Integrand size = 37, antiderivative size = 172 \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^3}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 e (d+e x)}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {e^{3/2} \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{5/2} d^{5/2}} \]
-2/3*(e*x+d)^3/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+e^(3/2)*arctanh (1/2*(2*c*d*e*x+a*e^2+c*d^2)/c^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)* x+c*d*e*x^2)^(1/2))/c^(5/2)/d^(5/2)-2*e*(e*x+d)/c^2/d^2/(a*d*e+(a*e^2+c*d^ 2)*x+c*d*e*x^2)^(1/2)
Time = 0.23 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {c} \sqrt {d} \left (3 a e^2+c d (d+4 e x)\right )}{(a e+c d x)^2}+\frac {3 e^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{3 c^{5/2} d^{5/2}} \]
(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[c]*Sqrt[d]*(3*a*e^2 + c*d*(d + 4 *e*x)))/(a*e + c*d*x)^2) + (3*e^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e* x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(3*c ^(5/2)*d^(5/2))
Time = 0.31 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1133, 1124, 1092, 219}
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.
\(\displaystyle \int \frac {(d+e x)^4}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1133 |
\(\displaystyle \frac {e \int \frac {(d+e x)^2}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{c d}-\frac {2 (d+e x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1124 |
\(\displaystyle \frac {e \left (\frac {e \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d}-\frac {2 (d+e x)}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c d}-\frac {2 (d+e x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {e \left (\frac {2 e \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{c d}-\frac {2 (d+e x)}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c d}-\frac {2 (d+e x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {e \left (\frac {\sqrt {e} \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c^{3/2} d^{3/2}}-\frac {2 (d+e x)}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{c d}-\frac {2 (d+e x)^3}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\) |
(-2*(d + e*x)^3)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + ( e*((-2*(d + e*x))/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (Sqr t[e]*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a *d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(c^(3/2)*d^(3/2))))/(c*d)
3.20.66.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))^(m_.)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[-2*e*(2*c*d - b*e)^(m - 2)*((d + e*x)/(c^(m - 1)*Sqrt[a + b*x + c*x^2])), x] + Simp[e^2/c^(m - 1) Int[(1/Sqrt[a + b*x + c*x^2])*Exp andToSum[((2*c*d - b*e)^(m - 1) - c^(m - 1)*(d + e*x)^(m - 1))/(c*d - b*e - c*e*x), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e ^2, 0] && IGtQ[m, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] - Simp[e^2*((m + p)/(c*(p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x ^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e ^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(2598\) vs. \(2(150)=300\).
Time = 3.37 (sec) , antiderivative size = 2599, normalized size of antiderivative = 15.11
d^4*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a *e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+16/3*c*d*e/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)^ 2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))+e^4*(-1 /3*x^3/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/2*(a*e^2+c*d^2)/c/d /e*(-x^2/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/2*(a*e^2+c*d^2)/c /d/e*(-1/2*x/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/4*(a*e^2+c*d^ 2)/c/d/e*(-1/3/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/2*(a*e^2+c* d^2)/c/d/e*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a *d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+16/3*c*d*e/(4*a*c*d^2*e^2-(a*e^2+c*d ^2)^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))) +1/2*a/c*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d *e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+16/3*c*d*e/(4*a*c*d^2*e^2-(a*e^2+c*d^2 )^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)))+2 *a/c*(-1/3/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-1/2*(a*e^2+c*d^2) /c/d/e*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e +(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+16/3*c*d*e/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^ 2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))))+1/ c/d/e*(-x/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^2)/ c/d/e*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/ e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2...
Time = 0.96 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.73 \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (c^{2} d^{2} e x^{2} + 2 \, a c d e^{2} x + a^{2} e^{3}\right )} \sqrt {\frac {e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \, {\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {e}{c d}}\right ) - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (4 \, c d e x + c d^{2} + 3 \, a e^{2}\right )}}{6 \, {\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}}, -\frac {3 \, {\left (c^{2} d^{2} e x^{2} + 2 \, a c d e^{2} x + a^{2} e^{3}\right )} \sqrt {-\frac {e}{c d}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-\frac {e}{c d}}}{2 \, {\left (c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x\right )}}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (4 \, c d e x + c d^{2} + 3 \, a e^{2}\right )}}{3 \, {\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}}\right ] \]
[1/6*(3*(c^2*d^2*e*x^2 + 2*a*c*d*e^2*x + a^2*e^3)*sqrt(e/(c*d))*log(8*c^2* d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3 )*x + 4*(2*c^2*d^2*e*x + c^2*d^3 + a*c*d*e^2)*sqrt(c*d*e*x^2 + a*d*e + (c* d^2 + a*e^2)*x)*sqrt(e/(c*d))) - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2 )*x)*(4*c*d*e*x + c*d^2 + 3*a*e^2))/(c^4*d^4*x^2 + 2*a*c^3*d^3*e*x + a^2*c ^2*d^2*e^2), -1/3*(3*(c^2*d^2*e*x^2 + 2*a*c*d*e^2*x + a^2*e^3)*sqrt(-e/(c* d))*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c* d^2 + a*e^2)*sqrt(-e/(c*d))/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(4*c*d*e*x + c*d^2 + 3*a* e^2))/(c^4*d^4*x^2 + 2*a*c^3*d^3*e*x + a^2*c^2*d^2*e^2)]
\[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{4}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?` f or more de
Exception generated. \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%{[%%%{1,[2,2,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[4,4] %%%}+%%%{
Timed out. \[ \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \]