3.17.12 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^4} \, dx\)
Optimal. Leaf size=169 \[ \frac {c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{e^{5/2}}-\frac {2 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 (d+e x)}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^3} \]
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Rubi [A] time = 0.09, antiderivative size = 169, 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 =
{662, 621, 206} \begin {gather*} \frac {c^{3/2} d^{3/2} \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 )}{e^{5/2}}-\frac {2 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2 (d+e x)}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^4,x]
[Out]
(-2*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^2*(d + e*x)) - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(3/2))/(3*e*(d + e*x)^3) + (c^(3/2)*d^(3/2)*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])])/e^(5/2)
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]
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 )^{3/2}}{(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 )^{3/2}}{3 e (d+e x)^3}+\frac {(c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx}{e}\\ &=-\frac {2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^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 )^{3/2}}{3 e (d+e x)^3}+\frac {\left (c^2 d^2\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{e^2}\\ &=-\frac {2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^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 )^{3/2}}{3 e (d+e x)^3}+\frac {\left (2 c^2 d^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 )}{e^2}\\ &=-\frac {2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^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 )^{3/2}}{3 e (d+e x)^3}+\frac {c^{3/2} d^{3/2} \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 )}{e^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 184, normalized size = 1.09 \begin {gather*} \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (\frac {3 c^{3/2} d^{3/2} \sqrt {c d} \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 )}{\sqrt {c d^2-a e^2} \sqrt {a e+c d x} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}-\frac {\sqrt {e} \left (a e^2+c d (3 d+4 e x)\right )}{(d+e x)^2}\right )}{3 e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^4,x]
[Out]
(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[e]*(a*e^2 + c*d*(3*d + 4*e*x)))/(d + e*x)^2) + (3*c^(3/2)*d^(3/2)*Sq
rt[c*d]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[c*d^2 - a*
e^2]*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(3*e^(5/2))
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IntegrateAlgebraic [A] time = 1.44, size = 297, normalized size = 1.76 \begin {gather*} -\frac {c d \sqrt {c d e} \log \left (a^2 e^4+8 c d e x \sqrt {c d e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-2 a c d^2 e^2-4 a c d e^3 x+c^2 d^4-4 c^2 d^3 e x-8 c^2 d^2 e^2 x^2\right )}{2 e^3}-\frac {c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} x \sqrt {c d e}}{a e^2+c d^2}-\frac {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}}{a e^2+c d^2}\right )}{e^{5/2}}-\frac {2 \left (a e^2+3 c d^2+4 c d e x\right ) \sqrt {a d e+a e^2 x+c d^2 x+c d e x^2}}{3 e^2 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^4,x]
[Out]
(-2*(3*c*d^2 + a*e^2 + 4*c*d*e*x)*Sqrt[a*d*e + c*d^2*x + a*e^2*x + c*d*e*x^2])/(3*e^2*(d + e*x)^2) - (c^(3/2)*
d^(3/2)*ArcTanh[(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[c*d*e]*x)/(c*d^2 + a*e^2) - (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d^2 + a*e^2)])/e^(5/2) - (c*d*Sqrt[c*d*e]*Log[c^2*d^4 - 2*a*c*d^2*e^2
+ a^2*e^4 - 4*c^2*d^3*e*x - 4*a*c*d*e^3*x - 8*c^2*d^2*e^2*x^2 + 8*c*d*e*Sqrt[c*d*e]*x*Sqrt[a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2]])/(2*e^3)
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fricas [A] time = 0.73, size = 426, normalized size = 2.52 \begin {gather*} \left [\frac {3 \, {\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\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 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (4 \, c d e x + 3 \, c d^{2} + a e^{2}\right )}}{6 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, -\frac {3 \, {\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\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 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (4 \, c d e x + 3 \, c d^{2} + a e^{2}\right )}}{3 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\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)^(3/2)/(e*x+d)^4,x, algorithm="fricas")
[Out]
[1/6*(3*(c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*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*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(4*c*d*e*x + 3*c*d^2 + a*e^2))/(e^4*x^2 + 2*d
*e^3*x + d^2*e^2), -1/3*(3*(c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*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*x^2 + a*c*d^2*e + (c^2*d^3 + a*c*d*e^
2)*x)) + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(4*c*d*e*x + 3*c*d^2 + a*e^2))/(e^4*x^2 + 2*d*e^3*x + d
^2*e^2)]
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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)^(3/2)/(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: 0.58Unable to divide, perhaps due to rounding error%%%{%%%{1,[0,0,4]%%%},[8,8,8]%%%}+%%%{%%%{8,[1
,2,4]%%%},[8,7,7]%%%}+%%%{%%%{-16,[1,2,6]%%%},[8,7,6]%%%}+%%%{%%%{28,[2,4,4]%%%},[8,6,6]%%%}+%%%{%%%{-96,[2,4,
6]%%%},[8,6,5]%%%}+%%%{%%%{96,[2,4,8]%%%},[8,6,4]%%%}+%%%{%%%{56,[3,6,4]%%%},[8,5,5]%%%}+%%%{%%%{-240,[3,6,6]%
%%},[8,5,4]%%%}+%%%{%%%{384,[3,6,8]%%%},[8,5,3]%%%}+%%%{%%%{-256,[3,6,10]%%%},[8,5,2]%%%}+%%%{%%%{70,[4,8,4]%%
%},[8,4,4]%%%}+%%%{%%%{-320,[4,8,6]%%%},[8,4,3]%%%}+%%%{%%%{576,[4,8,8]%%%},[8,4,2]%%%}+%%%{%%%{-512,[4,8,10]%
%%},[8,4,1]%%%}+%%%{%%%{256,[4,8,12]%%%},[8,4,0]%%%}+%%%{%%%{56,[5,10,4]%%%},[8,3,3]%%%}+%%%{%%%{-240,[5,10,6]
%%%},[8,3,2]%%%}+%%%{%%%{384,[5,10,8]%%%},[8,3,1]%%%}+%%%{%%%{-256,[5,10,10]%%%},[8,3,0]%%%}+%%%{%%%{28,[6,12,
4]%%%},[8,2,2]%%%}+%%%{%%%{-96,[6,12,6]%%%},[8,2,1]%%%}+%%%{%%%{96,[6,12,8]%%%},[8,2,0]%%%}+%%%{%%%{8,[7,14,4]
%%%},[8,1,1]%%%}+%%%{%%%{-16,[7,14,6]%%%},[8,1,0]%%%}+%%%{%%%{1,[8,16,4]%%%},[8,0,0]%%%}+%%%{%%{[%%%{-8,[0,1,3
]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,8,8]%%%}+%%%{%%{[%%%{-64,[1,3,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7
,7,7]%%%}+%%%{%%{[%%%{128,[1,3,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,7,6]%%%}+%%%{%%{[%%%{-224,[2,5,3]%%%},
0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,6,6]%%%}+%%%{%%{[%%%{768,[2,5,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,6,5]%
%%}+%%%{%%{[%%%{-768,[2,5,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,6,4]%%%}+%%%{%%{[%%%{-448,[3,7,3]%%%},0]:[1
,0,%%%{-1,[1,1,1]%%%}]%%},[7,5,5]%%%}+%%%{%%{[%%%{1920,[3,7,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,5,4]%%%}+
%%%{%%{[%%%{-3072,[3,7,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,5,3]%%%}+%%%{%%{[%%%{2048,[3,7,9]%%%},0]:[1,0,
%%%{-1,[1,1,1]%%%}]%%},[7,5,2]%%%}+%%%{%%{[%%%{-560,[4,9,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,4,4]%%%}+%%%
{%%{[%%%{2560,[4,9,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,4,3]%%%}+%%%{%%{[%%%{-4608,[4,9,7]%%%},0]:[1,0,%%%
{-1,[1,1,1]%%%}]%%},[7,4,2]%%%}+%%%{%%{[%%%{4096,[4,9,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,4,1]%%%}+%%%{%%
{[%%%{-2048,[4,9,11]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,4,0]%%%}+%%%{%%{[%%%{-448,[5,11,3]%%%},0]:[1,0,%%%{
-1,[1,1,1]%%%}]%%},[7,3,3]%%%}+%%%{%%{[%%%{1920,[5,11,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,3,2]%%%}+%%%{%%
{[%%%{-3072,[5,11,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,3,1]%%%}+%%%{%%{[%%%{2048,[5,11,9]%%%},0]:[1,0,%%%{
-1,[1,1,1]%%%}]%%},[7,3,0]%%%}+%%%{%%{[%%%{-224,[6,13,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,2,2]%%%}+%%%{%%
{[%%%{768,[6,13,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,2,1]%%%}+%%%{%%{[%%%{-768,[6,13,7]%%%},0]:[1,0,%%%{-1
,[1,1,1]%%%}]%%},[7,2,0]%%%}+%%%{%%{[%%%{-64,[7,15,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,1,1]%%%}+%%%{%%{[%
%%{128,[7,15,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[7,1,0]%%%}+%%%{%%{[%%%{-8,[8,17,3]%%%},0]:[1,0,%%%{-1,[1,1
,1]%%%}]%%},[7,0,0]%%%}+%%%{%%%{4,[0,1,3]%%%},[6,9,9]%%%}+%%%{%%%{-4,[0,1,5]%%%},[6,9,8]%%%}+%%%{%%%{60,[1,3,3
]%%%},[6,8,8]%%%}+%%%{%%%{-96,[1,3,5]%%%},[6,8,7]%%%}+%%%{%%%{64,[1,3,7]%%%},[6,8,6]%%%}+%%%{%%%{336,[2,5,3]%%
%},[6,7,7]%%%}+%%%{%%%{-944,[2,5,5]%%%},[6,7,6]%%%}+%%%{%%%{768,[2,5,7]%%%},[6,7,5]%%%}+%%%{%%%{-384,[2,5,9]%%
%},[6,7,4]%%%}+%%%{%%%{1008,[3,7,3]%%%},[6,6,6]%%%}+%%%{%%%{-3872,[3,7,5]%%%},[6,6,5]%%%}+%%%{%%%{5184,[3,7,7]
%%%},[6,6,4]%%%}+%%%{%%%{-2560,[3,7,9]%%%},[6,6,3]%%%}+%%%{%%%{1024,[3,7,11]%%%},[6,6,2]%%%}+%%%{%%%{1848,[4,9
,3]%%%},[6,5,5]%%%}+%%%{%%%{-8280,[4,9,5]%%%},[6,5,4]%%%}+%%%{%%%{14336,[4,9,7]%%%},[6,5,3]%%%}+%%%{%%%{-11520
,[4,9,9]%%%},[6,5,2]%%%}+%%%{%%%{3072,[4,9,11]%%%},[6,5,1]%%%}+%%%{%%%{-1024,[4,9,13]%%%},[6,5,0]%%%}+%%%{%%%{
2184,[5,11,3]%%%},[6,4,4]%%%}+%%%{%%%{-10144,[5,11,5]%%%},[6,4,3]%%%}+%%%{%%%{18624,[5,11,7]%%%},[6,4,2]%%%}+%
%%{%%%{-16896,[5,11,9]%%%},[6,4,1]%%%}+%%%{%%%{8192,[5,11,11]%%%},[6,4,0]%%%}+%%%{%%%{1680,[6,13,3]%%%},[6,3,3
]%%%}+%%%{%%%{-7216,[6,13,5]%%%},[6,3,2]%%%}+%%%{%%%{11520,[6,13,7]%%%},[6,3,1]%%%}+%%%{%%%{-7552,[6,13,9]%%%}
,[6,3,0]%%%}+%%%{%%%{816,[7,15,3]%%%},[6,2,2]%%%}+%%%{%%%{-2784,[7,15,5]%%%},[6,2,1]%%%}+%%%{%%%{2752,[7,15,7]
%%%},[6,2,0]%%%}+%%%{%%%{228,[8,17,3]%%%},[6,1,1]%%%}+%%%{%%%{-452,[8,17,5]%%%},[6,1,0]%%%}+%%%{%%%{28,[9,19,3
]%%%},[6,0,0]%%%}+%%%{%%{[%%%{-24,[0,2,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,9,9]%%%}+%%%{%%{[%%%{24,[0,2,4
]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,9,8]%%%}+%%%{%%{[%%%{-248,[1,4,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[
5,8,8]%%%}+%%%{%%{[%%%{576,[1,4,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,8,7]%%%}+%%%{%%{[%%%{-384,[1,4,6]%%%}
,0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,8,6]%%%}+%%%{%%{[%%%{-1120,[2,6,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,7,
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4,[3,10,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,6]%%%}+%%%{%%{[%%%{24384,[3,10,6]%%%},0]:[1,0,%%%{-1,[1,1,1
]%%%}]%%},[1,8,5]%%%}+%%%{%%{[%%%{-19584,[3,10,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,4]%%%}+%%%{%%{[%%%{9
216,[3,10,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,3]%%%}+%%%{%%{[%%%{-2048,[3,10,12]%%%},0]:[1,0,%%%{-1,[1
,1,1]%%%}]%%},[1,8,2]%%%}+%%%{%%{[%%%{-2640,[4,12,0]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,7]%%%}+%%%{%%{[%%
%{15792,[4,12,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,6]%%%}+%%%{%%{[%%%{-40656,[4,12,4]%%%},0]:[1,0,%%%{-1
,[1,1,1]%%%}]%%},[1,7,5]%%%}+%%%{%%{[%%%{58800,[4,12,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,4]%%%}+%%%{%%{
[%%%{-52224,[4,12,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,3]%%%}+%%%{%%{[%%%{29184,[4,12,10]%%%},0]:[1,0,%%
%{-1,[1,1,1]%%%}]%%},[1,7,2]%%%}+%%%{%%{[%%%{-10240,[4,12,12]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,1]%%%}+%
%%{%%{[%%%{2048,[4,12,14]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,0]%%%}+%%%{%%{[%%%{-3696,[5,14,0]%%%},0]:[1,
0,%%%{-1,[1,1,1]%%%}]%%},[1,6,6]%%%}+%%%{%%{[%%%{22176,[5,14,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,5]%%%}
+%%%{%%{[%%%{-56784,[5,14,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,4]%%%}+%%%{%%{[%%%{80448,[5,14,6]%%%},0]:
[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,3]%%%}+%%%{%%{[%%%{-67968,[5,14,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,2]
%%%}+%%%{%%{[%%%{33792,[5,14,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,1]%%%}+%%%{%%{[%%%{-8192,[5,14,12]%%%
},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,0]%%%}+%%%{%%{[%%%{-3696,[6,16,0]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,
5,5]%%%}+%%%{%%{[%%%{21168,[6,16,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,4]%%%}+%%%{%%{[%%%{-50400,[6,16,4]
%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,3]%%%}+%%%{%%{[%%%{63328,[6,16,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},
[1,5,2]%%%}+%%%{%%{[%%%{-43008,[6,16,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,1]%%%}+%%%{%%{[%%%{13056,[6,16
,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,0]%%%}+%%%{%%{[%%%{-2640,[7,18,0]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]
%%},[1,4,4]%%%}+%%%{%%{[%%%{13632,[7,18,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,3]%%%}+%%%{%%{[%%%{-27744,[
7,18,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,2]%%%}+%%%{%%{[%%%{26816,[7,18,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%
%}]%%},[1,4,1]%%%}+%%%{%%{[%%%{-10624,[7,18,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,0]%%%}+%%%{%%{[%%%{-132
0,[8,20,0]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,3]%%%}+%%%{%%{[%%%{5688,[8,20,2]%%%},0]:[1,0,%%%{-1,[1,1,1]
%%%}]%%},[1,3,2]%%%}+%%%{%%{[%%%{-8664,[8,20,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,1]%%%}+%%%{%%{[%%%{474
4,[8,20,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,0]%%%}+%%%{%%{[%%%{-440,[9,22,0]%%%},0]:[1,0,%%%{-1,[1,1,1]
%%%}]%%},[1,2,2]%%%}+%%%{%%{[%%%{1392,[9,22,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,2,1]%%%}+%%%{%%{[%%%{-117
6,[9,22,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,2,0]%%%}+%%%{%%{[%%%{-88,[10,24,0]%%%},0]:[1,0,%%%{-1,[1,1,1]
%%%}]%%},[1,1,1]%%%}+%%%{%%{[%%%{152,[10,24,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,1,0]%%%}+%%%{%%{[%%%{-8,[
11,26,0]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,0,0]%%%}+%%%{%%%{1,[0,4,0]%%%},[0,12,12]%%%}+%%%{%%%{-4,[0,4,2]
%%%},[0,12,11]%%%}+%%%{%%%{6,[0,4,4]%%%},[0,12,10]%%%}+%%%{%%%{-4,[0,4,6]%%%},[0,12,9]%%%}+%%%{%%%{1,[0,4,8]%%
%},[0,12,8]%%%}+%%%{%%%{12,[1,6,0]%%%},[0,11,11]%%%}+%%%{%%%{-60,[1,6,2]%%%},[0,11,10]%%%}+%%%{%%%{124,[1,6,4]
%%%},[0,11,9]%%%}+%%%{%%%{-132,[1,6,6]%%%},[0,11,8]%%%}+%%%{%%%{72,[1,6,8]%%%},[0,11,7]%%%}+%%%{%%%{-16,[1,6,1
0]%%%},[0,11,6]%%%}+%%%{%%%{66,[2,8,0]%%%},[0,10,10]%%%}+%%%{%%%{-380,[2,8,2]%%%},[0,10,9]%%%}+%%%{%%%{942,[2,
8,4]%%%},[0,10,8]%%%}+%%%{%%%{-1296,[2,8,6]%%%},[0,10,7]%%%}+%%%{%%%{1052,[2,8,8]%%%},[0,10,6]%%%}+%%%{%%%{-48
0,[2,8,10]%%%},[0,10,5]%%%}+%%%{%%%{96,[2,8,12]%%%},[0,10,4]%%%}+%%%{%%%{220,[3,10,0]%%%},[0,9,9]%%%}+%%%{%%%{
-1380,[3,10,2]%%%},[0,9,8]%%%}+%%%{%%%{3792,[3,10,4]%%%},[0,9,7]%%%}+%%%{%%%{-5968,[3,10,6]%%%},[0,9,6]%%%}+%%
%{%%%{5880,[3,10,8]%%%},[0,9,5]%%%}+%%%{%%%{-3696,[3,10,10]%%%},[0,9,4]%%%}+%%%{%%%{1408,[3,10,12]%%%},[0,9,3]
%%%}+%%%{%%%{-256,[3,10,14]%%%},[0,9,2]%%%}+%%%{%%%{495,[4,12,0]%%%},[0,8,8]%%%}+%%%{%%%{-3240,[4,12,2]%%%},[0
,8,7]%%%}+%%%{%%%{9324,[4,12,4]%%%},[0,8,6]%%%}+%%%{%%%{-15480,[4,12,6]%%%},[0,8,5]%%%}+%%%{%%%{16326,[4,12,8]
%%%},[0,8,4]%%%}+%%%{%%%{-11328,[4,12,10]%%%},[0,8,3]%%%}+%%%{%%%{5184,[4,12,12]%%%},[0,8,2]%%%}+%%%{%%%{-1536
,[4,12,14]%%%},[0,8,1]%%%}+%%%{%%%{256,[4,12,16]%%%},[0,8,0]%%%}+%%%{%%%{792,[5,14,0]%%%},[0,7,7]%%%}+%%%{%%%{
-5208,[5,14,2]%%%},[0,7,6]%%%}+%%%{%%%{14952,[5,14,4]%%%},[0,7,5]%%%}+%%%{%%%{-24504,[5,14,6]%%%},[0,7,4]%%%}+
%%%{%%%{25080,[5,14,8]%%%},[0,7,3]%%%}+%%%{%%%{-16368,[5,14,10]%%%},[0,7,2]%%%}+%%%{%%%{6528,[5,14,12]%%%},[0,
7,1]%%%}+%%%{%%%{-1280,[5,14,14]%%%},[0,7,0]%%%}+%%%{%%%{924,[6,16,0]%%%},[0,6,6]%%%}+%%%{%%%{-5880,[6,16,2]%%
%},[0,6,5]%%%}+%%%{%%%{16044,[6,16,4]%%%},[0,6,4]%%%}+%%%{%%%{-24272,[6,16,6]%%%},[0,6,3]%%%}+%%%{%%%{21788,[6
,16,8]%%%},[0,6,2]%%%}+%%%{%%%{-11232,[6,16,10]%%%},[0,6,1]%%%}+%%%{%%%{2656,[6,16,12]%%%},[0,6,0]%%%}+%%%{%%%
{792,[7,18,0]%%%},[0,5,5]%%%}+%%%{%%%{-4680,[7,18,2]%%%},[0,5,4]%%%}+%%%{%%%{11472,[7,18,4]%%%},[0,5,3]%%%}+%%
%{%%%{-14736,[7,18,6]%%%},[0,5,2]%%%}+%%%{%%%{10056,[7,18,8]%%%},[0,5,1]%%%}+%%%{%%%{-2960,[7,18,10]%%%},[0,5,
0]%%%}+%%%{%%%{495,[8,20,0]%%%},[0,4,4]%%%}+%%%{%%%{-2580,[8,20,2]%%%},[0,4,3]%%%}+%%%{%%%{5262,[8,20,4]%%%},[
0,4,2]%%%}+%%%{%%%{-5028,[8,20,6]%%%},[0,4,1]%%%}+%%%{%%%{1921,[8,20,8]%%%},[0,4,0]%%%}+%%%{%%%{220,[9,22,0]%%
%},[0,3,3]%%%}+%%%{%%%{-940,[9,22,2]%%%},[0,3,2]%%%}+%%%{%%%{1404,[9,22,4]%%%},[0,3,1]%%%}+%%%{%%%{-740,[9,22,
6]%%%},[0,3,0]%%%}+%%%{%%%{66,[10,24,0]%%%},[0,2,2]%%%}+%%%{%%%{-204,[10,24,2]%%%},[0,2,1]%%%}+%%%{%%%{166,[10
,24,4]%%%},[0,2,0]%%%}+%%%{%%%{12,[11,26,0]%%%},[0,1,1]%%%}+%%%{%%%{-20,[11,26,2]%%%},[0,1,0]%%%}+%%%{%%%{1,[1
2,28,0]%%%},[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[%%%{4
8,[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]%%%}+%%%{%%{poly
1[%%%{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]%%%}+%%%{%%{poly
1[%%%{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,[4,10,
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 = 0.05, size = 914, normalized size = 5.41 \begin {gather*} \frac {a^{3} c^{2} d^{2} e^{4} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+\left (x +\frac {d}{e}\right ) c d e}{\sqrt {c d e}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {c d e}}-\frac {3 a^{2} c^{3} d^{4} e^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+\left (x +\frac {d}{e}\right ) c d e}{\sqrt {c d e}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {c d e}}+\frac {3 a \,c^{4} d^{6} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+\left (x +\frac {d}{e}\right ) c d e}{\sqrt {c d e}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {c d e}}-\frac {c^{5} d^{8} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+\left (x +\frac {d}{e}\right ) c d e}{\sqrt {c d e}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {c d e}\, e^{2}}-\frac {4 \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, a \,c^{3} d^{3} e x}{\left (a \,e^{2}-c \,d^{2}\right )^{3}}+\frac {4 \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, c^{4} d^{5} x}{\left (a \,e^{2}-c \,d^{2}\right )^{3} e}-\frac {2 \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, a^{2} c^{2} d^{2} e^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{3}}+\frac {2 \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, c^{4} d^{6}}{\left (a \,e^{2}-c \,d^{2}\right )^{3} e^{2}}-\frac {16 \left (\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}} c^{3} d^{3}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} e}+\frac {16 \left (\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}} c^{2} d^{2}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (x +\frac {d}{e}\right )^{2} e^{2}}-\frac {4 \left (\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}} c d}{3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{3} e^{3}}-\frac {2 \left (\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4} e^{4}} \end {gather*}
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)^(3/2)/(e*x+d)^4,x)
[Out]
-2/3/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))^(5/2)-4/3/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))^(5/2)+16/3/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))^(5/2)-16/3/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))^(3/2)-4
*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))^(1/2)*x-2*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))^(1/2)+e^4*c^2*d^2/(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)-3*e^2*c^3*d^4/(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)+3*c^4*d^6/(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)+4/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))
^(1/2)*x+2/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))^(1/2)-1/e^2*c^5*d^8/(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)
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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)^(3/2)/(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 = -0.01 \begin {gather*} \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\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)^(3/2)/(d + e*x)^4,x)
[Out]
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(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 {3}{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)**(3/2)/(e*x+d)**4,x)
[Out]
Integral(((d + e*x)*(a*e + c*d*x))**(3/2)/(d + e*x)**4, x)
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