3.1938 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^3} \, dx\)
Optimal. Leaf size=244 \[ \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^2}-\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e^2}-\frac {5 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 \sqrt {c} \sqrt {d} e^{7/2}}+\frac {5 \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^3} \]
[Out]
-5/3*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^2+2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e/(e*x+d)^2-5/1
6*(-a*e^2+c*d^2)^3*arctanh(1/2*(2*c*d*e*x+a*e^2+c*d^2)/c^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2))/e^(7/2)/c^(1/2)/d^(1/2)+5/8*(-a*e^2+c*d^2)*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2)/e^3
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Rubi [A] time = 0.20, antiderivative size = 244, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used =
{662, 664, 612, 621, 206} \[ \frac {5 \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e^3}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^2}-\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e^2}-\frac {5 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 \sqrt {c} \sqrt {d} e^{7/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^3,x]
[Out]
(5*(c*d^2 - a*e^2)*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*e^3) - (5*c*d*(
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*e^2) + (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(e*(
d + e*x)^2) - (5*(c*d^2 - a*e^2)^3*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*Sqrt[c]*Sqrt[d]*e^(7/2))
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 612
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]
Rule 621
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 662
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2
)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && IntegerQ[2*p]
Rule 664
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[(p*(2*c*d - b*e))/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*
(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a
*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^3} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^2}-\frac {(5 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx}{e}\\ &=-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e^2}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^2}+\frac {\left (5 c d \left (c d^2-a e^2\right )\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{2 e^2}\\ &=\frac {5 \left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^3}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e^2}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^2}-\frac {\left (5 \left (c d^2-a e^2\right )^3\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 e^3}\\ &=\frac {5 \left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^3}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e^2}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^2}-\frac {\left (5 \left (c d^2-a e^2\right )^3\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 e^3}\\ &=\frac {5 \left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^3}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e^2}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e (d+e x)^2}-\frac {5 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 \sqrt {c} \sqrt {d} e^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 252, normalized size = 1.03 \[ \frac {\sqrt {e} (c d)^{3/2} (d+e x) \left (33 a^3 e^5+a^2 c d e^3 (59 e x-40 d)+a c^2 d^2 e \left (15 d^2-50 d e x+34 e^2 x^2\right )+c^3 d^3 x \left (15 d^2-10 d e x+8 e^2 x^2\right )\right )-15 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{7/2} \sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{24 e^{7/2} (c d)^{3/2} \sqrt {(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^3,x]
[Out]
((c*d)^(3/2)*Sqrt[e]*(d + e*x)*(33*a^3*e^5 + a^2*c*d*e^3*(-40*d + 59*e*x) + c^3*d^3*x*(15*d^2 - 10*d*e*x + 8*e
^2*x^2) + a*c^2*d^2*e*(15*d^2 - 50*d*e*x + 34*e^2*x^2)) - 15*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^(7/2)*Sqrt[a*e +
c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sq
rt[c*d^2 - a*e^2])])/(24*(c*d)^(3/2)*e^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
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fricas [A] time = 1.13, size = 534, normalized size = 2.19 \[ \left [\frac {15 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} + 15 \, c^{3} d^{5} e - 40 \, a c^{2} d^{3} e^{3} + 33 \, a^{2} c d e^{5} - 2 \, {\left (5 \, c^{3} d^{4} e^{2} - 13 \, a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{96 \, c d e^{4}}, \frac {15 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} + 15 \, c^{3} d^{5} e - 40 \, a c^{2} d^{3} e^{3} + 33 \, a^{2} c d e^{5} - 2 \, {\left (5 \, c^{3} d^{4} e^{2} - 13 \, a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, c d e^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^3,x, algorithm="fricas")
[Out]
[1/96*(15*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4
+ 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d
*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(8*c^3*d^3*e^3*x^2 + 15*c^3*d^5*e - 40*a*c^2*d^3*e^3 + 33*a^2*c*d*e^5 -
2*(5*c^3*d^4*e^2 - 13*a*c^2*d^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c*d*e^4), 1/48*(15*(c^3
*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)
) + 2*(8*c^3*d^3*e^3*x^2 + 15*c^3*d^5*e - 40*a*c^2*d^3*e^3 + 33*a^2*c*d*e^5 - 2*(5*c^3*d^4*e^2 - 13*a*c^2*d^2*
e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c*d*e^4)]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^3,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.61Unable to divide, perhaps due to rounding error%%%{%%%{16,[0,0,6]%%%},[6,6,6]%%%}+%%%{%%%{96,
[1,2,6]%%%},[6,5,5]%%%}+%%%{%%%{-192,[1,2,8]%%%},[6,5,4]%%%}+%%%{%%%{240,[2,4,6]%%%},[6,4,4]%%%}+%%%{%%%{-768,
[2,4,8]%%%},[6,4,3]%%%}+%%%{%%%{768,[2,4,10]%%%},[6,4,2]%%%}+%%%{%%%{320,[3,6,6]%%%},[6,3,3]%%%}+%%%{%%%{-1152
,[3,6,8]%%%},[6,3,2]%%%}+%%%{%%%{1536,[3,6,10]%%%},[6,3,1]%%%}+%%%{%%%{-1024,[3,6,12]%%%},[6,3,0]%%%}+%%%{%%%{
240,[4,8,6]%%%},[6,2,2]%%%}+%%%{%%%{-768,[4,8,8]%%%},[6,2,1]%%%}+%%%{%%%{768,[4,8,10]%%%},[6,2,0]%%%}+%%%{%%%{
96,[5,10,6]%%%},[6,1,1]%%%}+%%%{%%%{-192,[5,10,8]%%%},[6,1,0]%%%}+%%%{%%%{16,[6,12,6]%%%},[6,0,0]%%%}+%%%{%%{[
%%%{-96,[0,1,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,6,6]%%%}+%%%{%%{[%%%{-576,[1,3,5]%%%},0]:[1,0,%%%{-1,[1,
1,1]%%%}]%%},[5,5,5]%%%}+%%%{%%{[%%%{1152,[1,3,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,5,4]%%%}+%%%{%%{[%%%{-
1440,[2,5,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,4,4]%%%}+%%%{%%{[%%%{4608,[2,5,7]%%%},0]:[1,0,%%%{-1,[1,1,1
]%%%}]%%},[5,4,3]%%%}+%%%{%%{[%%%{-4608,[2,5,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,4,2]%%%}+%%%{%%{[%%%{-19
20,[3,7,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,3,3]%%%}+%%%{%%{[%%%{6912,[3,7,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%
%%}]%%},[5,3,2]%%%}+%%%{%%{[%%%{-9216,[3,7,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,3,1]%%%}+%%%{%%{[%%%{6144,
[3,7,11]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,3,0]%%%}+%%%{%%{[%%%{-1440,[4,9,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%
%}]%%},[5,2,2]%%%}+%%%{%%{[%%%{4608,[4,9,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,2,1]%%%}+%%%{%%{[%%%{-4608,[
4,9,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,2,0]%%%}+%%%{%%{[%%%{-576,[5,11,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}
]%%},[5,1,1]%%%}+%%%{%%{[%%%{1152,[5,11,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,1,0]%%%}+%%%{%%{[%%%{-96,[6,1
3,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,0,0]%%%}+%%%{%%%{48,[0,1,5]%%%},[4,7,7]%%%}+%%%{%%%{-48,[0,1,7]%%%}
,[4,7,6]%%%}+%%%{%%%{528,[1,3,5]%%%},[4,6,6]%%%}+%%%{%%%{-864,[1,3,7]%%%},[4,6,5]%%%}+%%%{%%%{576,[1,3,9]%%%},
[4,6,4]%%%}+%%%{%%%{2160,[2,5,5]%%%},[4,5,5]%%%}+%%%{%%%{-5904,[2,5,7]%%%},[4,5,4]%%%}+%%%{%%%{4608,[2,5,9]%%%
},[4,5,3]%%%}+%%%{%%%{-2304,[2,5,11]%%%},[4,5,2]%%%}+%%%{%%%{4560,[3,7,5]%%%},[4,4,4]%%%}+%%%{%%%{-15936,[3,7,
7]%%%},[4,4,3]%%%}+%%%{%%%{19584,[3,7,9]%%%},[4,4,2]%%%}+%%%{%%%{-7680,[3,7,11]%%%},[4,4,1]%%%}+%%%{%%%{3072,[
3,7,13]%%%},[4,4,0]%%%}+%%%{%%%{5520,[4,9,5]%%%},[4,3,3]%%%}+%%%{%%%{-20304,[4,9,7]%%%},[4,3,2]%%%}+%%%{%%%{27
648,[4,9,9]%%%},[4,3,1]%%%}+%%%{%%%{-17664,[4,9,11]%%%},[4,3,0]%%%}+%%%{%%%{3888,[5,11,5]%%%},[4,2,2]%%%}+%%%{
%%%{-12384,[5,11,7]%%%},[4,2,1]%%%}+%%%{%%%{12096,[5,11,9]%%%},[4,2,0]%%%}+%%%{%%%{1488,[6,13,5]%%%},[4,1,1]%%
%}+%%%{%%%{-2928,[6,13,7]%%%},[4,1,0]%%%}+%%%{%%%{240,[7,15,5]%%%},[4,0,0]%%%}+%%%{%%{[%%%{-192,[0,2,4]%%%},0]
:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,7,7]%%%}+%%%{%%{[%%%{192,[0,2,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,7,6]%%%
}+%%%{%%{[%%%{-1472,[1,4,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,6,6]%%%}+%%%{%%{[%%%{3456,[1,4,6]%%%},0]:[1,
0,%%%{-1,[1,1,1]%%%}]%%},[3,6,5]%%%}+%%%{%%{[%%%{-2304,[1,4,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,6,4]%%%}+
%%%{%%{[%%%{-4800,[2,6,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,5,5]%%%}+%%%{%%{[%%%{15936,[2,6,6]%%%},0]:[1,0
,%%%{-1,[1,1,1]%%%}]%%},[3,5,4]%%%}+%%%{%%{[%%%{-18432,[2,6,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,5,3]%%%}+
%%%{%%{[%%%{9216,[2,6,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,5,2]%%%}+%%%{%%{[%%%{-8640,[3,8,4]%%%},0]:[1,0
,%%%{-1,[1,1,1]%%%}]%%},[3,4,4]%%%}+%%%{%%{[%%%{33024,[3,8,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,4,3]%%%}+%
%%{%%{[%%%{-47616,[3,8,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,4,2]%%%}+%%%{%%{[%%%{30720,[3,8,10]%%%},0]:[1,
0,%%%{-1,[1,1,1]%%%}]%%},[3,4,1]%%%}+%%%{%%{[%%%{-12288,[3,8,12]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,4,0]%%%
}+%%%{%%{[%%%{-9280,[4,10,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,3,3]%%%}+%%%{%%{[%%%{35136,[4,10,6]%%%},0]:
[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,3,2]%%%}+%%%{%%{[%%%{-49152,[4,10,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,3,1]
%%%}+%%%{%%{[%%%{29696,[4,10,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,3,0]%%%}+%%%{%%{[%%%{-5952,[5,12,4]%%%}
,0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,2,2]%%%}+%%%{%%{[%%%{18816,[5,12,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,2
,1]%%%}+%%%{%%{[%%%{-17664,[5,12,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,2,0]%%%}+%%%{%%{[%%%{-2112,[6,14,4]%
%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,1,1]%%%}+%%%{%%{[%%%{4032,[6,14,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3
,1,0]%%%}+%%%{%%{[%%%{-320,[7,16,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,0,0]%%%}+%%%{%%%{48,[0,2,4]%%%},[2,8
,8]%%%}+%%%{%%%{-96,[0,2,6]%%%},[2,8,7]%%%}+%%%{%%%{48,[0,2,8]%%%},[2,8,6]%%%}+%%%{%%%{576,[1,4,4]%%%},[2,7,7]
%%%}+%%%{%%%{-1440,[1,4,6]%%%},[2,7,6]%%%}+%%%{%%%{1440,[1,4,8]%%%},[2,7,5]%%%}+%%%{%%%{-576,[1,4,10]%%%},[2,7
,4]%%%}+%%%{%%%{2688,[2,6,4]%%%},[2,6,6]%%%}+%%%{%%%{-8928,[2,6,6]%%%},[2,6,5]%%%}+%%%{%%%{11088,[2,6,8]%%%},[
2,6,4]%%%}+%%%{%%%{-6912,[2,6,10]%%%},[2,6,3]%%%}+%%%{%%%{2304,[2,6,12]%%%},[2,6,2]%%%}+%%%{%%%{6720,[3,8,4]%%
%},[2,5,5]%%%}+%%%{%%%{-26400,[3,8,6]%%%},[2,5,4]%%%}+%%%{%%%{40128,[3,8,8]%%%},[2,5,3]%%%}+%%%{%%%{-29568,[3,
8,10]%%%},[2,5,2]%%%}+%%%{%%%{10752,[3,8,12]%%%},[2,5,1]%%%}+%%%{%%%{-3072,[3,8,14]%%%},[2,5,0]%%%}+%%%{%%%{10
080,[4,10,4]%%%},[2,4,4]%%%}+%%%{%%%{-41760,[4,10,6]%%%},[2,4,3]%%%}+%%%{%%%{67536,[4,10,8]%%%},[2,4,2]%%%}+%%
%{%%%{-52992,[4,10,10]%%%},[2,4,1]%%%}+%%%{%%%{20736,[4,10,12]%%%},[2,4,0]%%%}+%%%{%%%{9408,[5,12,4]%%%},[2,3,
3]%%%}+%%%{%%%{-36576,[5,12,6]%%%},[2,3,2]%%%}+%%%{%%%{52128,[5,12,8]%%%},[2,3,1]%%%}+%%%{%%%{-29760,[5,12,10]
%%%},[2,3,0]%%%}+%%%{%%%{5376,[6,14,4]%%%},[2,2,2]%%%}+%%%{%%%{-16800,[6,14,6]%%%},[2,2,1]%%%}+%%%{%%%{15024,[
6,14,8]%%%},[2,2,0]%%%}+%%%{%%%{1728,[7,16,4]%%%},[2,1,1]%%%}+%%%{%%%{-3168,[7,16,6]%%%},[2,1,0]%%%}+%%%{%%%{2
40,[8,18,4]%%%},[2,0,0]%%%}+%%%{%%{[%%%{-96,[0,3,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,8]%%%}+%%%{%%{[%%%
{192,[0,3,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,7]%%%}+%%%{%%{[%%%{-96,[0,3,7]%%%},0]:[1,0,%%%{-1,[1,1,1]
%%%}]%%},[1,8,6]%%%}+%%%{%%{[%%%{-768,[1,5,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,7]%%%}+%%%{%%{[%%%{2496,
[1,5,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,6]%%%}+%%%{%%{[%%%{-2880,[1,5,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%
}]%%},[1,7,5]%%%}+%%%{%%{[%%%{1152,[1,5,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,4]%%%}+%%%{%%{[%%%{-2688,[2
,7,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,6]%%%}+%%%{%%{[%%%{10944,[2,7,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]
%%},[1,6,5]%%%}+%%%{%%{[%%%{-17568,[2,7,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,4]%%%}+%%%{%%{[%%%{13824,[2
,7,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,3]%%%}+%%%{%%{[%%%{-4608,[2,7,11]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}
]%%},[1,6,2]%%%}+%%%{%%{[%%%{-5376,[3,9,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,5]%%%}+%%%{%%{[%%%{24000,[3
,9,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,4]%%%}+%%%{%%{[%%%{-43392,[3,9,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}
]%%},[1,5,3]%%%}+%%%{%%{[%%%{40704,[3,9,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,2]%%%}+%%%{%%{[%%%{-21504,[
3,9,11]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,1]%%%}+%%%{%%{[%%%{6144,[3,9,13]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%
}]%%},[1,5,0]%%%}+%%%{%%{[%%%{-6720,[4,11,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,4]%%%}+%%%{%%{[%%%{29760,
[4,11,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,3]%%%}+%%%{%%{[%%%{-52128,[4,11,7]%%%},0]:[1,0,%%%{-1,[1,1,1]
%%%}]%%},[1,4,2]%%%}+%%%{%%{[%%%{44544,[4,11,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,1]%%%}+%%%{%%{[%%%{-16
896,[4,11,11]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,0]%%%}+%%%{%%{[%%%{-5376,[5,13,3]%%%},0]:[1,0,%%%{-1,[1,
1,1]%%%}]%%},[1,3,3]%%%}+%%%{%%{[%%%{21312,[5,13,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,2]%%%}+%%%{%%{[%%%
{-30528,[5,13,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,1]%%%}+%%%{%%{[%%%{16512,[5,13,9]%%%},0]:[1,0,%%%{-1,
[1,1,1]%%%}]%%},[1,3,0]%%%}+%%%{%%{[%%%{-2688,[6,15,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,2,2]%%%}+%%%{%%{[
%%%{8256,[6,15,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,2,1]%%%}+%%%{%%{[%%%{-7008,[6,15,7]%%%},0]:[1,0,%%%{-1
,[1,1,1]%%%}]%%},[1,2,0]%%%}+%%%{%%{[%%%{-768,[7,17,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,1,1]%%%}+%%%{%%{[
%%%{1344,[7,17,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,1,0]%%%}+%%%{%%{[%%%{-96,[8,19,3]%%%},0]:[1,0,%%%{-1,[
1,1,1]%%%}]%%},[1,0,0]%%%}+%%%{%%%{16,[0,3,3]%%%},[0,9,9]%%%}+%%%{%%%{-48,[0,3,5]%%%},[0,9,8]%%%}+%%%{%%%{48,[
0,3,7]%%%},[0,9,7]%%%}+%%%{%%%{-16,[0,3,9]%%%},[0,9,6]%%%}+%%%{%%%{144,[1,5,3]%%%},[0,8,8]%%%}+%%%{%%%{-576,[1
,5,5]%%%},[0,8,7]%%%}+%%%{%%%{912,[1,5,7]%%%},[0,8,6]%%%}+%%%{%%%{-672,[1,5,9]%%%},[0,8,5]%%%}+%%%{%%%{192,[1,
5,11]%%%},[0,8,4]%%%}+%%%{%%%{576,[2,7,3]%%%},[0,7,7]%%%}+%%%{%%%{-2688,[2,7,5]%%%},[0,7,6]%%%}+%%%{%%%{5232,[
2,7,7]%%%},[0,7,5]%%%}+%%%{%%%{-5424,[2,7,9]%%%},[0,7,4]%%%}+%%%{%%%{3072,[2,7,11]%%%},[0,7,3]%%%}+%%%{%%%{-76
8,[2,7,13]%%%},[0,7,2]%%%}+%%%{%%%{1344,[3,9,3]%%%},[0,6,6]%%%}+%%%{%%%{-6720,[3,9,5]%%%},[0,6,5]%%%}+%%%{%%%{
14160,[3,9,7]%%%},[0,6,4]%%%}+%%%{%%%{-16320,[3,9,9]%%%},[0,6,3]%%%}+%%%{%%%{11136,[3,9,11]%%%},[0,6,2]%%%}+%%
%{%%%{-4608,[3,9,13]%%%},[0,6,1]%%%}+%%%{%%%{1024,[3,9,15]%%%},[0,6,0]%%%}+%%%{%%%{2016,[4,11,3]%%%},[0,5,5]%%
%}+%%%{%%%{-10080,[4,11,5]%%%},[0,5,4]%%%}+%%%{%%%{20880,[4,11,7]%%%},[0,5,3]%%%}+%%%{%%%{-22896,[4,11,9]%%%},
[0,5,2]%%%}+%%%{%%%{13824,[4,11,11]%%%},[0,5,1]%%%}+%%%{%%%{-3840,[4,11,13]%%%},[0,5,0]%%%}+%%%{%%%{2016,[5,13
,3]%%%},[0,4,4]%%%}+%%%{%%%{-9408,[5,13,5]%%%},[0,4,3]%%%}+%%%{%%%{17328,[5,13,7]%%%},[0,4,2]%%%}+%%%{%%%{-152
64,[5,13,9]%%%},[0,4,1]%%%}+%%%{%%%{5568,[5,13,11]%%%},[0,4,0]%%%}+%%%{%%%{1344,[6,15,3]%%%},[0,3,3]%%%}+%%%{%
%%{-5376,[6,15,5]%%%},[0,3,2]%%%}+%%%{%%%{7632,[6,15,7]%%%},[0,3,1]%%%}+%%%{%%%{-3920,[6,15,9]%%%},[0,3,0]%%%}
+%%%{%%%{576,[7,17,3]%%%},[0,2,2]%%%}+%%%{%%%{-1728,[7,17,5]%%%},[0,2,1]%%%}+%%%{%%%{1392,[7,17,7]%%%},[0,2,0]
%%%}+%%%{%%%{144,[8,19,3]%%%},[0,1,1]%%%}+%%%{%%%{-240,[8,19,5]%%%},[0,1,0]%%%}+%%%{%%%{16,[9,21,3]%%%},[0,0,0
]%%%} / %%%{%%{poly1[%%%{-1,[1,1,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[6,0,0]%%%}+%%%{%%%{6,[2,3,4]%%%},[5,0,
0]%%%}+%%%{%%{poly1[%%%{-3,[1,2,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[4,1,1]%%%}+%%%{%%{poly1[%%%{3,[1,2,5]%%
%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[4,1,0]%%%}+%%%{%%{poly1[%%%{-15,[2,4,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},
[4,0,0]%%%}+%%%{%%%{12,[2,4,3]%%%},[3,1,1]%%%}+%%%{%%%{-12,[2,4,5]%%%},[3,1,0]%%%}+%%%{%%%{20,[3,6,3]%%%},[3,0
,0]%%%}+%%%{%%{poly1[%%%{-3,[1,3,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,2,2]%%%}+%%%{%%{poly1[%%%{6,[1,3,4]%
%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,2,1]%%%}+%%%{%%{poly1[%%%{-3,[1,3,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},
[2,2,0]%%%}+%%%{%%{poly1[%%%{-18,[2,5,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,1,1]%%%}+%%%{%%{poly1[%%%{18,[2
,5,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,1,0]%%%}+%%%{%%{poly1[%%%{-15,[3,7,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%
%}]%%},[2,0,0]%%%}+%%%{%%%{6,[2,5,2]%%%},[1,2,2]%%%}+%%%{%%%{-12,[2,5,4]%%%},[1,2,1]%%%}+%%%{%%%{6,[2,5,6]%%%}
,[1,2,0]%%%}+%%%{%%%{12,[3,7,2]%%%},[1,1,1]%%%}+%%%{%%%{-12,[3,7,4]%%%},[1,1,0]%%%}+%%%{%%%{6,[4,9,2]%%%},[1,0
,0]%%%}+%%%{%%{poly1[%%%{-1,[1,4,1]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[0,3,3]%%%}+%%%{%%{poly1[%%%{3,[1,4,3]%
%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[0,3,2]%%%}+%%%{%%{poly1[%%%{-3,[1,4,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},
[0,3,1]%%%}+%%%{%%{poly1[%%%{1,[1,4,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[0,3,0]%%%}+%%%{%%{poly1[%%%{-3,[2,6
,1]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[0,2,2]%%%}+%%%{%%{poly1[%%%{6,[2,6,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%
%},[0,2,1]%%%}+%%%{%%{poly1[%%%{-3,[2,6,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[0,2,0]%%%}+%%%{%%{poly1[%%%{-3,
[3,8,1]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[0,1,1]%%%}+%%%{%%{poly1[%%%{3,[3,8,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%
%}]%%},[0,1,0]%%%}+%%%{%%{poly1[%%%{-1,[4,10,1]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[0,0,0]%%%} Error: Bad Argu
ment Value
________________________________________________________________________________________
maple [B] time = 0.06, size = 1531, normalized size = 6.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)/(e*x+d)^3,x)
[Out]
2/e^3/(a*e^2-c*d^2)/(x+d/e)^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(7/2)-16/3/e^2*c*d/(a*e^2-c*d^2)^2/(x+d/
e)^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(7/2)+16/3/e*c^2*d^2/(a*e^2-c*d^2)^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^
2)*(x+d/e))^(5/2)-25/8*e*c^3*d^6/(a*e^2-c*d^2)^2*a^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+
d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+25/16/e*c^4*d^8/(a*e^2-c*d^2)^2*a*ln((1/2*a*e^2-1/2*c
*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-5/4*e^4*c*d/(a*
e^2-c*d^2)^2*a^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-15/4*c^3*d^5/(a*e^2-c*d^2)^2*a*((x+d/e)^2*c*d
*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+5/16*e^7/(a*e^2-c*d^2)^2*a^5*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^
(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+5/8/e^3*c^4*d^8/(a*e^2-c*d^2)^2*((x+d/e)^2*
c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-5/3/e^2*c^3*d^5/(a*e^2-c*d^2)^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/
2)-10/3/e*c^3*d^4/(a*e^2-c*d^2)^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x+5/4*e^3*c*d^2/(a*e^2-c*d^2)^
2*a^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+10/3*e*c^2*d^2/(a*e^2-c*d^2)^2*a*((x+d/e)^2*c*d*e+(a*e^2-c
*d^2)*(x+d/e))^(3/2)*x+5/3*e^2*c*d/(a*e^2-c*d^2)^2*a^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)+15/4*e^2*
c^2*d^3/(a*e^2-c*d^2)^2*a^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-5/8*e^5/(a*e^2-c*d^2)^2*a^4*((x+d/
e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+5/4/e^2*c^4*d^7/(a*e^2-c*d^2)^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e)
)^(1/2)*x-5/16/e^3*c^5*d^10/(a*e^2-c*d^2)^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*
d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+25/8*e^3*c^2*d^4/(a*e^2-c*d^2)^2*a^3*ln((1/2*a*e^2-1/2*c*d^2+(
x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-5/4/e*c^3*d^6/(a*e^2-
c*d^2)^2*a*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-25/16*e^5*c*d^2/(a*e^2-c*d^2)^2*a^4*ln((1/2*a*e^2-1/2
*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^3,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 details)Is a*e^2-c*d^2 zero or nonzero?
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^3,x)
[Out]
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^3, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**3,x)
[Out]
Integral(((d + e*x)*(a*e + c*d*x))**(5/2)/(d + e*x)**3, x)
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