3.2071 \(\int \frac {1}{(d+e x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=331 \[ -\frac {35 c^3 d^3 \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {35 c^3 d^3 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{8 \left (c d^2-a e^2\right )^{9/2}}+\frac {35 c^2 d^2}{24 \sqrt {d+e x} \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {7 c d}{12 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

-35/8*c^3*d^3*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))*e^(1/
2)/(-a*e^2+c*d^2)^(9/2)+1/3/(-a*e^2+c*d^2)/(e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+7/12*c*d/(-a*
e^2+c*d^2)^2/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/24*c^2*d^2/(-a*e^2+c*d^2)^3/(e*x+d)^(1/2
)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-35/8*c^3*d^3*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^4/(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)

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Rubi [A]  time = 0.25, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {672, 666, 660, 205} \[ -\frac {35 c^3 d^3 \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {35 c^2 d^2}{24 \sqrt {d+e x} \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {35 c^3 d^3 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{8 \left (c d^2-a e^2\right )^{9/2}}+\frac {7 c d}{12 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {1}{3 (d+e x)^{5/2} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

1/(3*(c*d^2 - a*e^2)*(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (7*c*d)/(12*(c*d^2 - a*e^2
)^2*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*c^2*d^2)/(24*(c*d^2 - a*e^2)^3*Sqrt[d +
e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (35*c^3*d^3*Sqrt[d + e*x])/(8*(c*d^2 - a*e^2)^4*Sqrt[a*d*
e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (35*c^3*d^3*Sqrt[e]*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])])/(8*(c*d^2 - a*e^2)^(9/2))

Rule 205

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

Rule 660

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 666

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 672

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)), I
nt[(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] && EqQ
[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 {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(7 c d) \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{6 \left (c d^2-a e^2\right )}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (35 c^2 d^2\right ) \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}{24 \left (c d^2-a e^2\right )^2}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2}{24 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (35 c^3 d^3\right ) \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}{16 \left (c d^2-a e^2\right )^3}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2}{24 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {35 c^3 d^3 \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (35 c^3 d^3 e\right ) \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}{16 \left (c d^2-a e^2\right )^4}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2}{24 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {35 c^3 d^3 \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (35 c^3 d^3 e^2\right ) \operatorname {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 )}{8 \left (c d^2-a e^2\right )^4}\\ &=\frac {1}{3 \left (c d^2-a e^2\right ) (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {7 c d}{12 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 c^2 d^2}{24 \left (c d^2-a e^2\right )^3 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {35 c^3 d^3 \sqrt {d+e x}}{8 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {35 c^3 d^3 \sqrt {e} \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 )}{8 \left (c d^2-a e^2\right )^{9/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 81, normalized size = 0.24 \[ -\frac {2 c^3 d^3 \sqrt {d+e x} \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{\left (c d^2-a e^2\right )^4 \sqrt {(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-2*c^3*d^3*Sqrt[d + e*x]*Hypergeometric2F1[-1/2, 4, 1/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/((c*d^2 - a*e
^2)^4*Sqrt[(a*e + c*d*x)*(d + e*x)])

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fricas [B]  time = 0.98, size = 1584, normalized size = 4.79 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/48*(105*(c^4*d^4*e^4*x^5 + a*c^3*d^7*e + (4*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^4 + 2*(3*c^4*d^6*e^2 + 2*a*c^3*d
^4*e^4)*x^3 + 2*(2*c^4*d^7*e + 3*a*c^3*d^5*e^3)*x^2 + (c^4*d^8 + 4*a*c^3*d^6*e^2)*x)*sqrt(-e/(c*d^2 - a*e^2))*
log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e
^2)*sqrt(e*x + d)*sqrt(-e/(c*d^2 - a*e^2)))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(105*c^3*d^3*e^3*x^3 + 48*c^3*d^6 +
87*a*c^2*d^4*e^2 - 38*a^2*c*d^2*e^4 + 8*a^3*e^6 + 35*(8*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + 7*(33*c^3*d^5*e +
14*a*c^2*d^3*e^3 - 2*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a*c^4*d^12*e
- 4*a^2*c^3*d^10*e^3 + 6*a^3*c^2*d^8*e^5 - 4*a^4*c*d^6*e^7 + a^5*d^4*e^9 + (c^5*d^9*e^4 - 4*a*c^4*d^7*e^6 + 6*
a^2*c^3*d^5*e^8 - 4*a^3*c^2*d^3*e^10 + a^4*c*d*e^12)*x^5 + (4*c^5*d^10*e^3 - 15*a*c^4*d^8*e^5 + 20*a^2*c^3*d^6
*e^7 - 10*a^3*c^2*d^4*e^9 + a^5*e^13)*x^4 + 2*(3*c^5*d^11*e^2 - 10*a*c^4*d^9*e^4 + 10*a^2*c^3*d^7*e^6 - 5*a^4*
c*d^3*e^10 + 2*a^5*d*e^12)*x^3 + 2*(2*c^5*d^12*e - 5*a*c^4*d^10*e^3 + 10*a^3*c^2*d^6*e^7 - 10*a^4*c*d^4*e^9 +
3*a^5*d^2*e^11)*x^2 + (c^5*d^13 - 10*a^2*c^3*d^9*e^4 + 20*a^3*c^2*d^7*e^6 - 15*a^4*c*d^5*e^8 + 4*a^5*d^3*e^10)
*x), -1/24*(105*(c^4*d^4*e^4*x^5 + a*c^3*d^7*e + (4*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^4 + 2*(3*c^4*d^6*e^2 + 2*a*
c^3*d^4*e^4)*x^3 + 2*(2*c^4*d^7*e + 3*a*c^3*d^5*e^3)*x^2 + (c^4*d^8 + 4*a*c^3*d^6*e^2)*x)*sqrt(e/(c*d^2 - a*e^
2))*arctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(e/(c*d^2 - a*e^2))/
(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) + (105*c^3*d^3*e^3*x^3 + 48*c^3*d^6 + 87*a*c^2*d^4*e^2 - 38*a^2
*c*d^2*e^4 + 8*a^3*e^6 + 35*(8*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + 7*(33*c^3*d^5*e + 14*a*c^2*d^3*e^3 - 2*a^2*c
*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a*c^4*d^12*e - 4*a^2*c^3*d^10*e^3 + 6*a
^3*c^2*d^8*e^5 - 4*a^4*c*d^6*e^7 + a^5*d^4*e^9 + (c^5*d^9*e^4 - 4*a*c^4*d^7*e^6 + 6*a^2*c^3*d^5*e^8 - 4*a^3*c^
2*d^3*e^10 + a^4*c*d*e^12)*x^5 + (4*c^5*d^10*e^3 - 15*a*c^4*d^8*e^5 + 20*a^2*c^3*d^6*e^7 - 10*a^3*c^2*d^4*e^9
+ a^5*e^13)*x^4 + 2*(3*c^5*d^11*e^2 - 10*a*c^4*d^9*e^4 + 10*a^2*c^3*d^7*e^6 - 5*a^4*c*d^3*e^10 + 2*a^5*d*e^12)
*x^3 + 2*(2*c^5*d^12*e - 5*a*c^4*d^10*e^3 + 10*a^3*c^2*d^6*e^7 - 10*a^4*c*d^4*e^9 + 3*a^5*d^2*e^11)*x^2 + (c^5
*d^13 - 10*a^2*c^3*d^9*e^4 + 20*a^3*c^2*d^7*e^6 - 15*a^4*c*d^5*e^8 + 4*a^5*d^3*e^10)*x)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.09, size = 559, normalized size = 1.69 \[ \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (105 \sqrt {c d x +a e}\, c^{3} d^{3} e^{4} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+315 \sqrt {c d x +a e}\, c^{3} d^{4} e^{3} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+315 \sqrt {c d x +a e}\, c^{3} d^{5} e^{2} x \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )-105 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{3} d^{3} e^{3} x^{3}-35 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{2} d^{2} e^{4} x^{2}+105 \sqrt {c d x +a e}\, c^{3} d^{6} e \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )-280 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{3} d^{4} e^{2} x^{2}+14 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c d \,e^{5} x -98 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{2} d^{3} e^{3} x -231 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{3} d^{5} e x -8 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{3} e^{6}+38 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c \,d^{2} e^{4}-87 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{2} d^{4} e^{2}-48 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{3} d^{6}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \left (c d x +a e \right ) \left (a \,e^{2}-c \,d^{2}\right )^{4} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2),x)

[Out]

1/24*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(105*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*(c*d*x+
a*e)^(1/2)*x^3*c^3*d^3*e^4+315*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*(c*d*x+a*e)^(1/2)*x^2*c^3*
d^4*e^3+315*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*(c*d*x+a*e)^(1/2)*x*c^3*d^5*e^2-105*((a*e^2-c
*d^2)*e)^(1/2)*x^3*c^3*d^3*e^3+105*arctanh((c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2)*e)*(c*d*x+a*e)^(1/2)*c^3*
d^6*e-35*((a*e^2-c*d^2)*e)^(1/2)*x^2*a*c^2*d^2*e^4-280*((a*e^2-c*d^2)*e)^(1/2)*x^2*c^3*d^4*e^2+14*((a*e^2-c*d^
2)*e)^(1/2)*x*a^2*c*d*e^5-98*((a*e^2-c*d^2)*e)^(1/2)*x*a*c^2*d^3*e^3-231*((a*e^2-c*d^2)*e)^(1/2)*x*c^3*d^5*e-8
*((a*e^2-c*d^2)*e)^(1/2)*a^3*e^6+38*((a*e^2-c*d^2)*e)^(1/2)*a^2*c*d^2*e^4-87*((a*e^2-c*d^2)*e)^(1/2)*a*c^2*d^4
*e^2-48*((a*e^2-c*d^2)*e)^(1/2)*c^3*d^6)/(e*x+d)^(7/2)/(c*d*x+a*e)/(a*e^2-c*d^2)^4/((a*e^2-c*d^2)*e)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^(5/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^(5/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)

[Out]

int(1/((d + e*x)^(5/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Integral(1/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**(5/2)), x)

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