3.20.72 \(\int \frac {1}{(d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [1972]

Optimal. Leaf size=252 \[ \frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 c d}{7 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {32 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {256 c^3 d^3 e \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

2/7/(-a*e^2+c*d^2)/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+4/7*c*d/(-a*e^2+c*d^2)^2/(e*x+d)/(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-32/21*c^2*d^2*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(3/2)+256/21*c^3*d^3*e*(2*c*d*e*x+a*e^2+c*d^2)/(-a*e^2+c*d^2)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {672, 628, 627} \begin {gather*} \frac {256 c^3 d^3 e \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {32 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {4 c d}{7 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {2}{7 (d+e x)^2 \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2/(7*(c*d^2 - a*e^2)*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (4*c*d)/(7*(c*d^2 - a*e^2)^2
*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (32*c^2*d^2*(c*d^2 + a*e^2 + 2*c*d*e*x))/(21*(c*d^
2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (256*c^3*d^3*e*(c*d^2 + a*e^2 + 2*c*d*e*x))/(21*
(c*d^2 - a*e^2)^6*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*} \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(10 c d) \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}{7 \left (c d^2-a e^2\right )}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 c d}{7 \left (c d^2-a e^2\right )^2 (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 {\left (16 c^2 d^2\right ) \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}{7 \left (c d^2-a e^2\right )^2}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 c d}{7 \left (c d^2-a e^2\right )^2 (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 {32 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {\left (128 c^3 d^3 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}{21 \left (c d^2-a e^2\right )^4}\\ &=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 c d}{7 \left (c d^2-a e^2\right )^2 (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 {32 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {256 c^3 d^3 e \left (c d^2+a e^2+2 c d e x\right )}{21 \left (c d^2-a e^2\right )^6 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.24, size = 181, normalized size = 0.72 \begin {gather*} -\frac {2 (a e+c d x)^7 \left (3 e^5-\frac {21 c d e^4 (d+e x)}{a e+c d x}+\frac {70 c^2 d^2 e^3 (d+e x)^2}{(a e+c d x)^2}-\frac {210 c^3 d^3 e^2 (d+e x)^3}{(a e+c d x)^3}-\frac {105 c^4 d^4 e (d+e x)^4}{(a e+c d x)^4}+\frac {7 c^5 d^5 (d+e x)^5}{(a e+c d x)^5}\right )}{21 \left (c d^2-a e^2\right )^6 ((a e+c d x) (d+e x))^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a*e + c*d*x)^7*(3*e^5 - (21*c*d*e^4*(d + e*x))/(a*e + c*d*x) + (70*c^2*d^2*e^3*(d + e*x)^2)/(a*e + c*d*x)
^2 - (210*c^3*d^3*e^2*(d + e*x)^3)/(a*e + c*d*x)^3 - (105*c^4*d^4*e*(d + e*x)^4)/(a*e + c*d*x)^4 + (7*c^5*d^5*
(d + e*x)^5)/(a*e + c*d*x)^5))/(21*(c*d^2 - a*e^2)^6*((a*e + c*d*x)*(d + e*x))^(7/2))

________________________________________________________________________________________

Maple [A]
time = 0.72, size = 323, normalized size = 1.28

method	result	size
default	\(\frac {-\frac {2}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {10 c d e \left (-\frac {2}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {8 c d e \left (-\frac {2 \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {16 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (e^{2} a -c \,d^{2}\right )}\right )}{7 \left (e^{2} a -c \,d^{2}\right )}}{e^{2}}\)	\(323\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-256 c^{5} d^{5} e^{5} x^{5}-384 a \,c^{4} d^{4} e^{6} x^{4}-896 c^{5} d^{6} e^{4} x^{4}-96 a^{2} c^{3} d^{3} e^{7} x^{3}-1344 a \,c^{4} d^{5} e^{5} x^{3}-1120 c^{5} d^{7} e^{3} x^{3}+16 a^{3} c^{2} d^{2} e^{8} x^{2}-336 a^{2} c^{3} d^{4} e^{6} x^{2}-1680 a \,c^{4} d^{6} e^{4} x^{2}-560 c^{5} d^{8} e^{2} x^{2}-6 a^{4} c d \,e^{9} x +56 a^{3} c^{2} d^{3} e^{7} x -420 a^{2} c^{3} d^{5} e^{5} x -840 a \,c^{4} d^{7} e^{3} x -70 c^{5} d^{9} e x +3 a^{5} e^{10}-21 a^{4} c \,d^{2} e^{8}+70 a^{3} c^{2} d^{4} e^{6}-210 a^{2} c^{3} d^{6} e^{4}-105 a \,c^{4} d^{8} e^{2}+7 c^{5} d^{10}\right )}{21 \left (e x +d \right ) \left (a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(412\)
trager	\(-\frac {2 \left (-256 c^{5} d^{5} e^{5} x^{5}-384 a \,c^{4} d^{4} e^{6} x^{4}-896 c^{5} d^{6} e^{4} x^{4}-96 a^{2} c^{3} d^{3} e^{7} x^{3}-1344 a \,c^{4} d^{5} e^{5} x^{3}-1120 c^{5} d^{7} e^{3} x^{3}+16 a^{3} c^{2} d^{2} e^{8} x^{2}-336 a^{2} c^{3} d^{4} e^{6} x^{2}-1680 a \,c^{4} d^{6} e^{4} x^{2}-560 c^{5} d^{8} e^{2} x^{2}-6 a^{4} c d \,e^{9} x +56 a^{3} c^{2} d^{3} e^{7} x -420 a^{2} c^{3} d^{5} e^{5} x -840 a \,c^{4} d^{7} e^{3} x -70 c^{5} d^{9} e x +3 a^{5} e^{10}-21 a^{4} c \,d^{2} e^{8}+70 a^{3} c^{2} d^{4} e^{6}-210 a^{2} c^{3} d^{6} e^{4}-105 a \,c^{4} d^{8} e^{2}+7 c^{5} d^{10}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{21 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{4}}\)	\(415\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e^2*(-2/7/(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))^(3/2)-10/7*c*d*e/(a*e^2-c*d^2)*(-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))))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(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(c*d^2-%e^2*a>0)', see `assume?
` for more d

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1074 vs. \(2 (239) = 478\).
time = 129.74, size = 1074, normalized size = 4.26 \begin {gather*} \frac {2 \, {\left (70 \, c^{5} d^{9} x e - 7 \, c^{5} d^{10} + 6 \, a^{4} c d x e^{9} - 3 \, a^{5} e^{10} - {\left (16 \, a^{3} c^{2} d^{2} x^{2} - 21 \, a^{4} c d^{2}\right )} e^{8} + 8 \, {\left (12 \, a^{2} c^{3} d^{3} x^{3} - 7 \, a^{3} c^{2} d^{3} x\right )} e^{7} + 2 \, {\left (192 \, a c^{4} d^{4} x^{4} + 168 \, a^{2} c^{3} d^{4} x^{2} - 35 \, a^{3} c^{2} d^{4}\right )} e^{6} + 4 \, {\left (64 \, c^{5} d^{5} x^{5} + 336 \, a c^{4} d^{5} x^{3} + 105 \, a^{2} c^{3} d^{5} x\right )} e^{5} + 14 \, {\left (64 \, c^{5} d^{6} x^{4} + 120 \, a c^{4} d^{6} x^{2} + 15 \, a^{2} c^{3} d^{6}\right )} e^{4} + 280 \, {\left (4 \, c^{5} d^{7} x^{3} + 3 \, a c^{4} d^{7} x\right )} e^{3} + 35 \, {\left (16 \, c^{5} d^{8} x^{2} + 3 \, a c^{4} d^{8}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{21 \, {\left (c^{8} d^{18} x^{2} + a^{8} x^{4} e^{18} + 2 \, {\left (a^{7} c d x^{5} + 2 \, a^{8} d x^{3}\right )} e^{17} + {\left (a^{6} c^{2} d^{2} x^{6} + 2 \, a^{7} c d^{2} x^{4} + 6 \, a^{8} d^{2} x^{2}\right )} e^{16} - 4 \, {\left (2 \, a^{6} c^{2} d^{3} x^{5} + 3 \, a^{7} c d^{3} x^{3} - a^{8} d^{3} x\right )} e^{15} - {\left (6 \, a^{5} c^{3} d^{4} x^{6} + 27 \, a^{6} c^{2} d^{4} x^{4} + 28 \, a^{7} c d^{4} x^{2} - a^{8} d^{4}\right )} e^{14} + 2 \, {\left (3 \, a^{5} c^{3} d^{5} x^{5} - 4 \, a^{6} c^{2} d^{5} x^{3} - 11 \, a^{7} c d^{5} x\right )} e^{13} + {\left (15 \, a^{4} c^{4} d^{6} x^{6} + 64 \, a^{5} c^{3} d^{6} x^{4} + 43 \, a^{6} c^{2} d^{6} x^{2} - 6 \, a^{7} c d^{6}\right )} e^{12} + 4 \, {\left (5 \, a^{4} c^{4} d^{7} x^{5} + 19 \, a^{5} c^{3} d^{7} x^{3} + 12 \, a^{6} c^{2} d^{7} x\right )} e^{11} - {\left (20 \, a^{3} c^{5} d^{8} x^{6} + 55 \, a^{4} c^{4} d^{8} x^{4} + 6 \, a^{5} c^{3} d^{8} x^{2} - 15 \, a^{6} c^{2} d^{8}\right )} e^{10} - 10 \, {\left (5 \, a^{3} c^{5} d^{9} x^{5} + 12 \, a^{4} c^{4} d^{9} x^{3} + 5 \, a^{5} c^{3} d^{9} x\right )} e^{9} + {\left (15 \, a^{2} c^{6} d^{10} x^{6} - 6 \, a^{3} c^{5} d^{10} x^{4} - 55 \, a^{4} c^{4} d^{10} x^{2} - 20 \, a^{5} c^{3} d^{10}\right )} e^{8} + 4 \, {\left (12 \, a^{2} c^{6} d^{11} x^{5} + 19 \, a^{3} c^{5} d^{11} x^{3} + 5 \, a^{4} c^{4} d^{11} x\right )} e^{7} - {\left (6 \, a c^{7} d^{12} x^{6} - 43 \, a^{2} c^{6} d^{12} x^{4} - 64 \, a^{3} c^{5} d^{12} x^{2} - 15 \, a^{4} c^{4} d^{12}\right )} e^{6} - 2 \, {\left (11 \, a c^{7} d^{13} x^{5} + 4 \, a^{2} c^{6} d^{13} x^{3} - 3 \, a^{3} c^{5} d^{13} x\right )} e^{5} + {\left (c^{8} d^{14} x^{6} - 28 \, a c^{7} d^{14} x^{4} - 27 \, a^{2} c^{6} d^{14} x^{2} - 6 \, a^{3} c^{5} d^{14}\right )} e^{4} + 4 \, {\left (c^{8} d^{15} x^{5} - 3 \, a c^{7} d^{15} x^{3} - 2 \, a^{2} c^{6} d^{15} x\right )} e^{3} + {\left (6 \, c^{8} d^{16} x^{4} + 2 \, a c^{7} d^{16} x^{2} + a^{2} c^{6} d^{16}\right )} e^{2} + 2 \, {\left (2 \, c^{8} d^{17} x^{3} + a c^{7} d^{17} x\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(70*c^5*d^9*x*e - 7*c^5*d^10 + 6*a^4*c*d*x*e^9 - 3*a^5*e^10 - (16*a^3*c^2*d^2*x^2 - 21*a^4*c*d^2)*e^8 + 8
*(12*a^2*c^3*d^3*x^3 - 7*a^3*c^2*d^3*x)*e^7 + 2*(192*a*c^4*d^4*x^4 + 168*a^2*c^3*d^4*x^2 - 35*a^3*c^2*d^4)*e^6
+ 4*(64*c^5*d^5*x^5 + 336*a*c^4*d^5*x^3 + 105*a^2*c^3*d^5*x)*e^5 + 14*(64*c^5*d^6*x^4 + 120*a*c^4*d^6*x^2 + 1
5*a^2*c^3*d^6)*e^4 + 280*(4*c^5*d^7*x^3 + 3*a*c^4*d^7*x)*e^3 + 35*(16*c^5*d^8*x^2 + 3*a*c^4*d^8)*e^2)*sqrt(c*d
^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)/(c^8*d^18*x^2 + a^8*x^4*e^18 + 2*(a^7*c*d*x^5 + 2*a^8*d*x^3)*e^17 + (a^6*c
^2*d^2*x^6 + 2*a^7*c*d^2*x^4 + 6*a^8*d^2*x^2)*e^16 - 4*(2*a^6*c^2*d^3*x^5 + 3*a^7*c*d^3*x^3 - a^8*d^3*x)*e^15
- (6*a^5*c^3*d^4*x^6 + 27*a^6*c^2*d^4*x^4 + 28*a^7*c*d^4*x^2 - a^8*d^4)*e^14 + 2*(3*a^5*c^3*d^5*x^5 - 4*a^6*c^
2*d^5*x^3 - 11*a^7*c*d^5*x)*e^13 + (15*a^4*c^4*d^6*x^6 + 64*a^5*c^3*d^6*x^4 + 43*a^6*c^2*d^6*x^2 - 6*a^7*c*d^6
)*e^12 + 4*(5*a^4*c^4*d^7*x^5 + 19*a^5*c^3*d^7*x^3 + 12*a^6*c^2*d^7*x)*e^11 - (20*a^3*c^5*d^8*x^6 + 55*a^4*c^4
*d^8*x^4 + 6*a^5*c^3*d^8*x^2 - 15*a^6*c^2*d^8)*e^10 - 10*(5*a^3*c^5*d^9*x^5 + 12*a^4*c^4*d^9*x^3 + 5*a^5*c^3*d
^9*x)*e^9 + (15*a^2*c^6*d^10*x^6 - 6*a^3*c^5*d^10*x^4 - 55*a^4*c^4*d^10*x^2 - 20*a^5*c^3*d^10)*e^8 + 4*(12*a^2
*c^6*d^11*x^5 + 19*a^3*c^5*d^11*x^3 + 5*a^4*c^4*d^11*x)*e^7 - (6*a*c^7*d^12*x^6 - 43*a^2*c^6*d^12*x^4 - 64*a^3
*c^5*d^12*x^2 - 15*a^4*c^4*d^12)*e^6 - 2*(11*a*c^7*d^13*x^5 + 4*a^2*c^6*d^13*x^3 - 3*a^3*c^5*d^13*x)*e^5 + (c^
8*d^14*x^6 - 28*a*c^7*d^14*x^4 - 27*a^2*c^6*d^14*x^2 - 6*a^3*c^5*d^14)*e^4 + 4*(c^8*d^15*x^5 - 3*a*c^7*d^15*x^
3 - 2*a^2*c^6*d^15*x)*e^3 + (6*c^8*d^16*x^4 + 2*a*c^7*d^16*x^2 + a^2*c^6*d^16)*e^2 + 2*(2*c^8*d^17*x^3 + a*c^7
*d^17*x)*e)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 10245 vs. \(2 (239) = 478\).
time = 1.22, size = 10245, normalized size = 40.65 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21*(256*c^4*d^4*e^3*sgn(1/(x*e + d))/(sqrt(c*d)*c^6*d^12*e^(1/2) - 6*sqrt(c*d)*a*c^5*d^10*e^(5/2) + 15*sqrt
(c*d)*a^2*c^4*d^8*e^(9/2) - 20*sqrt(c*d)*a^3*c^3*d^6*e^(13/2) + 15*sqrt(c*d)*a^4*c^2*d^4*e^(17/2) - 6*sqrt(c*d
)*a^5*c*d^2*e^(21/2) + sqrt(c*d)*a^6*e^(25/2)) - ((210*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*c^39*
d^75*e^51*sgn(1/(x*e + d))^6 - 70*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*c^38*d^74*e^50*sgn(1/(x*
e + d))^6 - 7560*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a*c^38*d^73*e^53*sgn(1/(x*e + d))^6 + 21*(c
*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*c^37*d^73*e^49*sgn(1/(x*e + d))^6 + 2520*(c*d*e - c*d^2*e/(x
*e + d) + a*e^3/(x*e + d))^(3/2)*a*c^37*d^72*e^52*sgn(1/(x*e + d))^6 - 3*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x
*e + d))^(7/2)*c^36*d^72*e^48*sgn(1/(x*e + d))^6 + 132300*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^
2*c^37*d^71*e^55*sgn(1/(x*e + d))^6 - 756*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a*c^36*d^71*e^51
*sgn(1/(x*e + d))^6 - 44100*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^2*c^36*d^70*e^54*sgn(1/(x*e
+ d))^6 + 108*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(7/2)*a*c^35*d^70*e^50*sgn(1/(x*e + d))^6 - 149940
0*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^3*c^36*d^69*e^57*sgn(1/(x*e + d))^6 + 13230*(c*d*e - c*d
^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^2*c^35*d^69*e^53*sgn(1/(x*e + d))^6 + 499800*(c*d*e - c*d^2*e/(x*e +
d) + a*e^3/(x*e + d))^(3/2)*a^3*c^35*d^68*e^56*sgn(1/(x*e + d))^6 - 1890*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(
x*e + d))^(7/2)*a^2*c^34*d^68*e^52*sgn(1/(x*e + d))^6 + 12370050*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e +
d))*a^4*c^35*d^67*e^59*sgn(1/(x*e + d))^6 - 149940*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^3*c^
34*d^67*e^55*sgn(1/(x*e + d))^6 - 4123350*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^4*c^34*d^66*e^
58*sgn(1/(x*e + d))^6 + 21420*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(7/2)*a^3*c^33*d^66*e^54*sgn(1/(x*
e + d))^6 - 79168320*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^5*c^34*d^65*e^61*sgn(1/(x*e + d))^6 +
1237005*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^4*c^33*d^65*e^57*sgn(1/(x*e + d))^6 + 26389440*
(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^5*c^33*d^64*e^60*sgn(1/(x*e + d))^6 - 176715*(c*d*e - c*
d^2*e/(x*e + d) + a*e^3/(x*e + d))^(7/2)*a^4*c^32*d^64*e^56*sgn(1/(x*e + d))^6 + 409036320*sqrt(c*d*e - c*d^2*
e/(x*e + d) + a*e^3/(x*e + d))*a^6*c^33*d^63*e^63*sgn(1/(x*e + d))^6 - 7916832*(c*d*e - c*d^2*e/(x*e + d) + a*
e^3/(x*e + d))^(5/2)*a^5*c^32*d^63*e^59*sgn(1/(x*e + d))^6 - 136345440*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e
+ d))^(3/2)*a^6*c^32*d^62*e^62*sgn(1/(x*e + d))^6 + 1130976*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(7/
2)*a^5*c^31*d^62*e^58*sgn(1/(x*e + d))^6 - 1753012800*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^7*c^
32*d^61*e^65*sgn(1/(x*e + d))^6 + 40903632*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^6*c^31*d^61*e
^61*sgn(1/(x*e + d))^6 + 584337600*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^7*c^31*d^60*e^64*sgn(
1/(x*e + d))^6 - 5843376*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(7/2)*a^6*c^30*d^60*e^60*sgn(1/(x*e + d
))^6 + 6354671400*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^8*c^31*d^59*e^67*sgn(1/(x*e + d))^6 - 17
5301280*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^7*c^30*d^59*e^63*sgn(1/(x*e + d))^6 - 2118223800
*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^8*c^30*d^58*e^66*sgn(1/(x*e + d))^6 + 25043040*(c*d*e -
c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(7/2)*a^7*c^29*d^58*e^62*sgn(1/(x*e + d))^6 - 19770088800*sqrt(c*d*e - c
*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^9*c^30*d^57*e^69*sgn(1/(x*e + d))^6 + 635467140*(c*d*e - c*d^2*e/(x*e +
d) + a*e^3/(x*e + d))^(5/2)*a^8*c^29*d^57*e^65*sgn(1/(x*e + d))^6 + 6590029600*(c*d*e - c*d^2*e/(x*e + d) + a*
e^3/(x*e + d))^(3/2)*a^9*c^29*d^56*e^68*sgn(1/(x*e + d))^6 - 90781020*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e
+ d))^(7/2)*a^8*c^28*d^56*e^64*sgn(1/(x*e + d))^6 + 53379239760*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e +
d))*a^10*c^29*d^55*e^71*sgn(1/(x*e + d))^6 - 1977008880*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^
9*c^28*d^55*e^67*sgn(1/(x*e + d))^6 - 17793079920*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^10*c^2
8*d^54*e^70*sgn(1/(x*e + d))^6 + 282429840*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(7/2)*a^9*c^27*d^54*e
^66*sgn(1/(x*e + d))^6 - 126169112160*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^11*c^28*d^53*e^73*sg
n(1/(x*e + d))^6 + 5337923976*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^10*c^27*d^53*e^69*sgn(1/(x
*e + d))^6 + 42056370720*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^11*c^27*d^52*e^72*sgn(1/(x*e +
d))^6 - 762560568*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(7/2)*a^10*c^26*d^52*e^68*sgn(1/(x*e + d))^6 +
262852317000*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^12*c^27*d^51*e^75*sgn(1/(x*e + d))^6 - 12616
911216*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e ...

________________________________________________________________________________________

Mupad [B]
time = 3.10, size = 2500, normalized size = 9.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((d*((12*c^3*d^4*e^4)/(7*(a*e^2 - c*d^2)^3*(5*a^3*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c*d^2*e^5)) -
(2*c^2*d^2*e^4*(19*a*e^2 - 7*c*d^2))/(7*(a*e^2 - c*d^2)^3*(5*a^3*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^
2*c*d^2*e^5))))/e + (e^3*(14*c^3*d^5 - 42*a*c^2*d^3*e^2 + 40*a^2*c*d*e^4))/(7*(a*e^2 - c*d^2)^3*(5*a^3*e^7 - 5
*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c*d^2*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3
- (((d*((24*c^4*d^5*e^4)/(35*(a*e^2 - c*d^2)^6*(3*a*e^3 - 3*c*d^2*e)) - (8*c^3*d^3*e^4*(11*a*e^2 - 5*c*d^2))/
(35*(a*e^2 - c*d^2)^6*(3*a*e^3 - 3*c*d^2*e))))/e + (2*c^2*d^2*e^3*(19*a^2*e^4 - 13*c^2*d^4 + 6*a*c*d^2*e^2))/(
35*(a*e^2 - c*d^2)^6*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + (((2
4*c^4*d^5*e^2)/(35*(a*e^2 - c*d^2)^7) - (4*c^3*d^3*e^2*(47*a*e^2 - 29*c*d^2))/(105*(a*e^2 - c*d^2)^7))*(x*(a*e
^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (2*e^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((d
+ e*x)^4*(7*a^3*e^7 - 7*c^3*d^6*e + 21*a*c^2*d^4*e^3 - 21*a^2*c*d^2*e^5)) - ((x*((a*(((a*e^2 + c*d^2)*((8*c^7*
d^7*e^5*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^7*d^7*e^5*
(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (4*c^6*d
^6*e^4*(13*a^2*e^4 - 31*c^2*d^4 + 42*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*
d*e^5)) - (16*a*c^7*d^8*e^6)/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^6*d^6*e
^4*(a*e^2 + c*d^2)*(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))
/c + ((a*e^2 + c*d^2)*((a*((8*c^7*d^7*e^5*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3
+ a^2*c*d*e^5)) - (16*c^7*d^7*e^5*(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 +
a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((8*c^7*d^7*e^5*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*
(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^7*d^7*e^5*(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c
^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (4*c^6*d^6*e^4*(13*a^2*e^4 - 31*c^2*d^4 + 42*a*c*d^2*e^
2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*a*c^7*d^8*e^6)/(35*(a*e^2 - c*d^2
)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^6*d^6*e^4*(a*e^2 + c*d^2)*(17*a*e^2 - 5*c*d^2))/(105*(
a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (8*c^5*d^5*e^3*(74*a^3*e^6 - 35*c^3*
d^6 + 198*a*c^2*d^4*e^2 - 261*a^2*c*d^2*e^4))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^
5)) + (2*c^5*d^5*e^3*(a*e^2 + c*d^2)*(13*a^2*e^4 - 31*c^2*d^4 + 42*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^
5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c^2*d^2*e^2*(70*c^6*d^10 - 420*a*c^5*d^8*e^2 + 1026*a^2*c
^4*d^6*e^4 - 1032*a^3*c^3*d^4*e^6 + 332*a^4*c^2*d^2*e^8))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3
+ a^2*c*d*e^5)) + (4*c^4*d^4*e^2*(a*e^2 + c*d^2)*(74*a^3*e^6 - 35*c^3*d^6 + 198*a*c^2*d^4*e^2 - 261*a^2*c*d^2*
e^4))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) + (a*((a*((8*c^7*d^7*e^5*(a*e^2 + c
*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^7*d^7*e^5*(17*a*e^2 - 5*c*d^
2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^
2)*((8*c^7*d^7*e^5*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c
^7*d^7*e^5*(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e)
+ (4*c^6*d^6*e^4*(13*a^2*e^4 - 31*c^2*d^4 + 42*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e
^3 + a^2*c*d*e^5)) - (16*a*c^7*d^8*e^6)/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (
8*c^6*d^6*e^4*(a*e^2 + c*d^2)*(17*a*e^2 - 5*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*
c*d*e^5))))/(c*d*e) - (8*c^5*d^5*e^3*(74*a^3*e^6 - 35*c^3*d^6 + 198*a*c^2*d^4*e^2 - 261*a^2*c*d^2*e^4))/(105*(
a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^5*d^5*e^3*(a*e^2 + c*d^2)*(13*a^2*e^4 - 3
1*c^2*d^4 + 42*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + (c*d*e*(
a*e^2 + c*d^2)*(70*c^6*d^10 - 420*a*c^5*d^8*e^2 + 1026*a^2*c^4*d^6*e^4 - 1032*a^3*c^3*d^4*e^6 + 332*a^4*c^2*d^
2*e^8))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*
e*x^2)^(1/2))/((a*e + c*d*x)^2*(d + e*x)^2) + ((x*(((a*e^2 + c*d^2)*((24*c^6*d^6*e^4*(a*e^2 + c*d^2))/(35*(a*e
^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (32*c^6*d^6*e^4*(7*a*e^2 - 4*c*d^2))/(35*(a*e^2 -
c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (4*c^5*d^5*e^3*(251*a^2*e^4 + 207*c^2*d^4 -
446*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (48*a*c^6*d^7*e^5)/(35
*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3...

________________________________________________________________________________________