\(\int \frac {1}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [1971]

Optimal result

Integrand size = 37, antiderivative size = 192 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {16 c d \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {128 c^2 d^2 e \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

2/5/(-a*e^2+c*d^2)/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-16/15*c*d*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c
*d^2)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+128/15*c^2*d^2*e*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c*d^2)^5/(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {672, 628, 627} \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {128 c^2 d^2 e \left (a e^2+c d^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {16 c d \left (a e^2+c d^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {2}{5 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[In]

Int[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

2/(5*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (16*c*d*(c*d^2 + a*e^2 + 2*c*d
*e*x))/(15*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (128*c^2*d^2*e*(c*d^2 + a*e^2 +
2*c*d*e*x))/(15*(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 672

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a
+ b*x + c*x^2)^(p + 1)/((m + p + 1)*(2*c*d - b*e))), x] + Dist[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d -
b*e))), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a
*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(8 c d) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx}{5 \left (c d^2-a e^2\right )} \\ & = \frac {2}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {16 c d \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {\left (64 c^2 d^2 e\right ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{15 \left (c d^2-a e^2\right )^3} \\ & = \frac {2}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {16 c d \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {128 c^2 d^2 e \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \left (3 a^4 e^8-4 a^3 c d e^6 (5 d+2 e x)+6 a^2 c^2 d^2 e^4 \left (15 d^2+20 d e x+8 e^2 x^2\right )+12 a c^3 d^3 e^2 \left (5 d^3+30 d^2 e x+40 d e^2 x^2+16 e^3 x^3\right )+c^4 d^4 \left (-5 d^4+40 d^3 e x+240 d^2 e^2 x^2+320 d e^3 x^3+128 e^4 x^4\right )\right )}{15 \left (c d^2-a e^2\right )^5 (d+e x) ((a e+c d x) (d+e x))^{3/2}} \]

[In]

Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]

[Out]

(2*(3*a^4*e^8 - 4*a^3*c*d*e^6*(5*d + 2*e*x) + 6*a^2*c^2*d^2*e^4*(15*d^2 + 20*d*e*x + 8*e^2*x^2) + 12*a*c^3*d^3
*e^2*(5*d^3 + 30*d^2*e*x + 40*d*e^2*x^2 + 16*e^3*x^3) + c^4*d^4*(-5*d^4 + 40*d^3*e*x + 240*d^2*e^2*x^2 + 320*d
*e^3*x^3 + 128*e^4*x^4)))/(15*(c*d^2 - a*e^2)^5*(d + e*x)*((a*e + c*d*x)*(d + e*x))^(3/2))

Maple [A] (verified)

Time = 3.11 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.26

method	result	size
default	\(\frac {-\frac {2}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {8 c d e \left (-\frac {2 \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (e^{2} a -c \,d^{2}\right )}}{e}\)	\(242\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (128 c^{4} d^{4} e^{4} x^{4}+192 a \,c^{3} d^{3} e^{5} x^{3}+320 c^{4} d^{5} e^{3} x^{3}+48 a^{2} c^{2} d^{2} e^{6} x^{2}+480 a \,c^{3} d^{4} e^{4} x^{2}+240 c^{4} d^{6} e^{2} x^{2}-8 a^{3} c d \,e^{7} x +120 a^{2} c^{2} d^{3} e^{5} x +360 a \,c^{3} d^{5} e^{3} x +40 c^{4} d^{7} e x +3 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}+90 a^{2} c^{2} d^{4} e^{4}+60 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right )}{15 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(300\)
trager	\(-\frac {2 \left (128 c^{4} d^{4} e^{4} x^{4}+192 a \,c^{3} d^{3} e^{5} x^{3}+320 c^{4} d^{5} e^{3} x^{3}+48 a^{2} c^{2} d^{2} e^{6} x^{2}+480 a \,c^{3} d^{4} e^{4} x^{2}+240 c^{4} d^{6} e^{2} x^{2}-8 a^{3} c d \,e^{7} x +120 a^{2} c^{2} d^{3} e^{5} x +360 a \,c^{3} d^{5} e^{3} x +40 c^{4} d^{7} e x +3 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}+90 a^{2} c^{2} d^{4} e^{4}+60 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{3}}\)	\(308\)

[In]

int(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/e*(-2/5/(a*e^2-c*d^2)/(x+d/e)/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-8/5*c*d*e/(a*e^2-c*d^2)*(-2/3*(2
*c*d*e*(x+d/e)+e^2*a-c*d^2)/(a*e^2-c*d^2)^2/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+16/3*c*d*e/(a*e^2-c*
d^2)^4*(2*c*d*e*(x+d/e)+e^2*a-c*d^2)/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 769 vs. \(2 (180) = 360\).

Time = 19.21 (sec) , antiderivative size = 769, normalized size of antiderivative = 4.01 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (128 \, c^{4} d^{4} e^{4} x^{4} - 5 \, c^{4} d^{8} + 60 \, a c^{3} d^{6} e^{2} + 90 \, a^{2} c^{2} d^{4} e^{4} - 20 \, a^{3} c d^{2} e^{6} + 3 \, a^{4} e^{8} + 64 \, {\left (5 \, c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 48 \, {\left (5 \, c^{4} d^{6} e^{2} + 10 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (5 \, c^{4} d^{7} e + 45 \, a c^{3} d^{5} e^{3} + 15 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15 \, {\left (a^{2} c^{5} d^{13} e^{2} - 5 \, a^{3} c^{4} d^{11} e^{4} + 10 \, a^{4} c^{3} d^{9} e^{6} - 10 \, a^{5} c^{2} d^{7} e^{8} + 5 \, a^{6} c d^{5} e^{10} - a^{7} d^{3} e^{12} + {\left (c^{7} d^{12} e^{3} - 5 \, a c^{6} d^{10} e^{5} + 10 \, a^{2} c^{5} d^{8} e^{7} - 10 \, a^{3} c^{4} d^{6} e^{9} + 5 \, a^{4} c^{3} d^{4} e^{11} - a^{5} c^{2} d^{2} e^{13}\right )} x^{5} + {\left (3 \, c^{7} d^{13} e^{2} - 13 \, a c^{6} d^{11} e^{4} + 20 \, a^{2} c^{5} d^{9} e^{6} - 10 \, a^{3} c^{4} d^{7} e^{8} - 5 \, a^{4} c^{3} d^{5} e^{10} + 7 \, a^{5} c^{2} d^{3} e^{12} - 2 \, a^{6} c d e^{14}\right )} x^{4} + {\left (3 \, c^{7} d^{14} e - 9 \, a c^{6} d^{12} e^{3} + a^{2} c^{5} d^{10} e^{5} + 25 \, a^{3} c^{4} d^{8} e^{7} - 35 \, a^{4} c^{3} d^{6} e^{9} + 17 \, a^{5} c^{2} d^{4} e^{11} - a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} x^{3} + {\left (c^{7} d^{15} + a c^{6} d^{13} e^{2} - 17 \, a^{2} c^{5} d^{11} e^{4} + 35 \, a^{3} c^{4} d^{9} e^{6} - 25 \, a^{4} c^{3} d^{7} e^{8} - a^{5} c^{2} d^{5} e^{10} + 9 \, a^{6} c d^{3} e^{12} - 3 \, a^{7} d e^{14}\right )} x^{2} + {\left (2 \, a c^{6} d^{14} e - 7 \, a^{2} c^{5} d^{12} e^{3} + 5 \, a^{3} c^{4} d^{10} e^{5} + 10 \, a^{4} c^{3} d^{8} e^{7} - 20 \, a^{5} c^{2} d^{6} e^{9} + 13 \, a^{6} c d^{4} e^{11} - 3 \, a^{7} d^{2} e^{13}\right )} x\right )}} \]

[In]

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

2/15*(128*c^4*d^4*e^4*x^4 - 5*c^4*d^8 + 60*a*c^3*d^6*e^2 + 90*a^2*c^2*d^4*e^4 - 20*a^3*c*d^2*e^6 + 3*a^4*e^8 +
64*(5*c^4*d^5*e^3 + 3*a*c^3*d^3*e^5)*x^3 + 48*(5*c^4*d^6*e^2 + 10*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 8*(5
*c^4*d^7*e + 45*a*c^3*d^5*e^3 + 15*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*
x)/(a^2*c^5*d^13*e^2 - 5*a^3*c^4*d^11*e^4 + 10*a^4*c^3*d^9*e^6 - 10*a^5*c^2*d^7*e^8 + 5*a^6*c*d^5*e^10 - a^7*d
^3*e^12 + (c^7*d^12*e^3 - 5*a*c^6*d^10*e^5 + 10*a^2*c^5*d^8*e^7 - 10*a^3*c^4*d^6*e^9 + 5*a^4*c^3*d^4*e^11 - a^
5*c^2*d^2*e^13)*x^5 + (3*c^7*d^13*e^2 - 13*a*c^6*d^11*e^4 + 20*a^2*c^5*d^9*e^6 - 10*a^3*c^4*d^7*e^8 - 5*a^4*c^
3*d^5*e^10 + 7*a^5*c^2*d^3*e^12 - 2*a^6*c*d*e^14)*x^4 + (3*c^7*d^14*e - 9*a*c^6*d^12*e^3 + a^2*c^5*d^10*e^5 +
25*a^3*c^4*d^8*e^7 - 35*a^4*c^3*d^6*e^9 + 17*a^5*c^2*d^4*e^11 - a^6*c*d^2*e^13 - a^7*e^15)*x^3 + (c^7*d^15 + a
*c^6*d^13*e^2 - 17*a^2*c^5*d^11*e^4 + 35*a^3*c^4*d^9*e^6 - 25*a^4*c^3*d^7*e^8 - a^5*c^2*d^5*e^10 + 9*a^6*c*d^3
*e^12 - 3*a^7*d*e^14)*x^2 + (2*a*c^6*d^14*e - 7*a^2*c^5*d^12*e^3 + 5*a^3*c^4*d^10*e^5 + 10*a^4*c^3*d^8*e^7 - 2
0*a^5*c^2*d^6*e^9 + 13*a^6*c*d^4*e^11 - 3*a^7*d^2*e^13)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),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(e*(a*e^2-c*d^2)>0)', see `assu
me?` for mor

Giac [F]

\[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} {\left (e x + d\right )}} \,d x } \]

[In]

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(e*x + d)), x)

Mupad [B] (verification not implemented)

Time = 10.95 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (3\,a^4\,e^8-20\,a^3\,c\,d^2\,e^6-8\,a^3\,c\,d\,e^7\,x+90\,a^2\,c^2\,d^4\,e^4+120\,a^2\,c^2\,d^3\,e^5\,x+48\,a^2\,c^2\,d^2\,e^6\,x^2+60\,a\,c^3\,d^6\,e^2+360\,a\,c^3\,d^5\,e^3\,x+480\,a\,c^3\,d^4\,e^4\,x^2+192\,a\,c^3\,d^3\,e^5\,x^3-5\,c^4\,d^8+40\,c^4\,d^7\,e\,x+240\,c^4\,d^6\,e^2\,x^2+320\,c^4\,d^5\,e^3\,x^3+128\,c^4\,d^4\,e^4\,x^4\right )}{15\,{\left (a\,e+c\,d\,x\right )}^2\,{\left (a\,e^2-c\,d^2\right )}^5\,{\left (d+e\,x\right )}^3} \]

[In]

int(1/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)),x)

[Out]

-(2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(3*a^4*e^8 - 5*c^4*d^8 + 60*a*c^3*d^6*e^2 - 20*a^3*c*d^2*e^6
+ 90*a^2*c^2*d^4*e^4 + 240*c^4*d^6*e^2*x^2 + 320*c^4*d^5*e^3*x^3 + 128*c^4*d^4*e^4*x^4 + 40*c^4*d^7*e*x - 8*a
^3*c*d*e^7*x + 48*a^2*c^2*d^2*e^6*x^2 + 360*a*c^3*d^5*e^3*x + 120*a^2*c^2*d^3*e^5*x + 480*a*c^3*d^4*e^4*x^2 +
192*a*c^3*d^3*e^5*x^3))/(15*(a*e + c*d*x)^2*(a*e^2 - c*d^2)^5*(d + e*x)^3)