Integrand size = 37, antiderivative size = 187 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {3}{4} \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 e (d+e x)}+\frac {3 \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {c} \sqrt {d} e^{5/2}} \]
1/2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e/(e*x+d)+3/8*(-a*e^2+c*d^2)^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))/e^(5/2)/c^(1/2)/d^(1/2)+3/4*(a-c*d^2/e^2)*(a*d *e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
Time = 0.34 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {\sqrt {e} (a e+c d x) (d+e x) \left (5 a e^2+c d (-3 d+2 e x)\right )+\frac {3 \left (c d^2-a e^2\right )^2 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {c} \sqrt {d}}}{4 e^{5/2} \sqrt {(a e+c d x) (d+e x)}} \]
(Sqrt[e]*(a*e + c*d*x)*(d + e*x)*(5*a*e^2 + c*d*(-3*d + 2*e*x)) + (3*(c*d^ 2 - a*e^2)^2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt [d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(Sqrt[c]*Sqrt[d]))/(4*e^(5/2)*Sqr t[(a*e + c*d*x)*(d + e*x)])
Time = 0.31 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1131, 1131, 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.
| \(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e (d+e x)}-\frac {3 \left (c d^2-a e^2\right ) \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{d+e x}dx}{4 e}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e (d+e x)}-\frac {3 \left (c d^2-a e^2\right ) \left (\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {\left (c d^2-a e^2\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 e}\right )}{4 e}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e (d+e x)}-\frac {3 \left (c d^2-a e^2\right ) \left (\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {\left (c d^2-a e^2\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{e}\right )}{4 e}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e (d+e x)}-\frac {3 \left (c d^2-a e^2\right ) \left (\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {\left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {c} \sqrt {d} e^{3/2}}\right )}{4 e}\) |
(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(2*e*(d + e*x)) - (3*(c*d^2 - a*e^2)*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/e - ((c*d^2 - a*e^2) *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])])/(2*Sqrt[c]*Sqrt[d]*e^(3/2))))/(4*e)
3.20.24.3.1 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)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1))) Int[(d + e*x)^(m + 1)*(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] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && Ne Q[m + 2*p + 1, 0] && IntegerQ[2*p]
Time = 2.80 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.66
| method | result | size |
| default | \(\frac {\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {6 c d e \left (\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (e^{2} a -c \,d^{2}\right )^{2} \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{2}\right )}{e^{2} a -c \,d^{2}}}{e^{2}}\) | \(311\) |
1/e^2*(2/(a*e^2-c*d^2)/(x+d/e)^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^( 5/2)-6*c*d*e/(a*e^2-c*d^2)*(1/3*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3 /2)+1/2*(a*e^2-c*d^2)*(1/4*(2*c*d*e*(x+d/e)+e^2*a-c*d^2)/c/d/e*(c*d*e*(x+d /e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/c/d/e*ln((1/2*e^2*a -1/2*c*d^2+c*d*e*(x+d/e))/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+ d/e))^(1/2))/(c*d*e)^(1/2))))
Time = 0.30 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.24 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^2} \, dx=\left [\frac {3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (2 \, c^{2} d^{2} e^{2} x - 3 \, c^{2} d^{3} e + 5 \, a c d e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, c d e^{3}}, -\frac {3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (2 \, c^{2} d^{2} e^{2} x - 3 \, c^{2} d^{3} e + 5 \, a c d e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, c d e^{3}}\right ] \]
[1/16*(3*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(c*d*e)*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 + 4*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(c*d*e) + 8*(c^2*d^3*e + a* c*d*e^3)*x) + 4*(2*c^2*d^2*e^2*x - 3*c^2*d^3*e + 5*a*c*d*e^3)*sqrt(c*d*e*x ^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c*d*e^3), -1/8*(3*(c^2*d^4 - 2*a*c*d^2*e ^2 + a^2*e^4)*sqrt(-c*d*e)*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(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^ 2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) - 2*(2*c^2*d^2*e^2*x - 3*c^2*d^3*e + 5 *a*c*d*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c*d*e^3)]
\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^2} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assume ?` for mor
Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (161) = 322\).
Time = 0.43 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.52 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^2} \, dx=-\frac {1}{4} \, {\left (\frac {3 \, {\left (c^{2} d^{4} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 2 \, a c d^{2} e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + a^{2} e^{4} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} \arctan \left (\frac {\sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}}}{\sqrt {-c d e}}\right )}{\sqrt {-c d e} e^{2} {\left | e \right |}} + \frac {3 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{3} d^{5} e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 6 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} a c^{2} d^{3} e^{3} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 3 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} a^{2} c d e^{5} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 5 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} c^{2} d^{4} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 10 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} a c d^{2} e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 5 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} a^{2} e^{4} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{{\left (\frac {c d^{2} e}{e x + d} - \frac {a e^{3}}{e x + d}\right )}^{2} e^{2} {\left | e \right |}}\right )} {\left | e \right |} \]
-1/4*(3*(c^2*d^4*sgn(1/(e*x + d))*sgn(e) - 2*a*c*d^2*e^2*sgn(1/(e*x + d))* sgn(e) + a^2*e^4*sgn(1/(e*x + d))*sgn(e))*arctan(sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))/sqrt(-c*d*e))/(sqrt(-c*d*e)*e^2*abs(e)) + (3*sqrt (c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^3*d^5*e*sgn(1/(e*x + d))*s gn(e) - 6*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a*c^2*d^3*e^3* sgn(1/(e*x + d))*sgn(e) + 3*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^2*c*d*e^5*sgn(1/(e*x + d))*sgn(e) - 5*(c*d*e - c*d^2*e/(e*x + d) + a *e^3/(e*x + d))^(3/2)*c^2*d^4*sgn(1/(e*x + d))*sgn(e) + 10*(c*d*e - c*d^2* e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a*c*d^2*e^2*sgn(1/(e*x + d))*sgn(e) - 5*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^2*e^4*sgn(1/(e*x + d))*sgn(e))/((c*d^2*e/(e*x + d) - a*e^3/(e*x + d))^2*e^2*abs(e)))*abs(e)
Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (d+e\,x\right )}^2} \,d x \]