\(\int \frac {(d+e x)^{9/2}}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx\) [2000]

Optimal result

Integrand size = 37, antiderivative size = 180 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^4 d^4}+\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3}+\frac {2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac {2 (d+e x)^{7/2}}{7 c d}-\frac {2 \left (c d^2-a e^2\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}} \]

[Out]

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)+2*(-a*e^2+c*d^2
)^3*(e*x+d)^(1/2)/c^4/d^4

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {640, 52, 65, 214} \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {2 \left (c d^2-a e^2\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac {2 \sqrt {d+e x} \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac {2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 c^3 d^3}+\frac {2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^2 d^2}+\frac {2 (d+e x)^{7/2}}{7 c d} \]

[In]

Int[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(2*(c*d^2 - a*e^2)^3*Sqrt[d + e*x])/(c^4*d^4) + (2*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2))/(3*c^3*d^3) + (2*(c*d^2
- a*e^2)*(d + e*x)^(5/2))/(5*c^2*d^2) + (2*(d + e*x)^(7/2))/(7*c*d) - (2*(c*d^2 - a*e^2)^(7/2)*ArcTanh[(Sqrt[c
]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(9/2)*d^(9/2))

Rule 52

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] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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]

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^{7/2}}{a e+c d x} \, dx \\ & = \frac {2 (d+e x)^{7/2}}{7 c d}+\frac {\left (c d^2-a e^2\right ) \int \frac {(d+e x)^{5/2}}{a e+c d x} \, dx}{c d} \\ & = \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 {\left (c d^2-a e^2\right )^2 \int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx}{c^2 d^2} \\ & = \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 {\left (c d^2-a e^2\right )^3 \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{c^3 d^3} \\ & = \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 {\left (c d^2-a e^2\right )^4 \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{c^4 d^4} \\ & = \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 {\left (2 \left (c d^2-a e^2\right )^4\right ) \text {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{c^4 d^4 e} \\ & = \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} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \sqrt {d+e x} \left (-105 a^3 e^6+35 a^2 c d e^4 (10 d+e x)-7 a c^2 d^2 e^2 \left (58 d^2+16 d e x+3 e^2 x^2\right )+c^3 d^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )}{105 c^4 d^4}+\frac {2 \left (-c d^2+a e^2\right )^{7/2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{9/2} d^{9/2}} \]

[In]

Integrate[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(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)*A
rcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/(c^(9/2)*d^(9/2))

Maple [A] (verified)

Time = 3.97 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.03

method	result	size
pseudoelliptic	\(\frac {2 \left (e^{2} a -c \,d^{2}\right )^{4} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )-2 \left (-\frac {176 \left (\frac {15}{176} e^{3} x^{3}+\frac {3}{8} d \,e^{2} x^{2}+\frac {61}{88} d^{2} e x +d^{3}\right ) d^{3} c^{3}}{105}+\frac {58 \left (\frac {3}{58} x^{2} e^{2}+\frac {8}{29} d e x +d^{2}\right ) e^{2} d^{2} a \,c^{2}}{15}-\frac {10 \left (\frac {e x}{10}+d \right ) e^{4} d \,a^{2} c}{3}+e^{6} a^{3}\right ) \sqrt {e x +d}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}{c^{4} d^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\)	\(185\)
derivativedivides	\(-\frac {2 \left (-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {\left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}} c d}{3}+\left (e^{2} a -c \,d^{2}\right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {e x +d}\right )}{c^{4} d^{4}}+\frac {2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{4} d^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\)	\(240\)
default	\(-\frac {2 \left (-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {\left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}} c d}{3}+\left (e^{2} a -c \,d^{2}\right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {e x +d}\right )}{c^{4} d^{4}}+\frac {2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{4} d^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\)	\(240\)
risch	\(-\frac {2 \left (-15 x^{3} c^{3} d^{3} e^{3}+21 x^{2} a \,c^{2} d^{2} e^{4}-66 x^{2} c^{3} d^{4} e^{2}-35 x \,a^{2} c d \,e^{5}+112 x a \,c^{2} d^{3} e^{3}-122 x \,c^{3} d^{5} e +105 e^{6} a^{3}-350 d^{2} e^{4} a^{2} c +406 d^{4} e^{2} c^{2} a -176 c^{3} d^{6}\right ) \sqrt {e x +d}}{105 c^{4} d^{4}}+\frac {2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{4} d^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\)	\(241\)

[In]

int((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x,method=_RETURNVERBOSE)

[Out]

2/((a*e^2-c*d^2)*c*d)^(1/2)*((a*e^2-c*d^2)^4*arctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2))-(-176/105*(15
/176*e^3*x^3+3/8*d*e^2*x^2+61/88*d^2*e*x+d^3)*d^3*c^3+58/15*(3/58*x^2*e^2+8/29*d*e*x+d^2)*e^2*d^2*a*c^2-10/3*(
1/10*e*x+d)*e^4*d*a^2*c+e^6*a^3)*(e*x+d)^(1/2)*((a*e^2-c*d^2)*c*d)^(1/2))/c^4/d^4

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.83 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\left [\frac {105 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, {\left (15 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 406 \, a c^{2} d^{4} e^{2} + 350 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \, {\left (22 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (122 \, c^{3} d^{5} e - 112 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{105 \, c^{4} d^{4}}, -\frac {2 \, {\left (105 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (15 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 406 \, a c^{2} d^{4} e^{2} + 350 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \, {\left (22 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (122 \, c^{3} d^{5} e - 112 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}\right )}}{105 \, c^{4} d^{4}}\right ] \]

[In]

integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas")

[Out]

[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 + 1
76*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 - 1
05*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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(9/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.66 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{4} d^{4}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{6} d^{6} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{6} d^{7} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{6} d^{8} + 105 \, \sqrt {e x + d} c^{6} d^{9} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} a c^{5} d^{5} e^{2} - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{5} d^{6} e^{2} - 315 \, \sqrt {e x + d} a c^{5} d^{7} e^{2} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} c^{4} d^{4} e^{4} + 315 \, \sqrt {e x + d} a^{2} c^{4} d^{5} e^{4} - 105 \, \sqrt {e x + d} a^{3} c^{3} d^{3} e^{6}\right )}}{105 \, c^{7} d^{7}} \]

[In]

integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

[Out]

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*sqr
t(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)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.15 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2\,{\left (d+e\,x\right )}^{7/2}}{7\,c\,d}+\frac {2\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,c^3\,d^3}-\frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,\sqrt {d+e\,x}}{c^4\,d^4}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,{\left (a\,e^2-c\,d^2\right )}^{7/2}\,\sqrt {d+e\,x}}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}\right )\,{\left (a\,e^2-c\,d^2\right )}^{7/2}}{c^{9/2}\,d^{9/2}}-\frac {2\,\left (a\,e^2-c\,d^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,c^2\,d^2} \]

[In]

int((d + e*x)^(9/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2),x)

[Out]

(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)