Integrand size = 28, antiderivative size = 371 \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\frac {2 \left (a b \left (b^2 d-a (3 c d+a f)\right )+\left (b^4 d+2 a^2 c (c d+a f)-a b^2 (4 c d+a f)\right ) x\right )}{c \left (b^2-4 a c\right ) \left (b^2 d f-(c d+a f)^2\right ) \sqrt {a+b x+c x^2}}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} f}+\frac {d^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2}}+\frac {d^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2}} \] Output:
2*(a*b*(b^2*d-a*(a*f+3*c*d))+(b^4*d+2*a^2*c*(a*f+c*d)-a*b^2*(a*f+4*c*d))*x )/c/(-4*a*c+b^2)/(b^2*d*f-(a*f+c*d)^2)/(c*x^2+b*x+a)^(1/2)-arctanh(1/2*(2* c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/f+1/2*d^(3/2)*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/(c*d-b*d^(1/2)*f^(1/2)+a*f)^(3/2)+1/2*d^(3/2) *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/(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 = 1.71 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.22 \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\frac {2 \left (b^4 d x+a b^2 d (b-4 c x)-a^3 f (b-2 c x)-a^2 \left (3 b c d-2 c^2 d x+b^2 f x\right )\right )}{c \left (-b^2+4 a c\right ) \left (c^2 d^2+2 a c d f+f \left (-b^2 d+a^2 f\right )\right ) \sqrt {a+x (b+c x)}}+\frac {\log \left (c f \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{c^{3/2} f}-\frac {d^2 \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b c d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+2 a b f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{3/2} d \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a \sqrt {c} f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-b f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{2 f \left (c^2 d^2+2 a c d f+f \left (-b^2 d+a^2 f\right )\right )} \] Input:
Integrate[x^4/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]
Output:
(2*(b^4*d*x + a*b^2*d*(b - 4*c*x) - a^3*f*(b - 2*c*x) - a^2*(3*b*c*d - 2*c ^2*d*x + b^2*f*x)))/(c*(-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)]) + Log[c*f*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]/(c^(3/2)*f) - (d^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 & , (b*c*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + 2*a*b*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - # 1] - 2*c^(3/2)*d*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*a*S qrt[c]*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - b*f*Log[-(Sqr t[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*f*(c^2*d^2 + 2*a*c*d*f + f*(-(b^2*d) + a^2*f)))
Time = 1.36 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.26, 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.
\(\displaystyle \int \frac {x^4}{\left (d-f x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {d^2}{f^2 \left (d-f x^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {d}{f^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {x^2}{f \left (a+b x+c x^2\right )^{3/2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} f}+\frac {d^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 f \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2}}+\frac {d^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 f \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2}}-\frac {2 d^2 \left (b \left (b^2 f-c (3 a f+c d)\right )-c x \left (2 a c f+b^2 (-f)+2 c^2 d\right )\right )}{f^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (b^2 d f-(a f+c d)^2\right )}+\frac {2 d (b+2 c x)}{f^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 b \sqrt {a+b x+c x^2}}{c f \left (b^2-4 a c\right )}-\frac {2 x (2 a+b x)}{f \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
Input:
Int[x^4/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)),x]
Output:
(-2*x*(2*a + b*x))/((b^2 - 4*a*c)*f*Sqrt[a + b*x + c*x^2]) + (2*d*(b + 2*c *x))/((b^2 - 4*a*c)*f^2*Sqrt[a + b*x + c*x^2]) - (2*d^2*(b*(b^2*f - c*(c*d + 3*a*f)) - c*(2*c^2*d - b^2*f + 2*a*c*f)*x))/((b^2 - 4*a*c)*f^2*(b^2*d*f - (c*d + a*f)^2)*Sqrt[a + b*x + c*x^2]) + (2*b*Sqrt[a + b*x + c*x^2])/(c* (b^2 - 4*a*c)*f) - ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])]/ (c^(3/2)*f) + (d^(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*(c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)) + (d^(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]*Sq rt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*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. \(1063\) vs. \(2(305)=610\).
Time = 2.57 (sec) , antiderivative size = 1064, normalized size of antiderivative = 2.87
Input:
int(x^4/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x,method=_RETURNVERBOSE)
Output:
-2*d/f^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-1/f*(-x/c/(c*x^2+b*x+a) ^(1/2)-1/2*b/c*(-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/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))-1/2/f ^2*d^2/(d*f)^(1/2)*(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/f^2*d^2/(d*f)^(1/2)*(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...
Timed out. \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(x^4/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=- \int \frac {x^{4}}{- a d \sqrt {a + b x + c x^{2}} + a f x^{2} \sqrt {a + b x + c x^{2}} - b d x \sqrt {a + b x + c x^{2}} + b f x^{3} \sqrt {a + b x + c x^{2}} - c d x^{2} \sqrt {a + b x + c x^{2}} + c f x^{4} \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate(x**4/(c*x**2+b*x+a)**(3/2)/(-f*x**2+d),x)
Output:
-Integral(x**4/(-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. \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^4/(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?
Exception generated. \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^4/(c*x^2+b*x+a)^(3/2)/(-f*x^2+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. \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\int \frac {x^4}{\left (d-f\,x^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:
int(x^4/((d - f*x^2)*(a + b*x + c*x^2)^(3/2)),x)
Output:
int(x^4/((d - f*x^2)*(a + b*x + c*x^2)^(3/2)), x)
\[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d-f x^2\right )} \, dx=\int \frac {x^{4}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (-f \,x^{2}+d \right )}d x \] Input:
int(x^4/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x)
Output:
int(x^4/(c*x^2+b*x+a)^(3/2)/(-f*x^2+d),x)