Integrand size = 28, antiderivative size = 307 \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=-\frac {2 \left (a \left (b^2 d-2 a (c d+a f)\right )+b \left (b^2 d-a (3 c d+a f)\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}-\frac {d \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {f} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}+\frac {d \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {f} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2}} \] Output:
(-2*a*(b^2*d-2*a*(a*f+c*d))-2*b*(b^2*d-a*(a*f+3*c*d))*x)/(-4*a*c+b^2)/(b^2 *d*f-(a*f+c*d)^2)/(c*x^2+b*x+a)^(1/2)-1/2*d*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))/f^(1/2)/(c*d-b*d^(1/2)*f^(1/2)+a*f)^(3/2)+1/2*d*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))/f^(1/2)/(c*d+b*d^(1/2)*f^(1/2)+a*f)^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.10 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.34 \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\frac {8 a^3 f-4 b^3 d x-4 a b d (b-3 c x)+4 a^2 (2 c d+b f x)-\left (b^2-4 a c\right ) d \sqrt {a+x (b+c x)} \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b^2 d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+a^2 f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 b \sqrt {c} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 \left (b^2-4 a c\right ) \left (-c^2 d^2-2 a c d f+f \left (b^2 d-a^2 f\right )\right ) \sqrt {a+x (b+c x)}} \] Input:
Integrate[x^3/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]
Output:
(8*a^3*f - 4*b^3*d*x - 4*a*b*d*(b - 3*c*x) + 4*a^2*(2*c*d + b*f*x) - (b^2 - 4*a*c)*d*Sqrt[a + x*(b + c*x)]*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 & , (b^2*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + a*c*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + a^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*b*Sqrt[c]*d*Log [-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - c*d*Log[-(Sqrt[c]*x) + Sq rt[a + b*x + c*x^2] - #1]*#1^2 - a*f*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) & ])/(2*(b^2 - 4*a*c)*(-(c^2*d^2) - 2*a*c*d*f + f*(b^2*d - a^2*f))*Sqrt[a + x*(b + c*x)])
Time = 1.07 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.11, 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.
| \(\displaystyle \int \frac {x^3}{\left (d-f x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {d x}{f \left (d-f x^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {x}{f \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {f} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2}}+\frac {d \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {f} \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2}}-\frac {2 d \left (a \left (2 a c f+b^2 (-f)+2 c^2 d\right )+b c x (c d-a f)\right )}{f \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}-\frac {2 (2 a+b x)}{f \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
Input:
Int[x^3/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]
Output:
(-2*(2*a + b*x))/((b^2 - 4*a*c)*f*Sqrt[a + b*x + c*x^2]) - (2*d*(a*(2*c^2* d - b^2*f + 2*a*c*f) + b*c*(c*d - a*f)*x))/((b^2 - 4*a*c)*f*(b^2*d*f - (c* d + a*f)^2)*Sqrt[a + b*x + c*x^2]) - (d*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*Sqrt[f]*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (d *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*Sqrt[f]*(c*d + b *Sqrt[d]*Sqrt[f] + a*f)^(3/2))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(959\) vs. \(2(250)=500\).
Time = 2.48 (sec) , antiderivative size = 960, normalized size of antiderivative = 3.13
| method | result | size |
| default | \(-\frac {-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}}{f}-\frac {d \left (\frac {f}{\left (b \sqrt {d f}+a f +c d \right ) \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}}}-\frac {\left (2 c \sqrt {d f}+f b \right ) \left (2 c \left (x -\frac {\sqrt {d f}}{f}\right )+\frac {2 c \sqrt {d f}+f b}{f}\right )}{\left (b \sqrt {d f}+a f +c d \right ) \left (\frac {4 c \left (b \sqrt {d f}+a f +c d \right )}{f}-\frac {\left (2 c \sqrt {d f}+f b \right )^{2}}{f^{2}}\right ) \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}}}-\frac {f \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 )}{\left (b \sqrt {d f}+a f +c d \right ) \sqrt {\frac {b \sqrt {d f}+a f +c d}{f}}}\right )}{2 f^{2}}-\frac {d \left (\frac {f}{\left (-b \sqrt {d f}+a f +c d \right ) \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}}}-\frac {\left (-2 c \sqrt {d f}+f b \right ) \left (2 c \left (x +\frac {\sqrt {d f}}{f}\right )+\frac {-2 c \sqrt {d f}+f b}{f}\right )}{\left (-b \sqrt {d f}+a f +c d \right ) \left (\frac {4 c \left (-b \sqrt {d f}+a f +c d \right )}{f}-\frac {\left (-2 c \sqrt {d f}+f b \right )^{2}}{f^{2}}\right ) \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}}}-\frac {f \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 )}{\left (-b \sqrt {d f}+a f +c d \right ) \sqrt {\frac {-b \sqrt {d f}+a f +c d}{f}}}\right )}{2 f^{2}}\) | \(960\) |
Input:
int(x^3/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x,method=_RETURNVERBOSE)
Output:
-1/f*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/ 2))-1/2*d/f^2*(1/(b*(d*f)^(1/2)+a*f+c*d)*f/(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)-(2*c*(d *f)^(1/2)+f*b)/(b*(d*f)^(1/2)+a*f+c*d)*(2*c*(x-(d*f)^(1/2)/f)+(2*c*(d*f)^( 1/2)+f*b)/f)/(4*c*(b*(d*f)^(1/2)+a*f+c*d)/f-(2*c*(d*f)^(1/2)+f*b)^2/f^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)-1/(b*(d*f)^(1/2)+a*f+c*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)))-1/2*d/f^2*(f/(-b*(d*f)^(1/2)+a*f+c*d)/(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)-(-2*c*(d*f)^(1/2)+f*b)/(-b*(d*f)^(1/2)+a*f+c*d)*(2*c*(x+(d*f)^ (1/2)/f)+1/f*(-2*c*(d*f)^(1/2)+f*b))/(4*c/f*(-b*(d*f)^(1/2)+a*f+c*d)-1/f^2 *(-2*c*(d*f)^(1/2)+f*b)^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)-f/(-b*(d*f)^(1/2) +a*f+c*d)/(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)))
Leaf count of result is larger than twice the leaf count of optimal. 17339 vs. \(2 (249) = 498\).
Time = 14.40 (sec) , antiderivative size = 17339, normalized size of antiderivative = 56.48 \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate(x^3/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=- \int \frac {x^{3}}{- a d \sqrt {a + b x + c x^{2}} + a f x^{2} \sqrt {a + b x + c x^{2}} - b d x \sqrt {a + b x + c x^{2}} + b f x^{3} \sqrt {a + b x + c x^{2}} - c d x^{2} \sqrt {a + b x + c x^{2}} + c f x^{4} \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate(x**3/(c*x**2+b*x+a)**(3/2)/(-f*x**2+d),x)
Output:
-Integral(x**3/(-a*d*sqrt(a + b*x + c*x**2) + a*f*x**2*sqrt(a + b*x + c*x* *2) - b*d*x*sqrt(a + b*x + c*x**2) + b*f*x**3*sqrt(a + b*x + c*x**2) - c*d *x**2*sqrt(a + b*x + c*x**2) + c*f*x**4*sqrt(a + b*x + c*x**2)), x)
Exception generated. \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="maxima")
Output:
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?
Timed out. \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(x^3/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\int \frac {x^3}{\left (d-f\,x^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:
int(x^3/((d - f*x^2)*(a + b*x + c*x^2)^(3/2)),x)
Output:
int(x^3/((d - f*x^2)*(a + b*x + c*x^2)^(3/2)), x)
Time = 1.06 (sec) , antiderivative size = 25446, normalized size of antiderivative = 82.89 \[ \int \frac {x^3}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx =\text {Too large to display} \] Input:
int(x^3/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x)
Output:
( - 8*sqrt(d)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*atan((sqrt(d)*sqrt(a + b *x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a*b*f - 2*sqrt(d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a*c*f*x + sqrt(d)*sqrt (a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*b**2*f*x - sqrt(d)* sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*b*c*d - 2*sqrt( d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*c**2*d*x + 2 *sqrt(f)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a**2*f + sqrt(f)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a*b* f*x + 2*sqrt(f)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d) *a*c*d - sqrt(f)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d )*b**2*d - sqrt(f)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c *d)*b*c*d*x)/(2*a**3*f**2 + 2*a**2*b*f**2*x + 4*a**2*c*d*f + 2*a**2*c*f**2 *x**2 - 2*a*b**2*d*f + 4*a*b*c*d*f*x + 2*a*c**2*d**2 + 4*a*c**2*d*f*x**2 - 2*b**3*d*f*x - 2*b**2*c*d*f*x**2 + 2*b*c**2*d**2*x + 2*c**3*d**2*x**2))*a **3*b*c*d*f**2 + 2*sqrt(d)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*atan((sqrt( d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a*b*f - 2*sq rt(d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*a*c*f*x + sqrt(d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*b**2*f *x - sqrt(d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c*d)*b* c*d - 2*sqrt(d)*sqrt(a + b*x + c*x**2)*sqrt(sqrt(f)*sqrt(d)*b - a*f - c...