\(\int \frac {(d+e x)^{9/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\) [2011]

Optimal result

Integrand size = 37, antiderivative size = 144 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {5 e \left (c d^2-a e^2\right ) \sqrt {d+e x}}{c^3 d^3}+\frac {5 e (d+e x)^{3/2}}{3 c^2 d^2}-\frac {(d+e x)^{5/2}}{c d (a e+c d x)}-\frac {5 e \left (c d^2-a e^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}} \]

[Out]

5/3*e*(e*x+d)^(3/2)/c^2/d^2-(e*x+d)^(5/2)/c/d/(c*d*x+a*e)-5*e*(-a*e^2+c*d^2)^(3/2)*arctanh(c^(1/2)*d^(1/2)*(e*
x+d)^(1/2)/(-a*e^2+c*d^2)^(1/2))/c^(7/2)/d^(7/2)+5*e*(-a*e^2+c*d^2)*(e*x+d)^(1/2)/c^3/d^3

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {5 e \left (c d^2-a e^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{7/2} d^{7/2}}+\frac {5 e \sqrt {d+e x} \left (c d^2-a e^2\right )}{c^3 d^3}-\frac {(d+e x)^{5/2}}{c d (a e+c d x)}+\frac {5 e (d+e x)^{3/2}}{3 c^2 d^2} \]

[In]

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

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {\sqrt {d+e x} \left (15 a^2 e^4+10 a c d e^2 (-2 d+e x)+c^2 d^2 \left (3 d^2-14 d e x-2 e^2 x^2\right )\right )}{3 c^3 d^3 (a e+c d x)}+\frac {5 e \left (-c d^2+a e^2\right )^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{7/2} d^{7/2}} \]

[In]

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

[Out]

-1/3*(Sqrt[d + e*x]*(15*a^2*e^4 + 10*a*c*d*e^2*(-2*d + e*x) + c^2*d^2*(3*d^2 - 14*d*e*x - 2*e^2*x^2)))/(c^3*d^
3*(a*e + c*d*x)) + (5*e*(-(c*d^2) + a*e^2)^(3/2)*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]
])/(c^(7/2)*d^(7/2))

Maple [A] (verified)

Time = 4.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.05

method	result	size
risch	\(-\frac {2 e \left (-x c d e +6 e^{2} a -7 c \,d^{2}\right ) \sqrt {e x +d}}{3 c^{3} d^{3}}+\frac {\left (2 a^{2} e^{4}-4 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right ) e \left (-\frac {\sqrt {e x +d}}{2 \left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )}+\frac {5 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{3} d^{3}}\)	\(151\)
pseudoelliptic	\(-\frac {5 \left (-e \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d x +a e \right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )+\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\, \sqrt {e x +d}\, \left (\frac {\left (-\frac {2}{3} x^{2} e^{2}-\frac {14}{3} d e x +d^{2}\right ) d^{2} c^{2}}{5}-\frac {4 \left (-\frac {e x}{2}+d \right ) e^{2} d a c}{3}+a^{2} e^{4}\right )\right )}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\, c^{3} d^{3} \left (c d x +a e \right )}\)	\(162\)
derivativedivides	\(2 e \left (-\frac {-\frac {c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a \,e^{2} \sqrt {e x +d}-2 c \,d^{2} \sqrt {e x +d}}{c^{3} d^{3}}+\frac {\frac {\left (-\frac {1}{2} a^{2} e^{4}+a c \,d^{2} e^{2}-\frac {1}{2} c^{2} d^{4}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {5 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{3} d^{3}}\right )\)	\(187\)
default	\(2 e \left (-\frac {-\frac {c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a \,e^{2} \sqrt {e x +d}-2 c \,d^{2} \sqrt {e x +d}}{c^{3} d^{3}}+\frac {\frac {\left (-\frac {1}{2} a^{2} e^{4}+a c \,d^{2} e^{2}-\frac {1}{2} c^{2} d^{4}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {5 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{3} d^{3}}\right )\)	\(187\)

[In]

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

[Out]

-2/3*e*(-c*d*e*x+6*a*e^2-7*c*d^2)*(e*x+d)^(1/2)/c^3/d^3+1/c^3/d^3*(2*a^2*e^4-4*a*c*d^2*e^2+2*c^2*d^4)*e*(-1/2*
(e*x+d)^(1/2)/(c*d*(e*x+d)+e^2*a-c*d^2)+5/2/((a*e^2-c*d^2)*c*d)^(1/2)*arctan(c*d*(e*x+d)^(1/2)/((a*e^2-c*d^2)*
c*d)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.92 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\left [\frac {15 \, {\left (a c d^{2} e^{2} - a^{2} e^{4} + {\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, {\left (2 \, c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} + 20 \, a c d^{2} e^{2} - 15 \, a^{2} e^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{6 \, {\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )}}, -\frac {15 \, {\left (a c d^{2} e^{2} - a^{2} e^{4} + {\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (2 \, c^{2} d^{2} e^{2} x^{2} - 3 \, c^{2} d^{4} + 20 \, a c d^{2} e^{2} - 15 \, a^{2} e^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 5 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{3 \, {\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )}}\right ] \]

[In]

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

[Out]

[1/6*(15*(a*c*d^2*e^2 - a^2*e^4 + (c^2*d^3*e - a*c*d*e^3)*x)*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*(2*c^2*d^2*e^2*x^2 - 3*c^2*d^4
+ 20*a*c*d^2*e^2 - 15*a^2*e^4 + 2*(7*c^2*d^3*e - 5*a*c*d*e^3)*x)*sqrt(e*x + d))/(c^4*d^4*x + a*c^3*d^3*e), -1
/3*(15*(a*c*d^2*e^2 - a^2*e^4 + (c^2*d^3*e - a*c*d*e^3)*x)*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)) - (2*c^2*d^2*e^2*x^2 - 3*c^2*d^4 + 20*a*c*d^2*e^2 - 15*a^2*e
^4 + 2*(7*c^2*d^3*e - 5*a*c*d*e^3)*x)*sqrt(e*x + d))/(c^4*d^4*x + a*c^3*d^3*e)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((e*x+d)^(9/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^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 = 0.27 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.51 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {5 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{3} d^{3}} - \frac {\sqrt {e x + d} c^{2} d^{4} e - 2 \, \sqrt {e x + d} a c d^{2} e^{3} + \sqrt {e x + d} a^{2} e^{5}}{{\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )} c^{3} d^{3}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{4} d^{4} e + 6 \, \sqrt {e x + d} c^{4} d^{5} e - 6 \, \sqrt {e x + d} a c^{3} d^{3} e^{3}\right )}}{3 \, c^{6} d^{6}} \]

[In]

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

[Out]

5*(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*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^3*d^3) - (sqrt(e*x + d)*c^2*d^4*e - 2*sqrt(e*x + d)*a*c*d^2*e^3 + sqrt(e*x + d)*a^2*e^5)/(((e*x +
d)*c*d - c*d^2 + a*e^2)*c^3*d^3) + 2/3*((e*x + d)^(3/2)*c^4*d^4*e + 6*sqrt(e*x + d)*c^4*d^5*e - 6*sqrt(e*x +
d)*a*c^3*d^3*e^3)/(c^6*d^6)

Mupad [B] (verification not implemented)

Time = 9.75 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.39 \[ \int \frac {(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {2\,e\,{\left (d+e\,x\right )}^{3/2}}{3\,c^2\,d^2}-\frac {\sqrt {d+e\,x}\,\left (a^2\,e^5-2\,a\,c\,d^2\,e^3+c^2\,d^4\,e\right )}{c^4\,d^4\,\left (d+e\,x\right )-c^4\,d^5+a\,c^3\,d^3\,e^2}+\frac {2\,e\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,\sqrt {d+e\,x}}{c^4\,d^4}+\frac {5\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e\,{\left (a\,e^2-c\,d^2\right )}^{3/2}\,\sqrt {d+e\,x}}{a^2\,e^5-2\,a\,c\,d^2\,e^3+c^2\,d^4\,e}\right )\,{\left (a\,e^2-c\,d^2\right )}^{3/2}}{c^{7/2}\,d^{7/2}} \]

[In]

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

[Out]

(2*e*(d + e*x)^(3/2))/(3*c^2*d^2) - ((d + e*x)^(1/2)*(a^2*e^5 + c^2*d^4*e - 2*a*c*d^2*e^3))/(c^4*d^4*(d + e*x)
- c^4*d^5 + a*c^3*d^3*e^2) + (2*e*(2*c^2*d^3 - 2*a*c*d*e^2)*(d + e*x)^(1/2))/(c^4*d^4) + (5*e*atan((c^(1/2)*d
^(1/2)*e*(a*e^2 - c*d^2)^(3/2)*(d + e*x)^(1/2))/(a^2*e^5 + c^2*d^4*e - 2*a*c*d^2*e^3))*(a*e^2 - c*d^2)^(3/2))/
(c^(7/2)*d^(7/2))