3.5.49 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{d+e x} \, dx\) [449]

Optimal. Leaf size=201 \[ \frac {1}{8} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e}+\frac {\left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 c^{3/2} d^{3/2} e^{5/2}} \]

[Out]

1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e+1/16*(-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))/c^(3/2)/d^(3/2)/e^(5/2)+1/8*(a/c/d-d/e^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)

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Rubi [A]
time = 0.07, antiderivative size = 201, 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 = {678, 626, 635, 212} \begin {gather*} \frac {\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 c^{3/2} d^{3/2} e^{5/2}}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e}+\frac {1}{8} \left (\frac {a}{c d}-\frac {d}{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} \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),x]

[Out]

((a/(c*d) - d/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 + (a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(3*e) + ((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*c^(3/2)*d^(3/2)*e^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

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 635

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 678

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} \, dx &=\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e}-\frac {\left (2 c d^2 e-e \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 {1}{8} \left (\frac {a}{c d}-\frac {d}{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}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e}+\frac {\left (c d^2-a e^2\right )^3 \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 c d e^2}\\ &=\frac {1}{8} \left (\frac {a}{c d}-\frac {d}{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}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e}+\frac {\left (c d^2-a e^2\right )^3 \text {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 c d e^2}\\ &=\frac {1}{8} \left (\frac {a}{c d}-\frac {d}{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}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e}+\frac {\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 c^{3/2} d^{3/2} e^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 180, normalized size = 0.90 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (3 a^2 e^4+2 a c d e^2 (4 d+7 e x)+c^2 d^2 \left (-3 d^2+2 d e x+8 e^2 x^2\right )\right )+\frac {3 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{24 c^{3/2} d^{3/2} 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),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(3*a^2*e^4 + 2*a*c*d*e^2*(4*d + 7*e*x) + c^2*d^2*(-3*d
^2 + 2*d*e*x + 8*e^2*x^2)) + (3*(c*d^2 - a*e^2)^3*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e +
c*d*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(24*c^(3/2)*d^(3/2)*e^(5/2))

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Maple [A]
time = 0.09, size = 230, normalized size = 1.14

method	result	size
default	\(\frac {\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{2}}{e}\)	\(230\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e*(1/3*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^2-c*d^2)*(1/4*(2*c*d*e*(x+d/e)+a*e^2-c*d^2)/c/
d/e*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/c/d/e*ln((1/2*a*e^2-1/2*c*d^2+c*d*e*(x+d
/e))/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (173) = 346\).
time = 0.30, size = 451, normalized size = 2.24 \begin {gather*} \frac {c^{3} d^{6} e^{\left (-\frac {5}{2}\right )} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{16 \, \left (c d\right )^{\frac {3}{2}}} - \frac {3 \, a c^{2} d^{4} e^{\left (-\frac {1}{2}\right )} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{16 \, \left (c d\right )^{\frac {3}{2}}} - \frac {1}{4} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} c d^{2} x e^{\left (-1\right )} - \frac {1}{8} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} c d^{3} e^{\left (-2\right )} + \frac {3 \, a^{2} c d^{2} e^{\frac {3}{2}} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{16 \, \left (c d\right )^{\frac {3}{2}}} + \frac {1}{4} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a x e - \frac {a^{3} e^{\frac {7}{2}} \log \left (c d^{2} e^{\left (-1\right )} + 2 \, c d x + a e + 2 \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} \sqrt {c d} e^{\left (-\frac {1}{2}\right )}\right )}{16 \, \left (c d\right )^{\frac {3}{2}}} + \frac {1}{3} \, {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{\frac {3}{2}} e^{\left (-1\right )} + \frac {\sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} a^{2} e^{2}}{8 \, c d} \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),x, algorithm="maxima")

[Out]

1/16*c^3*d^6*e^(-5/2)*log(c*d^2*e^(-1) + 2*c*d*x + a*e + 2*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*sqrt(c*
d)*e^(-1/2))/(c*d)^(3/2) - 3/16*a*c^2*d^4*e^(-1/2)*log(c*d^2*e^(-1) + 2*c*d*x + a*e + 2*sqrt(c*d*x^2*e + c*d^2
*x + a*x*e^2 + a*d*e)*sqrt(c*d)*e^(-1/2))/(c*d)^(3/2) - 1/4*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*c*d^2*
x*e^(-1) - 1/8*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*c*d^3*e^(-2) + 3/16*a^2*c*d^2*e^(3/2)*log(c*d^2*e^(
-1) + 2*c*d*x + a*e + 2*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*sqrt(c*d)*e^(-1/2))/(c*d)^(3/2) + 1/4*sqrt
(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*a*x*e - 1/16*a^3*e^(7/2)*log(c*d^2*e^(-1) + 2*c*d*x + a*e + 2*sqrt(c*d
*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*sqrt(c*d)*e^(-1/2))/(c*d)^(3/2) + 1/3*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d
*e)^(3/2)*e^(-1) + 1/8*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*a^2*e^2/(c*d)

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Fricas [A]
time = 2.71, size = 511, normalized size = 2.54 \begin {gather*} \left [-\frac {{\left (3 \, {\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^{\frac {1}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} - 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) - 4 \, {\left (2 \, c^{3} d^{4} x e^{2} - 3 \, c^{3} d^{5} e + 14 \, a c^{2} d^{2} x e^{4} + 3 \, a^{2} c d e^{5} + 8 \, {\left (c^{3} d^{3} x^{2} + a c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-3\right )}}{96 \, c^{2} d^{2}}, -\frac {{\left (3 \, {\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^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) - 2 \, {\left (2 \, c^{3} d^{4} x e^{2} - 3 \, c^{3} d^{5} e + 14 \, a c^{2} d^{2} x e^{4} + 3 \, a^{2} c d e^{5} + 8 \, {\left (c^{3} d^{3} x^{2} + a c^{2} d^{3}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-3\right )}}{48 \, c^{2} d^{2}}\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),x, algorithm="fricas")

[Out]

[-1/96*(3*(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^(1/2)*log(8*c^2*d^3*x*e + c^2*d^
4 + 8*a*c*d*x*e^3 + a^2*e^4 - 4*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(c
*d)*e^(1/2) + 2*(4*c^2*d^2*x^2 + 3*a*c*d^2)*e^2) - 4*(2*c^3*d^4*x*e^2 - 3*c^3*d^5*e + 14*a*c^2*d^2*x*e^4 + 3*a
^2*c*d*e^5 + 8*(c^3*d^3*x^2 + a*c^2*d^3)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-3)/(c^2*d^2), -
1/48*(3*(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^2*x + a*x*e^2
+ (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^3*x*e + a*c*d*x*e^3 + (c^2*d^2*x^2 + a*c
*d^2)*e^2)) - 2*(2*c^3*d^4*x*e^2 - 3*c^3*d^5*e + 14*a*c^2*d^2*x*e^4 + 3*a^2*c*d*e^5 + 8*(c^3*d^3*x^2 + a*c^2*d
^3)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-3)/(c^2*d^2)]

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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}}}{d + e x}\, 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),x)

[Out]

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

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Giac [A]
time = 2.46, size = 223, normalized size = 1.11 \begin {gather*} \frac {1}{24} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, c d x + \frac {{\left (c^{3} d^{4} e + 7 \, a c^{2} d^{2} e^{3}\right )} e^{\left (-2\right )}}{c^{2} d^{2}}\right )} x - \frac {{\left (3 \, c^{3} d^{5} - 8 \, a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4}\right )} e^{\left (-2\right )}}{c^{2} d^{2}}\right )} - \frac {{\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 )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{16 \, \sqrt {c d} c d} \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),x, algorithm="giac")

[Out]

1/24*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*c*d*x + (c^3*d^4*e + 7*a*c^2*d^2*e^3)*e^(-2)/(c^2*d^2))
*x - (3*c^3*d^5 - 8*a*c^2*d^3*e^2 - 3*a^2*c*d*e^4)*e^(-2)/(c^2*d^2)) - 1/16*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2
*c*d^2*e^4 - a^3*e^6)*e^(-5/2)*log(abs(-c*d^2 - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 +
a*d*e))*sqrt(c*d)*e^(1/2) - a*e^2))/(sqrt(c*d)*c*d)

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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 )}^{3/2}}{d+e\,x} \,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),x)

[Out]

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

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