3.21.75 \(\int \frac {(d+e x)^{3/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [2075]

Optimal. Leaf size=251 \[ -\frac {2 \sqrt {d+e x}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 e}{3 c d \left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]

[Out]

2*e^(3/2)*arctan(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^(1/2)/(e*x+d)^(1/2))/(-a*e^2+c
*d^2)^(5/2)-2/3*(e*x+d)^(1/2)/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-2/3*e/c/d/(-a*e^2+c*d^2)/(e*x+d)^(1/
2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+2*e*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^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 = 0.11, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {682, 686, 680, 674, 211} \begin {gather*} \frac {2 e^{3/2} \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}+\frac {2 e \sqrt {d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 e}{3 c d \sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \sqrt {d+e x}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*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]

Rule 680

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*c*d - b*e)*(d + e
*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(
b^2 - 4*a*c))), 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] && LtQ[p, -1] && LtQ[0, m, 1] && IntegerQ[2*p]

Rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] - Dist[e^2*((m + p)/(c*(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] &
& LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a
+ b*x + c*x^2)^(p + 1)/((m + p + 1)*(2*c*d - b*e))), x] + Dist[c*((m + 2*p + 2)/((m + p + 1)*(2*c*d - b*e))),
Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rubi steps

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

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Mathematica [A]
time = 0.18, size = 135, normalized size = 0.54 \begin {gather*} -\frac {2 (d+e x)^{3/2} \left (\sqrt {c d^2-a e^2} \left (-4 a e^2+c d (d-3 e x)\right )-3 e^{3/2} (a e+c d x)^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )\right )}{3 \left (c d^2-a e^2\right )^{5/2} ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

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

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Maple [A]
time = 0.74, size = 224, normalized size = 0.89

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c d \,e^{2} x +3 \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) a \,e^{3} \sqrt {c d x +a e}-3 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c d e x -4 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,e^{2}+\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c \,d^{2}\right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} \left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(224\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 3.93, size = 764, normalized size = 3.04 \begin {gather*} \left [\frac {3 \, {\left (c^{2} d^{3} x^{2} e + a^{2} x e^{4} + {\left (2 \, a c d x^{2} + a^{2} d\right )} e^{3} + {\left (c^{2} d^{2} x^{3} + 2 \, a c d^{2} x\right )} e^{2}\right )} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} \log \left (\frac {c d^{3} - 2 \, a x e^{3} - 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d^{2} - a e^{2}\right )} \sqrt {x e + d} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} - {\left (c d x^{2} + 2 \, a d\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (3 \, c d x e - c d^{2} + 4 \, a e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (c^{4} d^{7} x^{2} + a^{2} c^{2} d^{2} x^{3} e^{5} + a^{2} c^{2} d^{5} e^{2} + a^{4} x e^{7} + {\left (2 \, a^{3} c d x^{2} + a^{4} d\right )} e^{6} - {\left (3 \, a^{2} c^{2} d^{3} x^{2} + 2 \, a^{3} c d^{3}\right )} e^{4} - {\left (2 \, a c^{3} d^{4} x^{3} + 3 \, a^{2} c^{2} d^{4} x\right )} e^{3} + {\left (c^{4} d^{6} x^{3} + 2 \, a c^{3} d^{6} x\right )} e\right )}}, \frac {2 \, {\left (\frac {3 \, {\left (c^{2} d^{3} x^{2} e + a^{2} x e^{4} + {\left (2 \, a c d x^{2} + a^{2} d\right )} e^{3} + {\left (c^{2} d^{2} x^{3} + 2 \, a c d^{2} x\right )} e^{2}\right )} \arctan \left (-\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {c d^{2} - a e^{2}} \sqrt {x e + d} e^{\frac {1}{2}}}{c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}}\right ) e^{\frac {1}{2}}}{\sqrt {c d^{2} - a e^{2}}} + \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (3 \, c d x e - c d^{2} + 4 \, a e^{2}\right )} \sqrt {x e + d}\right )}}{3 \, {\left (c^{4} d^{7} x^{2} + a^{2} c^{2} d^{2} x^{3} e^{5} + a^{2} c^{2} d^{5} e^{2} + a^{4} x e^{7} + {\left (2 \, a^{3} c d x^{2} + a^{4} d\right )} e^{6} - {\left (3 \, a^{2} c^{2} d^{3} x^{2} + 2 \, a^{3} c d^{3}\right )} e^{4} - {\left (2 \, a c^{3} d^{4} x^{3} + 3 \, a^{2} c^{2} d^{4} x\right )} e^{3} + {\left (c^{4} d^{6} x^{3} + 2 \, a c^{3} d^{6} x\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/3*(3*(c^2*d^3*x^2*e + a^2*x*e^4 + (2*a*c*d*x^2 + a^2*d)*e^3 + (c^2*d^2*x^3 + 2*a*c*d^2*x)*e^2)*sqrt(-e/(c*d
^2 - a*e^2))*log((c*d^3 - 2*a*x*e^3 - 2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(c*d^2 - a*e^2)*sqrt(x*e +
d)*sqrt(-e/(c*d^2 - a*e^2)) - (c*d*x^2 + 2*a*d)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) + 2*sqrt(c*d^2*x + a*x*e^2 +
(c*d*x^2 + a*d)*e)*(3*c*d*x*e - c*d^2 + 4*a*e^2)*sqrt(x*e + d))/(c^4*d^7*x^2 + a^2*c^2*d^2*x^3*e^5 + a^2*c^2*d
^5*e^2 + a^4*x*e^7 + (2*a^3*c*d*x^2 + a^4*d)*e^6 - (3*a^2*c^2*d^3*x^2 + 2*a^3*c*d^3)*e^4 - (2*a*c^3*d^4*x^3 +
3*a^2*c^2*d^4*x)*e^3 + (c^4*d^6*x^3 + 2*a*c^3*d^6*x)*e), 2/3*(3*(c^2*d^3*x^2*e + a^2*x*e^4 + (2*a*c*d*x^2 + a^
2*d)*e^3 + (c^2*d^2*x^3 + 2*a*c*d^2*x)*e^2)*arctan(-sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(c*d^2 - a
*e^2)*sqrt(x*e + d)*e^(1/2)/(c*d^2*x*e + a*x*e^3 + (c*d*x^2 + a*d)*e^2))*e^(1/2)/sqrt(c*d^2 - a*e^2) + sqrt(c*
d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(3*c*d*x*e - c*d^2 + 4*a*e^2)*sqrt(x*e + d))/(c^4*d^7*x^2 + a^2*c^2*d^2*x
^3*e^5 + a^2*c^2*d^5*e^2 + a^4*x*e^7 + (2*a^3*c*d*x^2 + a^4*d)*e^6 - (3*a^2*c^2*d^3*x^2 + 2*a^3*c*d^3)*e^4 - (
2*a*c^3*d^4*x^3 + 3*a^2*c^2*d^4*x)*e^3 + (c^4*d^6*x^3 + 2*a*c^3*d^6*x)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Integral((d + e*x)**(3/2)/((d + e*x)*(a*e + c*d*x))**(5/2), x)

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Giac [A]
time = 3.23, size = 375, normalized size = 1.49 \begin {gather*} \frac {2}{3} \, {\left (\frac {3 \, \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {c d^{2} e^{2} - a e^{4} - 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} e}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}}\right )} e - \frac {2 \, {\left (3 \, \sqrt {-c d^{2} e + a e^{3}} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e^{2} + 4 \, \sqrt {c d^{2} e - a e^{3}} e^{2}\right )}}{3 \, {\left (\sqrt {c d^{2} e - a e^{3}} \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} - 2 \, \sqrt {c d^{2} e - a e^{3}} \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} + \sqrt {c d^{2} e - a e^{3}} \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

2/3*(3*arctan(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))*e/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2
*e^4)*sqrt(c*d^2*e - a*e^3)) - (c*d^2*e^2 - a*e^4 - 3*((x*e + d)*c*d*e - c*d^2*e + a*e^3)*e)/((c^2*d^4 - 2*a*c
*d^2*e^2 + a^2*e^4)*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)))*e - 2/3*(3*sqrt(-c*d^2*e + a*e^3)*arctan(sqrt(
-c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))*e^2 + 4*sqrt(c*d^2*e - a*e^3)*e^2)/(sqrt(c*d^2*e - a*e^3)*sqrt(-c*d^2
*e + a*e^3)*c^2*d^4 - 2*sqrt(c*d^2*e - a*e^3)*sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 + sqrt(c*d^2*e - a*e^3)*sqrt(
-c*d^2*e + a*e^3)*a^2*e^4)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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