Integrand size = 28, antiderivative size = 614 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}+\frac {f \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 \left (b^2+4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} d}-\frac {a^{3/2} f \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}+\frac {3 b \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {b \left (b^2-12 a c\right ) f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {b \left (24 c^2 d-b^2 f+12 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {\left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^2 \sqrt {f}}+\frac {\left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 d^2 \sqrt {f}} \]
-1/2*(c*x^2+b*x+a)^(3/2)/d/x^2-a^(3/2)*f*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c* x^2+b*x+a)^(1/2))/d^2-1/16*b*(-12*a*c+b^2)*f*arctanh(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^2-3/8*(4*a*c+b^2)*ar ctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/d/a^(1/2)+3/2*b*arctanh(1 /2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))*c^(1/2)/d-1/2*arctanh(1/2*(b*d^( 1/2)-2*a*f^(1/2)+x*(2*c*d^(1/2)-b*f^(1/2)))/(c*x^2+b*x+a)^(1/2)/(c*d+a*f-b *d^(1/2)*f^(1/2))^(1/2))*(c*d+a*f-b*d^(1/2)*f^(1/2))^(3/2)/d^2/f^(1/2)+1/2 *arctanh(1/2*(b*d^(1/2)+2*a*f^(1/2)+x*(2*c*d^(1/2)+b*f^(1/2)))/(c*x^2+b*x+ a)^(1/2)/(c*d+a*f+b*d^(1/2)*f^(1/2))^(1/2))*(c*d+a*f+b*d^(1/2)*f^(1/2))^(3 /2)/d^2/f^(1/2)-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^2-1/8*(2*b*c*f*x+8*a*c*f+b^2*f+8*c^2*d)*(c* x^2+b*x+a)^(1/2)/c/d^2
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.98 (sec) , antiderivative size = 587, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=-\frac {\frac {d (2 a+5 b x) \sqrt {a+x (b+c x)}}{x^2}+\frac {\left (3 b^2 d+4 a (3 c d+2 a f)\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}}+2 \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {2 b^2 c d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 a^2 c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^3 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-4 b c^{3/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a b \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a^2 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{4 d^2} \]
-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
Time = 1.73 (sec) , antiderivative size = 614, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle -\frac {a^{3/2} f \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}-\frac {b f \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}-\frac {b \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+b^2 (-f)+24 c^2 d\right )}{16 c^{3/2} d^2}-\frac {3 \left (4 a c+b^2\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a} d}-\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 d^2 \sqrt {f}}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 d^2 \sqrt {f}}+\frac {3 b \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 d}-\frac {\sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c d^2}+\frac {f \left (8 a c+b^2+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 d x^2}-\frac {3 (b-2 c x) \sqrt {a+b x+c x^2}}{4 d x}\) |
(-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])
3.1.90.3.1 Defintions of rubi rules used
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]
Time = 0.92 (sec) , antiderivative size = 549, normalized size of antiderivative = 0.89
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (5 b x +2 a \right )}{4 d \,x^{2}}-\frac {-\frac {\left (-8 a^{2} f -12 a c d -3 b^{2} d \right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}-\frac {\left (-8 \sqrt {d f}\, a b f -8 \sqrt {d f}\, b c d +4 a^{2} f^{2}+8 a c d f +4 b^{2} d f +4 c^{2} d^{2}\right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{d f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}-\frac {\left (8 \sqrt {d f}\, a b f +8 \sqrt {d f}\, b c d +4 a^{2} f^{2}+8 a c d f +4 b^{2} d f +4 c^{2} d^{2}\right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{d f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}}{8 d}\) | \(549\) |
| default | \(\text {Expression too large to display}\) | \(2139\) |
-1/4*(c*x^2+b*x+a)^(1/2)*(5*b*x+2*a)/d/x^2-1/8/d*(-(-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)-(-8*(d*f)^(1/ 2)*a*b*f-8*(d*f)^(1/2)*b*c*d+4*a^2*f^2+8*a*c*d*f+4*b^2*d*f+4*c^2*d^2)/d/f/ (1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*ln((2/f*(-b*(d*f)^(1/2)+f*a+c*d)+1/f* (-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+2*(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^ (1/2)*((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+ 1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2))/(x+(d*f)^(1/2)/f))-(8*(d*f)^(1/2)*a*b *f+8*(d*f)^(1/2)*b*c*d+4*a^2*f^2+8*a*c*d*f+4*b^2*d*f+4*c^2*d^2)/d/f/((b*(d *f)^(1/2)+f*a+c*d)/f)^(1/2)*ln((2*(b*(d*f)^(1/2)+f*a+c*d)/f+(2*c*(d*f)^(1/ 2)+b*f)/f*(x-(d*f)^(1/2)/f)+2*((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*((x-(d*f)^ (1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+(b*(d*f)^(1/2)+f*a+ c*d)/f)^(1/2))/(x-(d*f)^(1/2)/f)))
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=- \int \frac {a \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx - \int \frac {b x \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx - \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{- d x^{3} + f x^{5}}\, dx \]
-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)
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=\int { -\frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (f x^{2} - d\right )} x^{3}} \,d x } \]
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=\text {Exception raised: AttributeError} \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^3 \left (d-f x^2\right )} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^3\,\left (d-f\,x^2\right )} \,d x \]