3.7.64 \(\int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [664]

Optimal. Leaf size=280 \[ \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {5 c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{8 \sqrt {g} (c d f-a e g)^{7/2}} \]

[Out]

5/8*c^3*d^3*arctan(g^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^(1/2)/(e*x+d)^(1/2))/(-a*e*g
+c*d*f)^(7/2)/g^(1/2)+1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)^3/(e*x+d)^(1/2)+5/12*
c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^2/(g*x+f)^2/(e*x+d)^(1/2)+5/8*c^2*d^2*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^3/(g*x+f)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.28, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {886, 888, 211} \begin {gather*} \frac {5 c^3 d^3 \text {ArcTan}\left (\frac {\sqrt {g} \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 f-a e g}}\right )}{8 \sqrt {g} (c d f-a e g)^{7/2}}+\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 \sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {5 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) + (5*c*d*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^2) + (5*c^2*d^2*Sqrt[a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) + (5*c^3*d^3*ArcTan[(Sqrt[g]*Sqrt
[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])])/(8*Sqrt[g]*(c*d*f - a*e*g)^(7/2
))

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 886

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

Rule 888

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[
2*e^2, Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; Fre
eQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.51, size = 191, normalized size = 0.68 \begin {gather*} \frac {c^3 d^3 \sqrt {d+e x} \left (\frac {(a e+c d x) \left (8 a^2 e^2 g^2-2 a c d e g (13 f+5 g x)+c^2 d^2 \left (33 f^2+40 f g x+15 g^2 x^2\right )\right )}{c^3 d^3 (c d f-a e g)^3 (f+g x)^3}+\frac {15 \sqrt {a e+c d x} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{\sqrt {g} (c d f-a e g)^{7/2}}\right )}{24 \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(c^3*d^3*Sqrt[d + e*x]*(((a*e + c*d*x)*(8*a^2*e^2*g^2 - 2*a*c*d*e*g*(13*f + 5*g*x) + c^2*d^2*(33*f^2 + 40*f*g*
x + 15*g^2*x^2)))/(c^3*d^3*(c*d*f - a*e*g)^3*(f + g*x)^3) + (15*Sqrt[a*e + c*d*x]*ArcTan[(Sqrt[g]*Sqrt[a*e + c
*d*x])/Sqrt[c*d*f - a*e*g]])/(Sqrt[g]*(c*d*f - a*e*g)^(7/2))))/(24*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]
time = 0.16, size = 440, normalized size = 1.57

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} g^{3} x^{3}+45 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f \,g^{2} x^{2}+45 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{2} g x +15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{3}-15 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{2} x^{2}+10 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e \,g^{2} x -40 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f g x -8 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{2}+26 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e f g -33 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2}\right )}{24 \sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (a e g -c d f \right )^{3} \left (g x +f \right )^{3} \sqrt {\left (a e g -c d f \right ) g}}\)	\(440\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/24*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*g^3*x^3+45*a
rctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*f*g^2*x^2+45*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c
*d*f)*g)^(1/2))*c^3*d^3*f^2*g*x+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*f^3-15*((a*e*g
-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^2*g^2*x^2+10*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*g^2*x-
40*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^2*f*g*x-8*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^2
*g^2+26*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*f*g-33*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^2
*d^2*f^2)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/(a*e*g-c*d*f)^3/(g*x+f)^3/((a*e*g-c*d*f)*g)^(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)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x*e + d)/(sqrt(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^4), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1040 vs. \(2 (261) = 522\).
time = 3.32, size = 2119, normalized size = 7.57 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

[1/48*(15*(c^3*d^4*g^3*x^3 + 3*c^3*d^4*f*g^2*x^2 + 3*c^3*d^4*f^2*g*x + c^3*d^4*f^3 + (c^3*d^3*g^3*x^4 + 3*c^3*
d^3*f*g^2*x^3 + 3*c^3*d^3*f^2*g*x^2 + c^3*d^3*f^3*x)*e)*sqrt(-c*d*f*g + a*g^2*e)*log(-(c*d^2*g*x - c*d^2*f + 2
*a*g*x*e^2 + (c*d*g*x^2 - c*d*f*x + 2*a*d*g)*e + 2*sqrt(-c*d*f*g + a*g^2*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2
+ a*d)*e)*sqrt(x*e + d))/(d*g*x + d*f + (g*x^2 + f*x)*e)) + 2*(15*c^3*d^3*f*g^3*x^2 + 40*c^3*d^3*f^2*g^2*x + 3
3*c^3*d^3*f^3*g - 8*a^3*g^4*e^3 + 2*(5*a^2*c*d*g^4*x + 17*a^2*c*d*f*g^3)*e^2 - (15*a*c^2*d^2*g^4*x^2 + 50*a*c^
2*d^2*f*g^3*x + 59*a*c^2*d^2*f^2*g^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^4*d^5*f
^4*g^4*x^3 + 3*c^4*d^5*f^5*g^3*x^2 + 3*c^4*d^5*f^6*g^2*x + c^4*d^5*f^7*g + (a^4*g^8*x^4 + 3*a^4*f*g^7*x^3 + 3*
a^4*f^2*g^6*x^2 + a^4*f^3*g^5*x)*e^5 - (4*a^3*c*d*f*g^7*x^4 - a^4*d*f^3*g^5 + (12*a^3*c*d*f^2*g^6 - a^4*d*g^8)
*x^3 + 3*(4*a^3*c*d*f^3*g^5 - a^4*d*f*g^7)*x^2 + (4*a^3*c*d*f^4*g^4 - 3*a^4*d*f^2*g^6)*x)*e^4 + 2*(3*a^2*c^2*d
^2*f^2*g^6*x^4 - 2*a^3*c*d^2*f^4*g^4 + (9*a^2*c^2*d^2*f^3*g^5 - 2*a^3*c*d^2*f*g^7)*x^3 + 3*(3*a^2*c^2*d^2*f^4*
g^4 - 2*a^3*c*d^2*f^2*g^6)*x^2 + 3*(a^2*c^2*d^2*f^5*g^3 - 2*a^3*c*d^2*f^3*g^5)*x)*e^3 - 2*(2*a*c^3*d^3*f^3*g^5
*x^4 - 3*a^2*c^2*d^3*f^5*g^3 + 3*(2*a*c^3*d^3*f^4*g^4 - a^2*c^2*d^3*f^2*g^6)*x^3 + 3*(2*a*c^3*d^3*f^5*g^3 - 3*
a^2*c^2*d^3*f^3*g^5)*x^2 + (2*a*c^3*d^3*f^6*g^2 - 9*a^2*c^2*d^3*f^4*g^4)*x)*e^2 + (c^4*d^4*f^4*g^4*x^4 - 4*a*c
^3*d^4*f^6*g^2 + (3*c^4*d^4*f^5*g^3 - 4*a*c^3*d^4*f^3*g^5)*x^3 + 3*(c^4*d^4*f^6*g^2 - 4*a*c^3*d^4*f^4*g^4)*x^2
+ (c^4*d^4*f^7*g - 12*a*c^3*d^4*f^5*g^3)*x)*e), -1/24*(15*(c^3*d^4*g^3*x^3 + 3*c^3*d^4*f*g^2*x^2 + 3*c^3*d^4*
f^2*g*x + c^3*d^4*f^3 + (c^3*d^3*g^3*x^4 + 3*c^3*d^3*f*g^2*x^3 + 3*c^3*d^3*f^2*g*x^2 + c^3*d^3*f^3*x)*e)*sqrt(
c*d*f*g - a*g^2*e)*arctan(sqrt(c*d*f*g - a*g^2*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)/(c
*d^2*g*x + a*g*x*e^2 + (c*d*g*x^2 + a*d*g)*e)) - (15*c^3*d^3*f*g^3*x^2 + 40*c^3*d^3*f^2*g^2*x + 33*c^3*d^3*f^3
*g - 8*a^3*g^4*e^3 + 2*(5*a^2*c*d*g^4*x + 17*a^2*c*d*f*g^3)*e^2 - (15*a*c^2*d^2*g^4*x^2 + 50*a*c^2*d^2*f*g^3*x
+ 59*a*c^2*d^2*f^2*g^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^4*d^5*f^4*g^4*x^3 +
3*c^4*d^5*f^5*g^3*x^2 + 3*c^4*d^5*f^6*g^2*x + c^4*d^5*f^7*g + (a^4*g^8*x^4 + 3*a^4*f*g^7*x^3 + 3*a^4*f^2*g^6*x
^2 + a^4*f^3*g^5*x)*e^5 - (4*a^3*c*d*f*g^7*x^4 - a^4*d*f^3*g^5 + (12*a^3*c*d*f^2*g^6 - a^4*d*g^8)*x^3 + 3*(4*a
^3*c*d*f^3*g^5 - a^4*d*f*g^7)*x^2 + (4*a^3*c*d*f^4*g^4 - 3*a^4*d*f^2*g^6)*x)*e^4 + 2*(3*a^2*c^2*d^2*f^2*g^6*x^
4 - 2*a^3*c*d^2*f^4*g^4 + (9*a^2*c^2*d^2*f^3*g^5 - 2*a^3*c*d^2*f*g^7)*x^3 + 3*(3*a^2*c^2*d^2*f^4*g^4 - 2*a^3*c
*d^2*f^2*g^6)*x^2 + 3*(a^2*c^2*d^2*f^5*g^3 - 2*a^3*c*d^2*f^3*g^5)*x)*e^3 - 2*(2*a*c^3*d^3*f^3*g^5*x^4 - 3*a^2*
c^2*d^3*f^5*g^3 + 3*(2*a*c^3*d^3*f^4*g^4 - a^2*c^2*d^3*f^2*g^6)*x^3 + 3*(2*a*c^3*d^3*f^5*g^3 - 3*a^2*c^2*d^3*f
^3*g^5)*x^2 + (2*a*c^3*d^3*f^6*g^2 - 9*a^2*c^2*d^3*f^4*g^4)*x)*e^2 + (c^4*d^4*f^4*g^4*x^4 - 4*a*c^3*d^4*f^6*g^
2 + (3*c^4*d^4*f^5*g^3 - 4*a*c^3*d^4*f^3*g^5)*x^3 + 3*(c^4*d^4*f^6*g^2 - 4*a*c^3*d^4*f^4*g^4)*x^2 + (c^4*d^4*f
^7*g - 12*a*c^3*d^4*f^5*g^3)*x)*e)]

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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)**(1/2)/(g*x+f)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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Giac [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)^(1/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {d+e\,x}}{{\left (f+g\,x\right )}^4\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)/((f + g*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)),x)

[Out]

int((d + e*x)^(1/2)/((f + g*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)), x)

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