\(\int \frac {1}{x^2 \sqrt {c+d x} (a-b x^2)^{5/2}} \, dx\) [1587]

Optimal result

Integrand size = 25, antiderivative size = 677 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\frac {b (c-d x) \sqrt {c+d x}}{3 a \left (b c^2-a d^2\right ) x \left (a-b x^2\right )^{3/2}}+\frac {b \sqrt {c+d x} \left (4 c \left (2 b c^2-3 a d^2\right )-d \left (7 b c^2-11 a d^2\right ) x\right )}{6 a^2 \left (b c^2-a d^2\right )^2 x \sqrt {a-b x^2}}-\frac {\left (8 b^2 c^4-13 a b c^2 d^2+3 a^2 d^4\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{3 a^3 c \left (b c^2-a d^2\right )^2 x}+\frac {\sqrt {b} \left (8 b^2 c^4-13 a b c^2 d^2+3 a^2 d^4\right ) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 a^{5/2} c \left (b c^2-a d^2\right )^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {b} \left (16 b c^2-17 a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{6 a^{5/2} \left (b c^2-a d^2\right ) \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{a^2 c \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

1/3*b*(-d*x+c)*(d*x+c)^(1/2)/a/(-a*d^2+b*c^2)/x/(-b*x^2+a)^(3/2)+1/6*b*(d* 
x+c)^(1/2)*(4*c*(-3*a*d^2+2*b*c^2)-d*(-11*a*d^2+7*b*c^2)*x)/a^2/(-a*d^2+b* 
c^2)^2/x/(-b*x^2+a)^(1/2)-1/3*(3*a^2*d^4-13*a*b*c^2*d^2+8*b^2*c^4)*(d*x+c) 
^(1/2)*(-b*x^2+a)^(1/2)/a^3/c/(-a*d^2+b*c^2)^2/x+1/3*b^(1/2)*(3*a^2*d^4-13 
*a*b*c^2*d^2+8*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^(5/2)/c/(-a*d^2+b*c^2)^2/(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^ 
(1/2)/(-b*x^2+a)^(1/2)-1/6*b^(1/2)*(-17*a*d^2+16*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^( 
5/2)/(-a*d^2+b*c^2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)+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^2/c/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.90 (sec) , antiderivative size = 1493, normalized size of antiderivative = 2.21 \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[1/(x^2*Sqrt[c + d*x]*(a - b*x^2)^(5/2)),x]
 

Output:

(Sqrt[a - b*x^2]*(((c + d*x)*(-6/(c*x) + (2*a*b*(a*d - b*c*x))/((-(b*c^2) 
+ a*d^2)*(a - b*x^2)^2) + (b*(-11*a^2*d^3 - 10*b^2*c^3*x + 7*a*b*c*d*(c + 
2*d*x)))/((b*c^2 - a*d^2)^2*(-a + b*x^2))))/a^3 + (16*b^3*c^7*Sqrt[-c + (S 
qrt[a]*d)/Sqrt[b]] - 42*a*b^2*c^5*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 32* 
a^2*b*c^3*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 6*a^3*c*d^6*Sqrt[-c + (Sqrt 
[a]*d)/Sqrt[b]] - 32*b^3*c^6*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 52 
*a*b^2*c^4*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 12*a^2*b*c^2*d^4 
*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 16*b^3*c^5*Sqrt[-c + (Sqrt[a]* 
d)/Sqrt[b]]*(c + d*x)^2 - 26*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 + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - ( 
2*I)*Sqrt[b]*c*(8*b^(5/2)*c^5 - 8*Sqrt[a]*b^2*c^4*d - 13*a*b^(3/2)*c^3*d^2 
 + 13*a^(3/2)*b*c^2*d^3 + 3*a^2*Sqrt[b]*c*d^4 - 3*a^(5/2)*d^5)*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)*EllipticE[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)] - I*Sqrt[a]* 
d*(16*b^(5/2)*c^5 - Sqrt[a]*b^2*c^4*d - 26*a*b^(3/2)*c^3*d^2 - 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)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqr 
t[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + (6*I)*a*b^2*c^4*d^2*Sqrt...
 

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.

\(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{5/2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 638

\(\displaystyle \int \frac {1}{x^2 \left (a-b x^2\right )^{5/2} \sqrt {c+d x}}dx\)

Input:

Int[1/(x^2*Sqrt[c + d*x]*(a - b*x^2)^(5/2)),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 = 13.99 (sec) , antiderivative size = 1147, normalized size of antiderivative = 1.69

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1147\)
risch	\(\text {Expression too large to display}\)	\(1756\)
default	\(\text {Expression too large to display}\)	\(5272\)

Input:

int(1/x^2/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

((d*x+c)*(-b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((-1/3*c/(a*d^2- 
b*c^2)/a^2*x+1/3*d/(a*d^2-b*c^2)/b/a)*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/( 
x^2-a/b)^2-2*(-b*d*x-b*c)*(-1/6*b*c*(7*a*d^2-5*b*c^2)/a^3/(a*d^2-b*c^2)^2* 
x+1/12*d*(11*a*d^2-7*b*c^2)/a^2/(a*d^2-b*c^2)^2)/((x^2-a/b)*(-b*d*x-b*c))^ 
(1/2)-1/a^3/c*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/x+2*(1/6*b/(a*d^2-b*c^2)* 
(11*a*d^2-10*b*c^2)/a^3-1/12*b*d^2*(11*a*d^2-7*b*c^2)/a^2/(a*d^2-b*c^2)^2+ 
1/3*b^2*c^2*(7*a*d^2-5*b*c^2)/a^3/(a*d^2-b*c^2)^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*(1/6*c*d*b^2*(7*a 
*d^2-5*b*c^2)/a^3/(a*d^2-b*c^2)^2-1/2/c*d*b/a^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)))+1/c^2*d^2/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)))^...
 

Fricas [F]

\[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c} x^{2}} \,d x } \] Input:

integrate(1/x^2/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

integral(-sqrt(-b*x^2 + a)*sqrt(d*x + c)/(b^3*d*x^9 + b^3*c*x^8 - 3*a*b^2* 
d*x^7 - 3*a*b^2*c*x^6 + 3*a^2*b*d*x^5 + 3*a^2*b*c*x^4 - a^3*d*x^3 - a^3*c* 
x^2), x)
 

Sympy [F]

\[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \left (a - b x^{2}\right )^{\frac {5}{2}} \sqrt {c + d x}}\, dx \] Input:

integrate(1/x**2/(d*x+c)**(1/2)/(-b*x**2+a)**(5/2),x)
 

Output:

Integral(1/(x**2*(a - b*x**2)**(5/2)*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c} x^{2}} \,d x } \] Input:

integrate(1/x^2/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)*x^2), x)
 

Giac [F]

\[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x + c} x^{2}} \,d x } \] Input:

integrate(1/x^2/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

integrate(1/((-b*x^2 + a)^(5/2)*sqrt(d*x + c)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {1}{x^2\,{\left (a-b\,x^2\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:

int(1/(x^2*(a - b*x^2)^(5/2)*(c + d*x)^(1/2)),x)
 

Output:

int(1/(x^2*(a - b*x^2)^(5/2)*(c + d*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^2 \sqrt {c+d x} \left (a-b x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \sqrt {d x +c}\, \left (-b \,x^{2}+a \right )^{\frac {5}{2}}}d x \] Input:

int(1/x^2/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)
 

Output:

int(1/x^2/(d*x+c)^(1/2)/(-b*x^2+a)^(5/2),x)