3.17.25 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^4} \, dx\)
Optimal. Leaf size=235 \[ -\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{e (d+e x)^3}+\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e^2 (d+e x)}+\frac {15 c d \left (a-\frac {c d^2}{e^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}+\frac {15 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 e^{7/2}} \]
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Rubi [A] time = 0.21, antiderivative size = 235, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used =
{662, 664, 621, 206} \begin {gather*} -\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)^3}+\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}}{2 e^2 (d+e x)}+\frac {15 c d \left (a-\frac {c d^2}{e^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 e}+\frac {15 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^2 \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 )}{8 e^{7/2}} \end {gather*}
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)^4,x]
[Out]
(15*c*d*(a - (c*d^2)/e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*e) + (5*c*d*(a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2)^(3/2))/(2*e^2*(d + e*x)) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(e*(d + e*x)^3) +
(15*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^2*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])])/(8*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 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)^4} \, 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)^3}+\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)^2} \, 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}}{2 e^2 (d+e x)}-\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)^3}-\frac {\left (15 c d \left (c d^2-a e^2\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx}{4 e^2}\\ &=-\frac {15 c d \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 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}}{2 e^2 (d+e x)}-\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)^3}+\frac {\left (15 c d \left (c d^2-a e^2\right )^2\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 e^3}\\ &=-\frac {15 c d \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 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}}{2 e^2 (d+e x)}-\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)^3}+\frac {\left (15 c d \left (c d^2-a e^2\right )^2\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 )}{4 e^3}\\ &=-\frac {15 c d \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 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}}{2 e^2 (d+e x)}-\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)^3}+\frac {15 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^2 \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 )}{8 e^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 108, normalized size = 0.46 \begin {gather*} \frac {2 c d (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{7 \left (c d^2-a e^2\right )^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}} \end {gather*}
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)^4,x]
[Out]
(2*c*d*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[3/2, 7/2, 9/2, (e*(a*e + c*d*x))/(-(c*d
^2) + a*e^2)])/(7*(c*d^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])
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IntegrateAlgebraic [F] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^4,x]
[Out]
$Aborted
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fricas [A] time = 0.65, size = 554, normalized size = 2.36 \begin {gather*} \left [\frac {15 \, {\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} + {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt {\frac {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 \, {\left (2 \, c d e^{2} x + c d^{2} e + a e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {c d}{e}} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (2 \, c^{2} d^{2} e^{2} x^{2} - 15 \, c^{2} d^{4} + 25 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4} - {\left (5 \, c^{2} d^{3} e - 9 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left (e^{4} x + d e^{3}\right )}}, -\frac {15 \, {\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} + {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt {-\frac {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 {-\frac {c d}{e}}}{2 \, {\left (c^{2} d^{2} e x^{2} + a c d^{2} e + {\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, c^{2} d^{2} e^{2} x^{2} - 15 \, c^{2} d^{4} + 25 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4} - {\left (5 \, c^{2} d^{3} e - 9 \, a c d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left (e^{4} x + d e^{3}\right )}}\right ] \end {gather*}
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)^4,x, algorithm="fricas")
[Out]
[1/16*(15*(c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*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*(2*c*d*e^2*x + c*d^2*e + a*e^3)*sqrt(c*d*e*x^2 + a*d*e +
(c*d^2 + a*e^2)*x)*sqrt(c*d/e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(2*c^2*d^2*e^2*x^2 - 15*c^2*d^4 + 25*a*c*d^
2*e^2 - 8*a^2*e^4 - (5*c^2*d^3*e - 9*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^4*x + d*e^3
), -1/8*(15*(c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(-c*d/e)*arcta
n(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*x^2 + a*
c*d^2*e + (c^2*d^3 + a*c*d*e^2)*x)) - 2*(2*c^2*d^2*e^2*x^2 - 15*c^2*d^4 + 25*a*c*d^2*e^2 - 8*a^2*e^4 - (5*c^2*
d^3*e - 9*a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^4*x + d*e^3)]
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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)^4,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.81Unable to divide, perhaps due to rounding error%%%{%%%{8,[0,0,7]%%%},[8,8,8]%%%}+%%%{%%%{64,[
1,2,7]%%%},[8,7,7]%%%}+%%%{%%%{-128,[1,2,9]%%%},[8,7,6]%%%}+%%%{%%%{224,[2,4,7]%%%},[8,6,6]%%%}+%%%{%%%{-768,[
2,4,9]%%%},[8,6,5]%%%}+%%%{%%%{768,[2,4,11]%%%},[8,6,4]%%%}+%%%{%%%{448,[3,6,7]%%%},[8,5,5]%%%}+%%%{%%%{-1920,
[3,6,9]%%%},[8,5,4]%%%}+%%%{%%%{3072,[3,6,11]%%%},[8,5,3]%%%}+%%%{%%%{-2048,[3,6,13]%%%},[8,5,2]%%%}+%%%{%%%{5
60,[4,8,7]%%%},[8,4,4]%%%}+%%%{%%%{-2560,[4,8,9]%%%},[8,4,3]%%%}+%%%{%%%{4608,[4,8,11]%%%},[8,4,2]%%%}+%%%{%%%
{-4096,[4,8,13]%%%},[8,4,1]%%%}+%%%{%%%{2048,[4,8,15]%%%},[8,4,0]%%%}+%%%{%%%{448,[5,10,7]%%%},[8,3,3]%%%}+%%%
{%%%{-1920,[5,10,9]%%%},[8,3,2]%%%}+%%%{%%%{3072,[5,10,11]%%%},[8,3,1]%%%}+%%%{%%%{-2048,[5,10,13]%%%},[8,3,0]
%%%}+%%%{%%%{224,[6,12,7]%%%},[8,2,2]%%%}+%%%{%%%{-768,[6,12,9]%%%},[8,2,1]%%%}+%%%{%%%{768,[6,12,11]%%%},[8,2
,0]%%%}+%%%{%%%{64,[7,14,7]%%%},[8,1,1]%%%}+%%%{%%%{-128,[7,14,9]%%%},[8,1,0]%%%}+%%%{%%%{8,[8,16,7]%%%},[8,0,
0]%%%}+%%%{%%{[%%%{-64,[0,1,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,8,8]%%%}+%%%{%%{[%%%{-512,[1,3,6]%%%},0]:
[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,7,7]%%%}+%%%{%%{[%%%{1024,[1,3,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,7,6]%%%
}+%%%{%%{[%%%{-1792,[2,5,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,6,6]%%%}+%%%{%%{[%%%{6144,[2,5,8]%%%},0]:[1,
0,%%%{-1,[1,1,1]%%%}]%%},[7,6,5]%%%}+%%%{%%{[%%%{-6144,[2,5,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,6,4]%%%}
+%%%{%%{[%%%{-3584,[3,7,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,5,5]%%%}+%%%{%%{[%%%{15360,[3,7,8]%%%},0]:[1,
0,%%%{-1,[1,1,1]%%%}]%%},[7,5,4]%%%}+%%%{%%{[%%%{-24576,[3,7,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,5,3]%%%
}+%%%{%%{[%%%{16384,[3,7,12]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,5,2]%%%}+%%%{%%{[%%%{-4480,[4,9,6]%%%},0]:[
1,0,%%%{-1,[1,1,1]%%%}]%%},[7,4,4]%%%}+%%%{%%{[%%%{20480,[4,9,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,4,3]%%%
}+%%%{%%{[%%%{-36864,[4,9,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,4,2]%%%}+%%%{%%{[%%%{32768,[4,9,12]%%%},0]
:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,4,1]%%%}+%%%{%%{[%%%{-16384,[4,9,14]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,4,0
]%%%}+%%%{%%{[%%%{-3584,[5,11,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,3,3]%%%}+%%%{%%{[%%%{15360,[5,11,8]%%%}
,0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,3,2]%%%}+%%%{%%{[%%%{-24576,[5,11,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7
,3,1]%%%}+%%%{%%{[%%%{16384,[5,11,12]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,3,0]%%%}+%%%{%%{[%%%{-1792,[6,13,6
]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,2,2]%%%}+%%%{%%{[%%%{6144,[6,13,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},
[7,2,1]%%%}+%%%{%%{[%%%{-6144,[6,13,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,2,0]%%%}+%%%{%%{[%%%{-512,[7,15,
6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,1,1]%%%}+%%%{%%{[%%%{1024,[7,15,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%}
,[7,1,0]%%%}+%%%{%%{[%%%{-64,[8,17,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,0,0]%%%}+%%%{%%%{32,[0,1,6]%%%},[6
,9,9]%%%}+%%%{%%%{-32,[0,1,8]%%%},[6,9,8]%%%}+%%%{%%%{480,[1,3,6]%%%},[6,8,8]%%%}+%%%{%%%{-768,[1,3,8]%%%},[6,
8,7]%%%}+%%%{%%%{512,[1,3,10]%%%},[6,8,6]%%%}+%%%{%%%{2688,[2,5,6]%%%},[6,7,7]%%%}+%%%{%%%{-7552,[2,5,8]%%%},[
6,7,6]%%%}+%%%{%%%{6144,[2,5,10]%%%},[6,7,5]%%%}+%%%{%%%{-3072,[2,5,12]%%%},[6,7,4]%%%}+%%%{%%%{8064,[3,7,6]%%
%},[6,6,6]%%%}+%%%{%%%{-30976,[3,7,8]%%%},[6,6,5]%%%}+%%%{%%%{41472,[3,7,10]%%%},[6,6,4]%%%}+%%%{%%%{-20480,[3
,7,12]%%%},[6,6,3]%%%}+%%%{%%%{8192,[3,7,14]%%%},[6,6,2]%%%}+%%%{%%%{14784,[4,9,6]%%%},[6,5,5]%%%}+%%%{%%%{-66
240,[4,9,8]%%%},[6,5,4]%%%}+%%%{%%%{114688,[4,9,10]%%%},[6,5,3]%%%}+%%%{%%%{-92160,[4,9,12]%%%},[6,5,2]%%%}+%%
%{%%%{24576,[4,9,14]%%%},[6,5,1]%%%}+%%%{%%%{-8192,[4,9,16]%%%},[6,5,0]%%%}+%%%{%%%{17472,[5,11,6]%%%},[6,4,4]
%%%}+%%%{%%%{-81152,[5,11,8]%%%},[6,4,3]%%%}+%%%{%%%{148992,[5,11,10]%%%},[6,4,2]%%%}+%%%{%%%{-135168,[5,11,12
]%%%},[6,4,1]%%%}+%%%{%%%{65536,[5,11,14]%%%},[6,4,0]%%%}+%%%{%%%{13440,[6,13,6]%%%},[6,3,3]%%%}+%%%{%%%{-5772
8,[6,13,8]%%%},[6,3,2]%%%}+%%%{%%%{92160,[6,13,10]%%%},[6,3,1]%%%}+%%%{%%%{-60416,[6,13,12]%%%},[6,3,0]%%%}+%%
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,[1,11,8]%%%}+%%%{%%{[%%%{-704,[1,6,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,10,10]%%%}+%%%{%%{[%%%{2944,[1,6,
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},[1,10,8]%%%}+%%%{%%{[%%%{3584,[1,6,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,10,7]%%%}+%%%{%%{[%%%{-1024,[1,6
,11]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,10,6]%%%}+%%%{%%{[%%%{-3520,[2,8,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]
%%},[1,9,9]%%%}+%%%{%%{[%%%{17856,[2,8,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,9,8]%%%}+%%%{%%{[%%%{-37632,[2
,8,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,9,7]%%%}+%%%{%%{[%%%{41728,[2,8,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]
%%},[1,9,6]%%%}+%%%{%%{[%%%{-24576,[2,8,11]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,9,5]%%%}+%%%{%%{[%%%{6144,[2
,8,13]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,9,4]%%%}+%%%{%%{[%%%{-10560,[3,10,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%
%}]%%},[1,8,8]%%%}+%%%{%%{[%%%{59904,[3,10,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,7]%%%}+%%%{%%{[%%%{-1451
52,[3,10,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,6]%%%}+%%%{%%{[%%%{195072,[3,10,9]%%%},0]:[1,0,%%%{-1,[1,1
,1]%%%}]%%},[1,8,5]%%%}+%%%{%%{[%%%{-156672,[3,10,11]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,4]%%%}+%%%{%%{[%
%%{73728,[3,10,13]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,3]%%%}+%%%{%%{[%%%{-16384,[3,10,15]%%%},0]:[1,0,%%%
{-1,[1,1,1]%%%}]%%},[1,8,2]%%%}+%%%{%%{[%%%{-21120,[4,12,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,7]%%%}+%%%
{%%{[%%%{126336,[4,12,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,6]%%%}+%%%{%%{[%%%{-325248,[4,12,7]%%%},0]:[1
,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,5]%%%}+%%%{%%{[%%%{470400,[4,12,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,4]%%
%}+%%%{%%{[%%%{-417792,[4,12,11]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,3]%%%}+%%%{%%{[%%%{233472,[4,12,13]%%
%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,2]%%%}+%%%{%%{[%%%{-81920,[4,12,15]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},
[1,7,1]%%%}+%%%{%%{[%%%{16384,[4,12,17]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,0]%%%}+%%%{%%{[%%%{-29568,[5,1
4,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,6]%%%}+%%%{%%{[%%%{177408,[5,14,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}
]%%},[1,6,5]%%%}+%%%{%%{[%%%{-454272,[5,14,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,4]%%%}+%%%{%%{[%%%{64358
4,[5,14,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,3]%%%}+%%%{%%{[%%%{-543744,[5,14,11]%%%},0]:[1,0,%%%{-1,[1,
1,1]%%%}]%%},[1,6,2]%%%}+%%%{%%{[%%%{270336,[5,14,13]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,1]%%%}+%%%{%%{[%
%%{-65536,[5,14,15]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,0]%%%}+%%%{%%{[%%%{-29568,[6,16,3]%%%},0]:[1,0,%%%
{-1,[1,1,1]%%%}]%%},[1,5,5]%%%}+%%%{%%{[%%%{169344,[6,16,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,4]%%%}+%%%
{%%{[%%%{-403200,[6,16,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,3]%%%}+%%%{%%{[%%%{506624,[6,16,9]%%%},0]:[1
,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,2]%%%}+%%%{%%{[%%%{-344064,[6,16,11]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,1]
%%%}+%%%{%%{[%%%{104448,[6,16,13]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,0]%%%}+%%%{%%{[%%%{-21120,[7,18,3]%%
%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,4]%%%}+%%%{%%{[%%%{109056,[7,18,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[
1,4,3]%%%}+%%%{%%{[%%%{-221952,[7,18,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,2]%%%}+%%%{%%{[%%%{214528,[7,1
8,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,1]%%%}+%%%{%%{[%%%{-84992,[7,18,11]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%
}]%%},[1,4,0]%%%}+%%%{%%{[%%%{-10560,[8,20,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,3]%%%}+%%%{%%{[%%%{45504
,[8,20,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,2]%%%}+%%%{%%{[%%%{-69312,[8,20,7]%%%},0]:[1,0,%%%{-1,[1,1,1
]%%%}]%%},[1,3,1]%%%}+%%%{%%{[%%%{37952,[8,20,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,0]%%%}+%%%{%%{[%%%{-3
520,[9,22,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,2,2]%%%}+%%%{%%{[%%%{11136,[9,22,5]%%%},0]:[1,0,%%%{-1,[1,1
,1]%%%}]%%},[1,2,1]%%%}+%%%{%%{[%%%{-9408,[9,22,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,2,0]%%%}+%%%{%%{[%%%{
-704,[10,24,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,1,1]%%%}+%%%{%%{[%%%{1216,[10,24,5]%%%},0]:[1,0,%%%{-1,[1
,1,1]%%%}]%%},[1,1,0]%%%}+%%%{%%{[%%%{-64,[11,26,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,0,0]%%%}+%%%{%%%{8,[
0,4,3]%%%},[0,12,12]%%%}+%%%{%%%{-32,[0,4,5]%%%},[0,12,11]%%%}+%%%{%%%{48,[0,4,7]%%%},[0,12,10]%%%}+%%%{%%%{-3
2,[0,4,9]%%%},[0,12,9]%%%}+%%%{%%%{8,[0,4,11]%%%},[0,12,8]%%%}+%%%{%%%{96,[1,6,3]%%%},[0,11,11]%%%}+%%%{%%%{-4
80,[1,6,5]%%%},[0,11,10]%%%}+%%%{%%%{992,[1,6,7]%%%},[0,11,9]%%%}+%%%{%%%{-1056,[1,6,9]%%%},[0,11,8]%%%}+%%%{%
%%{576,[1,6,11]%%%},[0,11,7]%%%}+%%%{%%%{-128,[1,6,13]%%%},[0,11,6]%%%}+%%%{%%%{528,[2,8,3]%%%},[0,10,10]%%%}+
%%%{%%%{-3040,[2,8,5]%%%},[0,10,9]%%%}+%%%{%%%{7536,[2,8,7]%%%},[0,10,8]%%%}+%%%{%%%{-10368,[2,8,9]%%%},[0,10,
7]%%%}+%%%{%%%{8416,[2,8,11]%%%},[0,10,6]%%%}+%%%{%%%{-3840,[2,8,13]%%%},[0,10,5]%%%}+%%%{%%%{768,[2,8,15]%%%}
,[0,10,4]%%%}+%%%{%%%{1760,[3,10,3]%%%},[0,9,9]%%%}+%%%{%%%{-11040,[3,10,5]%%%},[0,9,8]%%%}+%%%{%%%{30336,[3,1
0,7]%%%},[0,9,7]%%%}+%%%{%%%{-47744,[3,10,9]%%%},[0,9,6]%%%}+%%%{%%%{47040,[3,10,11]%%%},[0,9,5]%%%}+%%%{%%%{-
29568,[3,10,13]%%%},[0,9,4]%%%}+%%%{%%%{11264,[3,10,15]%%%},[0,9,3]%%%}+%%%{%%%{-2048,[3,10,17]%%%},[0,9,2]%%%
}+%%%{%%%{3960,[4,12,3]%%%},[0,8,8]%%%}+%%%{%%%{-25920,[4,12,5]%%%},[0,8,7]%%%}+%%%{%%%{74592,[4,12,7]%%%},[0,
8,6]%%%}+%%%{%%%{-123840,[4,12,9]%%%},[0,8,5]%%%}+%%%{%%%{130608,[4,12,11]%%%},[0,8,4]%%%}+%%%{%%%{-90624,[4,1
2,13]%%%},[0,8,3]%%%}+%%%{%%%{41472,[4,12,15]%%%},[0,8,2]%%%}+%%%{%%%{-12288,[4,12,17]%%%},[0,8,1]%%%}+%%%{%%%
{2048,[4,12,19]%%%},[0,8,0]%%%}+%%%{%%%{6336,[5,14,3]%%%},[0,7,7]%%%}+%%%{%%%{-41664,[5,14,5]%%%},[0,7,6]%%%}+
%%%{%%%{119616,[5,14,7]%%%},[0,7,5]%%%}+%%%{%%%{-196032,[5,14,9]%%%},[0,7,4]%%%}+%%%{%%%{200640,[5,14,11]%%%},
[0,7,3]%%%}+%%%{%%%{-130944,[5,14,13]%%%},[0,7,2]%%%}+%%%{%%%{52224,[5,14,15]%%%},[0,7,1]%%%}+%%%{%%%{-10240,[
5,14,17]%%%},[0,7,0]%%%}+%%%{%%%{7392,[6,16,3]%%%},[0,6,6]%%%}+%%%{%%%{-47040,[6,16,5]%%%},[0,6,5]%%%}+%%%{%%%
{128352,[6,16,7]%%%},[0,6,4]%%%}+%%%{%%%{-194176,[6,16,9]%%%},[0,6,3]%%%}+%%%{%%%{174304,[6,16,11]%%%},[0,6,2]
%%%}+%%%{%%%{-89856,[6,16,13]%%%},[0,6,1]%%%}+%%%{%%%{21248,[6,16,15]%%%},[0,6,0]%%%}+%%%{%%%{6336,[7,18,3]%%%
},[0,5,5]%%%}+%%%{%%%{-37440,[7,18,5]%%%},[0,5,4]%%%}+%%%{%%%{91776,[7,18,7]%%%},[0,5,3]%%%}+%%%{%%%{-117888,[
7,18,9]%%%},[0,5,2]%%%}+%%%{%%%{80448,[7,18,11]%%%},[0,5,1]%%%}+%%%{%%%{-23680,[7,18,13]%%%},[0,5,0]%%%}+%%%{%
%%{3960,[8,20,3]%%%},[0,4,4]%%%}+%%%{%%%{-20640,[8,20,5]%%%},[0,4,3]%%%}+%%%{%%%{42096,[8,20,7]%%%},[0,4,2]%%%
}+%%%{%%%{-40224,[8,20,9]%%%},[0,4,1]%%%}+%%%{%%%{15368,[8,20,11]%%%},[0,4,0]%%%}+%%%{%%%{1760,[9,22,3]%%%},[0
,3,3]%%%}+%%%{%%%{-7520,[9,22,5]%%%},[0,3,2]%%%}+%%%{%%%{11232,[9,22,7]%%%},[0,3,1]%%%}+%%%{%%%{-5920,[9,22,9]
%%%},[0,3,0]%%%}+%%%{%%%{528,[10,24,3]%%%},[0,2,2]%%%}+%%%{%%%{-1632,[10,24,5]%%%},[0,2,1]%%%}+%%%{%%%{1328,[1
0,24,7]%%%},[0,2,0]%%%}+%%%{%%%{96,[11,26,3]%%%},[0,1,1]%%%}+%%%{%%%{-160,[11,26,5]%%%},[0,1,0]%%%}+%%%{%%%{8,
[12,28,3]%%%},[0,0,0]%%%} / %%%{%%%{1,[2,2,6]%%%},[8,0,0]%%%}+%%%{%%{poly1[%%%{-8,[2,3,5]%%%},0]:[1,0,%%%{-1,[
1,1,1]%%%}]%%},[7,0,0]%%%}+%%%{%%%{4,[2,3,5]%%%},[6,1,1]%%%}+%%%{%%%{-4,[2,3,7]%%%},[6,1,0]%%%}+%%%{%%%{28,[3,
5,5]%%%},[6,0,0]%%%}+%%%{%%{poly1[%%%{-24,[2,4,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,1,1]%%%}+%%%{%%{poly1[
%%%{24,[2,4,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,1,0]%%%}+%%%{%%{poly1[%%%{-56,[3,6,4]%%%},0]:[1,0,%%%{-1,
[1,1,1]%%%}]%%},[5,0,0]%%%}+%%%{%%%{6,[2,4,4]%%%},[4,2,2]%%%}+%%%{%%%{-12,[2,4,6]%%%},[4,2,1]%%%}+%%%{%%%{6,[2
,4,8]%%%},[4,2,0]%%%}+%%%{%%%{60,[3,6,4]%%%},[4,1,1]%%%}+%%%{%%%{-60,[3,6,6]%%%},[4,1,0]%%%}+%%%{%%%{70,[4,8,4
]%%%},[4,0,0]%%%}+%%%{%%{poly1[%%%{-24,[2,5,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,2,2]%%%}+%%%{%%{poly1[%%%
{48,[2,5,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,2,1]%%%}+%%%{%%{poly1[%%%{-24,[2,5,7]%%%},0]:[1,0,%%%{-1,[1,
1,1]%%%}]%%},[3,2,0]%%%}+%%%{%%{poly1[%%%{-80,[3,7,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,1,1]%%%}+%%%{%%{po
ly1[%%%{80,[3,7,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,1,0]%%%}+%%%{%%{poly1[%%%{-56,[4,9,3]%%%},0]:[1,0,%%%
{-1,[1,1,1]%%%}]%%},[3,0,0]%%%}+%%%{%%%{4,[2,5,3]%%%},[2,3,3]%%%}+%%%{%%%{-12,[2,5,5]%%%},[2,3,2]%%%}+%%%{%%%{
12,[2,5,7]%%%},[2,3,1]%%%}+%%%{%%%{-4,[2,5,9]%%%},[2,3,0]%%%}+%%%{%%%{36,[3,7,3]%%%},[2,2,2]%%%}+%%%{%%%{-72,[
3,7,5]%%%},[2,2,1]%%%}+%%%{%%%{36,[3,7,7]%%%},[2,2,0]%%%}+%%%{%%%{60,[4,9,3]%%%},[2,1,1]%%%}+%%%{%%%{-60,[4,9,
5]%%%},[2,1,0]%%%}+%%%{%%%{28,[5,11,3]%%%},[2,0,0]%%%}+%%%{%%{poly1[%%%{-8,[2,6,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%
%%}]%%},[1,3,3]%%%}+%%%{%%{poly1[%%%{24,[2,6,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,2]%%%}+%%%{%%{poly1[%%
%{-24,[2,6,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,1]%%%}+%%%{%%{poly1[%%%{8,[2,6,8]%%%},0]:[1,0,%%%{-1,[1,
1,1]%%%}]%%},[1,3,0]%%%}+%%%{%%{poly1[%%%{-24,[3,8,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,2,2]%%%}+%%%{%%{po
ly1[%%%{48,[3,8,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,2,1]%%%}+%%%{%%{poly1[%%%{-24,[3,8,6]%%%},0]:[1,0,%%%
{-1,[1,1,1]%%%}]%%},[1,2,0]%%%}+%%%{%%{poly1[%%%{-24,[4,10,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,1,1]%%%}+%
%%{%%{poly1[%%%{24,[4,10,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,1,0]%%%}+%%%{%%{poly1[%%%{-8,[5,12,2]%%%},0]
:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,0,0]%%%}+%%%{%%%{1,[2,6,2]%%%},[0,4,4]%%%}+%%%{%%%{-4,[2,6,4]%%%},[0,4,3]%%%}+
%%%{%%%{6,[2,6,6]%%%},[0,4,2]%%%}+%%%{%%%{-4,[2,6,8]%%%},[0,4,1]%%%}+%%%{%%%{1,[2,6,10]%%%},[0,4,0]%%%}+%%%{%%
%{4,[3,8,2]%%%},[0,3,3]%%%}+%%%{%%%{-12,[3,8,4]%%%},[0,3,2]%%%}+%%%{%%%{12,[3,8,6]%%%},[0,3,1]%%%}+%%%{%%%{-4,
[3,8,8]%%%},[0,3,0]%%%}+%%%{%%%{6,[4,10,2]%%%},[0,2,2]%%%}+%%%{%%%{-12,[4,10,4]%%%},[0,2,1]%%%}+%%%{%%%{6,[4,1
0,6]%%%},[0,2,0]%%%}+%%%{%%%{4,[5,12,2]%%%},[0,1,1]%%%}+%%%{%%%{-4,[5,12,4]%%%},[0,1,0]%%%}+%%%{%%%{1,[6,14,2]
%%%},[0,0,0]%%%} Error: Bad Argument Value
________________________________________________________________________________________
maple [B] time = 0.06, size = 1617, normalized size = 6.88
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)^4,x)
[Out]
-2/e^4/(a*e^2-c*d^2)/(x+d/e)^4*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(7/2)+12/e^3*c*d/(a*e^2-c*d^2)^2/(x+d/e
)^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(7/2)-32/e^2*c^2*d^2/(a*e^2-c*d^2)^3/(x+d/e)^2*((x+d/e)^2*c*d*e+(a
*e^2-c*d^2)*(x+d/e))^(7/2)+32/e*c^3*d^3/(a*e^2-c*d^2)^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(5/2)+15/4/e^3
*c^5*d^9/(a*e^2-c*d^2)^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-20/e*c^4*d^5/(a*e^2-c*d^2)^3*((x+d/e)^2
*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x-10/e^2*c^4*d^6/(a*e^2-c*d^2)^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(
3/2)+20*e*c^3*d^3/(a*e^2-c*d^2)^3*a*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x+15/2*e^3*c^2*d^3/(a*e^2-c*
d^2)^3*a^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+75/8/e*c^5*d^9/(a*e^2-c*d^2)^3*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)-45/2*c^4*d^6/(a*
e^2-c*d^2)^3*a*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+15/8*e^7*c*d/(a*e^2-c*d^2)^3*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)-75/4*e*c^4
*d^7/(a*e^2-c*d^2)^3*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)-15/2*e^4*c^2*d^2/(a*e^2-c*d^2)^3*a^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2
)*x+10*e^2*c^2*d^2/(a*e^2-c*d^2)^3*a^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)+45/2*e^2*c^3*d^4/(a*e^2-c
*d^2)^3*a^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-15/4*e^5*c*d/(a*e^2-c*d^2)^3*a^4*((x+d/e)^2*c*d*e+
(a*e^2-c*d^2)*(x+d/e))^(1/2)-15/2/e*c^4*d^7/(a*e^2-c*d^2)^3*a*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-15
/8/e^3*c^6*d^11/(a*e^2-c*d^2)^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)+15/2/e^2*c^5*d^8/(a*e^2-c*d^2)^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(
1/2)*x-75/8*e^5*c^2*d^3/(a*e^2-c*d^2)^3*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)+75/4*e^3*c^3*d^5/(a*e^2-c*d^2)^3*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)
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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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^4,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 \begin {gather*} \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 )}^4} \,d x \end {gather*}
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)^4,x)
[Out]
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^4, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \end {gather*}
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)**4,x)
[Out]
Integral(((d + e*x)*(a*e + c*d*x))**(5/2)/(d + e*x)**4, x)
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