3.841 \(\int \frac{a+b x+c x^2}{(d+e x)^{9/2} \sqrt{f+g x}} \, dx\)
Optimal. Leaf size=281 \[ -\frac{4 g \sqrt{f+g x} \left (4 e g (-6 a e g-b d g+7 b e f)-c \left (3 d^2 g^2-14 d e f g+35 e^2 f^2\right )\right )}{105 e^2 \sqrt{d+e x} (e f-d g)^4}+\frac{2 \sqrt{f+g x} \left (4 e g (-6 a e g-b d g+7 b e f)-c \left (3 d^2 g^2-14 d e f g+35 e^2 f^2\right )\right )}{105 e^2 (d+e x)^{3/2} (e f-d g)^3}-\frac{2 \sqrt{f+g x} \left (a+\frac{d (c d-b e)}{e^2}\right )}{7 (d+e x)^{7/2} (e f-d g)}+\frac{2 \sqrt{f+g x} (2 c d (7 e f-4 d g)-e (-6 a e g-b d g+7 b e f))}{35 e^2 (d+e x)^{5/2} (e f-d g)^2} \]
[Out]
(-2*(a + (d*(c*d - b*e))/e^2)*Sqrt[f + g*x])/(7*(e*f - d*g)*(d + e*x)^(7/2)) + (2*(2*c*d*(7*e*f - 4*d*g) - e*(
7*b*e*f - b*d*g - 6*a*e*g))*Sqrt[f + g*x])/(35*e^2*(e*f - d*g)^2*(d + e*x)^(5/2)) + (2*(4*e*g*(7*b*e*f - b*d*g
- 6*a*e*g) - c*(35*e^2*f^2 - 14*d*e*f*g + 3*d^2*g^2))*Sqrt[f + g*x])/(105*e^2*(e*f - d*g)^3*(d + e*x)^(3/2))
- (4*g*(4*e*g*(7*b*e*f - b*d*g - 6*a*e*g) - c*(35*e^2*f^2 - 14*d*e*f*g + 3*d^2*g^2))*Sqrt[f + g*x])/(105*e^2*(
e*f - d*g)^4*Sqrt[d + e*x])
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Rubi [A] time = 0.291033, antiderivative size = 281, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used =
{949, 78, 45, 37} \[ -\frac{4 g \sqrt{f+g x} \left (4 e g (-6 a e g-b d g+7 b e f)-c \left (3 d^2 g^2-14 d e f g+35 e^2 f^2\right )\right )}{105 e^2 \sqrt{d+e x} (e f-d g)^4}+\frac{2 \sqrt{f+g x} \left (4 e g (-6 a e g-b d g+7 b e f)-c \left (3 d^2 g^2-14 d e f g+35 e^2 f^2\right )\right )}{105 e^2 (d+e x)^{3/2} (e f-d g)^3}-\frac{2 \sqrt{f+g x} \left (a+\frac{d (c d-b e)}{e^2}\right )}{7 (d+e x)^{7/2} (e f-d g)}+\frac{2 \sqrt{f+g x} (2 c d (7 e f-4 d g)-e (-6 a e g-b d g+7 b e f))}{35 e^2 (d+e x)^{5/2} (e f-d g)^2} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x + c*x^2)/((d + e*x)^(9/2)*Sqrt[f + g*x]),x]
[Out]
(-2*(a + (d*(c*d - b*e))/e^2)*Sqrt[f + g*x])/(7*(e*f - d*g)*(d + e*x)^(7/2)) + (2*(2*c*d*(7*e*f - 4*d*g) - e*(
7*b*e*f - b*d*g - 6*a*e*g))*Sqrt[f + g*x])/(35*e^2*(e*f - d*g)^2*(d + e*x)^(5/2)) + (2*(4*e*g*(7*b*e*f - b*d*g
- 6*a*e*g) - c*(35*e^2*f^2 - 14*d*e*f*g + 3*d^2*g^2))*Sqrt[f + g*x])/(105*e^2*(e*f - d*g)^3*(d + e*x)^(3/2))
- (4*g*(4*e*g*(7*b*e*f - b*d*g - 6*a*e*g) - c*(35*e^2*f^2 - 14*d*e*f*g + 3*d^2*g^2))*Sqrt[f + g*x])/(105*e^2*(
e*f - d*g)^4*Sqrt[d + e*x])
Rule 949
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p,
d + e*x, x]}, Simp[(R*(d + e*x)^(m + 1)*(f + g*x)^(n + 1))/((m + 1)*(e*f - d*g)), x] + Dist[1/((m + 1)*(e*f -
d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& IGtQ[p, 0] && LtQ[m, -1]
Rule 78
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ
[p, n]))))
Rule 45
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
Rule 37
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]
Rubi steps
\begin{align*} \int \frac{a+b x+c x^2}{(d+e x)^{9/2} \sqrt{f+g x}} \, dx &=-\frac{2 \left (a+\frac{d (c d-b e)}{e^2}\right ) \sqrt{f+g x}}{7 (e f-d g) (d+e x)^{7/2}}-\frac{2 \int \frac{\frac{c d (7 e f-d g)-e (7 b e f-b d g-6 a e g)}{2 e^2}-\frac{7}{2} c \left (f-\frac{d g}{e}\right ) x}{(d+e x)^{7/2} \sqrt{f+g x}} \, dx}{7 (e f-d g)}\\ &=-\frac{2 \left (a+\frac{d (c d-b e)}{e^2}\right ) \sqrt{f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac{2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt{f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}-\frac{\left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \int \frac{1}{(d+e x)^{5/2} \sqrt{f+g x}} \, dx}{35 e^2 (e f-d g)^2}\\ &=-\frac{2 \left (a+\frac{d (c d-b e)}{e^2}\right ) \sqrt{f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac{2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt{f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}+\frac{2 \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt{f+g x}}{105 e^2 (e f-d g)^3 (d+e x)^{3/2}}+\frac{\left (2 g \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right )\right ) \int \frac{1}{(d+e x)^{3/2} \sqrt{f+g x}} \, dx}{105 e^2 (e f-d g)^3}\\ &=-\frac{2 \left (a+\frac{d (c d-b e)}{e^2}\right ) \sqrt{f+g x}}{7 (e f-d g) (d+e x)^{7/2}}+\frac{2 (2 c d (7 e f-4 d g)-e (7 b e f-b d g-6 a e g)) \sqrt{f+g x}}{35 e^2 (e f-d g)^2 (d+e x)^{5/2}}+\frac{2 \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt{f+g x}}{105 e^2 (e f-d g)^3 (d+e x)^{3/2}}-\frac{4 g \left (4 e g (7 b e f-b d g-6 a e g)-c \left (35 e^2 f^2-14 d e f g+3 d^2 g^2\right )\right ) \sqrt{f+g x}}{105 e^2 (e f-d g)^4 \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.365036, size = 332, normalized size = 1.18 \[ \frac{2 \sqrt{f+g x} \left (3 a \left (-35 d^2 e g^2 (f-2 g x)+35 d^3 g^3+7 d e^2 g \left (3 f^2-4 f g x+8 g^2 x^2\right )+e^3 \left (6 f^2 g x-5 f^3-8 f g^2 x^2+16 g^3 x^3\right )\right )+b \left (7 d^2 e g \left (4 f^2-37 f g x+4 g^2 x^2\right )+35 d^3 g^2 (g x-2 f)+d e^2 \left (101 f^2 g x-6 f^3-200 f g^2 x^2+8 g^3 x^3\right )-7 e^3 f x \left (3 f^2-4 f g x+8 g^2 x^2\right )\right )+c \left (d^2 e \left (200 f^2 g x-8 f^3-101 f g^2 x^2+6 g^3 x^3\right )+7 d^3 g \left (8 f^2-4 f g x+3 g^2 x^2\right )-7 d e^2 f x \left (4 f^2-37 f g x+4 g^2 x^2\right )-35 e^3 f^2 x^2 (f-2 g x)\right )\right )}{105 (d+e x)^{7/2} (e f-d g)^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x + c*x^2)/((d + e*x)^(9/2)*Sqrt[f + g*x]),x]
[Out]
(2*Sqrt[f + g*x]*(c*(-35*e^3*f^2*x^2*(f - 2*g*x) + 7*d^3*g*(8*f^2 - 4*f*g*x + 3*g^2*x^2) - 7*d*e^2*f*x*(4*f^2
- 37*f*g*x + 4*g^2*x^2) + d^2*e*(-8*f^3 + 200*f^2*g*x - 101*f*g^2*x^2 + 6*g^3*x^3)) + b*(35*d^3*g^2*(-2*f + g*
x) + 7*d^2*e*g*(4*f^2 - 37*f*g*x + 4*g^2*x^2) - 7*e^3*f*x*(3*f^2 - 4*f*g*x + 8*g^2*x^2) + d*e^2*(-6*f^3 + 101*
f^2*g*x - 200*f*g^2*x^2 + 8*g^3*x^3)) + 3*a*(35*d^3*g^3 - 35*d^2*e*g^2*(f - 2*g*x) + 7*d*e^2*g*(3*f^2 - 4*f*g*
x + 8*g^2*x^2) + e^3*(-5*f^3 + 6*f^2*g*x - 8*f*g^2*x^2 + 16*g^3*x^3))))/(105*(e*f - d*g)^4*(d + e*x)^(7/2))
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Maple [A] time = 0.056, size = 468, normalized size = 1.7 \begin{align*}{\frac{96\,a{e}^{3}{g}^{3}{x}^{3}+16\,bd{e}^{2}{g}^{3}{x}^{3}-112\,b{e}^{3}f{g}^{2}{x}^{3}+12\,c{d}^{2}e{g}^{3}{x}^{3}-56\,cd{e}^{2}f{g}^{2}{x}^{3}+140\,c{e}^{3}{f}^{2}g{x}^{3}+336\,ad{e}^{2}{g}^{3}{x}^{2}-48\,a{e}^{3}f{g}^{2}{x}^{2}+56\,b{d}^{2}e{g}^{3}{x}^{2}-400\,bd{e}^{2}f{g}^{2}{x}^{2}+56\,b{e}^{3}{f}^{2}g{x}^{2}+42\,c{d}^{3}{g}^{3}{x}^{2}-202\,c{d}^{2}ef{g}^{2}{x}^{2}+518\,cd{e}^{2}{f}^{2}g{x}^{2}-70\,c{e}^{3}{f}^{3}{x}^{2}+420\,a{d}^{2}e{g}^{3}x-168\,ad{e}^{2}f{g}^{2}x+36\,a{e}^{3}{f}^{2}gx+70\,b{d}^{3}{g}^{3}x-518\,b{d}^{2}ef{g}^{2}x+202\,bd{e}^{2}{f}^{2}gx-42\,b{e}^{3}{f}^{3}x-56\,c{d}^{3}f{g}^{2}x+400\,c{d}^{2}e{f}^{2}gx-56\,cd{e}^{2}{f}^{3}x+210\,a{d}^{3}{g}^{3}-210\,a{d}^{2}ef{g}^{2}+126\,ad{e}^{2}{f}^{2}g-30\,a{e}^{3}{f}^{3}-140\,b{d}^{3}f{g}^{2}+56\,b{d}^{2}e{f}^{2}g-12\,bd{e}^{2}{f}^{3}+112\,c{d}^{3}{f}^{2}g-16\,c{d}^{2}e{f}^{3}}{105\,{g}^{4}{d}^{4}-420\,e{g}^{3}f{d}^{3}+630\,{d}^{2}{e}^{2}{f}^{2}{g}^{2}-420\,d{e}^{3}{f}^{3}g+105\,{e}^{4}{f}^{4}}\sqrt{gx+f} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*x^2+b*x+a)/(e*x+d)^(9/2)/(g*x+f)^(1/2),x)
[Out]
2/105*(g*x+f)^(1/2)*(48*a*e^3*g^3*x^3+8*b*d*e^2*g^3*x^3-56*b*e^3*f*g^2*x^3+6*c*d^2*e*g^3*x^3-28*c*d*e^2*f*g^2*
x^3+70*c*e^3*f^2*g*x^3+168*a*d*e^2*g^3*x^2-24*a*e^3*f*g^2*x^2+28*b*d^2*e*g^3*x^2-200*b*d*e^2*f*g^2*x^2+28*b*e^
3*f^2*g*x^2+21*c*d^3*g^3*x^2-101*c*d^2*e*f*g^2*x^2+259*c*d*e^2*f^2*g*x^2-35*c*e^3*f^3*x^2+210*a*d^2*e*g^3*x-84
*a*d*e^2*f*g^2*x+18*a*e^3*f^2*g*x+35*b*d^3*g^3*x-259*b*d^2*e*f*g^2*x+101*b*d*e^2*f^2*g*x-21*b*e^3*f^3*x-28*c*d
^3*f*g^2*x+200*c*d^2*e*f^2*g*x-28*c*d*e^2*f^3*x+105*a*d^3*g^3-105*a*d^2*e*f*g^2+63*a*d*e^2*f^2*g-15*a*e^3*f^3-
70*b*d^3*f*g^2+28*b*d^2*e*f^2*g-6*b*d*e^2*f^3+56*c*d^3*f^2*g-8*c*d^2*e*f^3)/(e*x+d)^(7/2)/(d^4*g^4-4*d^3*e*f*g
^3+6*d^2*e^2*f^2*g^2-4*d*e^3*f^3*g+e^4*f^4)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+b*x+a)/(e*x+d)^(9/2)/(g*x+f)^(1/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+b*x+a)/(e*x+d)^(9/2)/(g*x+f)^(1/2),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x**2+b*x+a)/(e*x+d)**(9/2)/(g*x+f)**(1/2),x)
[Out]
Timed out
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Giac [B] time = 2.01607, size = 2522, normalized size = 8.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+b*x+a)/(e*x+d)^(9/2)/(g*x+f)^(1/2),x, algorithm="giac")
[Out]
8/105*(3*c*d^5*g^(13/2)*e^(11/2) + 21*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2*
c*d^4*g^(11/2)*e^(9/2) - 42*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c*d^3*g^(9
/2)*e^(7/2) + 210*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c*d^2*g^(7/2)*e^(5/2
) - 105*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*c*d*g^(5/2)*e^(3/2) + 105*(sqr
t(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^10*c*g^(3/2)*e^(1/2) - 23*c*d^4*f*g^(11/2)*e
^(13/2) + 4*b*d^4*g^(13/2)*e^(13/2) - 140*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2)
)^2*c*d^3*f*g^(9/2)*e^(11/2) + 28*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2*b*d^
3*g^(11/2)*e^(11/2) - 42*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c*d^2*f*g^(7/
2)*e^(9/2) + 84*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*b*d^2*g^(9/2)*e^(9/2)
- 140*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c*d*f*g^(5/2)*e^(7/2) - 140*(sqr
t(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*b*d*g^(7/2)*e^(7/2) - 455*(sqrt(x*e + d)*s
qrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*c*f*g^(3/2)*e^(5/2) + 280*(sqrt(x*e + d)*sqrt(g)*e^(1/
2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^8*b*g^(5/2)*e^(5/2) + 86*c*d^3*f^2*g^(9/2)*e^(15/2) - 40*b*d^3*f*g^(
11/2)*e^(15/2) + 24*a*d^3*g^(13/2)*e^(15/2) + 462*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e
+ f*e^2))^2*c*d^2*f^2*g^(7/2)*e^(13/2) - 252*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e
^2))^2*b*d^2*f*g^(9/2)*e^(13/2) + 168*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2*
a*d^2*g^(11/2)*e^(13/2) + 714*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*c*d*f^2*
g^(5/2)*e^(11/2) - 672*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*b*d*f*g^(7/2)*e
^(11/2) + 504*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a*d*g^(9/2)*e^(11/2) + 7
70*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*c*f^2*g^(3/2)*e^(9/2) - 700*(sqrt(x
*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*b*f*g^(5/2)*e^(9/2) + 840*(sqrt(x*e + d)*sqrt
(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^6*a*g^(7/2)*e^(9/2) - 150*c*d^2*f^3*g^(7/2)*e^(17/2) + 96*b
*d^2*f^2*g^(9/2)*e^(17/2) - 72*a*d^2*f*g^(11/2)*e^(17/2) - 588*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)
*g*e - d*g*e + f*e^2))^2*c*d*f^3*g^(5/2)*e^(15/2) + 420*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e -
d*g*e + f*e^2))^2*b*d*f^2*g^(7/2)*e^(15/2) - 336*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e +
f*e^2))^2*a*d*f*g^(9/2)*e^(15/2) - 630*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^
4*c*f^3*g^(3/2)*e^(13/2) + 588*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*b*f^2*g
^(5/2)*e^(13/2) - 504*(sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^4*a*f*g^(7/2)*e^(1
3/2) + 119*c*d*f^4*g^(5/2)*e^(19/2) - 88*b*d*f^3*g^(7/2)*e^(19/2) + 72*a*d*f^2*g^(9/2)*e^(19/2) + 245*(sqrt(x*
e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2*c*f^4*g^(3/2)*e^(17/2) - 196*(sqrt(x*e + d)*sq
rt(g)*e^(1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2*b*f^3*g^(5/2)*e^(17/2) + 168*(sqrt(x*e + d)*sqrt(g)*e^(
1/2) - sqrt((x*e + d)*g*e - d*g*e + f*e^2))^2*a*f^2*g^(7/2)*e^(17/2) - 35*c*f^5*g^(3/2)*e^(21/2) + 28*b*f^4*g^
(5/2)*e^(21/2) - 24*a*f^3*g^(7/2)*e^(21/2))*e^(-1)/(d*g*e + (sqrt(x*e + d)*sqrt(g)*e^(1/2) - sqrt((x*e + d)*g*
e - d*g*e + f*e^2))^2 - f*e^2)^7