\(\int \frac {(a+b x+c x^2)^{3/2}}{x^3 (d-f x^2)} \, dx\) [57]

Optimal result

Integrand size = 28, antiderivative size = 632 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=-\frac {\left (8 c^2 d+b^2 f+8 a c f\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {b f x \sqrt {a+b x+c x^2}}{4 d^2}-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 \left (b^2+4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} d}-\frac {a^{3/2} f \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}+\frac {3 b \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {b \left (b^2-12 a c\right ) f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {b \left (24 c^2 d-b^2 f+12 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^2 \sqrt {f}}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^2 \sqrt {f}} \] Output:

-1/8*(8*a*c*f+b^2*f+8*c^2*d)*(c*x^2+b*x+a)^(1/2)/c/d^2-1/4*b*f*x*(c*x^2+b* 
x+a)^(1/2)/d^2-3/4*(-2*c*x+b)*(c*x^2+b*x+a)^(1/2)/d/x+1/8*f*(2*b*c*x+8*a*c 
+b^2)*(c*x^2+b*x+a)^(1/2)/c/d^2-1/2*(c*x^2+b*x+a)^(3/2)/d/x^2-3/8*(4*a*c+b 
^2)*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(1/2)/d-a^(3/2)*f 
*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/d^2+3/2*b*c^(1/2)*arct 
anh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/d-1/16*b*(-12*a*c+b^2)*f*ar 
ctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/d^2-1/16*b*(12*a* 
c*f-b^2*f+24*c^2*d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^( 
3/2)/d^2-1/2*(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^2/f^(1/2)+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^2/f^(1/2)
 

Mathematica [C] (verified)

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

Time = 1.46 (sec) , antiderivative size = 587, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=-\frac {\frac {d (2 a+5 b x) \sqrt {a+x (b+c x)}}{x^2}+\frac {\left (3 b^2 d+4 a (3 c d+2 a f)\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}}+2 \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {2 b^2 c d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 a^2 c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^3 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-4 b c^{3/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a b \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a^2 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{4 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 614, normalized size of antiderivative = 0.97, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^{3/2} f \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}-\frac {b f \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+b^2 (-f)+24 c^2 d\right )}{16 c^{3/2} d^2}-\frac {3 \left (4 a c+b^2\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} d}-\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^2 \sqrt {f}}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^2 \sqrt {f}}+\frac {3 b \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c d^2}+\frac {f \left (8 a c+b^2+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}\)

Input:

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

Output:

(-3*(b - 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d*x) + (f*(b^2 + 8*a*c + 2*b*c*x 
)*Sqrt[a + b*x + c*x^2])/(8*c*d^2) - ((8*c^2*d + b^2*f + 8*a*c*f + 2*b*c*f 
*x)*Sqrt[a + b*x + c*x^2])/(8*c*d^2) - (a + b*x + c*x^2)^(3/2)/(2*d*x^2) - 
 (3*(b^2 + 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/ 
(8*Sqrt[a]*d) - (a^(3/2)*f*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c 
*x^2])])/d^2 + (3*b*Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + 
c*x^2])])/(2*d) - (b*(b^2 - 12*a*c)*f*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[ 
a + b*x + c*x^2])])/(16*c^(3/2)*d^2) - (b*(24*c^2*d - b^2*f + 12*a*c*f)*Ar 
cTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^2) - ( 
(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*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a 
+ b*x + c*x^2])])/(2*d^2*Sqrt[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*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*d^2*Sqrt[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 = 2.66 (sec) , antiderivative size = 545, normalized size of antiderivative = 0.86

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (5 b x +2 a \right )}{4 d \,x^{2}}-\frac {\frac {4 \left (2 \sqrt {d f}\, a b f +2 \sqrt {d f}\, b c d -a^{2} f^{2}-2 a c d f -b^{2} d f -c^{2} d^{2}\right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x +\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+a f +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{d f \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}-\frac {4 \left (2 \sqrt {d f}\, a b f +2 \sqrt {d f}\, b c d +a^{2} f^{2}+2 a c d f +b^{2} d f +c^{2} d^{2}\right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 a f +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}\, \sqrt {c \left (x -\frac {\sqrt {d f}}{f}\right )^{2}+\frac {\left (2 c \sqrt {d f}+f b \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+a f +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{d f \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}+\frac {\left (8 f \,a^{2}+12 a c d +3 b^{2} d \right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}}{8 d}\)	\(545\)
default	\(\text {Expression too large to display}\)	\(2139\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=- \int \frac {a \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx - \int \frac {b x \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx - \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx \] Input:

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

Output:

-Integral(a*sqrt(a + b*x + c*x**2)/(-d*x**3 + f*x**5), x) - Integral(b*x*s 
qrt(a + b*x + c*x**2)/(-d*x**3 + f*x**5), x) - Integral(c*x**2*sqrt(a + b* 
x + c*x**2)/(-d*x**3 + f*x**5), x)
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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