b^2*d^2/(a*d^2-b*c^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)*((-c/d-1/b*(a*b)^(1/2))*Ellip 
ticE(((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)))-3/4* 
(a*d^2+4*b*c^2)/a^2/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)*d 
*EllipticPi(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),-(-c/d+1/b*(a*b)^(1/2))/ 
c*d,((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))
 

Fricas [F]

\[ \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c} x^{3}} \,d x } \] Input:

integrate(1/x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-b*x^2 + a)*sqrt(d*x + c)/(b^2*d*x^8 + b^2*c*x^7 - 2*a*b*d*x 
^6 - 2*a*b*c*x^5 + a^2*d*x^4 + a^2*c*x^3), x)
 

Sympy [F]

\[ \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \] Input:

integrate(1/x**3/(d*x+c)**(1/2)/(-b*x**2+a)**(3/2),x)
 

Output:

Integral(1/(x**3*(a - b*x**2)**(3/2)*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c} x^{3}} \,d x } \] Input:

integrate(1/x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((-b*x^2 + a)^(3/2)*sqrt(d*x + c)*x^3), x)
 

Giac [F]

\[ \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x + c} x^{3}} \,d x } \] Input:

integrate(1/x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((-b*x^2 + a)^(3/2)*sqrt(d*x + c)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int(1/(x^3*(a - b*x^2)^(3/2)*(c + d*x)^(1/2)),x)
 

Output:

int(1/(x^3*(a - b*x^2)^(3/2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^3 \sqrt {c+d x} \left (a-b x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{\frac {3}{2}}}d x \] Input:

int(1/x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x)
 

Output:

int(1/x^3/(d*x+c)^(1/2)/(-b*x^2+a)^(3/2),x)