3.17.11 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=175 \[ -\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{e (d+e x)^2}+\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2}-\frac {3 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 e^{5/2}} \]

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Rubi [A]  time = 0.10, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 4, 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 )^{3/2}}{e (d+e x)^2}+\frac {3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2}-\frac {3 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right ) \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 )}{2 e^{5/2}} \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)^3,x]

[Out]

(3*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/e^2 - (2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(e
*(d + e*x)^2) - (3*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^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])])/(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]

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 )^{3/2}}{(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 )^{3/2}}{e (d+e x)^2}+\frac {(3 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} \, dx}{e}\\ &=\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{e (d+e x)^2}-\frac {\left (3 c d \left (c d^2-a e^2\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 e^2}\\ &=\frac {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{e (d+e x)^2}-\frac {\left (3 c d \left (c d^2-a e^2\right )\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 {3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{e (d+e x)^2}-\frac {3 \sqrt {c} \sqrt {d} \left (c d^2-a e^2\right ) \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 )}{2 e^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 108, normalized size = 0.62 \begin {gather*} \frac {2 c d (a e+c d x)^2 \sqrt {(d+e x) (a e+c d x)} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{5 \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)^(3/2)/(d + e*x)^3,x]

[Out]

(2*c*d*(a*e + c*d*x)^2*Sqrt[(a*e + c*d*x)*(d + e*x)]*Hypergeometric2F1[3/2, 5/2, 7/2, (e*(a*e + c*d*x))/(-(c*d
^2) + a*e^2)])/(5*(c*d^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])

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IntegrateAlgebraic [A]  time = 1.38, size = 298, normalized size = 1.70 \begin {gather*} \frac {3 \sqrt {c d e} \left (c d^2-a e^2\right ) \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 )}{4 e^3}-\frac {3 \left (c^{3/2} d^{5/2}-a \sqrt {c} \sqrt {d} e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-2 x \sqrt {c d e}\right )}{a e^2+c d^2}\right )}{2 e^{5/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (-2 a e^2+3 c d^2+c d e x\right )}{e^2 (d+e x)} \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)^3,x]

[Out]

((3*c*d^2 - 2*a*e^2 + c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^2*(d + e*x)) - (3*(c^(3/2)*d^(5
/2) - a*Sqrt[c]*Sqrt[d]*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-2*Sqrt[c*d*e]*x + 2*Sqrt[a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2]))/(c*d^2 + a*e^2)])/(2*e^(5/2)) + (3*Sqrt[c*d*e]*(c*d^2 - a*e^2)*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]])/(4*e^3)

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fricas [A]  time = 0.50, size = 414, normalized size = 2.37 \begin {gather*} \left [\frac {3 \, {\left (c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\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 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d e x + 3 \, c d^{2} - 2 \, a e^{2}\right )}}{4 \, {\left (e^{3} x + d e^{2}\right )}}, \frac {3 \, {\left (c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\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 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d e x + 3 \, c d^{2} - 2 \, a e^{2}\right )}}{2 \, {\left (e^{3} x + d 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)^3,x, algorithm="fricas")

[Out]

[1/4*(3*(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*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*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*e*x + 3*c*d^2 - 2*a*e^2))/(e^3*x + d
*e^2), 1/2*(3*(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*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)*(c*d*e*x + 3*c*d^2 - 2*a*e^2))/(e^3*x + d*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)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.46Unable to divide, perhaps due to rounding error%%%{%%%{2,[0,0,5]%%%},[6,6,6]%%%}+%%%{%%%{12,[
1,2,5]%%%},[6,5,5]%%%}+%%%{%%%{-24,[1,2,7]%%%},[6,5,4]%%%}+%%%{%%%{30,[2,4,5]%%%},[6,4,4]%%%}+%%%{%%%{-96,[2,4
,7]%%%},[6,4,3]%%%}+%%%{%%%{96,[2,4,9]%%%},[6,4,2]%%%}+%%%{%%%{40,[3,6,5]%%%},[6,3,3]%%%}+%%%{%%%{-144,[3,6,7]
%%%},[6,3,2]%%%}+%%%{%%%{192,[3,6,9]%%%},[6,3,1]%%%}+%%%{%%%{-128,[3,6,11]%%%},[6,3,0]%%%}+%%%{%%%{30,[4,8,5]%
%%},[6,2,2]%%%}+%%%{%%%{-96,[4,8,7]%%%},[6,2,1]%%%}+%%%{%%%{96,[4,8,9]%%%},[6,2,0]%%%}+%%%{%%%{12,[5,10,5]%%%}
,[6,1,1]%%%}+%%%{%%%{-24,[5,10,7]%%%},[6,1,0]%%%}+%%%{%%%{2,[6,12,5]%%%},[6,0,0]%%%}+%%%{%%{[%%%{-12,[0,1,4]%%
%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,6,6]%%%}+%%%{%%{[%%%{-72,[1,3,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,5,
5]%%%}+%%%{%%{[%%%{144,[1,3,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,5,4]%%%}+%%%{%%{[%%%{-180,[2,5,4]%%%},0]:
[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,4,4]%%%}+%%%{%%{[%%%{576,[2,5,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,4,3]%%%}
+%%%{%%{[%%%{-576,[2,5,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,4,2]%%%}+%%%{%%{[%%%{-240,[3,7,4]%%%},0]:[1,0,
%%%{-1,[1,1,1]%%%}]%%},[5,3,3]%%%}+%%%{%%{[%%%{864,[3,7,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,3,2]%%%}+%%%{
%%{[%%%{-1152,[3,7,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,3,1]%%%}+%%%{%%{[%%%{768,[3,7,10]%%%},0]:[1,0,%%%{
-1,[1,1,1]%%%}]%%},[5,3,0]%%%}+%%%{%%{[%%%{-180,[4,9,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,2,2]%%%}+%%%{%%{
[%%%{576,[4,9,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,2,1]%%%}+%%%{%%{[%%%{-576,[4,9,8]%%%},0]:[1,0,%%%{-1,[1
,1,1]%%%}]%%},[5,2,0]%%%}+%%%{%%{[%%%{-72,[5,11,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,1,1]%%%}+%%%{%%{[%%%{
144,[5,11,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[5,1,0]%%%}+%%%{%%{[%%%{-12,[6,13,4]%%%},0]:[1,0,%%%{-1,[1,1,1
]%%%}]%%},[5,0,0]%%%}+%%%{%%%{6,[0,1,4]%%%},[4,7,7]%%%}+%%%{%%%{-6,[0,1,6]%%%},[4,7,6]%%%}+%%%{%%%{66,[1,3,4]%
%%},[4,6,6]%%%}+%%%{%%%{-108,[1,3,6]%%%},[4,6,5]%%%}+%%%{%%%{72,[1,3,8]%%%},[4,6,4]%%%}+%%%{%%%{270,[2,5,4]%%%
},[4,5,5]%%%}+%%%{%%%{-738,[2,5,6]%%%},[4,5,4]%%%}+%%%{%%%{576,[2,5,8]%%%},[4,5,3]%%%}+%%%{%%%{-288,[2,5,10]%%
%},[4,5,2]%%%}+%%%{%%%{570,[3,7,4]%%%},[4,4,4]%%%}+%%%{%%%{-1992,[3,7,6]%%%},[4,4,3]%%%}+%%%{%%%{2448,[3,7,8]%
%%},[4,4,2]%%%}+%%%{%%%{-960,[3,7,10]%%%},[4,4,1]%%%}+%%%{%%%{384,[3,7,12]%%%},[4,4,0]%%%}+%%%{%%%{690,[4,9,4]
%%%},[4,3,3]%%%}+%%%{%%%{-2538,[4,9,6]%%%},[4,3,2]%%%}+%%%{%%%{3456,[4,9,8]%%%},[4,3,1]%%%}+%%%{%%%{-2208,[4,9
,10]%%%},[4,3,0]%%%}+%%%{%%%{486,[5,11,4]%%%},[4,2,2]%%%}+%%%{%%%{-1548,[5,11,6]%%%},[4,2,1]%%%}+%%%{%%%{1512,
[5,11,8]%%%},[4,2,0]%%%}+%%%{%%%{186,[6,13,4]%%%},[4,1,1]%%%}+%%%{%%%{-366,[6,13,6]%%%},[4,1,0]%%%}+%%%{%%%{30
,[7,15,4]%%%},[4,0,0]%%%}+%%%{%%{[%%%{-24,[0,2,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,7,7]%%%}+%%%{%%{[%%%{2
4,[0,2,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,7,6]%%%}+%%%{%%{[%%%{-184,[1,4,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%
%}]%%},[3,6,6]%%%}+%%%{%%{[%%%{432,[1,4,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,6,5]%%%}+%%%{%%{[%%%{-288,[1,
4,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,6,4]%%%}+%%%{%%{[%%%{-600,[2,6,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%
},[3,5,5]%%%}+%%%{%%{[%%%{1992,[2,6,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,5,4]%%%}+%%%{%%{[%%%{-2304,[2,6,7
]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,5,3]%%%}+%%%{%%{[%%%{1152,[2,6,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[
3,5,2]%%%}+%%%{%%{[%%%{-1080,[3,8,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,4,4]%%%}+%%%{%%{[%%%{4128,[3,8,5]%%
%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,4,3]%%%}+%%%{%%{[%%%{-5952,[3,8,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,
4,2]%%%}+%%%{%%{[%%%{3840,[3,8,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,4,1]%%%}+%%%{%%{[%%%{-1536,[3,8,11]%%%
},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,4,0]%%%}+%%%{%%{[%%%{-1160,[4,10,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,
3,3]%%%}+%%%{%%{[%%%{4392,[4,10,5]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,3,2]%%%}+%%%{%%{[%%%{-6144,[4,10,7]%%
%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,3,1]%%%}+%%%{%%{[%%%{3712,[4,10,9]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,
3,0]%%%}+%%%{%%{[%%%{-744,[5,12,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,2,2]%%%}+%%%{%%{[%%%{2352,[5,12,5]%%%
},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,2,1]%%%}+%%%{%%{[%%%{-2208,[5,12,7]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,
2,0]%%%}+%%%{%%{[%%%{-264,[6,14,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,1,1]%%%}+%%%{%%{[%%%{504,[6,14,5]%%%}
,0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,1,0]%%%}+%%%{%%{[%%%{-40,[7,16,3]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[3,0,0
]%%%}+%%%{%%%{6,[0,2,3]%%%},[2,8,8]%%%}+%%%{%%%{-12,[0,2,5]%%%},[2,8,7]%%%}+%%%{%%%{6,[0,2,7]%%%},[2,8,6]%%%}+
%%%{%%%{72,[1,4,3]%%%},[2,7,7]%%%}+%%%{%%%{-180,[1,4,5]%%%},[2,7,6]%%%}+%%%{%%%{180,[1,4,7]%%%},[2,7,5]%%%}+%%
%{%%%{-72,[1,4,9]%%%},[2,7,4]%%%}+%%%{%%%{336,[2,6,3]%%%},[2,6,6]%%%}+%%%{%%%{-1116,[2,6,5]%%%},[2,6,5]%%%}+%%
%{%%%{1386,[2,6,7]%%%},[2,6,4]%%%}+%%%{%%%{-864,[2,6,9]%%%},[2,6,3]%%%}+%%%{%%%{288,[2,6,11]%%%},[2,6,2]%%%}+%
%%{%%%{840,[3,8,3]%%%},[2,5,5]%%%}+%%%{%%%{-3300,[3,8,5]%%%},[2,5,4]%%%}+%%%{%%%{5016,[3,8,7]%%%},[2,5,3]%%%}+
%%%{%%%{-3696,[3,8,9]%%%},[2,5,2]%%%}+%%%{%%%{1344,[3,8,11]%%%},[2,5,1]%%%}+%%%{%%%{-384,[3,8,13]%%%},[2,5,0]%
%%}+%%%{%%%{1260,[4,10,3]%%%},[2,4,4]%%%}+%%%{%%%{-5220,[4,10,5]%%%},[2,4,3]%%%}+%%%{%%%{8442,[4,10,7]%%%},[2,
4,2]%%%}+%%%{%%%{-6624,[4,10,9]%%%},[2,4,1]%%%}+%%%{%%%{2592,[4,10,11]%%%},[2,4,0]%%%}+%%%{%%%{1176,[5,12,3]%%
%},[2,3,3]%%%}+%%%{%%%{-4572,[5,12,5]%%%},[2,3,2]%%%}+%%%{%%%{6516,[5,12,7]%%%},[2,3,1]%%%}+%%%{%%%{-3720,[5,1
2,9]%%%},[2,3,0]%%%}+%%%{%%%{672,[6,14,3]%%%},[2,2,2]%%%}+%%%{%%%{-2100,[6,14,5]%%%},[2,2,1]%%%}+%%%{%%%{1878,
[6,14,7]%%%},[2,2,0]%%%}+%%%{%%%{216,[7,16,3]%%%},[2,1,1]%%%}+%%%{%%%{-396,[7,16,5]%%%},[2,1,0]%%%}+%%%{%%%{30
,[8,18,3]%%%},[2,0,0]%%%}+%%%{%%{[%%%{-12,[0,3,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,8]%%%}+%%%{%%{[%%%{2
4,[0,3,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,7]%%%}+%%%{%%{[%%%{-12,[0,3,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%
}]%%},[1,8,6]%%%}+%%%{%%{[%%%{-96,[1,5,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,7]%%%}+%%%{%%{[%%%{312,[1,5,
4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,6]%%%}+%%%{%%{[%%%{-360,[1,5,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},
[1,7,5]%%%}+%%%{%%{[%%%{144,[1,5,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,4]%%%}+%%%{%%{[%%%{-336,[2,7,2]%%%
},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,6]%%%}+%%%{%%{[%%%{1368,[2,7,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,
5]%%%}+%%%{%%{[%%%{-2196,[2,7,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,4]%%%}+%%%{%%{[%%%{1728,[2,7,8]%%%},0
]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,3]%%%}+%%%{%%{[%%%{-576,[2,7,10]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,2]
%%%}+%%%{%%{[%%%{-672,[3,9,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,5]%%%}+%%%{%%{[%%%{3000,[3,9,4]%%%},0]:[
1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,4]%%%}+%%%{%%{[%%%{-5424,[3,9,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,3]%%%
}+%%%{%%{[%%%{5088,[3,9,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,2]%%%}+%%%{%%{[%%%{-2688,[3,9,10]%%%},0]:[1
,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,1]%%%}+%%%{%%{[%%%{768,[3,9,12]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,0]%%%}+
%%%{%%{[%%%{-840,[4,11,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,4]%%%}+%%%{%%{[%%%{3720,[4,11,4]%%%},0]:[1,0
,%%%{-1,[1,1,1]%%%}]%%},[1,4,3]%%%}+%%%{%%{[%%%{-6516,[4,11,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,2]%%%}+
%%%{%%{[%%%{5568,[4,11,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,1]%%%}+%%%{%%{[%%%{-2112,[4,11,10]%%%},0]:[1
,0,%%%{-1,[1,1,1]%%%}]%%},[1,4,0]%%%}+%%%{%%{[%%%{-672,[5,13,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,3]%%%}
+%%%{%%{[%%%{2664,[5,13,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,2]%%%}+%%%{%%{[%%%{-3816,[5,13,6]%%%},0]:[1
,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,1]%%%}+%%%{%%{[%%%{2064,[5,13,8]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,3,0]%%%}
+%%%{%%{[%%%{-336,[6,15,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,2,2]%%%}+%%%{%%{[%%%{1032,[6,15,4]%%%},0]:[1,
0,%%%{-1,[1,1,1]%%%}]%%},[1,2,1]%%%}+%%%{%%{[%%%{-876,[6,15,6]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,2,0]%%%}+
%%%{%%{[%%%{-96,[7,17,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,1,1]%%%}+%%%{%%{[%%%{168,[7,17,4]%%%},0]:[1,0,%
%%{-1,[1,1,1]%%%}]%%},[1,1,0]%%%}+%%%{%%{[%%%{-12,[8,19,2]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,0,0]%%%}+%%%{
%%%{2,[0,3,2]%%%},[0,9,9]%%%}+%%%{%%%{-6,[0,3,4]%%%},[0,9,8]%%%}+%%%{%%%{6,[0,3,6]%%%},[0,9,7]%%%}+%%%{%%%{-2,
[0,3,8]%%%},[0,9,6]%%%}+%%%{%%%{18,[1,5,2]%%%},[0,8,8]%%%}+%%%{%%%{-72,[1,5,4]%%%},[0,8,7]%%%}+%%%{%%%{114,[1,
5,6]%%%},[0,8,6]%%%}+%%%{%%%{-84,[1,5,8]%%%},[0,8,5]%%%}+%%%{%%%{24,[1,5,10]%%%},[0,8,4]%%%}+%%%{%%%{72,[2,7,2
]%%%},[0,7,7]%%%}+%%%{%%%{-336,[2,7,4]%%%},[0,7,6]%%%}+%%%{%%%{654,[2,7,6]%%%},[0,7,5]%%%}+%%%{%%%{-678,[2,7,8
]%%%},[0,7,4]%%%}+%%%{%%%{384,[2,7,10]%%%},[0,7,3]%%%}+%%%{%%%{-96,[2,7,12]%%%},[0,7,2]%%%}+%%%{%%%{168,[3,9,2
]%%%},[0,6,6]%%%}+%%%{%%%{-840,[3,9,4]%%%},[0,6,5]%%%}+%%%{%%%{1770,[3,9,6]%%%},[0,6,4]%%%}+%%%{%%%{-2040,[3,9
,8]%%%},[0,6,3]%%%}+%%%{%%%{1392,[3,9,10]%%%},[0,6,2]%%%}+%%%{%%%{-576,[3,9,12]%%%},[0,6,1]%%%}+%%%{%%%{128,[3
,9,14]%%%},[0,6,0]%%%}+%%%{%%%{252,[4,11,2]%%%},[0,5,5]%%%}+%%%{%%%{-1260,[4,11,4]%%%},[0,5,4]%%%}+%%%{%%%{261
0,[4,11,6]%%%},[0,5,3]%%%}+%%%{%%%{-2862,[4,11,8]%%%},[0,5,2]%%%}+%%%{%%%{1728,[4,11,10]%%%},[0,5,1]%%%}+%%%{%
%%{-480,[4,11,12]%%%},[0,5,0]%%%}+%%%{%%%{252,[5,13,2]%%%},[0,4,4]%%%}+%%%{%%%{-1176,[5,13,4]%%%},[0,4,3]%%%}+
%%%{%%%{2166,[5,13,6]%%%},[0,4,2]%%%}+%%%{%%%{-1908,[5,13,8]%%%},[0,4,1]%%%}+%%%{%%%{696,[5,13,10]%%%},[0,4,0]
%%%}+%%%{%%%{168,[6,15,2]%%%},[0,3,3]%%%}+%%%{%%%{-672,[6,15,4]%%%},[0,3,2]%%%}+%%%{%%%{954,[6,15,6]%%%},[0,3,
1]%%%}+%%%{%%%{-490,[6,15,8]%%%},[0,3,0]%%%}+%%%{%%%{72,[7,17,2]%%%},[0,2,2]%%%}+%%%{%%%{-216,[7,17,4]%%%},[0,
2,1]%%%}+%%%{%%%{174,[7,17,6]%%%},[0,2,0]%%%}+%%%{%%%{18,[8,19,2]%%%},[0,1,1]%%%}+%%%{%%%{-30,[8,19,4]%%%},[0,
1,0]%%%}+%%%{%%%{2,[9,21,2]%%%},[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[%%%{-1
5,[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 Argument Value

________________________________________________________________________________________

maple [B]  time = 0.06, size = 837, normalized size = 4.78 \begin {gather*} \frac {3 a^{3} c d \,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 )}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {c d e}}-\frac {9 a^{2} c^{2} d^{3} 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 )}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {c d e}}+\frac {9 a \,c^{3} d^{5} \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 )}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {c d e}}-\frac {3 c^{4} d^{7} \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 )}{2 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {c d e}\, e^{2}}-\frac {6 \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^{2} d^{2} e x}{\left (a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {6 \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^{3} d^{4} x}{\left (a \,e^{2}-c \,d^{2}\right )^{2} e}-\frac {3 \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 d \,e^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {3 \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^{3} d^{5}}{\left (a \,e^{2}-c \,d^{2}\right )^{2} e^{2}}-\frac {8 \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^{2} d^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{2} e}+\frac {8 \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}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{2} e^{2}}-\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}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3} e^{3}} \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)^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))^(5/2)+8/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))^(5/2)-8/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))^(3/2)-6*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))^(1/2)*x-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))^(1/2)+3/2*e^4*c*d/(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)-9/2*e^2*c^2*d^3/(
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)+9/2*c^3*d^5/(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)+6/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))^(1/2)*x+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))^(1/2)-3/2/e^2*c^4
*d^7/(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)

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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)^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 = -0.01 \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 )}^3} \,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)^3,x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(d + e*x)^3, 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 )^{3}}\, 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)**3,x)

[Out]

Integral(((d + e*x)*(a*e + c*d*x))**(3/2)/(d + e*x)**3, x)

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