Integrand size = 25, antiderivative size = 257 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 (d+e x)} \, dx=-\frac {a \sqrt {a+b x+c x^2}}{2 d x^2}-\frac {(5 b d-4 a e) \sqrt {a+b x+c x^2}}{4 d^2 x}-\frac {\left (3 b^2 d^2+12 a c d^2-12 a b d e+8 a^2 e^2\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} d^3}+\frac {c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{e}-\frac {\left (c d^2-b d e+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d^3 e} \] Output:
-1/2*a*(c*x^2+b*x+a)^(1/2)/d/x^2-1/4*(-4*a*e+5*b*d)*(c*x^2+b*x+a)^(1/2)/d^ 2/x-1/8*(8*a^2*e^2-12*a*b*d*e+12*a*c*d^2+3*b^2*d^2)*arctanh(1/2*(b*x+2*a)/ a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(1/2)/d^3+c^(3/2)*arctanh(1/2*(2*c*x+b)/c^( 1/2)/(c*x^2+b*x+a)^(1/2))/e-(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^3/e
Time = 1.72 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 (d+e x)} \, dx=\frac {\frac {d (-2 a d-5 b d x+4 a e x) \sqrt {a+x (b+c x)}}{x^2}+\frac {8 \left (-c d^2+e (b d-a e)\right )^{3/2} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{e}+8 a^{3/2} e^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-\frac {3 d \left (b^2 d+4 a c d-4 a b e\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {4 c^{3/2} d^3 \log \left (e \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{e}}{4 d^3} \] Input:
Integrate[(a + b*x + c*x^2)^(3/2)/(x^3*(d + e*x)),x]
Output:
((d*(-2*a*d - 5*b*d*x + 4*a*e*x)*Sqrt[a + x*(b + c*x)])/x^2 + (8*(-(c*d^2) + e*(b*d - a*e))^(3/2)*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x) ])/Sqrt[-(c*d^2) + e*(b*d - a*e)]])/e + 8*a^(3/2)*e^2*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] - (3*d*(b^2*d + 4*a*c*d - 4*a*b*e)*ArcTan h[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/Sqrt[a] - (4*c^(3/2)*d^ 3*Log[e*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/e)/(4*d^3)
Leaf count is larger than twice the leaf count of optimal. \(672\) vs. \(2(257)=514\).
Time = 1.10 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.61, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle \int \left (-\frac {e^3 \left (a+b x+c x^2\right )^{3/2}}{d^3 (d+e x)}+\frac {e^2 \left (a+b x+c x^2\right )^{3/2}}{d^3 x}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{d^2 x^2}+\frac {\left (a+b x+c x^2\right )^{3/2}}{d x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^{3/2} e^2 \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^3}-\frac {b e^2 \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^3}+\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right )}{16 c^{3/2} d^3 e}-\frac {3 e \left (4 a c+b^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} d^2}-\frac {3 \left (4 a c+b^2\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} d}+\frac {3 \sqrt {a} b e \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 d^2}-\frac {\left (a e^2-b d e+c d^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{d^3 e}+\frac {3 b \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x (2 c d-b e)+8 c^2 d^2\right )}{8 c d^3}+\frac {e^2 \left (8 a c+b^2+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^3}+\frac {e \left (a+b x+c x^2\right )^{3/2}}{d^2 x}-\frac {3 e (3 b+2 c x) \sqrt {a+b x+c x^2}}{4 d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}\) |
Input:
Int[(a + b*x + c*x^2)^(3/2)/(x^3*(d + e*x)),x]
Output:
(-3*(b - 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d*x) - (3*e*(3*b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d^2) + (e^2*(b^2 + 8*a*c + 2*b*c*x)*Sqrt[a + b*x + c*x ^2])/(8*c*d^3) - ((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^3) - (a + b*x + c*x^2)^(3/2)/( 2*d*x^2) + (e*(a + b*x + c*x^2)^(3/2))/(d^2*x) - (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) + (3*Sqrt[a] *b*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*d^2) - (a^ (3/2)*e^2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d^3 + (3 *b*Sqrt[c]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*d) - (3*(b^2 + 4*a*c)*e*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])] )/(8*Sqrt[c]*d^2) - (b*(b^2 - 12*a*c)*e^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*S qrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^3) + ((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^3*e) - ((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^3*e)
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]))
Time = 1.60 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.39
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-4 a e x +5 b d x +2 a d \right )}{4 d^{2} x^{2}}+\frac {-\frac {\left (8 e^{2} a^{2}-12 a b d e +12 a \,d^{2} c +3 b^{2} d^{2}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}+\frac {8 \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} d \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {8 c^{\frac {3}{2}} d^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e}}{8 d^{2}}\) | \(356\) |
| default | \(\text {Expression too large to display}\) | \(1600\) |
Input:
int((c*x^2+b*x+a)^(3/2)/x^3/(e*x+d),x,method=_RETURNVERBOSE)
Output:
-1/4*(c*x^2+b*x+a)^(1/2)*(-4*a*e*x+5*b*d*x+2*a*d)/d^2/x^2+1/8/d^2*(-(8*a^2 *e^2-12*a*b*d*e+12*a*c*d^2+3*b^2*d^2)/d/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x ^2+b*x+a)^(1/2))/x)+8/e^2*(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))+8*c^(3 /2)*d^2/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 (d+e x)} \, dx=\text {Timed out} \] Input:
integrate((c*x^2+b*x+a)^(3/2)/x^3/(e*x+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 (d+e x)} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{3} \left (d + e x\right )}\, dx \] Input:
integrate((c*x**2+b*x+a)**(3/2)/x**3/(e*x+d),x)
Output:
Integral((a + b*x + c*x**2)**(3/2)/(x**3*(d + e*x)), x)
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 (d+e x)} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )} x^{3}} \,d x } \] Input:
integrate((c*x^2+b*x+a)^(3/2)/x^3/(e*x+d),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x + a)^(3/2)/((e*x + d)*x^3), x)
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 (d+e x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c*x^2+b*x+a)^(3/2)/x^3/(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
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 (d+e x)} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^3\,\left (d+e\,x\right )} \,d x \] Input:
int((a + b*x + c*x^2)^(3/2)/(x^3*(d + e*x)),x)
Output:
int((a + b*x + c*x^2)^(3/2)/(x^3*(d + e*x)), x)
Time = 0.53 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.53 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 (d+e x)} \, dx=\frac {8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (-2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) a^{2} e^{2} x^{2}-8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (-2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) a b d e \,x^{2}+8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (-2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) a c \,d^{2} x^{2}-8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) a^{2} e^{2} x^{2}+8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) a b d e \,x^{2}-8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) a c \,d^{2} x^{2}-4 \sqrt {c \,x^{2}+b x +a}\, a^{2} d^{2} e +8 \sqrt {c \,x^{2}+b x +a}\, a^{2} d \,e^{2} x -10 \sqrt {c \,x^{2}+b x +a}\, a b \,d^{2} e x +8 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} e^{3} x^{2}-12 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a b d \,e^{2} x^{2}+12 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a c \,d^{2} e \,x^{2}+3 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{2} d^{2} e \,x^{2}-8 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} e^{3} x^{2}+12 \sqrt {a}\, \mathrm {log}\left (x \right ) a b d \,e^{2} x^{2}-12 \sqrt {a}\, \mathrm {log}\left (x \right ) a c \,d^{2} e \,x^{2}-3 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{2} d^{2} e \,x^{2}+8 \sqrt {c}\, \mathrm {log}\left (-2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}-b -2 c x \right ) a c \,d^{3} x^{2}}{8 a \,d^{3} e \,x^{2}} \] Input:
int((c*x^2+b*x+a)^(3/2)/x^3/(e*x+d),x)
Output:
(8*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**2*e**2*x**2 - 8*s qrt(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*b*d*e*x**2 + 8*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*c*d**2*x**2 - 8*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*a**2*e**2*x**2 + 8*sqrt(a*e**2 - b*d*e + c* d**2)*log(d + e*x)*a*b*d*e*x**2 - 8*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*a*c*d**2*x**2 - 4*sqrt(a + b*x + c*x**2)*a**2*d**2*e + 8*sqrt(a + b*x + c*x**2)*a**2*d*e**2*x - 10*sqrt(a + b*x + c*x**2)*a*b*d**2*e*x + 8*sqrt (a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*e**3*x**2 - 12* sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b*d*e**2*x**2 + 12*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*c*d**2*e* x**2 + 3*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**2*d* *2*e*x**2 - 8*sqrt(a)*log(x)*a**2*e**3*x**2 + 12*sqrt(a)*log(x)*a*b*d*e**2 *x**2 - 12*sqrt(a)*log(x)*a*c*d**2*e*x**2 - 3*sqrt(a)*log(x)*b**2*d**2*e*x **2 + 8*sqrt(c)*log( - 2*sqrt(c)*sqrt(a + b*x + c*x**2) - b - 2*c*x)*a*c*d **3*x**2)/(8*a*d**3*e*x**2)