3.7.40 \(\int \frac {(d+e x)^{3/2}}{(a-c x^2)^3} \, dx\) [640]

Optimal. Leaf size=268 \[ \frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {(a e-6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a-c x^2\right )}-\frac {3 \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {3 \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {c} d+\sqrt {a} e}} \]

[Out]

1/4*(c*d*x+a*e)*(e*x+d)^(1/2)/a/c/(-c*x^2+a)^2-1/16*(-6*c*d*x+a*e)*(e*x+d)^(1/2)/a^2/c/(-c*x^2+a)-3/32*arctanh
(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(4*c*d^2-a*e^2-2*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(5/4)/(-e
*a^(1/2)+d*c^(1/2))^(1/2)+3/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(4*c*d^2-a*e^2+2*d*e
*a^(1/2)*c^(1/2))/a^(5/2)/c^(5/4)/(e*a^(1/2)+d*c^(1/2))^(1/2)

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Rubi [A]
time = 0.32, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {753, 837, 841, 1180, 214} \begin {gather*} -\frac {3 \left (-2 \sqrt {a} \sqrt {c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {3 \left (2 \sqrt {a} \sqrt {c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\sqrt {d+e x} (a e-6 c d x)}{16 a^2 c \left (a-c x^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(a - c*x^2)^3,x]

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(4*a*c*(a - c*x^2)^2) - ((a*e - 6*c*d*x)*Sqrt[d + e*x])/(16*a^2*c*(a - c*x^2)) -
(3*(4*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(3
2*a^(5/2)*c^(5/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + (3*(4*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)
*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(5/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e])

Rule 214

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

Rule 753

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a*e - c*d*x)*((a
+ c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 841

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (-6 c d^2+a e^2\right )-\frac {5}{2} c d e x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx}{4 a c}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {(a e-6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a-c x^2\right )}+\frac {\int \frac {\frac {3}{4} c \left (c d^2-a e^2\right ) \left (4 c d^2-a e^2\right )+\frac {3}{2} c^2 d e \left (c d^2-a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c^2 \left (c d^2-a e^2\right )}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {(a e-6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a-c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{2} c^2 d^2 e \left (c d^2-a e^2\right )+\frac {3}{4} c e \left (c d^2-a e^2\right ) \left (4 c d^2-a e^2\right )+\frac {3}{2} c^2 d e \left (c d^2-a e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{4 a^2 c^2 \left (c d^2-a e^2\right )}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {(a e-6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a-c x^2\right )}-\frac {\left (3 \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{32 a^{5/2} \sqrt {c}}+\frac {\left (3 \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{32 a^{5/2} \sqrt {c}}\\ &=\frac {(a e+c d x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {(a e-6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a-c x^2\right )}-\frac {3 \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {3 \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt {\sqrt {c} d+\sqrt {a} e}}\\ \end {align*}

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Mathematica [A]
time = 1.28, size = 268, normalized size = 1.00 \begin {gather*} \frac {-\frac {2 \sqrt {a} \sqrt {d+e x} \left (-3 a^2 e+6 c^2 d x^3-a c x (10 d+e x)\right )}{\left (a-c x^2\right )^2}+\frac {3 \left (4 c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {3 \left (4 c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{32 a^{5/2} c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Sqrt[a]*Sqrt[d + e*x]*(-3*a^2*e + 6*c^2*d*x^3 - a*c*x*(10*d + e*x)))/(a - c*x^2)^2 + (3*(4*c*d^2 + 2*Sqrt
[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/Sqr
t[-(c*d) - Sqrt[a]*Sqrt[c]*e] - (3*(4*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqr
t[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e])/(32*a^(5/2)*c)

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Maple [A]
time = 0.44, size = 320, normalized size = 1.19

method	result	size
derivativedivides	\(-2 e^{5} \left (-\frac {-\frac {3 c d \left (e x +d \right )^{\frac {7}{2}}}{16 a^{2} e^{4}}+\frac {\left (e^{2} a +18 c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{4}}+\frac {d \left (4 e^{2} a -9 c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 a^{2} e^{4}}+\frac {3 \left (e^{2} a -c \,d^{2}\right ) \left (e^{2} a -2 c \,d^{2}\right ) \sqrt {e x +d}}{32 a^{2} e^{4} c}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}-\frac {3 \left (\frac {\left (-e^{2} a +4 c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a -4 c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 a^{2} e^{4}}\right )\)	\(320\)
default	\(-2 e^{5} \left (-\frac {-\frac {3 c d \left (e x +d \right )^{\frac {7}{2}}}{16 a^{2} e^{4}}+\frac {\left (e^{2} a +18 c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{4}}+\frac {d \left (4 e^{2} a -9 c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 a^{2} e^{4}}+\frac {3 \left (e^{2} a -c \,d^{2}\right ) \left (e^{2} a -2 c \,d^{2}\right ) \sqrt {e x +d}}{32 a^{2} e^{4} c}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}-\frac {3 \left (\frac {\left (-e^{2} a +4 c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (e^{2} a -4 c \,d^{2}-2 \sqrt {a c \,e^{2}}\, d \right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{32 a^{2} e^{4}}\right )\)	\(320\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*e^5*(-(-3/16*c*d/a^2/e^4*(e*x+d)^(7/2)+1/32*(a*e^2+18*c*d^2)/a^2/e^4*(e*x+d)^(5/2)+1/16*d*(4*a*e^2-9*c*d^2)
/a^2/e^4*(e*x+d)^(3/2)+3/32*(a*e^2-c*d^2)*(a*e^2-2*c*d^2)/a^2/e^4/c*(e*x+d)^(1/2))/(-c*(e*x+d)^2+2*c*d*(e*x+d)
+e^2*a-c*d^2)^2-3/32/a^2/e^4*(1/2*(-e^2*a+4*c*d^2-2*(a*c*e^2)^(1/2)*d)/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))
*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))-1/2*(e^2*a-4*c*d^2-2*(a*c*e^2)^(1/2)*d)/(a*
c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(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)/(-c*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1628 vs. \(2 (221) = 442\).
time = 2.23, size = 1628, normalized size = 6.07 \begin {gather*} \frac {3 \, {\left (a^{2} c^{3} x^{4} - 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4} + \frac {{\left (a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}}{a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} - 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {x e + d} + 27 \, {\left (2 \, a^{3} c^{2} d^{2} e^{6} - a^{4} c e^{8} - \frac {{\left (4 \, a^{5} c^{6} d^{5} - 7 \, a^{6} c^{5} d^{3} e^{2} + 3 \, a^{7} c^{4} d e^{4}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}\right )} \sqrt {\frac {16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4} + \frac {{\left (a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}}{a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}}}\right ) - 3 \, {\left (a^{2} c^{3} x^{4} - 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4} + \frac {{\left (a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}}{a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} - 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {x e + d} - 27 \, {\left (2 \, a^{3} c^{2} d^{2} e^{6} - a^{4} c e^{8} - \frac {{\left (4 \, a^{5} c^{6} d^{5} - 7 \, a^{6} c^{5} d^{3} e^{2} + 3 \, a^{7} c^{4} d e^{4}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}\right )} \sqrt {\frac {16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4} + \frac {{\left (a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}}{a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}}}\right ) + 3 \, {\left (a^{2} c^{3} x^{4} - 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4} - \frac {{\left (a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}}{a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} - 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {x e + d} + 27 \, {\left (2 \, a^{3} c^{2} d^{2} e^{6} - a^{4} c e^{8} + \frac {{\left (4 \, a^{5} c^{6} d^{5} - 7 \, a^{6} c^{5} d^{3} e^{2} + 3 \, a^{7} c^{4} d e^{4}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}\right )} \sqrt {\frac {16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4} - \frac {{\left (a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}}{a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}}}\right ) - 3 \, {\left (a^{2} c^{3} x^{4} - 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt {\frac {16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4} - \frac {{\left (a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}}{a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}}} \log \left (27 \, {\left (16 \, c^{2} d^{4} e^{5} - 12 \, a c d^{2} e^{7} + a^{2} e^{9}\right )} \sqrt {x e + d} - 27 \, {\left (2 \, a^{3} c^{2} d^{2} e^{6} - a^{4} c e^{8} + \frac {{\left (4 \, a^{5} c^{6} d^{5} - 7 \, a^{6} c^{5} d^{3} e^{2} + 3 \, a^{7} c^{4} d e^{4}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}\right )} \sqrt {\frac {16 \, c^{2} d^{5} - 20 \, a c d^{3} e^{2} + 5 \, a^{2} d e^{4} - \frac {{\left (a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}\right )} e^{5}}{\sqrt {a^{5} c^{7} d^{4} - 2 \, a^{6} c^{6} d^{2} e^{2} + a^{7} c^{5} e^{4}}}}{a^{5} c^{3} d^{2} - a^{6} c^{2} e^{2}}}\right ) - 4 \, {\left (6 \, c^{2} d x^{3} - 10 \, a c d x - {\left (a c x^{2} + 3 \, a^{2}\right )} e\right )} \sqrt {x e + d}}{64 \, {\left (a^{2} c^{3} x^{4} - 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 -
a^6*c^2*e^2)*e^5/sqrt(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(27*(16
*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(x*e + d) + 27*(2*a^3*c^2*d^2*e^6 - a^4*c*e^8 - (4*a^5*c^6*d^5 -
7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*e^5/sqrt(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4))*sqrt((16*c^2*d^5
- 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*e^5/sqrt(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c
^5*e^4))/(a^5*c^3*d^2 - a^6*c^2*e^2))) - 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 20*a*c*d^3
*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*e^5/sqrt(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4))/(a^5
*c^3*d^2 - a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(x*e + d) - 27*(2*a^3*c^2*d^2*
e^6 - a^4*c*e^8 - (4*a^5*c^6*d^5 - 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*e^5/sqrt(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e
^2 + a^7*c^5*e^4))*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*e^5/sqrt(a^5*
c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4))/(a^5*c^3*d^2 - a^6*c^2*e^2))) + 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a
^4*c)*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 - a^6*c^2*e^2)*e^5/sqrt(a^5*c^7*d^4 - 2*a
^6*c^6*d^2*e^2 + a^7*c^5*e^4))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)
*sqrt(x*e + d) + 27*(2*a^3*c^2*d^2*e^6 - a^4*c*e^8 + (4*a^5*c^6*d^5 - 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*e^5
/sqrt(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4))*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c
^3*d^2 - a^6*c^2*e^2)*e^5/sqrt(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4))/(a^5*c^3*d^2 - a^6*c^2*e^2))) -
3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 - a^6*
c^2*e^2)*e^5/sqrt(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(27*(16*c^2*
d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(x*e + d) - 27*(2*a^3*c^2*d^2*e^6 - a^4*c*e^8 + (4*a^5*c^6*d^5 - 7*a^6
*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*e^5/sqrt(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4))*sqrt((16*c^2*d^5 - 20
*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 - a^6*c^2*e^2)*e^5/sqrt(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^
4))/(a^5*c^3*d^2 - a^6*c^2*e^2))) - 4*(6*c^2*d*x^3 - 10*a*c*d*x - (a*c*x^2 + 3*a^2)*e)*sqrt(x*e + d))/(a^2*c^3
*x^4 - 2*a^3*c^2*x^2 + a^4*c)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (221) = 442\).
time = 4.84, size = 509, normalized size = 1.90 \begin {gather*} -\frac {3 \, {\left (4 \, \sqrt {a c} c^{3} d^{3} - 3 \, \sqrt {a c} a c^{2} d e^{2} - {\left (2 \, a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d + \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{3} d - \sqrt {a c} a^{3} c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e}} + \frac {3 \, {\left (4 \, \sqrt {a c} c^{3} d^{3} - 3 \, \sqrt {a c} a c^{2} d e^{2} + {\left (2 \, a c^{2} d^{2} e - a^{2} c e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d - \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{3} d + \sqrt {a c} a^{3} c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e}} - \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} d e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d^{2} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{3} e - 6 \, \sqrt {x e + d} c^{2} d^{4} e - {\left (x e + d\right )}^{\frac {5}{2}} a c e^{3} - 8 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d e^{3} + 9 \, \sqrt {x e + d} a c d^{2} e^{3} - 3 \, \sqrt {x e + d} a^{2} e^{5}}{16 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )}^{2} a^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/32*(4*sqrt(a*c)*c^3*d^3 - 3*sqrt(a*c)*a*c^2*d*e^2 - (2*a*c^2*d^2*e - a^2*c*e^3)*abs(c))*arctan(sqrt(x*e + d
)/sqrt(-(a^2*c^2*d + sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*e^2)*a^2*c^2))/(a^2*c^2)))/((a^3*c^3*d - sqrt(a*c
)*a^3*c^2*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)) + 3/32*(4*sqrt(a*c)*c^3*d^3 - 3*sqrt(a*c)*a*c^2*d*e^2 + (2*a*c^2*d^
2*e - a^2*c*e^3)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^2*d - sqrt(a^4*c^4*d^2 - (a^2*c^2*d^2 - a^3*c*e^2)*
a^2*c^2))/(a^2*c^2)))/((a^3*c^3*d + sqrt(a*c)*a^3*c^2*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)) - 1/16*(6*(x*e + d)^(7/
2)*c^2*d*e - 18*(x*e + d)^(5/2)*c^2*d^2*e + 18*(x*e + d)^(3/2)*c^2*d^3*e - 6*sqrt(x*e + d)*c^2*d^4*e - (x*e +
d)^(5/2)*a*c*e^3 - 8*(x*e + d)^(3/2)*a*c*d*e^3 + 9*sqrt(x*e + d)*a*c*d^2*e^3 - 3*sqrt(x*e + d)*a^2*e^5)/(((x*e
+ d)^2*c - 2*(x*e + d)*c*d + c*d^2 - a*e^2)^2*a^2*c)

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Mupad [B]
time = 2.74, size = 2500, normalized size = 9.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(a - c*x^2)^3,x)

[Out]

(((4*a*d*e^3 - 9*c*d^3*e)*(d + e*x)^(3/2))/(8*a^2) + (e*(a*e^2 + 18*c*d^2)*(d + e*x)^(5/2))/(16*a^2) + (3*(d +
e*x)^(1/2)*(a^2*e^5 + 2*c^2*d^4*e - 3*a*c*d^2*e^3))/(16*a^2*c) - (3*c*d*e*(d + e*x)^(7/2))/(8*a^2))/(c^2*(d +
e*x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2*d^2 - 2*a*c*e^2)*(d + e*x)^2 - (4*c^2*d^3 - 4*a*c*d*e^2)*(d + e*x) - 4*c^
2*d*(d + e*x)^3 - 2*a*c*d^2*e^2) + atan(((((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048*a^6) - 64*a*c^4
*d*e^2*(d + e*x)^(1/2)*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4
096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*
a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^
2 - 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^
3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*1i - (((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048
*a^6) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*
c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7
*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c*e^6
+ 144*c^3*d^4*e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4
+ 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*1i)/((3*(9*a^2*d*e^7 + 144*c^2*d^5*e^3 - 1
08*a*c*d^3*e^5))/(1024*a^6) + (((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048*a^6) - 64*a*c^4*d*e^2*(d +
e*x)^(1/2)*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c
^6*d^2 - a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3
*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 - 36*a*c^
2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(40
96*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + (((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048*a^6) + 64*a*c
^4*d*e^2*(d + e*x)^(1/2)*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/
(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 2
0*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*
e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^3*d*e^4 + 20*a^6*c^4*
d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)))*(-(9*(e^5*(a^15*c^5)^(1/2) - 16*a^5*c^5*d^5 - 5*a^7*c^
3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*2i + atan(((((3*(2048*a^6*c^2*e^5 -
4096*a^5*c^3*d^2*e^3))/(2048*a^6) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5*c^5*d^5
+ 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*((9*(e^5*(a^15*c^5)^(1/
2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + ((d
+ e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5
*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*1i - (((3*(2048*
a^6*c^2*e^5 - 4096*a^5*c^3*d^2*e^3))/(2048*a^6) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((9*(e^5*(a^15*c^5)^(1/2) + 1
6*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*((9*(e^5*(
a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2))
)^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*((9*(e^5*(a^15*c^5)^(
1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2)*1i)
/((3*(9*a^2*d*e^7 + 144*c^2*d^5*e^3 - 108*a*c*d^3*e^5))/(1024*a^6) + (((3*(2048*a^6*c^2*e^5 - 4096*a^5*c^3*d^2
*e^3))/(2048*a^6) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^
4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5*c^5*d
^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*
a^2*c*e^6 + 144*c^3*d^4*e^2 - 36*a*c^2*d^2*e^4))/(64*a^4))*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*
c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) + (((3*(2048*a^6*c^2*e^5 - 4096*a
^5*c^3*d^2*e^3))/(2048*a^6) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((9*(e^5*(a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^
7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2))*((9*(e^5*(a^15*c^5)^(1/2) + 16
*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 - 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 - a^11*c^5*e^2)))^(1/2) - ((d + e*x)
^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 - 36*a*c^...

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