Integrand size = 28, antiderivative size = 501 \[ \int \frac {x^3 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=-\frac {3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^3 f}-\frac {d \left (8 c^2 d+b^2 f+8 a c f+2 b c f x\right ) \sqrt {a+b x+c x^2}}{8 c f^3}-\frac {d \left (a+b x+c x^2\right )^{3/2}}{3 f^2}+\frac {b (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2 f}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 c f}+\frac {3 b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2} f}-\frac {b d \left (24 c^2 d-b^2 f+12 a c f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} f^3}-\frac {d \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{7/2}}+\frac {d \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^{7/2}} \]
-1/3*d*(c*x^2+b*x+a)^(3/2)/f^2+1/16*b*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^2/f- 1/5*(c*x^2+b*x+a)^(5/2)/c/f+3/256*b*(-4*a*c+b^2)^2*arctanh(1/2*(2*c*x+b)/c ^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)/f-1/16*b*d*(12*a*c*f-b^2*f+24*c^2*d)*a rctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/f^3-1/2*d*arctan h(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)/f^( 7/2)+1/2*d*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)/f^(7/2)-3/128*b*(-4*a*c+b^2)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c ^3/f-1/8*d*(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/f^3
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 2.02 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.47 \[ \int \frac {x^3 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\frac {-2 \sqrt {c} \sqrt {a+x (b+c x)} \left (45 b^4 f^2-30 b^2 c f^2 (10 a+b x)+16 c^3 f \left (160 a d+70 b d x+48 a f x^2+33 b f x^3\right )+128 c^4 \left (15 d^2+5 d f x^2+3 f^2 x^4\right )+24 c^2 f \left (16 a^2 f+7 a b f x+b^2 \left (10 d+f x^2\right )\right )\right )-15 b \left (-384 c^4 d^2-192 a c^3 d f+3 b^4 f^2-24 a b^2 c f^2+16 c^2 f \left (b^2 d+3 a^2 f\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )-1920 c^{7/2} d \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {2 b^2 c d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 a^2 c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^3 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-4 b c^{3/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a b \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+b^2 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a^2 f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{3840 c^{7/2} f^3} \]
(-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(45*b^4*f^2 - 30*b^2*c*f^2*(10*a + b*x) + 16*c^3*f*(160*a*d + 70*b*d*x + 48*a*f*x^2 + 33*b*f*x^3) + 128*c^4*(15*d^ 2 + 5*d*f*x^2 + 3*f^2*x^4) + 24*c^2*f*(16*a^2*f + 7*a*b*f*x + b^2*(10*d + f*x^2))) - 15*b*(-384*c^4*d^2 - 192*a*c^3*d*f + 3*b^4*f^2 - 24*a*b^2*c*f^2 + 16*c^2*f*(b^2*d + 3*a^2*f))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c *x)]] - 1920*c^(7/2)*d*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*L og[-(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) + Sqrt[a + b*x + c*x^2] - #1]*#1^2)/(b*Sqrt[c]*d - 2*c*d*#1 - a*f*#1 + f*#1^3) & ])/(3840*c^(7/2)*f^3)
Time = 1.62 (sec) , antiderivative size = 501, 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.
| \(\displaystyle \int \frac {x^3 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {d x \left (a+b x+c x^2\right )^{3/2}}{f \left (d-f x^2\right )}-\frac {x \left (a+b x+c x^2\right )^{3/2}}{f}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{7/2} f}-\frac {b d \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (12 a c f+b^2 (-f)+24 c^2 d\right )}{16 c^{3/2} f^3}-\frac {d \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 f^{7/2}}+\frac {d \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 f^{7/2}}-\frac {3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^3 f}-\frac {d \sqrt {a+b x+c x^2} \left (8 a c f+b^2 f+2 b c f x+8 c^2 d\right )}{8 c f^3}+\frac {b (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{16 c^2 f}-\frac {d \left (a+b x+c x^2\right )^{3/2}}{3 f^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{5 c f}\) |
(-3*b*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(128*c^3*f) - (d*(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*f^3) - ( d*(a + b*x + c*x^2)^(3/2))/(3*f^2) + (b*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2 ))/(16*c^2*f) - (a + b*x + c*x^2)^(5/2)/(5*c*f) + (3*b*(b^2 - 4*a*c)^2*Arc Tanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(7/2)*f) - (b* d*(24*c^2*d - b^2*f + 12*a*c*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b* x + c*x^2])])/(16*c^(3/2)*f^3) - (d*(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*f^(7/2)) + (d*(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*f^(7/2))
3.1.84.3.1 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 = 0.80 (sec) , antiderivative size = 758, normalized size of antiderivative = 1.51
| method | result | size |
| risch | \(-\frac {\left (384 c^{4} f^{2} x^{4}+528 b \,f^{2} c^{3} x^{3}+768 a \,c^{3} f^{2} x^{2}+24 b^{2} c^{2} f^{2} x^{2}+640 c^{4} d f \,x^{2}+168 a b \,c^{2} f^{2} x -30 b^{3} c \,f^{2} x +1120 b \,c^{3} d f x +384 a^{2} c^{2} f^{2}-300 a \,b^{2} c \,f^{2}+2560 a \,c^{3} d f +45 b^{4} f^{2}+240 b^{2} c^{2} d f +1920 c^{4} d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{3} f^{3}}+\frac {\frac {b \left (48 a^{2} c^{2} f^{2}-24 a \,b^{2} c \,f^{2}-192 a \,c^{3} d f +3 b^{4} f^{2}+16 b^{2} c^{2} d f -384 c^{4} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+\frac {128 c^{3} d \left (\sqrt {d f}\, a^{2} f^{2}+2 \sqrt {d f}\, a c d f +\sqrt {d f}\, b^{2} d f +\sqrt {d f}\, c^{2} d^{2}+2 a b d \,f^{2}+2 b c \,d^{2} f \right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}+\frac {128 c^{3} d \left (\sqrt {d f}\, a^{2} f^{2}+2 \sqrt {d f}\, a c d f +\sqrt {d f}\, b^{2} d f +\sqrt {d f}\, c^{2} d^{2}-2 a b d \,f^{2}-2 b c \,d^{2} f \right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}}{256 f^{3} c^{3}}\) | \(758\) |
| default | \(\text {Expression too large to display}\) | \(1607\) |
-1/1920*(384*c^4*f^2*x^4+528*b*c^3*f^2*x^3+768*a*c^3*f^2*x^2+24*b^2*c^2*f^ 2*x^2+640*c^4*d*f*x^2+168*a*b*c^2*f^2*x-30*b^3*c*f^2*x+1120*b*c^3*d*f*x+38 4*a^2*c^2*f^2-300*a*b^2*c*f^2+2560*a*c^3*d*f+45*b^4*f^2+240*b^2*c^2*d*f+19 20*c^4*d^2)*(c*x^2+b*x+a)^(1/2)/c^3/f^3+1/256/f^3/c^3*(b*(48*a^2*c^2*f^2-2 4*a*b^2*c*f^2-192*a*c^3*d*f+3*b^4*f^2+16*b^2*c^2*d*f-384*c^4*d^2)*ln((1/2* b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+128*c^3*d*((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*a*b*d*f^2 +2*b*c*d^2*f)/(d*f)^(1/2)/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))+128*c^ 3*d*((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*a*b*d*f^2-2*b*c*d^2*f)/(d*f)^(1/2)/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)))
Timed out. \[ \int \frac {x^3 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^3 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^3 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(((c*sqrt(4*d*f))/(2*f^2)>0)', se e `assume?
Exception generated. \[ \int \frac {x^3 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {x^3 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\int \frac {x^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d-f\,x^2} \,d x \]