Integrand size = 37, antiderivative size = 180 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^4 d^4}+\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3}+\frac {2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac {2 (d+e x)^{7/2}}{7 c d}-\frac {2 \left (c d^2-a e^2\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}} \] Output:
2*(-a*e^2+c*d^2)^3*(e*x+d)^(1/2)/c^4/d^4+2/3*(-a*e^2+c*d^2)^2*(e*x+d)^(3/2 )/c^3/d^3+2/5*(-a*e^2+c*d^2)*(e*x+d)^(5/2)/c^2/d^2+2/7*(e*x+d)^(7/2)/c/d-2 *(-a*e^2+c*d^2)^(7/2)*arctanh(c^(1/2)*d^(1/2)*(e*x+d)^(1/2)/(-a*e^2+c*d^2) ^(1/2))/c^(9/2)/d^(9/2)
Time = 0.21 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \sqrt {d+e x} \left (-105 a^3 e^6+35 a^2 c d e^4 (10 d+e x)-7 a c^2 d^2 e^2 \left (58 d^2+16 d e x+3 e^2 x^2\right )+c^3 d^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )}{105 c^4 d^4}+\frac {2 \left (-c d^2+a e^2\right )^{7/2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{9/2} d^{9/2}} \] Input:
Integrate[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
Output:
(2*Sqrt[d + e*x]*(-105*a^3*e^6 + 35*a^2*c*d*e^4*(10*d + e*x) - 7*a*c^2*d^2 *e^2*(58*d^2 + 16*d*e*x + 3*e^2*x^2) + c^3*d^3*(176*d^3 + 122*d^2*e*x + 66 *d*e^2*x^2 + 15*e^3*x^3)))/(105*c^4*d^4) + (2*(-(c*d^2) + a*e^2)^(7/2)*Arc Tan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(c^(9/2)*d^(9 /2))
Time = 0.53 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {1121, 60, 60, 60, 60, 73, 221}
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.
| \(\displaystyle \int \frac {(d+e x)^{9/2}}{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \, dx\) |
| \(\Big \downarrow \) 1121 |
| \(\displaystyle \int \frac {(d+e x)^{7/2}}{a e+c d x}dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {(d+e x)^{5/2}}{a e+c d x}dx}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {(d+e x)^{3/2}}{a e+c d x}dx}{d}+\frac {2 (d+e x)^{5/2}}{5 c d}\right )}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {\sqrt {d+e x}}{a e+c d x}dx}{d}+\frac {2 (d+e x)^{3/2}}{3 c d}\right )}{d}+\frac {2 (d+e x)^{5/2}}{5 c d}\right )}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}}dx}{d}+\frac {2 \sqrt {d+e x}}{c d}\right )}{d}+\frac {2 (d+e x)^{3/2}}{3 c d}\right )}{d}+\frac {2 (d+e x)^{5/2}}{5 c d}\right )}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{-\frac {c d^2}{e}+\frac {c (d+e x) d}{e}+a e}d\sqrt {d+e x}}{d e}+\frac {2 \sqrt {d+e x}}{c d}\right )}{d}+\frac {2 (d+e x)^{3/2}}{3 c d}\right )}{d}+\frac {2 (d+e x)^{5/2}}{5 c d}\right )}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \sqrt {d+e x}}{c d}-\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} d^{3/2} \sqrt {c d^2-a e^2}}\right )}{d}+\frac {2 (d+e x)^{3/2}}{3 c d}\right )}{d}+\frac {2 (d+e x)^{5/2}}{5 c d}\right )}{d}+\frac {2 (d+e x)^{7/2}}{7 c d}\) |
Input:
Int[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
Output:
(2*(d + e*x)^(7/2))/(7*c*d) + ((d^2 - (a*e^2)/c)*((2*(d + e*x)^(5/2))/(5*c *d) + ((d^2 - (a*e^2)/c)*((2*(d + e*x)^(3/2))/(3*c*d) + ((d^2 - (a*e^2)/c) *((2*Sqrt[d + e*x])/(c*d) - (2*(d^2 - (a*e^2)/c)*ArcTanh[(Sqrt[c]*Sqrt[d]* Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt[c]*d^(3/2)*Sqrt[c*d^2 - a*e^2]) ))/d))/d))/d
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Time = 4.47 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.03
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (a \,e^{2}-c \,d^{2}\right )^{4} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )-2 \sqrt {e x +d}\, \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}\, \left (-\frac {176 d^{3} \left (\frac {15}{176} e^{3} x^{3}+\frac {3}{8} d \,e^{2} x^{2}+\frac {61}{88} d^{2} e x +d^{3}\right ) c^{3}}{105}+\frac {58 \left (\frac {3}{58} e^{2} x^{2}+\frac {8}{29} d e x +d^{2}\right ) e^{2} a \,d^{2} c^{2}}{15}-\frac {10 e^{4} \left (\frac {e x}{10}+d \right ) a^{2} d c}{3}+e^{6} a^{3}\right )}{d^{4} c^{4} \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\) | \(185\) |
| derivativedivides | \(-\frac {2 \left (-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}} c d}{3}+\left (a \,e^{2}-c \,d^{2}\right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {e x +d}\right )}{c^{4} d^{4}}+\frac {2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{c^{4} d^{4} \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\) | \(240\) |
| default | \(-\frac {2 \left (-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}} c d}{3}+\left (a \,e^{2}-c \,d^{2}\right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {e x +d}\right )}{c^{4} d^{4}}+\frac {2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{c^{4} d^{4} \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\) | \(240\) |
| risch | \(-\frac {2 \left (-15 c^{3} d^{3} e^{3} x^{3}+21 x^{2} a \,c^{2} d^{2} e^{4}-66 c^{3} d^{4} e^{2} x^{2}-35 x \,a^{2} c d \,e^{5}+112 x a \,c^{2} d^{3} e^{3}-122 c^{3} d^{5} e x +105 e^{6} a^{3}-350 d^{2} e^{4} a^{2} c +406 d^{4} e^{2} a \,c^{2}-176 d^{6} c^{3}\right ) \sqrt {e x +d}}{105 d^{4} c^{4}}+\frac {2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{c^{4} d^{4} \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\) | \(241\) |
Input:
int((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e),x,method=_RETURNVERBOS E)
Output:
2*((a*e^2-c*d^2)^4*arctan(c*d*(e*x+d)^(1/2)/(c*d*(a*e^2-c*d^2))^(1/2))-(e* x+d)^(1/2)*(c*d*(a*e^2-c*d^2))^(1/2)*(-176/105*d^3*(15/176*e^3*x^3+3/8*d*e ^2*x^2+61/88*d^2*e*x+d^3)*c^3+58/15*(3/58*e^2*x^2+8/29*d*e*x+d^2)*e^2*a*d^ 2*c^2-10/3*e^4*(1/10*e*x+d)*a^2*d*c+e^6*a^3))/(c*d*(a*e^2-c*d^2))^(1/2)/d^ 4/c^4
Time = 0.10 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.83 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\left [-\frac {105 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} + 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) - 2 \, {\left (15 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 406 \, a c^{2} d^{4} e^{2} + 350 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \, {\left (22 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (122 \, c^{3} d^{5} e - 112 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{105 \, c^{4} d^{4}}, -\frac {2 \, {\left (105 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (15 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 406 \, a c^{2} d^{4} e^{2} + 350 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \, {\left (22 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (122 \, c^{3} d^{5} e - 112 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}\right )}}{105 \, c^{4} d^{4}}\right ] \] Input:
integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fr icas")
Output:
[-1/105*(105*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*sqrt( (c*d^2 - a*e^2)/(c*d))*log((c*d*e*x + 2*c*d^2 - a*e^2 + 2*sqrt(e*x + d)*c* d*sqrt((c*d^2 - a*e^2)/(c*d)))/(c*d*x + a*e)) - 2*(15*c^3*d^3*e^3*x^3 + 17 6*c^3*d^6 - 406*a*c^2*d^4*e^2 + 350*a^2*c*d^2*e^4 - 105*a^3*e^6 + 3*(22*c^ 3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 + (122*c^3*d^5*e - 112*a*c^2*d^3*e^3 + 35 *a^2*c*d*e^5)*x)*sqrt(e*x + d))/(c^4*d^4), -2/105*(105*(c^3*d^6 - 3*a*c^2* d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*sqrt(-(c*d^2 - a*e^2)/(c*d))*arctan(- sqrt(e*x + d)*c*d*sqrt(-(c*d^2 - a*e^2)/(c*d))/(c*d^2 - a*e^2)) - (15*c^3* d^3*e^3*x^3 + 176*c^3*d^6 - 406*a*c^2*d^4*e^2 + 350*a^2*c*d^2*e^4 - 105*a^ 3*e^6 + 3*(22*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 + (122*c^3*d^5*e - 112*a* c^2*d^3*e^3 + 35*a^2*c*d*e^5)*x)*sqrt(e*x + d))/(c^4*d^4)]
Time = 161.60 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\begin {cases} \frac {2 \left (\frac {e \left (d + e x\right )^{\frac {7}{2}}}{7 c d} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- a e^{3} + c d^{2} e\right )}{5 c^{2} d^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{2} e^{5} - 2 a c d^{2} e^{3} + c^{2} d^{4} e\right )}{3 c^{3} d^{3}} + \frac {\sqrt {d + e x} \left (- a^{3} e^{7} + 3 a^{2} c d^{2} e^{5} - 3 a c^{2} d^{4} e^{3} + c^{3} d^{6} e\right )}{c^{4} d^{4}} + \frac {e \left (a e^{2} - c d^{2}\right )^{4} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e^{2} - c d^{2}}{c d}}} \right )}}{c^{5} d^{5} \sqrt {\frac {a e^{2} - c d^{2}}{c d}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {d^{\frac {5}{2}} \log {\left (x \right )}}{c} & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**(9/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
Output:
Piecewise((2*(e*(d + e*x)**(7/2)/(7*c*d) + (d + e*x)**(5/2)*(-a*e**3 + c*d **2*e)/(5*c**2*d**2) + (d + e*x)**(3/2)*(a**2*e**5 - 2*a*c*d**2*e**3 + c** 2*d**4*e)/(3*c**3*d**3) + sqrt(d + e*x)*(-a**3*e**7 + 3*a**2*c*d**2*e**5 - 3*a*c**2*d**4*e**3 + c**3*d**6*e)/(c**4*d**4) + e*(a*e**2 - c*d**2)**4*at an(sqrt(d + e*x)/sqrt((a*e**2 - c*d**2)/(c*d)))/(c**5*d**5*sqrt((a*e**2 - c*d**2)/(c*d))))/e, Ne(e, 0)), (d**(5/2)*log(x)/c, True))
Exception generated. \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="ma xima")
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(a*e^2-c*d^2>0)', see `assume?` f or more de
Time = 0.15 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.66 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{4} d^{4}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{6} d^{6} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{6} d^{7} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{6} d^{8} + 105 \, \sqrt {e x + d} c^{6} d^{9} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} a c^{5} d^{5} e^{2} - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{5} d^{6} e^{2} - 315 \, \sqrt {e x + d} a c^{5} d^{7} e^{2} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} c^{4} d^{4} e^{4} + 315 \, \sqrt {e x + d} a^{2} c^{4} d^{5} e^{4} - 105 \, \sqrt {e x + d} a^{3} c^{3} d^{3} e^{6}\right )}}{105 \, c^{7} d^{7}} \] Input:
integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="gi ac")
Output:
2*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e ^8)*arctan(sqrt(e*x + d)*c*d/sqrt(-c^2*d^3 + a*c*d*e^2))/(sqrt(-c^2*d^3 + a*c*d*e^2)*c^4*d^4) + 2/105*(15*(e*x + d)^(7/2)*c^6*d^6 + 21*(e*x + d)^(5/ 2)*c^6*d^7 + 35*(e*x + d)^(3/2)*c^6*d^8 + 105*sqrt(e*x + d)*c^6*d^9 - 21*( e*x + d)^(5/2)*a*c^5*d^5*e^2 - 70*(e*x + d)^(3/2)*a*c^5*d^6*e^2 - 315*sqrt (e*x + d)*a*c^5*d^7*e^2 + 35*(e*x + d)^(3/2)*a^2*c^4*d^4*e^4 + 315*sqrt(e* x + d)*a^2*c^4*d^5*e^4 - 105*sqrt(e*x + d)*a^3*c^3*d^3*e^6)/(c^7*d^7)
Time = 0.04 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.15 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2\,{\left (d+e\,x\right )}^{7/2}}{7\,c\,d}+\frac {2\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,c^3\,d^3}-\frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,\sqrt {d+e\,x}}{c^4\,d^4}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,{\left (a\,e^2-c\,d^2\right )}^{7/2}\,\sqrt {d+e\,x}}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}\right )\,{\left (a\,e^2-c\,d^2\right )}^{7/2}}{c^{9/2}\,d^{9/2}}-\frac {2\,\left (a\,e^2-c\,d^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,c^2\,d^2} \] Input:
int((d + e*x)^(9/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2),x)
Output:
(2*(d + e*x)^(7/2))/(7*c*d) + (2*(a*e^2 - c*d^2)^2*(d + e*x)^(3/2))/(3*c^3 *d^3) - (2*(a*e^2 - c*d^2)^3*(d + e*x)^(1/2))/(c^4*d^4) + (2*atan((c^(1/2) *d^(1/2)*(a*e^2 - c*d^2)^(7/2)*(d + e*x)^(1/2))/(a^4*e^8 + c^4*d^8 - 4*a*c ^3*d^6*e^2 - 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4))*(a*e^2 - c*d^2)^(7/2))/ (c^(9/2)*d^(9/2)) - (2*(a*e^2 - c*d^2)*(d + e*x)^(5/2))/(5*c^2*d^2)
Time = 0.23 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.41 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{3} e^{6}-6 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} c \,d^{2} e^{4}+6 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,c^{2} d^{4} e^{2}-2 \sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{3} d^{6}-2 \sqrt {e x +d}\, a^{3} c d \,e^{6}+\frac {20 \sqrt {e x +d}\, a^{2} c^{2} d^{3} e^{4}}{3}+\frac {2 \sqrt {e x +d}\, a^{2} c^{2} d^{2} e^{5} x}{3}-\frac {116 \sqrt {e x +d}\, a \,c^{3} d^{5} e^{2}}{15}-\frac {32 \sqrt {e x +d}\, a \,c^{3} d^{4} e^{3} x}{15}-\frac {2 \sqrt {e x +d}\, a \,c^{3} d^{3} e^{4} x^{2}}{5}+\frac {352 \sqrt {e x +d}\, c^{4} d^{7}}{105}+\frac {244 \sqrt {e x +d}\, c^{4} d^{6} e x}{105}+\frac {44 \sqrt {e x +d}\, c^{4} d^{5} e^{2} x^{2}}{35}+\frac {2 \sqrt {e x +d}\, c^{4} d^{4} e^{3} x^{3}}{7}}{c^{5} d^{5}} \] Input:
int((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x)
Output:
(2*(105*sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sq rt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*a**3*e**6 - 315*sqrt(d)*sqrt(c)*sqrt (a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*a**2*c*d**2*e**4 + 315*sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)*ata n((sqrt(d + e*x)*c*d)/(sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*a*c**2*d**4 *e**2 - 105*sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d) /(sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*c**3*d**6 - 105*sqrt(d + e*x)*a* *3*c*d*e**6 + 350*sqrt(d + e*x)*a**2*c**2*d**3*e**4 + 35*sqrt(d + e*x)*a** 2*c**2*d**2*e**5*x - 406*sqrt(d + e*x)*a*c**3*d**5*e**2 - 112*sqrt(d + e*x )*a*c**3*d**4*e**3*x - 21*sqrt(d + e*x)*a*c**3*d**3*e**4*x**2 + 176*sqrt(d + e*x)*c**4*d**7 + 122*sqrt(d + e*x)*c**4*d**6*e*x + 66*sqrt(d + e*x)*c** 4*d**5*e**2*x**2 + 15*sqrt(d + e*x)*c**4*d**4*e**3*x**3))/(105*c**5*d**5)