\(\int \frac {(a+c x^2)^{3/2}}{x^2 (d+e x+f x^2)} \, dx\) [29]

Optimal result

Integrand size = 27, antiderivative size = 531 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {a \sqrt {a+c x^2}}{d x}+\frac {3 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d}+\frac {\sqrt {c} (2 c d-3 a f) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d f}-\frac {\left (e (c d-a f) (c d+a f) \left (e-\sqrt {e^2-4 d f}\right )-2 f \left (c^2 d^3-2 a c d^2 f-a^2 f \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (e (c d-a f) (c d+a f) \left (e+\sqrt {e^2-4 d f}\right )-2 f \left (c^2 d^3-2 a c d^2 f-a^2 f \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}}+\frac {a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2} \] Output:

-a*(c*x^2+a)^(1/2)/d/x+3/2*a*c^(1/2)*arctanh(c^(1/2)*x/(c*x^2+a)^(1/2))/d+ 
1/2*c^(1/2)*(-3*a*f+2*c*d)*arctanh(c^(1/2)*x/(c*x^2+a)^(1/2))/d/f-1/2*(e*( 
-a*f+c*d)*(a*f+c*d)*(e-(-4*d*f+e^2)^(1/2))-2*f*(c^2*d^3-2*a*c*d^2*f-a^2*f* 
(-d*f+e^2)))*arctanh(1/2*(2*a*f-c*(e-(-4*d*f+e^2)^(1/2))*x)*2^(1/2)/(2*a*f 
^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2)/(c*x^2+a)^(1/2))*2^(1/2)/d^2/ 
f/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2)+1/ 
2*(e*(-a*f+c*d)*(a*f+c*d)*(e+(-4*d*f+e^2)^(1/2))-2*f*(c^2*d^3-2*a*c*d^2*f- 
a^2*f*(-d*f+e^2)))*arctanh(1/2*(2*a*f-c*(e+(-4*d*f+e^2)^(1/2))*x)*2^(1/2)/ 
(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2)/(c*x^2+a)^(1/2))*2^(1/2 
)/d^2/f/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1 
/2)+a^(3/2)*e*arctanh((c*x^2+a)^(1/2)/a^(1/2))/d^2
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.83 (sec) , antiderivative size = 497, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {2 a^{3/2} e f x \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )+d \left (a f \sqrt {a+c x^2}+c^{3/2} d x \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )\right )+x \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {-a c^2 d^2 e \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+a^3 e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-2 c^{5/2} d^3 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a c^{3/2} d^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a^2 \sqrt {c} e^2 f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^2 \sqrt {c} d f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^2 e \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 e f^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{d^2 f x} \] Input:

Integrate[(a + c*x^2)^(3/2)/(x^2*(d + e*x + f*x^2)),x]
 

Output:

-((2*a^(3/2)*e*f*x*ArcTanh[(Sqrt[c]*x - Sqrt[a + c*x^2])/Sqrt[a]] + d*(a*f 
*Sqrt[a + c*x^2] + c^(3/2)*d*x*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]]) + x*Ro 
otSum[a^2*f + 2*a*Sqrt[c]*e*#1 + 4*c*d*#1^2 - 2*a*f*#1^2 - 2*Sqrt[c]*e*#1^ 
3 + f*#1^4 & , (-(a*c^2*d^2*e*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]) + 
a^3*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1] - 2*c^(5/2)*d^3*Log[-(S 
qrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1 + 4*a*c^(3/2)*d^2*f*Log[-(Sqrt[c]*x) 
+ Sqrt[a + c*x^2] - #1]*#1 + 2*a^2*Sqrt[c]*e^2*f*Log[-(Sqrt[c]*x) + Sqrt[a 
 + c*x^2] - #1]*#1 - 2*a^2*Sqrt[c]*d*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2 
] - #1]*#1 + c^2*d^2*e*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2 - a^2 
*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2)/(a*Sqrt[c]*e + 4*c*d 
*#1 - 2*a*f*#1 - 3*Sqrt[c]*e*#1^2 + 2*f*#1^3) & ])/(d^2*f*x))
 

Rubi [A] (verified)

Time = 2.62 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {7279, 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.

\(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx\)

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^{3/2} e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^2}-\frac {\left (a^2 f^2 \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )+4 a c d^2 f^2+c^2 d^2 \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (a^2 f^2 \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )+4 a c d^2 f^2+c^2 d^2 \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )\right ) \text {arctanh}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) (2 c d-3 a f)}{2 d f}+\frac {3 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 d}-\frac {a e \sqrt {a+c x^2}}{d^2}+\frac {\sqrt {a+c x^2} (2 a e-c d x)}{2 d^2}-\frac {\left (a+c x^2\right )^{3/2}}{d x}+\frac {3 c x \sqrt {a+c x^2}}{2 d}\)

Input:

Int[(a + c*x^2)^(3/2)/(x^2*(d + e*x + f*x^2)),x]
 

Output:

-((a*e*Sqrt[a + c*x^2])/d^2) + (3*c*x*Sqrt[a + c*x^2])/(2*d) + ((2*a*e - c 
*d*x)*Sqrt[a + c*x^2])/(2*d^2) - (a + c*x^2)^(3/2)/(d*x) + (3*a*Sqrt[c]*Ar 
cTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*d) + (Sqrt[c]*(2*c*d - 3*a*f)*ArcTa 
nh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*d*f) - ((4*a*c*d^2*f^2 + c^2*d^2*(e^2 
- 2*d*f - e*Sqrt[e^2 - 4*d*f]) + a^2*f^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f 
]))*ArcTanh[(2*a*f - c*(e - Sqrt[e^2 - 4*d*f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + 
c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]*Sqrt[a + c*x^2])])/(Sqrt[2]*d^2*f*S 
qrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])]) + 
((4*a*c*d^2*f^2 + a^2*f^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + c^2*d^2*(e 
^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]))*ArcTanh[(2*a*f - c*(e + Sqrt[e^2 - 4*d* 
f])*x)/(Sqrt[2]*Sqrt[2*a*f^2 + c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f])]*Sqrt 
[a + c*x^2])])/(Sqrt[2]*d^2*f*Sqrt[e^2 - 4*d*f]*Sqrt[2*a*f^2 + c*(e^2 - 2* 
d*f + e*Sqrt[e^2 - 4*d*f])]) + (a^(3/2)*e*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]] 
)/d^2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(949\) vs. \(2(470)=940\).

Time = 2.22 (sec) , antiderivative size = 950, normalized size of antiderivative = 1.79

method	result	size
risch	\(-\frac {a \sqrt {c \,x^{2}+a}}{d x}+\frac {\frac {c^{\frac {3}{2}} d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{f}+\frac {\left (a^{2} f^{3} \sqrt {-4 d f +e^{2}}-2 a c d \,f^{2} \sqrt {-4 d f +e^{2}}+c^{2} d^{2} f \sqrt {-4 d f +e^{2}}-\sqrt {-4 d f +e^{2}}\, c^{2} d \,e^{2}+a^{2} e \,f^{3}+2 a c d \,f^{2} e -3 c^{2} d^{2} e f +c^{2} d \,e^{3}\right ) \sqrt {2}\, \ln \left (\frac {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}-\frac {c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}}\, \sqrt {4 c {\left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}^{2}-\frac {4 c \left (e -\sqrt {-4 d f +e^{2}}\right ) \left (x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {-2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 d f c +2 c \,e^{2}}{f^{2}}}}{2}}{x -\frac {-e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{f^{2} \sqrt {-4 d f +e^{2}}\, \left (-e +\sqrt {-4 d f +e^{2}}\right ) \sqrt {\frac {-\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}}}-\frac {\left (a^{2} f^{3} \sqrt {-4 d f +e^{2}}-2 a c d \,f^{2} \sqrt {-4 d f +e^{2}}+c^{2} d^{2} f \sqrt {-4 d f +e^{2}}-\sqrt {-4 d f +e^{2}}\, c^{2} d \,e^{2}-a^{2} e \,f^{3}-2 a c d \,f^{2} e +3 c^{2} d^{2} e f -c^{2} d \,e^{3}\right ) \sqrt {2}\, \ln \left (\frac {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}-\frac {c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {\sqrt {2}\, \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}}\, \sqrt {4 c {\left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}^{2}-\frac {4 c \left (e +\sqrt {-4 d f +e^{2}}\right ) \left (x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}\right )}{f}+\frac {2 \sqrt {-4 d f +e^{2}}\, c e +4 a \,f^{2}-4 d f c +2 c \,e^{2}}{f^{2}}}}{2}}{x +\frac {e +\sqrt {-4 d f +e^{2}}}{2 f}}\right )}{f^{2} \sqrt {-4 d f +e^{2}}\, \left (e +\sqrt {-4 d f +e^{2}}\right ) \sqrt {\frac {\sqrt {-4 d f +e^{2}}\, c e +2 a \,f^{2}-2 d f c +c \,e^{2}}{f^{2}}}}-\frac {4 f \,a^{\frac {3}{2}} e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{\left (-e +\sqrt {-4 d f +e^{2}}\right ) \left (e +\sqrt {-4 d f +e^{2}}\right )}}{d}\)	\(950\)
default	\(\text {Expression too large to display}\)	\(2494\)

Input:

int((c*x^2+a)^(3/2)/x^2/(f*x^2+e*x+d),x,method=_RETURNVERBOSE)
 

Output:

-a*(c*x^2+a)^(1/2)/d/x+1/d*(c^(3/2)*d/f*ln(c^(1/2)*x+(c*x^2+a)^(1/2))+1/f^ 
2*(a^2*f^3*(-4*d*f+e^2)^(1/2)-2*a*c*d*f^2*(-4*d*f+e^2)^(1/2)+c^2*d^2*f*(-4 
*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c^2*d*e^2+a^2*e*f^3+2*a*c*d*f^2*e-3*c^2 
*d^2*e*f+c^2*d*e^3)/(-4*d*f+e^2)^(1/2)/(-e+(-4*d*f+e^2)^(1/2))*2^(1/2)/((- 
(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2)*ln(((-(-4*d*f+e^2 
)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f 
*(-e+(-4*d*f+e^2)^(1/2)))+1/2*2^(1/2)*((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2* 
d*f*c+c*e^2)/f^2)^(1/2)*(4*c*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2-4*c*(e-(- 
4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(-(-4*d*f+e^2)^(1/ 
2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2))/(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)) 
))-1/f^2*(a^2*f^3*(-4*d*f+e^2)^(1/2)-2*a*c*d*f^2*(-4*d*f+e^2)^(1/2)+c^2*d^ 
2*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c^2*d*e^2-a^2*e*f^3-2*a*c*d*f^2* 
e+3*c^2*d^2*e*f-c^2*d*e^3)/(-4*d*f+e^2)^(1/2)/(e+(-4*d*f+e^2)^(1/2))*2^(1/ 
2)/(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2)*ln((((-4*d*f 
+e^2)^(1/2)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1 
/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2- 
2*d*f*c+c*e^2)/f^2)^(1/2)*(4*c*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2-4*c*(e+( 
-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*((-4*d*f+e^2)^(1/2 
)*c*e+2*a*f^2-2*d*f*c+c*e^2)/f^2)^(1/2))/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)) 
-4*f*a^(3/2)*e/(-e+(-4*d*f+e^2)^(1/2))/(e+(-4*d*f+e^2)^(1/2))*ln((2*a+2...
 

Fricas [F(-1)]

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

integrate((c*x^2+a)^(3/2)/x^2/(f*x^2+e*x+d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((c*x**2+a)**(3/2)/x**2/(f*x**2+e*x+d),x)
 

Output:

Integral((a + c*x**2)**(3/2)/(x**2*(d + e*x + f*x**2)), x)
 

Maxima [F]

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

integrate((c*x^2+a)^(3/2)/x^2/(f*x^2+e*x+d),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + a)^(3/2)/((f*x^2 + e*x + d)*x^2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c*x^2+a)^(3/2)/x^2/(f*x^2+e*x+d),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((a + c*x^2)^(3/2)/(x^2*(d + e*x + f*x^2)),x)
 

Output:

int((a + c*x^2)^(3/2)/(x^2*(d + e*x + f*x^2)), x)
 

Reduce [F]

\[ \int \frac {\left (a+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\frac {-2 \sqrt {c \,x^{2}+a}\, a d f -\sqrt {a}\, \mathrm {log}\left (\sqrt {c \,x^{2}+a}-\sqrt {a}\right ) a e f x +\sqrt {a}\, \mathrm {log}\left (\sqrt {c \,x^{2}+a}+\sqrt {a}\right ) a e f x -\sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{2}+a}-\sqrt {c}\, x \right ) c \,d^{2} x +\sqrt {c}\, \mathrm {log}\left (\sqrt {c \,x^{2}+a}+\sqrt {c}\, x \right ) c \,d^{2} x -2 \left (\int \frac {\sqrt {c \,x^{2}+a}}{c f \,x^{4}+c e \,x^{3}+a f \,x^{2}+c d \,x^{2}+a e x +a d}d x \right ) a^{2} d \,f^{2} x +2 \left (\int \frac {\sqrt {c \,x^{2}+a}}{c f \,x^{4}+c e \,x^{3}+a f \,x^{2}+c d \,x^{2}+a e x +a d}d x \right ) a^{2} e^{2} f x +4 \left (\int \frac {\sqrt {c \,x^{2}+a}}{c f \,x^{4}+c e \,x^{3}+a f \,x^{2}+c d \,x^{2}+a e x +a d}d x \right ) a c \,d^{2} f x -2 \left (\int \frac {\sqrt {c \,x^{2}+a}}{c f \,x^{4}+c e \,x^{3}+a f \,x^{2}+c d \,x^{2}+a e x +a d}d x \right ) c^{2} d^{3} x +2 \left (\int \frac {\sqrt {c \,x^{2}+a}\, x}{c f \,x^{4}+c e \,x^{3}+a f \,x^{2}+c d \,x^{2}+a e x +a d}d x \right ) a^{2} e \,f^{2} x -2 \left (\int \frac {\sqrt {c \,x^{2}+a}\, x}{c f \,x^{4}+c e \,x^{3}+a f \,x^{2}+c d \,x^{2}+a e x +a d}d x \right ) c^{2} d^{2} e x}{2 d^{2} f x} \] Input:

int((c*x^2+a)^(3/2)/x^2/(f*x^2+e*x+d),x)
 

Output:

( - 2*sqrt(a + c*x**2)*a*d*f - sqrt(a)*log(sqrt(a + c*x**2) - sqrt(a))*a*e 
*f*x + sqrt(a)*log(sqrt(a + c*x**2) + sqrt(a))*a*e*f*x - sqrt(c)*log(sqrt( 
a + c*x**2) - sqrt(c)*x)*c*d**2*x + sqrt(c)*log(sqrt(a + c*x**2) + sqrt(c) 
*x)*c*d**2*x - 2*int(sqrt(a + c*x**2)/(a*d + a*e*x + a*f*x**2 + c*d*x**2 + 
 c*e*x**3 + c*f*x**4),x)*a**2*d*f**2*x + 2*int(sqrt(a + c*x**2)/(a*d + a*e 
*x + a*f*x**2 + c*d*x**2 + c*e*x**3 + c*f*x**4),x)*a**2*e**2*f*x + 4*int(s 
qrt(a + c*x**2)/(a*d + a*e*x + a*f*x**2 + c*d*x**2 + c*e*x**3 + c*f*x**4), 
x)*a*c*d**2*f*x - 2*int(sqrt(a + c*x**2)/(a*d + a*e*x + a*f*x**2 + c*d*x** 
2 + c*e*x**3 + c*f*x**4),x)*c**2*d**3*x + 2*int((sqrt(a + c*x**2)*x)/(a*d 
+ a*e*x + a*f*x**2 + c*d*x**2 + c*e*x**3 + c*f*x**4),x)*a**2*e*f**2*x - 2* 
int((sqrt(a + c*x**2)*x)/(a*d + a*e*x + a*f*x**2 + c*d*x**2 + c*e*x**3 + c 
*f*x**4),x)*c**2*d**2*e*x)/(2*d**2*f*x)