3.20.63 \(\int \frac {1}{(d+e x)^4 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [1963]

Optimal. Leaf size=301 \[ \frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {64 c^3 d^3}{63 \left (c d^2-a e^2\right )^4 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {256 c^4 d^4 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^6 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

2/9/(-a*e^2+c*d^2)/(e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+20/63*c*d/(-a*e^2+c*d^2)^2/(e*x+d)^3/(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+32/63*c^2*d^2/(-a*e^2+c*d^2)^3/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2)+64/63*c^3*d^3/(-a*e^2+c*d^2)^4/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-256/63*c^4*d^4*(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)

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Rubi [A]
time = 0.11, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 627} \begin {gather*} -\frac {256 c^4 d^4 \left (a e^2+c d^2+2 c d e x\right )}{63 \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 {64 c^3 d^3}{63 (d+e x) \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {32 c^2 d^2}{63 (d+e x)^2 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {20 c d}{63 (d+e x)^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {2}{9 (d+e x)^4 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

2/(9*(c*d^2 - a*e^2)*(d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (20*c*d)/(63*(c*d^2 - a*e^2)^2
*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (32*c^2*d^2)/(63*(c*d^2 - a*e^2)^3*(d + e*x)^2*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (64*c^3*d^3)/(63*(c*d^2 - a*e^2)^4*(d + e*x)*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2]) - (256*c^4*d^4*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(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 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)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(10 c d) \int \frac {1}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{9 \left (c d^2-a e^2\right )}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (80 c^2 d^2\right ) \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 )^{3/2}} \, dx}{63 \left (c d^2-a e^2\right )^2}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (32 c^3 d^3\right ) \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 )^{3/2}} \, dx}{21 \left (c d^2-a e^2\right )^3}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {64 c^3 d^3}{63 \left (c d^2-a e^2\right )^4 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (128 c^4 d^4\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}{63 \left (c d^2-a e^2\right )^4}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {64 c^3 d^3}{63 \left (c d^2-a e^2\right )^4 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {256 c^4 d^4 \left (c d^2+a e^2+2 c d e x\right )}{63 \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*}

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Mathematica [A]
time = 0.22, size = 188, normalized size = 0.62 \begin {gather*} -\frac {2 (a e+c d x)^6 \left (7 e^5-\frac {45 c d e^4 (d+e x)}{a e+c d x}+\frac {126 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 {315 c^4 d^4 e (d+e x)^4}{(a e+c d x)^4}+\frac {63 c^5 d^5 (d+e x)^5}{(a e+c d x)^5}\right )}{63 \left (c d^2-a e^2\right )^6 (d+e x)^3 ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-2*(a*e + c*d*x)^6*(7*e^5 - (45*c*d*e^4*(d + e*x))/(a*e + c*d*x) + (126*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 + (315*c^4*d^4*e*(d + e*x)^4)/(a*e + c*d*x)^4 + (63*c^5*d^
5*(d + e*x)^5)/(a*e + c*d*x)^5))/(63*(c*d^2 - a*e^2)^6*(d + e*x)^3*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]
time = 0.79, size = 389, normalized size = 1.29

method	result	size
default	\(\frac {-\frac {2}{9 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {10 c d e \left (-\frac {2}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {8 c d e \left (-\frac {2}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 c d e \left (-\frac {2}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (e^{2} a -c \,d^{2}\right )}\right )}{7 \left (e^{2} a -c \,d^{2}\right )}\right )}{9 \left (e^{2} a -c \,d^{2}\right )}}{e^{4}}\)	\(389\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (256 c^{5} d^{5} e^{5} x^{5}+128 a \,c^{4} d^{4} e^{6} x^{4}+1152 c^{5} d^{6} e^{4} x^{4}-32 a^{2} c^{3} d^{3} e^{7} x^{3}+576 a \,c^{4} d^{5} e^{5} x^{3}+2016 c^{5} d^{7} e^{3} x^{3}+16 a^{3} c^{2} d^{2} e^{8} x^{2}-144 a^{2} c^{3} d^{4} e^{6} x^{2}+1008 a \,c^{4} d^{6} e^{4} x^{2}+1680 c^{5} d^{8} e^{2} x^{2}-10 a^{4} c d \,e^{9} x +72 a^{3} c^{2} d^{3} e^{7} x -252 a^{2} c^{3} d^{5} e^{5} x +840 a \,c^{4} d^{7} e^{3} x +630 c^{5} d^{9} e x +7 a^{5} e^{10}-45 a^{4} c \,d^{2} e^{8}+126 a^{3} c^{2} d^{4} e^{6}-210 a^{2} c^{3} d^{6} e^{4}+315 a \,c^{4} d^{8} e^{2}+63 c^{5} d^{10}\right )}{63 \left (e x +d \right )^{3} \left (a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(412\)
trager	\(-\frac {2 \left (256 c^{5} d^{5} e^{5} x^{5}+128 a \,c^{4} d^{4} e^{6} x^{4}+1152 c^{5} d^{6} e^{4} x^{4}-32 a^{2} c^{3} d^{3} e^{7} x^{3}+576 a \,c^{4} d^{5} e^{5} x^{3}+2016 c^{5} d^{7} e^{3} x^{3}+16 a^{3} c^{2} d^{2} e^{8} x^{2}-144 a^{2} c^{3} d^{4} e^{6} x^{2}+1008 a \,c^{4} d^{6} e^{4} x^{2}+1680 c^{5} d^{8} e^{2} x^{2}-10 a^{4} c d \,e^{9} x +72 a^{3} c^{2} d^{3} e^{7} x -252 a^{2} c^{3} d^{5} e^{5} x +840 a \,c^{4} d^{7} e^{3} x +630 c^{5} d^{9} e x +7 a^{5} e^{10}-45 a^{4} c \,d^{2} e^{8}+126 a^{3} c^{2} d^{4} e^{6}-210 a^{2} c^{3} d^{6} e^{4}+315 a \,c^{4} d^{8} e^{2}+63 c^{5} d^{10}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{63 \left (c d x +a e \right ) \left (e^{2} a -c \,d^{2}\right ) \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (e x +d \right )^{5}}\)	\(415\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/e^4*(-2/9/(a*e^2-c*d^2)/(x+d/e)^4/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-10/9*c*d*e/(a*e^2-c*d^2)*(-2
/7/(a*e^2-c*d^2)/(x+d/e)^3/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-8/7*c*d*e/(a*e^2-c*d^2)*(-2/5/(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))^(1/2)-6/5*c*d*e/(a*e^2-c*d^2)*(-2/3/(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))^(1/2)+8/3*c*d*e/(a*e^2-c*d^2)^3*(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)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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. 1019 vs. \(2 (285) = 570\).
time = 136.06, size = 1019, normalized size = 3.39 \begin {gather*} -\frac {2 \, {\left (630 \, c^{5} d^{9} x e + 63 \, c^{5} d^{10} - 10 \, a^{4} c d x e^{9} + 7 \, a^{5} e^{10} + {\left (16 \, a^{3} c^{2} d^{2} x^{2} - 45 \, a^{4} c d^{2}\right )} e^{8} - 8 \, {\left (4 \, a^{2} c^{3} d^{3} x^{3} - 9 \, a^{3} c^{2} d^{3} x\right )} e^{7} + 2 \, {\left (64 \, a c^{4} d^{4} x^{4} - 72 \, a^{2} c^{3} d^{4} x^{2} + 63 \, a^{3} c^{2} d^{4}\right )} e^{6} + 4 \, {\left (64 \, c^{5} d^{5} x^{5} + 144 \, a c^{4} d^{5} x^{3} - 63 \, a^{2} c^{3} d^{5} x\right )} e^{5} + 6 \, {\left (192 \, c^{5} d^{6} x^{4} + 168 \, a c^{4} d^{6} x^{2} - 35 \, a^{2} c^{3} d^{6}\right )} e^{4} + 168 \, {\left (12 \, c^{5} d^{7} x^{3} + 5 \, a c^{4} d^{7} x\right )} e^{3} + 105 \, {\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}}{63 \, {\left (c^{7} d^{18} x + a^{7} x^{5} e^{18} + {\left (a^{6} c d x^{6} + 5 \, a^{7} d x^{4}\right )} e^{17} - {\left (a^{6} c d^{2} x^{5} - 10 \, a^{7} d^{2} x^{3}\right )} e^{16} - 2 \, {\left (3 \, a^{5} c^{2} d^{3} x^{6} + 10 \, a^{6} c d^{3} x^{4} - 5 \, a^{7} d^{3} x^{2}\right )} e^{15} - 5 \, {\left (3 \, a^{5} c^{2} d^{4} x^{5} + 10 \, a^{6} c d^{4} x^{3} - a^{7} d^{4} x\right )} e^{14} + {\left (15 \, a^{4} c^{3} d^{5} x^{6} + 15 \, a^{5} c^{2} d^{5} x^{4} - 55 \, a^{6} c d^{5} x^{2} + a^{7} d^{5}\right )} e^{13} + {\left (55 \, a^{4} c^{3} d^{6} x^{5} + 90 \, a^{5} c^{2} d^{6} x^{3} - 29 \, a^{6} c d^{6} x\right )} e^{12} - 2 \, {\left (10 \, a^{3} c^{4} d^{7} x^{6} - 25 \, a^{4} c^{3} d^{7} x^{4} - 60 \, a^{5} c^{2} d^{7} x^{2} + 3 \, a^{6} c d^{7}\right )} e^{11} - {\left (85 \, a^{3} c^{4} d^{8} x^{5} + 50 \, a^{4} c^{3} d^{8} x^{3} - 69 \, a^{5} c^{2} d^{8} x\right )} e^{10} + 5 \, {\left (3 \, a^{2} c^{5} d^{9} x^{6} - 25 \, a^{3} c^{4} d^{9} x^{4} - 25 \, a^{4} c^{3} d^{9} x^{2} + 3 \, a^{5} c^{2} d^{9}\right )} e^{9} + {\left (69 \, a^{2} c^{5} d^{10} x^{5} - 50 \, a^{3} c^{4} d^{10} x^{3} - 85 \, a^{4} c^{3} d^{10} x\right )} e^{8} - 2 \, {\left (3 \, a c^{6} d^{11} x^{6} - 60 \, a^{2} c^{5} d^{11} x^{4} - 25 \, a^{3} c^{4} d^{11} x^{2} + 10 \, a^{4} c^{3} d^{11}\right )} e^{7} - {\left (29 \, a c^{6} d^{12} x^{5} - 90 \, a^{2} c^{5} d^{12} x^{3} - 55 \, a^{3} c^{4} d^{12} x\right )} e^{6} + {\left (c^{7} d^{13} x^{6} - 55 \, a c^{6} d^{13} x^{4} + 15 \, a^{2} c^{5} d^{13} x^{2} + 15 \, a^{3} c^{4} d^{13}\right )} e^{5} + 5 \, {\left (c^{7} d^{14} x^{5} - 10 \, a c^{6} d^{14} x^{3} - 3 \, a^{2} c^{5} d^{14} x\right )} e^{4} + 2 \, {\left (5 \, c^{7} d^{15} x^{4} - 10 \, a c^{6} d^{15} x^{2} - 3 \, a^{2} c^{5} d^{15}\right )} e^{3} + {\left (10 \, c^{7} d^{16} x^{3} - a c^{6} d^{16} x\right )} e^{2} + {\left (5 \, c^{7} d^{17} x^{2} + a c^{6} d^{17}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

-2/63*(630*c^5*d^9*x*e + 63*c^5*d^10 - 10*a^4*c*d*x*e^9 + 7*a^5*e^10 + (16*a^3*c^2*d^2*x^2 - 45*a^4*c*d^2)*e^8
- 8*(4*a^2*c^3*d^3*x^3 - 9*a^3*c^2*d^3*x)*e^7 + 2*(64*a*c^4*d^4*x^4 - 72*a^2*c^3*d^4*x^2 + 63*a^3*c^2*d^4)*e^
6 + 4*(64*c^5*d^5*x^5 + 144*a*c^4*d^5*x^3 - 63*a^2*c^3*d^5*x)*e^5 + 6*(192*c^5*d^6*x^4 + 168*a*c^4*d^6*x^2 - 3
5*a^2*c^3*d^6)*e^4 + 168*(12*c^5*d^7*x^3 + 5*a*c^4*d^7*x)*e^3 + 105*(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^7*d^18*x + a^7*x^5*e^18 + (a^6*c*d*x^6 + 5*a^7*d*x^4)*e^17 - (a^6*c*d
^2*x^5 - 10*a^7*d^2*x^3)*e^16 - 2*(3*a^5*c^2*d^3*x^6 + 10*a^6*c*d^3*x^4 - 5*a^7*d^3*x^2)*e^15 - 5*(3*a^5*c^2*d
^4*x^5 + 10*a^6*c*d^4*x^3 - a^7*d^4*x)*e^14 + (15*a^4*c^3*d^5*x^6 + 15*a^5*c^2*d^5*x^4 - 55*a^6*c*d^5*x^2 + a^
7*d^5)*e^13 + (55*a^4*c^3*d^6*x^5 + 90*a^5*c^2*d^6*x^3 - 29*a^6*c*d^6*x)*e^12 - 2*(10*a^3*c^4*d^7*x^6 - 25*a^4
*c^3*d^7*x^4 - 60*a^5*c^2*d^7*x^2 + 3*a^6*c*d^7)*e^11 - (85*a^3*c^4*d^8*x^5 + 50*a^4*c^3*d^8*x^3 - 69*a^5*c^2*
d^8*x)*e^10 + 5*(3*a^2*c^5*d^9*x^6 - 25*a^3*c^4*d^9*x^4 - 25*a^4*c^3*d^9*x^2 + 3*a^5*c^2*d^9)*e^9 + (69*a^2*c^
5*d^10*x^5 - 50*a^3*c^4*d^10*x^3 - 85*a^4*c^3*d^10*x)*e^8 - 2*(3*a*c^6*d^11*x^6 - 60*a^2*c^5*d^11*x^4 - 25*a^3
*c^4*d^11*x^2 + 10*a^4*c^3*d^11)*e^7 - (29*a*c^6*d^12*x^5 - 90*a^2*c^5*d^12*x^3 - 55*a^3*c^4*d^12*x)*e^6 + (c^
7*d^13*x^6 - 55*a*c^6*d^13*x^4 + 15*a^2*c^5*d^13*x^2 + 15*a^3*c^4*d^13)*e^5 + 5*(c^7*d^14*x^5 - 10*a*c^6*d^14*
x^3 - 3*a^2*c^5*d^14*x)*e^4 + 2*(5*c^7*d^15*x^4 - 10*a*c^6*d^15*x^2 - 3*a^2*c^5*d^15)*e^3 + (10*c^7*d^16*x^3 -
a*c^6*d^16*x)*e^2 + (5*c^7*d^17*x^2 + a*c^6*d^17)*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)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),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)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.36, size = 2500, normalized size = 8.31 \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)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)

[Out]

(((64*c^5*d^6*e)/(315*(a*e^2 - c*d^2)^7) + (16*c^4*d^4*e*(11*a*e^2 - 15*c*d^2))/(315*(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) - (((e^2*(18*c^2*d^3 - 34*a*c*d*e^2))/(9*(a*e^2 - c*d^2)^2
*(7*a^2*e^5 + 7*c^2*d^4*e - 14*a*c*d^2*e^3)) + (16*c^2*d^3*e^2)/(9*(a*e^2 - c*d^2)^2*(7*a^2*e^5 + 7*c^2*d^4*e
- 14*a*c*d^2*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 - (((d*((64*c^5*d^6*e^3)/(315*(
a*e^2 - c*d^2)^6*(3*a*e^3 - 3*c*d^2*e)) + (8*c^4*d^4*e^3*(35*a*e^2 - 51*c*d^2))/(315*(a*e^2 - c*d^2)^6*(3*a*e^
3 - 3*c*d^2*e))))/e + (4*c^3*d^3*e^2*(47*c^2*d^4 - 39*a^2*e^4 + 8*a*c*d^2*e^2))/(315*(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 + (((d*((32*c^4*d^5*e^3)/(63*(a*e^
2 - c*d^2)^4*(5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2*e^3)) + (4*c^3*d^3*e^3*(39*a*e^2 - 55*c*d^2))/(63*(a*e^2 -
c*d^2)^4*(5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2*e^3))))/e - (e^2*(126*c^4*d^6 - 472*a*c^3*d^4*e^2 + 314*a^2*c^2
*d^2*e^4))/(63*(a*e^2 - c*d^2)^4*(5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d
*e*x^2)^(1/2))/(d + e*x)^3 + (((32*c^3*d^4*e^2)/(63*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)) - (16*c^2*d^2*e^2
*(a*e^2 + c*d^2))/(63*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))
/(d + e*x)^3 + (((d*((d*((128*c^7*d^8*e^3)/(945*(a*e^2 - c*d^2)^9) + (16*c^6*d^6*e^3*(23*a*e^2 - 47*c*d^2))/(9
45*(a*e^2 - c*d^2)^9)))/e + (16*c^5*d^5*e^2*(109*a^2*e^4 + 179*c^2*d^4 - 264*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2
)^9)))/e - (4*c^4*d^4*e*(245*a^3*e^6 + 315*c^3*d^6 - 229*a*c^2*d^4*e^2 - 299*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d
^2)^9))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((d*((64*c^6*d^7*e^4)/(315*(a*e^2 - c*
d^2)^6*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3)) + (8*c^5*d^5*e^4*(9*a*e^2 - 17*c*d^2))/(105*(a*e^2 - c*d^2)^
6*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3))))/e + (4*c^4*d^4*e^3*(245*a^2*e^4 + 401*c^2*d^4 - 598*a*c*d^2*e^2
))/(315*(a*e^2 - c*d^2)^6*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3))))/e + (e^2*(630*c^6*d^9 - 3494*a*c^5*d^7*
e^2 + 4690*a^2*c^4*d^5*e^4 - 1890*a^3*c^3*d^3*e^6))/(315*(a*e^2 - c*d^2)^6*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^
2*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (2*e^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*
e*x^2)^(1/2))/((d + e*x)^5*(9*a^2*e^5 + 9*c^2*d^4*e - 18*a*c*d^2*e^3)) - ((x*((a*(((a*e^2 + c*d^2)*((128*c^9*d
^9*e^5*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^9*d^9*e^5*
(15*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (32*c^8
*d^8*e^4*(98*a^2*e^4 + 191*c^2*d^4 - 241*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a
^2*c*d*e^5)) - (256*a*c^9*d^10*e^6)/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*
c^8*d^8*e^4*(a*e^2 + c*d^2)*(15*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^8*(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*((128*c^9*d^9*e^5*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a
*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^9*d^9*e^5*(15*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^8*(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)*((128*c^9*d^9*e^5*(a*e^2 + c*d^2))/(945*(
a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^9*d^9*e^5*(15*a*e^2 - 47*c*d^2))/(945*(a
*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (32*c^8*d^8*e^4*(98*a^2*e^4 + 191*c^2
*d^4 - 241*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (256*a*c^9*d^10
*e^6)/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^8*d^8*e^4*(a*e^2 + c*d^2)*(1
5*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (16*c^7*d
^7*e^3*(213*a^3*e^6 - 759*c^3*d^6 + 1513*a*c^2*d^4*e^2 - 1031*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d^2)^8*(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^3*(a*e^2 + c*d^2)*(98*a^2*e^4 + 191*c^2*d^4 - 241*a*c*d^2*
e^2))/(945*(a*e^2 - c*d^2)^8*(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*(1890*c^8
*d^12 - 13632*a*c^7*d^10*e^2 + 26500*a^2*c^6*d^8*e^4 - 20416*a^3*c^5*d^6*e^6 + 5530*a^4*c^4*d^4*e^8))/(945*(a*
e^2 - c*d^2)^8*(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^2*(a*e^2 + c*d^2)*(213*a^3*e^6 - 75
9*c^3*d^6 + 1513*a*c^2*d^4*e^2 - 1031*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^
2*c*d*e^5))) + (a*((a*((128*c^9*d^9*e^5*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 +
a^2*c*d*e^5)) + (32*c^9*d^9*e^5*(15*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^8*(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)*((128*c^9*d^9*e^5*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)
^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^9*d^9*e^5*(15*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^
8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))...

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