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

Optimal. Leaf size=192 \[ \frac {2}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {16 c d \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {128 c^2 d^2 e \left (c d^2+a e^2+2 c d e x\right )}{15 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[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)

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Rubi [A]
time = 0.04, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {(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*}

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Mathematica [A]
time = 0.19, size = 153, normalized size = 0.80 \begin {gather*} -\frac {2 (a e+c d x)^5 \left (-3 e^4+\frac {20 c d e^3 (d+e x)}{a e+c d x}-\frac {90 c^2 d^2 e^2 (d+e x)^2}{(a e+c d x)^2}-\frac {60 c^3 d^3 e (d+e x)^3}{(a e+c d x)^3}+\frac {5 c^4 d^4 (d+e x)^4}{(a e+c d x)^4}\right )}{15 \left (c d^2-a e^2\right )^5 ((a e+c d x) (d+e x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[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*(a*e + c*d*x)^5*(-3*e^4 + (20*c*d*e^3*(d + e*x))/(a*e + c*d*x) - (90*c^2*d^2*e^2*(d + e*x)^2)/(a*e + c*d*x
)^2 - (60*c^3*d^3*e*(d + e*x)^3)/(a*e + c*d*x)^3 + (5*c^4*d^4*(d + e*x)^4)/(a*e + c*d*x)^4))/(15*(c*d^2 - a*e^
2)^5*((a*e + c*d*x)*(d + e*x))^(5/2))

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Maple [A]
time = 0.72, size = 242, normalized size = 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	\(-\frac {2 \left (128 c^{4} d^{4} e^{4} x^{4}+192 a \,c^{3} d^{3} e^{5} x^{3}+320 c^{4} d^{5} e^{3} x^{3}+48 a^{2} c^{2} d^{2} e^{6} x^{2}+480 a \,c^{3} d^{4} e^{4} x^{2}+240 c^{4} d^{6} e^{2} x^{2}-8 a^{3} c d \,e^{7} x +120 a^{2} c^{2} d^{3} e^{5} x +360 a \,c^{3} d^{5} e^{3} x +40 c^{4} d^{7} e x +3 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}+90 a^{2} c^{2} d^{4} e^{4}+60 a \,c^{3} d^{6} e^{2}-5 c^{4} d^{8}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{3}}\)	\(308\)

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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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)/(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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 780 vs. \(2 (182) = 364\).
time = 61.17, size = 780, normalized size = 4.06 \begin {gather*} \frac {2 \, {\left (40 \, c^{4} d^{7} x e - 5 \, c^{4} d^{8} - 8 \, a^{3} c d x e^{7} + 3 \, a^{4} e^{8} + 4 \, {\left (12 \, a^{2} c^{2} d^{2} x^{2} - 5 \, a^{3} c d^{2}\right )} e^{6} + 24 \, {\left (8 \, a c^{3} d^{3} x^{3} + 5 \, a^{2} c^{2} d^{3} x\right )} e^{5} + 2 \, {\left (64 \, c^{4} d^{4} x^{4} + 240 \, a c^{3} d^{4} x^{2} + 45 \, a^{2} c^{2} d^{4}\right )} e^{4} + 40 \, {\left (8 \, c^{4} d^{5} x^{3} + 9 \, a c^{3} d^{5} x\right )} e^{3} + 60 \, {\left (4 \, c^{4} d^{6} x^{2} + a c^{3} d^{6}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{15 \, {\left (c^{7} d^{15} x^{2} - a^{7} x^{3} e^{15} - {\left (2 \, a^{6} c d x^{4} + 3 \, a^{7} d x^{2}\right )} e^{14} - {\left (a^{5} c^{2} d^{2} x^{5} + a^{6} c d^{2} x^{3} + 3 \, a^{7} d^{2} x\right )} e^{13} + {\left (7 \, a^{5} c^{2} d^{3} x^{4} + 9 \, a^{6} c d^{3} x^{2} - a^{7} d^{3}\right )} e^{12} + {\left (5 \, a^{4} c^{3} d^{4} x^{5} + 17 \, a^{5} c^{2} d^{4} x^{3} + 13 \, a^{6} c d^{4} x\right )} e^{11} - {\left (5 \, a^{4} c^{3} d^{5} x^{4} + a^{5} c^{2} d^{5} x^{2} - 5 \, a^{6} c d^{5}\right )} e^{10} - 5 \, {\left (2 \, a^{3} c^{4} d^{6} x^{5} + 7 \, a^{4} c^{3} d^{6} x^{3} + 4 \, a^{5} c^{2} d^{6} x\right )} e^{9} - 5 \, {\left (2 \, a^{3} c^{4} d^{7} x^{4} + 5 \, a^{4} c^{3} d^{7} x^{2} + 2 \, a^{5} c^{2} d^{7}\right )} e^{8} + 5 \, {\left (2 \, a^{2} c^{5} d^{8} x^{5} + 5 \, a^{3} c^{4} d^{8} x^{3} + 2 \, a^{4} c^{3} d^{8} x\right )} e^{7} + 5 \, {\left (4 \, a^{2} c^{5} d^{9} x^{4} + 7 \, a^{3} c^{4} d^{9} x^{2} + 2 \, a^{4} c^{3} d^{9}\right )} e^{6} - {\left (5 \, a c^{6} d^{10} x^{5} - a^{2} c^{5} d^{10} x^{3} - 5 \, a^{3} c^{4} d^{10} x\right )} e^{5} - {\left (13 \, a c^{6} d^{11} x^{4} + 17 \, a^{2} c^{5} d^{11} x^{2} + 5 \, a^{3} c^{4} d^{11}\right )} e^{4} + {\left (c^{7} d^{12} x^{5} - 9 \, a c^{6} d^{12} x^{3} - 7 \, a^{2} c^{5} d^{12} x\right )} e^{3} + {\left (3 \, c^{7} d^{13} x^{4} + a c^{6} d^{13} x^{2} + a^{2} c^{5} d^{13}\right )} e^{2} + {\left (3 \, c^{7} d^{14} x^{3} + 2 \, a c^{6} d^{14} x\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*(40*c^4*d^7*x*e - 5*c^4*d^8 - 8*a^3*c*d*x*e^7 + 3*a^4*e^8 + 4*(12*a^2*c^2*d^2*x^2 - 5*a^3*c*d^2)*e^6 + 24
*(8*a*c^3*d^3*x^3 + 5*a^2*c^2*d^3*x)*e^5 + 2*(64*c^4*d^4*x^4 + 240*a*c^3*d^4*x^2 + 45*a^2*c^2*d^4)*e^4 + 40*(8
*c^4*d^5*x^3 + 9*a*c^3*d^5*x)*e^3 + 60*(4*c^4*d^6*x^2 + a*c^3*d^6)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*
d)*e)/(c^7*d^15*x^2 - a^7*x^3*e^15 - (2*a^6*c*d*x^4 + 3*a^7*d*x^2)*e^14 - (a^5*c^2*d^2*x^5 + a^6*c*d^2*x^3 + 3
*a^7*d^2*x)*e^13 + (7*a^5*c^2*d^3*x^4 + 9*a^6*c*d^3*x^2 - a^7*d^3)*e^12 + (5*a^4*c^3*d^4*x^5 + 17*a^5*c^2*d^4*
x^3 + 13*a^6*c*d^4*x)*e^11 - (5*a^4*c^3*d^5*x^4 + a^5*c^2*d^5*x^2 - 5*a^6*c*d^5)*e^10 - 5*(2*a^3*c^4*d^6*x^5 +
7*a^4*c^3*d^6*x^3 + 4*a^5*c^2*d^6*x)*e^9 - 5*(2*a^3*c^4*d^7*x^4 + 5*a^4*c^3*d^7*x^2 + 2*a^5*c^2*d^7)*e^8 + 5*
(2*a^2*c^5*d^8*x^5 + 5*a^3*c^4*d^8*x^3 + 2*a^4*c^3*d^8*x)*e^7 + 5*(4*a^2*c^5*d^9*x^4 + 7*a^3*c^4*d^9*x^2 + 2*a
^4*c^3*d^9)*e^6 - (5*a*c^6*d^10*x^5 - a^2*c^5*d^10*x^3 - 5*a^3*c^4*d^10*x)*e^5 - (13*a*c^6*d^11*x^4 + 17*a^2*c
^5*d^11*x^2 + 5*a^3*c^4*d^11)*e^4 + (c^7*d^12*x^5 - 9*a*c^6*d^12*x^3 - 7*a^2*c^5*d^12*x)*e^3 + (3*c^7*d^13*x^4
+ a*c^6*d^13*x^2 + a^2*c^5*d^13)*e^2 + (3*c^7*d^14*x^3 + 2*a*c^6*d^14*x)*e)

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Mupad [B]
time = 1.65, size = 253, normalized size = 1.32 \begin {gather*} -\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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