3.7.38 \(\int \frac {(d+e x)^{7/2}}{(a-c x^2)^3} \, dx\) [638]

Optimal. Leaf size=294 \[ \frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{9/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{9/4}} \]

[Out]

1/4*(c*d*x+a*e)*(e*x+d)^(5/2)/a/c/(-c*x^2+a)^2+1/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))
*(e*a^(1/2)+d*c^(1/2))^(3/2)*(12*c*d^2+5*a*e^2-18*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(9/4)-1/32*arctanh(c^(1/4)*(e
*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(-e*a^(1/2)+d*c^(1/2))^(3/2)*(12*c*d^2+5*a*e^2+18*d*e*a^(1/2)*c^(1/2
))/a^(5/2)/c^(9/4)+1/16*(a*e*(-5*a*e^2+7*c*d^2)+2*c*d*(-2*a*e^2+3*c*d^2)*x)*(e*x+d)^(1/2)/a^2/c^2/(-c*x^2+a)

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Rubi [A]
time = 0.36, antiderivative size = 294, 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, 833, 841, 1180, 214} \begin {gather*} -\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 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^{9/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \left (-18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 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^{9/4}}+\frac {\sqrt {d+e x} \left (2 c d x \left (3 c d^2-2 a e^2\right )+a e \left (7 c d^2-5 a e^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {(d+e x)^{5/2} (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)^(7/2)/(a - c*x^2)^3,x]

[Out]

((a*e + c*d*x)*(d + e*x)^(5/2))/(4*a*c*(a - c*x^2)^2) + (Sqrt[d + e*x]*(a*e*(7*c*d^2 - 5*a*e^2) + 2*c*d*(3*c*d
^2 - 2*a*e^2)*x))/(16*a^2*c^2*(a - c*x^2)) - ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e
+ 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4)) + ((Sqrt[c]*d +
Sqrt[a]*e)^(3/2)*(12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d
+ Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4))

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 833

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

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)^{7/2}}{\left (a-c x^2\right )^3} \, dx &=\frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (-6 c d^2+5 a e^2\right )-\frac {1}{2} c d e x\right )}{\left (a-c x^2\right )^2} \, dx}{4 a c}\\ &=\frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\int \frac {\frac {1}{4} \left (4 c d^2-5 a e^2\right ) \left (3 c d^2-a e^2\right )+\frac {1}{2} c d e \left (3 c d^2-4 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c^2}\\ &=\frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} c d^2 e \left (3 c d^2-4 a e^2\right )+\frac {1}{4} e \left (4 c d^2-5 a e^2\right ) \left (3 c d^2-a e^2\right )+\frac {1}{2} c d e \left (3 c d^2-4 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}\\ &=\frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\left (\left (\sqrt {c} d+\sqrt {a} e\right )^2 \left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 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} c^{3/2}}-\frac {\left (\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 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} c^{3/2}}\\ &=\frac {(a e+c d x) (d+e x)^{5/2}}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 c d^2-5 a e^2\right )+2 c d \left (3 c d^2-2 a e^2\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 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^{9/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 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^{9/4}}\\ \end {align*}

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Mathematica [A]
time = 1.87, size = 346, normalized size = 1.18 \begin {gather*} \frac {-\frac {2 \sqrt {a} \sqrt {d+e x} \left (5 a^3 e^3+6 c^3 d^3 x^3-a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )-a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )\right )}{\left (a-c x^2\right )^2}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^2 \left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 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 {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 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^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Sqrt[a]*Sqrt[d + e*x]*(5*a^3*e^3 + 6*c^3*d^3*x^3 - a*c^2*d*x*(10*d^2 + d*e*x + 8*e^2*x^2) - a^2*c*e*(11*d
^2 + 4*d*e*x + 9*e^2*x^2)))/(a - c*x^2)^2 + ((Sqrt[c]*d + Sqrt[a]*e)^2*(12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5*
a*e^2)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/Sqrt[-(c*d) - Sqrt[a]
*Sqrt[c]*e] - ((Sqrt[c]*d - Sqrt[a]*e)^2*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTan[(Sqrt[-(c*d) + S
qrt[a]*Sqrt[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^2)

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Maple [A]
time = 0.45, size = 450, normalized size = 1.53

method	result	size
derivativedivides	\(-2 e^{5} \left (-\frac {\frac {d \left (4 e^{2} a -3 c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{16 a^{2} e^{4}}+\frac {\left (9 a^{2} e^{4}-23 a c \,d^{2} e^{2}+18 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{4} c}-\frac {d \left (7 a^{2} e^{4}-16 a c \,d^{2} e^{2}+9 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 a^{2} e^{4} c}-\frac {\left (e^{2} a -c \,d^{2}\right ) \left (5 a^{2} e^{4}-11 a c \,d^{2} e^{2}+6 c^{2} d^{4}\right ) \sqrt {e x +d}}{32 a^{2} e^{4} c^{2}}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}-\frac {-\frac {\left (-5 a^{2} e^{4}+19 a c \,d^{2} e^{2}-12 c^{2} d^{4}+8 \sqrt {a c \,e^{2}}\, a d \,e^{2}-6 \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (5 a^{2} e^{4}-19 a c \,d^{2} e^{2}+12 c^{2} d^{4}+8 \sqrt {a c \,e^{2}}\, a d \,e^{2}-6 \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{32 a^{2} e^{4} c}\right )\)	\(450\)
default	\(-2 e^{5} \left (-\frac {\frac {d \left (4 e^{2} a -3 c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{16 a^{2} e^{4}}+\frac {\left (9 a^{2} e^{4}-23 a c \,d^{2} e^{2}+18 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{4} c}-\frac {d \left (7 a^{2} e^{4}-16 a c \,d^{2} e^{2}+9 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 a^{2} e^{4} c}-\frac {\left (e^{2} a -c \,d^{2}\right ) \left (5 a^{2} e^{4}-11 a c \,d^{2} e^{2}+6 c^{2} d^{4}\right ) \sqrt {e x +d}}{32 a^{2} e^{4} c^{2}}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}-\frac {-\frac {\left (-5 a^{2} e^{4}+19 a c \,d^{2} e^{2}-12 c^{2} d^{4}+8 \sqrt {a c \,e^{2}}\, a d \,e^{2}-6 \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctanh \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (5 a^{2} e^{4}-19 a c \,d^{2} e^{2}+12 c^{2} d^{4}+8 \sqrt {a c \,e^{2}}\, a d \,e^{2}-6 \sqrt {a c \,e^{2}}\, c \,d^{3}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{32 a^{2} e^{4} c}\right )\)	\(450\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(-c*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

-2*e^5*(-(1/16*d*(4*a*e^2-3*c*d^2)/a^2/e^4*(e*x+d)^(7/2)+1/32*(9*a^2*e^4-23*a*c*d^2*e^2+18*c^2*d^4)/a^2/e^4/c*
(e*x+d)^(5/2)-1/16*d*(7*a^2*e^4-16*a*c*d^2*e^2+9*c^2*d^4)/a^2/e^4/c*(e*x+d)^(3/2)-1/32*(a*e^2-c*d^2)*(5*a^2*e^
4-11*a*c*d^2*e^2+6*c^2*d^4)/a^2/e^4/c^2*(e*x+d)^(1/2))/(-c*(e*x+d)^2+2*c*d*(e*x+d)+e^2*a-c*d^2)^2-1/32/a^2/e^4
/c*(-1/2*(-5*a^2*e^4+19*a*c*d^2*e^2-12*c^2*d^4+8*(a*c*e^2)^(1/2)*a*d*e^2-6*(a*c*e^2)^(1/2)*c*d^3)/(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))+1/2*(5*a^2*e^4-19
*a*c*d^2*e^2+12*c^2*d^4+8*(a*c*e^2)^(1/2)*a*d*e^2-6*(a*c*e^2)^(1/2)*c*d^3)/(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))))

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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)^(7/2)/(-c*x^2+a)^3,x, algorithm="maxima")

[Out]

-integrate((x*e + d)^(7/2)/(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. 1638 vs. \(2 (244) = 488\).
time = 2.70, size = 1638, normalized size = 5.57 \begin {gather*} \frac {{\left (a^{2} c^{4} x^{4} - 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt {\frac {144 \, c^{3} d^{7} + a^{5} c^{4} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}} - 420 \, a c^{2} d^{5} e^{2} + 385 \, a^{2} c d^{3} e^{4} - 105 \, a^{3} d e^{6}}{a^{5} c^{4}}} \log \left ({\left (3024 \, c^{4} d^{8} e^{5} - 10908 \, a c^{3} d^{6} e^{7} + 13509 \, a^{2} c^{2} d^{4} e^{9} - 6250 \, a^{3} c d^{2} e^{11} + 625 \, a^{4} e^{13}\right )} \sqrt {x e + d} + {\left (126 \, a^{3} c^{4} d^{4} e^{6} - 255 \, a^{4} c^{3} d^{2} e^{8} + 125 \, a^{5} c^{2} e^{10} - {\left (12 \, a^{5} c^{8} d^{3} - 13 \, a^{6} c^{7} d e^{2}\right )} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}}\right )} \sqrt {\frac {144 \, c^{3} d^{7} + a^{5} c^{4} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}} - 420 \, a c^{2} d^{5} e^{2} + 385 \, a^{2} c d^{3} e^{4} - 105 \, a^{3} d e^{6}}{a^{5} c^{4}}}\right ) - {\left (a^{2} c^{4} x^{4} - 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt {\frac {144 \, c^{3} d^{7} + a^{5} c^{4} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}} - 420 \, a c^{2} d^{5} e^{2} + 385 \, a^{2} c d^{3} e^{4} - 105 \, a^{3} d e^{6}}{a^{5} c^{4}}} \log \left ({\left (3024 \, c^{4} d^{8} e^{5} - 10908 \, a c^{3} d^{6} e^{7} + 13509 \, a^{2} c^{2} d^{4} e^{9} - 6250 \, a^{3} c d^{2} e^{11} + 625 \, a^{4} e^{13}\right )} \sqrt {x e + d} - {\left (126 \, a^{3} c^{4} d^{4} e^{6} - 255 \, a^{4} c^{3} d^{2} e^{8} + 125 \, a^{5} c^{2} e^{10} - {\left (12 \, a^{5} c^{8} d^{3} - 13 \, a^{6} c^{7} d e^{2}\right )} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}}\right )} \sqrt {\frac {144 \, c^{3} d^{7} + a^{5} c^{4} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}} - 420 \, a c^{2} d^{5} e^{2} + 385 \, a^{2} c d^{3} e^{4} - 105 \, a^{3} d e^{6}}{a^{5} c^{4}}}\right ) + {\left (a^{2} c^{4} x^{4} - 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt {\frac {144 \, c^{3} d^{7} - a^{5} c^{4} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}} - 420 \, a c^{2} d^{5} e^{2} + 385 \, a^{2} c d^{3} e^{4} - 105 \, a^{3} d e^{6}}{a^{5} c^{4}}} \log \left ({\left (3024 \, c^{4} d^{8} e^{5} - 10908 \, a c^{3} d^{6} e^{7} + 13509 \, a^{2} c^{2} d^{4} e^{9} - 6250 \, a^{3} c d^{2} e^{11} + 625 \, a^{4} e^{13}\right )} \sqrt {x e + d} + {\left (126 \, a^{3} c^{4} d^{4} e^{6} - 255 \, a^{4} c^{3} d^{2} e^{8} + 125 \, a^{5} c^{2} e^{10} + {\left (12 \, a^{5} c^{8} d^{3} - 13 \, a^{6} c^{7} d e^{2}\right )} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}}\right )} \sqrt {\frac {144 \, c^{3} d^{7} - a^{5} c^{4} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}} - 420 \, a c^{2} d^{5} e^{2} + 385 \, a^{2} c d^{3} e^{4} - 105 \, a^{3} d e^{6}}{a^{5} c^{4}}}\right ) - {\left (a^{2} c^{4} x^{4} - 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt {\frac {144 \, c^{3} d^{7} - a^{5} c^{4} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}} - 420 \, a c^{2} d^{5} e^{2} + 385 \, a^{2} c d^{3} e^{4} - 105 \, a^{3} d e^{6}}{a^{5} c^{4}}} \log \left ({\left (3024 \, c^{4} d^{8} e^{5} - 10908 \, a c^{3} d^{6} e^{7} + 13509 \, a^{2} c^{2} d^{4} e^{9} - 6250 \, a^{3} c d^{2} e^{11} + 625 \, a^{4} e^{13}\right )} \sqrt {x e + d} - {\left (126 \, a^{3} c^{4} d^{4} e^{6} - 255 \, a^{4} c^{3} d^{2} e^{8} + 125 \, a^{5} c^{2} e^{10} + {\left (12 \, a^{5} c^{8} d^{3} - 13 \, a^{6} c^{7} d e^{2}\right )} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}}\right )} \sqrt {\frac {144 \, c^{3} d^{7} - a^{5} c^{4} \sqrt {\frac {441 \, c^{2} d^{4} e^{10} - 1050 \, a c d^{2} e^{12} + 625 \, a^{2} e^{14}}{a^{5} c^{9}}} - 420 \, a c^{2} d^{5} e^{2} + 385 \, a^{2} c d^{3} e^{4} - 105 \, a^{3} d e^{6}}{a^{5} c^{4}}}\right ) - 4 \, {\left (6 \, c^{3} d^{3} x^{3} - 10 \, a c^{2} d^{3} x - {\left (9 \, a^{2} c x^{2} - 5 \, a^{3}\right )} e^{3} - 4 \, {\left (2 \, a c^{2} d x^{3} + a^{2} c d x\right )} e^{2} - {\left (a c^{2} d^{2} x^{2} + 11 \, a^{2} c d^{2}\right )} e\right )} \sqrt {x e + d}}{64 \, {\left (a^{2} c^{4} x^{4} - 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(-c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/64*((a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 + a^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^
2*e^12 + 625*a^2*e^14)/(a^5*c^9)) - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6)/(a^5*c^4))*log((302
4*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 625*a^4*e^13)*sqrt(x*e + d
) + (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - (12*a^5*c^8*d^3 - 13*a^6*c^7*d*e^2)*sqrt((
441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 + a^5*c^4*sqrt((441*c^2*d^4
*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)) - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6)/
(a^5*c^4))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 + a^5*c^4*sqrt((441*c^2*d^4*e^10 - 105
0*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)) - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6)/(a^5*c^4))*
log((3024*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 625*a^4*e^13)*sqrt
(x*e + d) - (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - (12*a^5*c^8*d^3 - 13*a^6*c^7*d*e^2
)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 + a^5*c^4*sqrt((441
*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)) - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3
*d*e^6)/(a^5*c^4))) + (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - a^5*c^4*sqrt((441*c^2*d^4*e^
10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)) - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*e^6)/(a^
5*c^4))*log((3024*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 625*a^4*e^
13)*sqrt(x*e + d) + (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + (12*a^5*c^8*d^3 - 13*a^6*c
^7*d*e^2)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 - a^5*c^4*s
qrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)) - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 -
105*a^3*d*e^6)/(a^5*c^4))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*c^3*d^7 - a^5*c^4*sqrt((441*c^
2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)) - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 - 105*a^3*d*
e^6)/(a^5*c^4))*log((3024*c^4*d^8*e^5 - 10908*a*c^3*d^6*e^7 + 13509*a^2*c^2*d^4*e^9 - 6250*a^3*c*d^2*e^11 + 62
5*a^4*e^13)*sqrt(x*e + d) - (126*a^3*c^4*d^4*e^6 - 255*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + (12*a^5*c^8*d^3 -
13*a^6*c^7*d*e^2)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)))*sqrt((144*c^3*d^7 - a
^5*c^4*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9)) - 420*a*c^2*d^5*e^2 + 385*a^2*c*d
^3*e^4 - 105*a^3*d*e^6)/(a^5*c^4))) - 4*(6*c^3*d^3*x^3 - 10*a*c^2*d^3*x - (9*a^2*c*x^2 - 5*a^3)*e^3 - 4*(2*a*c
^2*d*x^3 + a^2*c*d*x)*e^2 - (a*c^2*d^2*x^2 + 11*a^2*c*d^2)*e)*sqrt(x*e + d))/(a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^
4*c^2)

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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)**(7/2)/(-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. 535 vs. \(2 (244) = 488\).
time = 2.61, size = 535, normalized size = 1.82 \begin {gather*} -\frac {{\left (6 \, a c d^{2} e - 12 \, \sqrt {a c} c d^{3} + 13 \, \sqrt {a c} a d e^{2} - 5 \, a^{2} e^{3}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d + \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, a^{3} c^{4}} - \frac {{\left (6 \, a c d^{2} e + 12 \, \sqrt {a c} c d^{3} - 13 \, \sqrt {a c} a d e^{2} - 5 \, a^{2} e^{3}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d - \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, a^{3} c^{4}} - \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{3} e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{4} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{5} e - 6 \, \sqrt {x e + d} c^{3} d^{6} e - 8 \, {\left (x e + d\right )}^{\frac {7}{2}} a c^{2} d e^{3} + 23 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} d^{2} e^{3} - 32 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{3} + 17 \, \sqrt {x e + d} a c^{2} d^{4} e^{3} - 9 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} c e^{5} + 14 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c d e^{5} - 16 \, \sqrt {x e + d} a^{2} c d^{2} e^{5} + 5 \, \sqrt {x e + d} a^{3} e^{7}}{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^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(-c*x^2+a)^3,x, algorithm="giac")

[Out]

-1/32*(6*a*c*d^2*e - 12*sqrt(a*c)*c*d^3 + 13*sqrt(a*c)*a*d*e^2 - 5*a^2*e^3)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(c
)*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d + sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 - a^3*c^2*e^2)*a^2*c^3))/(a^2*c^3)))
/(a^3*c^4) - 1/32*(6*a*c*d^2*e + 12*sqrt(a*c)*c*d^3 - 13*sqrt(a*c)*a*d*e^2 - 5*a^2*e^3)*sqrt(-c^2*d + sqrt(a*c
)*c*e)*abs(c)*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d - sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 - a^3*c^2*e^2)*a^2*c^3))
/(a^2*c^3)))/(a^3*c^4) - 1/16*(6*(x*e + d)^(7/2)*c^3*d^3*e - 18*(x*e + d)^(5/2)*c^3*d^4*e + 18*(x*e + d)^(3/2)
*c^3*d^5*e - 6*sqrt(x*e + d)*c^3*d^6*e - 8*(x*e + d)^(7/2)*a*c^2*d*e^3 + 23*(x*e + d)^(5/2)*a*c^2*d^2*e^3 - 32
*(x*e + d)^(3/2)*a*c^2*d^3*e^3 + 17*sqrt(x*e + d)*a*c^2*d^4*e^3 - 9*(x*e + d)^(5/2)*a^2*c*e^5 + 14*(x*e + d)^(
3/2)*a^2*c*d*e^5 - 16*sqrt(x*e + d)*a^2*c*d^2*e^5 + 5*sqrt(x*e + d)*a^3*e^7)/(((x*e + d)^2*c - 2*(x*e + d)*c*d
+ c*d^2 - a*e^2)^2*a^2*c^2)

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Mupad [B]
time = 0.74, size = 2500, normalized size = 8.50 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(7/2)/(a - c*x^2)^3,x)

[Out]

- ((e*(3*c*d^3 - 4*a*d*e^2)*(d + e*x)^(7/2))/(8*a^2) + ((d + e*x)^(3/2)*(7*a^2*d*e^5 + 9*c^2*d^5*e - 16*a*c*d^
3*e^3))/(8*a^2*c) + ((d + e*x)^(1/2)*(5*a^3*e^7 - 6*c^3*d^6*e + 17*a*c^2*d^4*e^3 - 16*a^2*c*d^2*e^5))/(16*a^2*
c^2) - (e*(d + e*x)^(5/2)*(9*a^2*e^4 + 18*c^2*d^4 - 23*a*c*d^2*e^2))/(16*a^2*c))/(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) - 2*atanh((25*e^10*(d + e*x)^(1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e
^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) + (21*d^2*e^5*(a^
15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((825*d^5*e^9)/(2048*a^3) + (325*d*e^13)/(2048*a*c^2) - (63*c*d^7*e
^7)/(512*a^4) - (449*d^3*e^11)/(1024*a^2*c) + (125*e^14*(a^15*c^9)^(1/2))/(2048*a^8*c^7) - (95*d^2*e^12*(a^15*
c^9)^(1/2))/(512*a^9*c^6) + (381*d^4*e^10*(a^15*c^9)^(1/2))/(2048*a^10*c^5) - (63*d^6*e^8*(a^15*c^9)^(1/2))/(1
024*a^11*c^4))) - (21*d^2*e^8*(d + e*x)^(1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e^4
)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) + (21*d^2*e^5*(a^15
*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((325*d*e^13)/(2048*c^3) - (63*d^7*e^7)/(512*a^3) - (449*d^3*e^11)/(1
024*a*c^2) + (825*d^5*e^9)/(2048*a^2*c) + (125*e^14*(a^15*c^9)^(1/2))/(2048*a^7*c^8) - (95*d^2*e^12*(a^15*c^9)
^(1/2))/(512*a^8*c^7) + (381*d^4*e^10*(a^15*c^9)^(1/2))/(2048*a^9*c^6) - (63*d^6*e^8*(a^15*c^9)^(1/2))/(1024*a
^10*c^5))) + (25*d*e^9*(a^15*c^9)^(1/2)*(d + e*x)^(1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (3
85*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) + (21*d^2
*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((125*e^14*(a^15*c^9)^(1/2))/(2048*c^3) + (325*a^7*c^2*d*e^
13)/2048 - (63*a^4*c^5*d^7*e^7)/512 + (825*a^5*c^4*d^5*e^9)/2048 - (449*a^6*c^3*d^3*e^11)/1024 - (63*d^6*e^8*(
a^15*c^9)^(1/2))/(1024*a^3) - (95*d^2*e^12*(a^15*c^9)^(1/2))/(512*a*c^2) + (381*d^4*e^10*(a^15*c^9)^(1/2))/(20
48*a^2*c))) + (21*d^3*e^7*(a^15*c^9)^(1/2)*(d + e*x)^(1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) +
(385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) - (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) + (21*
d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((63*a^5*c^4*d^7*e^7)/512 - (825*a^6*c^3*d^5*e^9)/2048 +
(449*a^7*c^2*d^3*e^11)/1024 - (125*a*e^14*(a^15*c^9)^(1/2))/(2048*c^4) - (325*a^8*c*d*e^13)/2048 + (95*d^2*e^
12*(a^15*c^9)^(1/2))/(512*c^3) - (381*d^4*e^10*(a^15*c^9)^(1/2))/(2048*a*c^2) + (63*d^6*e^8*(a^15*c^9)^(1/2))/
(1024*a^2*c))))*((144*a^5*c^8*d^7 - 25*a*e^7*(a^15*c^9)^(1/2) - 105*a^8*c^5*d*e^6 - 420*a^6*c^7*d^5*e^2 + 385*
a^7*c^6*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^9))^(1/2) - 2*atanh((25*e^10*(d + e*x)^(1/2)*((9
*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) +
(25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) - (21*d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((825*d^
5*e^9)/(2048*a^3) + (325*d*e^13)/(2048*a*c^2) - (63*c*d^7*e^7)/(512*a^4) - (449*d^3*e^11)/(1024*a^2*c) - (125*
e^14*(a^15*c^9)^(1/2))/(2048*a^8*c^7) + (95*d^2*e^12*(a^15*c^9)^(1/2))/(512*a^9*c^6) - (381*d^4*e^10*(a^15*c^9
)^(1/2))/(2048*a^10*c^5) + (63*d^6*e^8*(a^15*c^9)^(1/2))/(1024*a^11*c^4))) - (21*d^2*e^8*(d + e*x)^(1/2)*((9*d
^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^4*c^2) + (
25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) - (21*d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*((325*d*e^
13)/(2048*c^3) - (63*d^7*e^7)/(512*a^3) - (449*d^3*e^11)/(1024*a*c^2) + (825*d^5*e^9)/(2048*a^2*c) - (125*e^14
*(a^15*c^9)^(1/2))/(2048*a^7*c^8) + (95*d^2*e^12*(a^15*c^9)^(1/2))/(512*a^8*c^7) - (381*d^4*e^10*(a^15*c^9)^(1
/2))/(2048*a^9*c^6) + (63*d^6*e^8*(a^15*c^9)^(1/2))/(1024*a^10*c^5))) + (25*d*e^9*(a^15*c^9)^(1/2)*(d + e*x)^(
1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024*a^
4*c^2) + (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) - (21*d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(32*
((125*e^14*(a^15*c^9)^(1/2))/(2048*c^3) - (325*a^7*c^2*d*e^13)/2048 + (63*a^4*c^5*d^7*e^7)/512 - (825*a^5*c^4*
d^5*e^9)/2048 + (449*a^6*c^3*d^3*e^11)/1024 - (63*d^6*e^8*(a^15*c^9)^(1/2))/(1024*a^3) - (95*d^2*e^12*(a^15*c^
9)^(1/2))/(512*a*c^2) + (381*d^4*e^10*(a^15*c^9)^(1/2))/(2048*a^2*c))) - (21*d^3*e^7*(a^15*c^9)^(1/2)*(d + e*x
)^(1/2)*((9*d^7)/(256*a^5*c) - (105*d*e^6)/(4096*a^2*c^4) + (385*d^3*e^4)/(4096*a^3*c^3) - (105*d^5*e^2)/(1024
*a^4*c^2) + (25*e^7*(a^15*c^9)^(1/2))/(4096*a^9*c^9) - (21*d^2*e^5*(a^15*c^9)^(1/2))/(4096*a^10*c^8))^(1/2))/(
32*((63*a^5*c^4*d^7*e^7)/512 - (825*a^6*c^3*d^5*e^9)/2048 + (449*a^7*c^2*d^3*e^11)/1024 + (125*a*e^14*(a^15*c^
9)^(1/2))/(2048*c^4) - (325*a^8*c*d*e^13)/2048 - (95*d^2*e^12*(a^15*c^9)^(1/2))/(512*c^3) + (381*d^4*e^10*(a^1
5*c^9)^(1/2))/(2048*a*c^2) - (63*d^6*e^8*(a^15*...

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