3.7.87 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^4} \, dx\) [687]

Optimal. Leaf size=277 \[ -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}+\frac {c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{8 g^{3/2} (c d f-a e g)^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {876, 886, 888, 211} \begin {gather*} \frac {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 g^{3/2} (c d f-a e g)^{5/2}}+\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 g \sqrt {d+e x} (f+g x) (c d f-a e g)^2}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 876

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

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 {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^4} \, dx &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}+\frac {(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 g}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {\left (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 g (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 g \sqrt {d+e x} (f+g x)^3}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}+\frac {\left (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 g (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 g \sqrt {d+e x} (f+g x)^3}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}+\frac {\left (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 g (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 g \sqrt {d+e x} (f+g x)^3}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}+\frac {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 g^{3/2} (c d f-a e g)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.81, size = 202, normalized size = 0.73 \begin {gather*} \frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {c d f-a e g} \sqrt {a e+c d x} \left (-8 a^2 e^2 g^2-2 a c d e g (-7 f+g x)+c^2 d^2 \left (-3 f^2+8 f g x+3 g^2 x^2\right )\right )+3 c^3 d^3 (f+g x)^3 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{24 g^{3/2} (c d f-a e g)^{5/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 443, normalized size = 1.60

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} g^{3} x^{3}+9 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f \,g^{2} x^{2}+9 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{2} g x +3 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{3}-3 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{2} x^{2}+2 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e \,g^{2} x -8 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f g x +8 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{2}-14 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e f g +3 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2}\right )}{24 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{3} g \left (a e g -c d f \right )^{2} \sqrt {c d x +a e}}\)	\(443\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((c*d*x+a*e)*(e*x+d))^(1/2)*(3*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+9*ar
ctanh(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+9*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+3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*f^3-3*((a*e*g-c*d
*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^2*g^2*x^2+2*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*g^2*x-8*((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^2-1
4*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*f*g+3*((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)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)^3/g/(a*e*g-c*d*f)^2/(c*d*x+a*e)^(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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^4/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (257) = 514\).
time = 1.32, size = 1801, normalized size = 6.50 \begin {gather*} \left [-\frac {3 \, {\left (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} + {\left (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\right )} e\right )} \sqrt {-c d f g + a g^{2} e} \log \left (-\frac {c d^{2} g x - c d^{2} f + 2 \, a g x e^{2} + {\left (c d g x^{2} - c d f x + 2 \, a d g\right )} e - 2 \, \sqrt {-c d f g + a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{d g x + d f + {\left (g x^{2} + f x\right )} e}\right ) - 2 \, {\left (3 \, c^{3} d^{3} f g^{3} x^{2} + 8 \, c^{3} d^{3} f^{2} g^{2} x - 3 \, c^{3} d^{3} f^{3} g + 8 \, a^{3} g^{4} e^{3} + 2 \, {\left (a^{2} c d g^{4} x - 11 \, a^{2} c d f g^{3}\right )} e^{2} - {\left (3 \, a c^{2} d^{2} g^{4} x^{2} + 10 \, a c^{2} d^{2} f g^{3} x - 17 \, a c^{2} d^{2} f^{2} g^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{48 \, {\left (c^{3} d^{4} f^{3} g^{5} x^{3} + 3 \, c^{3} d^{4} f^{4} g^{4} x^{2} + 3 \, c^{3} d^{4} f^{5} g^{3} x + c^{3} d^{4} f^{6} g^{2} - {\left (a^{3} g^{8} x^{4} + 3 \, a^{3} f g^{7} x^{3} + 3 \, a^{3} f^{2} g^{6} x^{2} + a^{3} f^{3} g^{5} x\right )} e^{4} + {\left (3 \, a^{2} c d f g^{7} x^{4} - a^{3} d f^{3} g^{5} + {\left (9 \, a^{2} c d f^{2} g^{6} - a^{3} d g^{8}\right )} x^{3} + 3 \, {\left (3 \, a^{2} c d f^{3} g^{5} - a^{3} d f g^{7}\right )} x^{2} + 3 \, {\left (a^{2} c d f^{4} g^{4} - a^{3} d f^{2} g^{6}\right )} x\right )} e^{3} - 3 \, {\left (a c^{2} d^{2} f^{2} g^{6} x^{4} - a^{2} c d^{2} f^{4} g^{4} + {\left (3 \, a c^{2} d^{2} f^{3} g^{5} - a^{2} c d^{2} f g^{7}\right )} x^{3} + 3 \, {\left (a c^{2} d^{2} f^{4} g^{4} - a^{2} c d^{2} f^{2} g^{6}\right )} x^{2} + {\left (a c^{2} d^{2} f^{5} g^{3} - 3 \, a^{2} c d^{2} f^{3} g^{5}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{3} g^{5} x^{4} - 3 \, a c^{2} d^{3} f^{5} g^{3} + 3 \, {\left (c^{3} d^{3} f^{4} g^{4} - a c^{2} d^{3} f^{2} g^{6}\right )} x^{3} + 3 \, {\left (c^{3} d^{3} f^{5} g^{3} - 3 \, a c^{2} d^{3} f^{3} g^{5}\right )} x^{2} + {\left (c^{3} d^{3} f^{6} g^{2} - 9 \, a c^{2} d^{3} f^{4} g^{4}\right )} x\right )} e\right )}}, -\frac {3 \, {\left (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} + {\left (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\right )} e\right )} \sqrt {c d f g - a g^{2} e} \arctan \left (\frac {\sqrt {c d f g - a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{c d^{2} g x + a g x e^{2} + {\left (c d g x^{2} + a d g\right )} e}\right ) - {\left (3 \, c^{3} d^{3} f g^{3} x^{2} + 8 \, c^{3} d^{3} f^{2} g^{2} x - 3 \, c^{3} d^{3} f^{3} g + 8 \, a^{3} g^{4} e^{3} + 2 \, {\left (a^{2} c d g^{4} x - 11 \, a^{2} c d f g^{3}\right )} e^{2} - {\left (3 \, a c^{2} d^{2} g^{4} x^{2} + 10 \, a c^{2} d^{2} f g^{3} x - 17 \, a c^{2} d^{2} f^{2} g^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{24 \, {\left (c^{3} d^{4} f^{3} g^{5} x^{3} + 3 \, c^{3} d^{4} f^{4} g^{4} x^{2} + 3 \, c^{3} d^{4} f^{5} g^{3} x + c^{3} d^{4} f^{6} g^{2} - {\left (a^{3} g^{8} x^{4} + 3 \, a^{3} f g^{7} x^{3} + 3 \, a^{3} f^{2} g^{6} x^{2} + a^{3} f^{3} g^{5} x\right )} e^{4} + {\left (3 \, a^{2} c d f g^{7} x^{4} - a^{3} d f^{3} g^{5} + {\left (9 \, a^{2} c d f^{2} g^{6} - a^{3} d g^{8}\right )} x^{3} + 3 \, {\left (3 \, a^{2} c d f^{3} g^{5} - a^{3} d f g^{7}\right )} x^{2} + 3 \, {\left (a^{2} c d f^{4} g^{4} - a^{3} d f^{2} g^{6}\right )} x\right )} e^{3} - 3 \, {\left (a c^{2} d^{2} f^{2} g^{6} x^{4} - a^{2} c d^{2} f^{4} g^{4} + {\left (3 \, a c^{2} d^{2} f^{3} g^{5} - a^{2} c d^{2} f g^{7}\right )} x^{3} + 3 \, {\left (a c^{2} d^{2} f^{4} g^{4} - a^{2} c d^{2} f^{2} g^{6}\right )} x^{2} + {\left (a c^{2} d^{2} f^{5} g^{3} - 3 \, a^{2} c d^{2} f^{3} g^{5}\right )} x\right )} e^{2} + {\left (c^{3} d^{3} f^{3} g^{5} x^{4} - 3 \, a c^{2} d^{3} f^{5} g^{3} + 3 \, {\left (c^{3} d^{3} f^{4} g^{4} - a c^{2} d^{3} f^{2} g^{6}\right )} x^{3} + 3 \, {\left (c^{3} d^{3} f^{5} g^{3} - 3 \, a c^{2} d^{3} f^{3} g^{5}\right )} x^{2} + {\left (c^{3} d^{3} f^{6} g^{2} - 9 \, a c^{2} d^{3} f^{4} g^{4}\right )} x\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(3*(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*(3*c^3*d^3*f*g^3*x^2 + 8*c^3*d^3*f^2*g^2*x - 3*c
^3*d^3*f^3*g + 8*a^3*g^4*e^3 + 2*(a^2*c*d*g^4*x - 11*a^2*c*d*f*g^3)*e^2 - (3*a*c^2*d^2*g^4*x^2 + 10*a*c^2*d^2*
f*g^3*x - 17*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^3*d^4*f^3*g^5
*x^3 + 3*c^3*d^4*f^4*g^4*x^2 + 3*c^3*d^4*f^5*g^3*x + c^3*d^4*f^6*g^2 - (a^3*g^8*x^4 + 3*a^3*f*g^7*x^3 + 3*a^3*
f^2*g^6*x^2 + a^3*f^3*g^5*x)*e^4 + (3*a^2*c*d*f*g^7*x^4 - a^3*d*f^3*g^5 + (9*a^2*c*d*f^2*g^6 - a^3*d*g^8)*x^3
+ 3*(3*a^2*c*d*f^3*g^5 - a^3*d*f*g^7)*x^2 + 3*(a^2*c*d*f^4*g^4 - a^3*d*f^2*g^6)*x)*e^3 - 3*(a*c^2*d^2*f^2*g^6*
x^4 - a^2*c*d^2*f^4*g^4 + (3*a*c^2*d^2*f^3*g^5 - a^2*c*d^2*f*g^7)*x^3 + 3*(a*c^2*d^2*f^4*g^4 - a^2*c*d^2*f^2*g
^6)*x^2 + (a*c^2*d^2*f^5*g^3 - 3*a^2*c*d^2*f^3*g^5)*x)*e^2 + (c^3*d^3*f^3*g^5*x^4 - 3*a*c^2*d^3*f^5*g^3 + 3*(c
^3*d^3*f^4*g^4 - a*c^2*d^3*f^2*g^6)*x^3 + 3*(c^3*d^3*f^5*g^3 - 3*a*c^2*d^3*f^3*g^5)*x^2 + (c^3*d^3*f^6*g^2 - 9
*a*c^2*d^3*f^4*g^4)*x)*e), -1/24*(3*(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)*arct
an(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)) - (3*c^3*d^3*f*g^3*x^2 + 8*c^3*d^3*f^2*g^2*x - 3*c^3*d^3*f^3*g + 8*a^3*g^4*e^3 + 2*(a^
2*c*d*g^4*x - 11*a^2*c*d*f*g^3)*e^2 - (3*a*c^2*d^2*g^4*x^2 + 10*a*c^2*d^2*f*g^3*x - 17*a*c^2*d^2*f^2*g^2)*e)*s
qrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^3*d^4*f^3*g^5*x^3 + 3*c^3*d^4*f^4*g^4*x^2 + 3*c^3
*d^4*f^5*g^3*x + c^3*d^4*f^6*g^2 - (a^3*g^8*x^4 + 3*a^3*f*g^7*x^3 + 3*a^3*f^2*g^6*x^2 + a^3*f^3*g^5*x)*e^4 + (
3*a^2*c*d*f*g^7*x^4 - a^3*d*f^3*g^5 + (9*a^2*c*d*f^2*g^6 - a^3*d*g^8)*x^3 + 3*(3*a^2*c*d*f^3*g^5 - a^3*d*f*g^7
)*x^2 + 3*(a^2*c*d*f^4*g^4 - a^3*d*f^2*g^6)*x)*e^3 - 3*(a*c^2*d^2*f^2*g^6*x^4 - a^2*c*d^2*f^4*g^4 + (3*a*c^2*d
^2*f^3*g^5 - a^2*c*d^2*f*g^7)*x^3 + 3*(a*c^2*d^2*f^4*g^4 - a^2*c*d^2*f^2*g^6)*x^2 + (a*c^2*d^2*f^5*g^3 - 3*a^2
*c*d^2*f^3*g^5)*x)*e^2 + (c^3*d^3*f^3*g^5*x^4 - 3*a*c^2*d^3*f^5*g^3 + 3*(c^3*d^3*f^4*g^4 - a*c^2*d^3*f^2*g^6)*
x^3 + 3*(c^3*d^3*f^5*g^3 - 3*a*c^2*d^3*f^3*g^5)*x^2 + (c^3*d^3*f^6*g^2 - 9*a*c^2*d^3*f^4*g^4)*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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(g*x+f)**4/(e*x+d)**(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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^4/(e*x+d)^(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 {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (f+g\,x\right )}^4\,\sqrt {d+e\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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