\(\int \frac {(a+b x+c x^2)^{3/2}}{x^2 (d-f x^2)} \, dx\) [56]

Optimal result

Integrand size = 28, antiderivative size = 463 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d-f x^2\right )} \, dx=\frac {3 (3 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {\left (a+b x+c x^2\right )^{3/2}}{d x}-\frac {3 \sqrt {a} b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 d}+\frac {3 \left (b^2+4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} d}-\frac {\left (8 c^2 d+3 b^2 f+12 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} d f}+\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} f}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^{3/2} f} \] Output:

3/4*(2*c*x+3*b)*(c*x^2+b*x+a)^(1/2)/d-1/4*(2*c*x+5*b)*(c*x^2+b*x+a)^(1/2)/ 
d-(c*x^2+b*x+a)^(3/2)/d/x-3/2*a^(1/2)*b*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x 
^2+b*x+a)^(1/2))/d+3/8*(4*a*c+b^2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b* 
x+a)^(1/2))/c^(1/2)/d-1/8*(12*a*c*f+3*b^2*f+8*c^2*d)*arctanh(1/2*(2*c*x+b) 
/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(1/2)/d/f+1/2*(c*d-b*d^(1/2)*f^(1/2)+a*f)^ 
(3/2)*arctanh(1/2*(b*d^(1/2)-2*a*f^(1/2)+(2*c*d^(1/2)-b*f^(1/2))*x)/(c*d-b 
*d^(1/2)*f^(1/2)+a*f)^(1/2)/(c*x^2+b*x+a)^(1/2))/d^(3/2)/f+1/2*(c*d+b*d^(1 
/2)*f^(1/2)+a*f)^(3/2)*arctanh(1/2*(b*d^(1/2)+2*a*f^(1/2)+(2*c*d^(1/2)+b*f 
^(1/2))*x)/(c*d+b*d^(1/2)*f^(1/2)+a*f)^(1/2)/(c*x^2+b*x+a)^(1/2))/d^(3/2)/ 
f
 

Mathematica [C] (verified)

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

Time = 0.94 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d-f x^2\right )} \, dx=\frac {-2 a f \sqrt {a+x (b+c x)}+6 \sqrt {a} b f x \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+2 c^{3/2} d x \log \left (f \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )-x \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+b^3 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^2 b f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{5/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 b^2 \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a c^{3/2} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^2 \sqrt {c} f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 b c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a b f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 d f x} \] Input:

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

Output:

(-2*a*f*Sqrt[a + x*(b + c*x)] + 6*Sqrt[a]*b*f*x*ArcTanh[(Sqrt[c]*x - Sqrt[ 
a + x*(b + c*x)])/Sqrt[a]] + 2*c^(3/2)*d*x*Log[f*(b + 2*c*x - 2*Sqrt[c]*Sq 
rt[a + x*(b + c*x)])] - x*RootSum[b^2*d - a^2*f - 4*b*Sqrt[c]*d*#1 + 4*c*d 
*#1^2 + 2*a*f*#1^2 - f*#1^4 & , (b*c^2*d^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x 
 + c*x^2] - #1] + b^3*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 
 a^2*b*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*c^(5/2)*d^2* 
Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*b^2*Sqrt[c]*d*f*Log[ 
-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 4*a*c^(3/2)*d*f*Log[-(Sqrt 
[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*a^2*Sqrt[c]*f^2*Log[-(Sqrt[c]* 
x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + 2*b*c*d*f*Log[-(Sqrt[c]*x) + Sqrt[a 
+ b*x + c*x^2] - #1]*#1^2 + 2*a*b*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c* 
x^2] - #1]*#1^2)/(b*Sqrt[c]*d - 2*c*d*#1 - a*f*#1 + f*#1^3) & ])/(2*d*f*x)
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {7276, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+3 b^2 f+8 c^2 d\right )}{8 \sqrt {c} d f}+\frac {3 \left (4 a c+b^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} d}+\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^{3/2} f}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^{3/2} f}-\frac {3 \sqrt {a} b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {\left (a+b x+c x^2\right )^{3/2}}{d x}+\frac {3 (3 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {(5 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}\)

Input:

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

Output:

(3*(3*b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d) - ((5*b + 2*c*x)*Sqrt[a + b* 
x + c*x^2])/(4*d) - (a + b*x + c*x^2)^(3/2)/(d*x) - (3*Sqrt[a]*b*ArcTanh[( 
2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*d) + (3*(b^2 + 4*a*c)*Ar 
cTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*d) - ((8* 
c^2*d + 3*b^2*f + 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + 
c*x^2])])/(8*Sqrt[c]*d*f) + ((c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh 
[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*S 
qrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^(3/2)*f) + ((c*d + b*S 
qrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sqrt[d 
] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c* 
x^2])])/(2*d^(3/2)*f)
 

Defintions of rubi rules used

rule 2009

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

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 2.95 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.25

method	result	size
risch	\(-\frac {a \sqrt {c \,x^{2}+b x +a}}{d x}-\frac {\frac {\left (\sqrt {d f}\, a^{2} f^{2}+2 \sqrt {d f}\, a c d f +\sqrt {d f}\, b^{2} d f +\sqrt {d f}\, c^{2} d^{2}-2 d \,f^{2} a b -2 d^{2} f b c \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{f^{2} d \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}-\frac {\left (\sqrt {d f}\, a^{2} f^{2}+2 \sqrt {d f}\, a c d f +\sqrt {d f}\, b^{2} d f +\sqrt {d f}\, c^{2} d^{2}+2 d \,f^{2} a b +2 d^{2} f b c \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{f^{2} d \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}+3 \sqrt {a}\, b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )+\frac {2 c^{\frac {3}{2}} d \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{f}}{2 d}\)	\(581\)
default	\(\text {Expression too large to display}\)	\(1787\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

-Integral(a*sqrt(a + b*x + c*x**2)/(-d*x**2 + f*x**4), x) - Integral(b*x*s 
qrt(a + b*x + c*x**2)/(-d*x**2 + f*x**4), x) - Integral(c*x**2*sqrt(a + b* 
x + c*x**2)/(-d*x**2 + f*x**4), x)
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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