\(\int \frac {(a+b x+c x^2)^{3/2}}{x^2 (d+e x)} \, dx\) [25]

Optimal result

Integrand size = 25, antiderivative size = 231 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 (d+e x)} \, dx=\frac {c \sqrt {a+b x+c x^2}}{e}-\frac {a \sqrt {a+b x+c x^2}}{d x}-\frac {\sqrt {a} (3 b d-2 a e) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 d^2}-\frac {\sqrt {c} (2 c d-3 b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 e^2}+\frac {\left (c d^2-b d e+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d^2 e^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.41 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 (d+e x)} \, dx=\frac {-4 \left (-c d^2+e (b d-a e)\right )^{3/2} x \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )+2 \sqrt {a} e^2 (-3 b d+2 a e) x \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+d \left (2 e (-a e+c d x) \sqrt {a+x (b+c x)}+\sqrt {c} d (2 c d-3 b e) x \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{2 d^2 e^2 x} \] Input:

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

Output:

(-4*(-(c*d^2) + e*(b*d - a*e))^(3/2)*x*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[ 
a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]] + 2*Sqrt[a]*e^2*(-3*b*d 
+ 2*a*e)*x*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]] + d*(2* 
e*(-(a*e) + c*d*x)*Sqrt[a + x*(b + c*x)] + Sqrt[c]*d*(2*c*d - 3*b*e)*x*Log 
[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(2*d^2*e^2*x)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(516\) vs. \(2(231)=462\).

Time = 0.97 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.23, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1289, 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.

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

\(\Big \downarrow \) 1289

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^{3/2} e \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}+\frac {b e \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right )}{16 c^{3/2} d^2 e^2}+\frac {3 \left (4 a c+b^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} d}+\frac {\left (a e^2-b d e+c d^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{d^2 e^2}-\frac {3 \sqrt {a} b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 d}+\frac {\sqrt {a+b x+c x^2} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x (2 c d-b e)+8 c^2 d^2\right )}{8 c d^2 e}-\frac {e \left (8 a c+b^2+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{d x}+\frac {3 (3 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}\)

Input:

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

Output:

(3*(3*b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d) - (e*(b^2 + 8*a*c + 2*b*c*x) 
*Sqrt[a + b*x + c*x^2])/(8*c*d^2) + ((8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 
 4*a*e) - 2*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*c*d^2*e) - (a + 
 b*x + c*x^2)^(3/2)/(d*x) - (3*Sqrt[a]*b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sq 
rt[a + b*x + c*x^2])])/(2*d) + (a^(3/2)*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*S 
qrt[a + b*x + c*x^2])])/d^2 + (3*(b^2 + 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt 
[c]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[c]*d) + (b*(b^2 - 12*a*c)*e*ArcTanh[( 
b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^2) - ((2*c*d 
- b*e)*(8*c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/( 
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^2*e^2) + ((c*d^2 - b*d*e 
+ a*e^2)^(3/2)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d 
*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(d^2*e^2)
 

Defintions of rubi rules used

rule 1289

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( 
IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(405\) vs. \(2(199)=398\).

Time = 1.54 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.76

method	result	size
risch	\(-\frac {a \sqrt {c \,x^{2}+b x +a}}{d x}+\frac {-\frac {2 \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{3} d \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {\sqrt {a}\, \left (2 a e -3 b d \right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d}+\frac {2 c d \left (c e \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+\frac {2 b e \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\sqrt {c}\, d \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )\right )}{e^{2}}}{2 d}\)	\(406\)
default	\(\text {Expression too large to display}\)	\(1097\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((c*x^2+b*x+a)^(3/2)/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:Error: Bad Argument Type
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 513, normalized size of antiderivative = 2.22 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 (d+e x)} \, dx=\frac {2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) a \,e^{2} x -2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) b d e x +2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) c \,d^{2} x -2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) a \,e^{2} x +2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) b d e x -2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) c \,d^{2} x -2 \sqrt {c \,x^{2}+b x +a}\, a d \,e^{2}+2 \sqrt {c \,x^{2}+b x +a}\, c \,d^{2} e x +2 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,e^{3} x -3 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b d \,e^{2} x -2 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,e^{3} x +3 \sqrt {a}\, \mathrm {log}\left (x \right ) b d \,e^{2} x +3 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) b \,d^{2} e x -2 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) c \,d^{3} x}{2 d^{2} e^{2} x} \] Input:

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

Output:

(2*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 
- b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*e**2*x - 2*sqrt(a*e** 
2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d 
**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*b*d*e*x + 2*sqrt(a*e**2 - b*d*e + c* 
d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + 
 b*d - b*e*x + 2*c*d*x)*c*d**2*x - 2*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + 
 e*x)*a*e**2*x + 2*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*b*d*e*x - 2* 
sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*c*d**2*x - 2*sqrt(a + b*x + c*x 
**2)*a*d*e**2 + 2*sqrt(a + b*x + c*x**2)*c*d**2*e*x + 2*sqrt(a)*log( - 2*s 
qrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*e**3*x - 3*sqrt(a)*log( - 2*s 
qrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b*d*e**2*x - 2*sqrt(a)*log(x)*a 
*e**3*x + 3*sqrt(a)*log(x)*b*d*e**2*x + 3*sqrt(c)*log( - 2*sqrt(c)*sqrt(a 
+ b*x + c*x**2) - b - 2*c*x)*b*d**2*e*x - 2*sqrt(c)*log( - 2*sqrt(c)*sqrt( 
a + b*x + c*x**2) - b - 2*c*x)*c*d**3*x)/(2*d**2*e**2*x)