3.7.86 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^3} \, dx\) [686]

Optimal. Leaf size=207 \[ -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g) \sqrt {d+e x} (f+g x)}+\frac {c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 g^{3/2} (c d f-a e g)^{3/2}} \]

[Out]

1/4*c^2*d^2*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)^(3/2)-1/2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g/(g*x+f)^2/(e*x+d)^(1/2)+1/4*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)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 4, 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^2 d^2 \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 )}{4 g^{3/2} (c d f-a e g)^{3/2}}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x) (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}}{2 g \sqrt {d+e x} (f+g x)^2} \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)^3),x]

[Out]

-1/2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(g*Sqrt[d + e*x]*(f + g*x)^2) + (c*d*Sqrt[a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2])/(4*g*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)) + (c^2*d^2*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])])/(4*g^(3/2)*(c*d*f - a*e*g)^(3/2))

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

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Mathematica [A]
time = 0.54, size = 165, normalized size = 0.80 \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} (2 a e g+c d (-f+g x))+c^2 d^2 (f+g x)^2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{4 g^{3/2} (c d f-a e g)^{3/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)^2} \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)^3),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]*(2*a*e*g + c*d*(-f + g*x)) + c
^2*d^2*(f + g*x)^2*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]]))/(4*g^(3/2)*(c*d*f - a*e*g)^(3/2)*
Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^2)

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Maple [A]
time = 0.14, size = 275, normalized size = 1.33

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (\arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} g^{2} x^{2}+2 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f g x +\arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} f^{2}-\sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c d g x -2 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a e g +\sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c d f \right )}{4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (a e g -c d f \right ) g \left (g x +f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\)	\(275\)

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)^3/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*((c*d*x+a*e)*(e*x+d))^(1/2)*(arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^2*g^2*x^2+2*arctan
h(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^2*f*g*x+arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/
2))*c^2*d^2*f^2-((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c*d*g*x-2*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*
a*e*g+((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c*d*f)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/(a*e*g-c*d*f)/g/(g*x+f)
^2/((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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^3/(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)^3*sqrt(x*e + d)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (190) = 380\).
time = 1.61, size = 1095, normalized size = 5.29 \begin {gather*} \left [\frac {{\left (c^{2} d^{3} g^{2} x^{2} + 2 \, c^{2} d^{3} f g x + c^{2} d^{3} f^{2} + {\left (c^{2} d^{2} g^{2} x^{3} + 2 \, c^{2} d^{2} f g x^{2} + c^{2} d^{2} f^{2} 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 (c^{2} d^{2} f g^{2} x - c^{2} d^{2} f^{2} g - 2 \, a^{2} g^{3} e^{2} - {\left (a c d g^{3} x - 3 \, a c d f 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}}{8 \, {\left (c^{2} d^{3} f^{2} g^{4} x^{2} + 2 \, c^{2} d^{3} f^{3} g^{3} x + c^{2} d^{3} f^{4} g^{2} + {\left (a^{2} g^{6} x^{3} + 2 \, a^{2} f g^{5} x^{2} + a^{2} f^{2} g^{4} x\right )} e^{3} - {\left (2 \, a c d f g^{5} x^{3} - a^{2} d f^{2} g^{4} + {\left (4 \, a c d f^{2} g^{4} - a^{2} d g^{6}\right )} x^{2} + 2 \, {\left (a c d f^{3} g^{3} - a^{2} d f g^{5}\right )} x\right )} e^{2} + {\left (c^{2} d^{2} f^{2} g^{4} x^{3} - 2 \, a c d^{2} f^{3} g^{3} + 2 \, {\left (c^{2} d^{2} f^{3} g^{3} - a c d^{2} f g^{5}\right )} x^{2} + {\left (c^{2} d^{2} f^{4} g^{2} - 4 \, a c d^{2} f^{2} g^{4}\right )} x\right )} e\right )}}, -\frac {{\left (c^{2} d^{3} g^{2} x^{2} + 2 \, c^{2} d^{3} f g x + c^{2} d^{3} f^{2} + {\left (c^{2} d^{2} g^{2} x^{3} + 2 \, c^{2} d^{2} f g x^{2} + c^{2} d^{2} f^{2} 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 (c^{2} d^{2} f g^{2} x - c^{2} d^{2} f^{2} g - 2 \, a^{2} g^{3} e^{2} - {\left (a c d g^{3} x - 3 \, a c d f 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}}{4 \, {\left (c^{2} d^{3} f^{2} g^{4} x^{2} + 2 \, c^{2} d^{3} f^{3} g^{3} x + c^{2} d^{3} f^{4} g^{2} + {\left (a^{2} g^{6} x^{3} + 2 \, a^{2} f g^{5} x^{2} + a^{2} f^{2} g^{4} x\right )} e^{3} - {\left (2 \, a c d f g^{5} x^{3} - a^{2} d f^{2} g^{4} + {\left (4 \, a c d f^{2} g^{4} - a^{2} d g^{6}\right )} x^{2} + 2 \, {\left (a c d f^{3} g^{3} - a^{2} d f g^{5}\right )} x\right )} e^{2} + {\left (c^{2} d^{2} f^{2} g^{4} x^{3} - 2 \, a c d^{2} f^{3} g^{3} + 2 \, {\left (c^{2} d^{2} f^{3} g^{3} - a c d^{2} f g^{5}\right )} x^{2} + {\left (c^{2} d^{2} f^{4} g^{2} - 4 \, a c d^{2} f^{2} 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)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[1/8*((c^2*d^3*g^2*x^2 + 2*c^2*d^3*f*g*x + c^2*d^3*f^2 + (c^2*d^2*g^2*x^3 + 2*c^2*d^2*f*g*x^2 + c^2*d^2*f^2*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*s
qrt(-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*(c^2*d^2*f*g^2*x - c^2*d^2*f^2*g - 2*a^2*g^3*e^2 - (a*c*d*g^3*x - 3*a*c*d*f*g^2)*e)*sqrt(c*d^2*x + a
*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^2*d^3*f^2*g^4*x^2 + 2*c^2*d^3*f^3*g^3*x + c^2*d^3*f^4*g^2 + (a^2
*g^6*x^3 + 2*a^2*f*g^5*x^2 + a^2*f^2*g^4*x)*e^3 - (2*a*c*d*f*g^5*x^3 - a^2*d*f^2*g^4 + (4*a*c*d*f^2*g^4 - a^2*
d*g^6)*x^2 + 2*(a*c*d*f^3*g^3 - a^2*d*f*g^5)*x)*e^2 + (c^2*d^2*f^2*g^4*x^3 - 2*a*c*d^2*f^3*g^3 + 2*(c^2*d^2*f^
3*g^3 - a*c*d^2*f*g^5)*x^2 + (c^2*d^2*f^4*g^2 - 4*a*c*d^2*f^2*g^4)*x)*e), -1/4*((c^2*d^3*g^2*x^2 + 2*c^2*d^3*f
*g*x + c^2*d^3*f^2 + (c^2*d^2*g^2*x^3 + 2*c^2*d^2*f*g*x^2 + c^2*d^2*f^2*x)*e)*sqrt(c*d*f*g - a*g^2*e)*arctan(s
qrt(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)) - (c^2*d^2*f*g^2*x - c^2*d^2*f^2*g - 2*a^2*g^3*e^2 - (a*c*d*g^3*x - 3*a*c*d*f*g^2)*e)*sqrt
(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c^2*d^3*f^2*g^4*x^2 + 2*c^2*d^3*f^3*g^3*x + c^2*d^3*f^
4*g^2 + (a^2*g^6*x^3 + 2*a^2*f*g^5*x^2 + a^2*f^2*g^4*x)*e^3 - (2*a*c*d*f*g^5*x^3 - a^2*d*f^2*g^4 + (4*a*c*d*f^
2*g^4 - a^2*d*g^6)*x^2 + 2*(a*c*d*f^3*g^3 - a^2*d*f*g^5)*x)*e^2 + (c^2*d^2*f^2*g^4*x^3 - 2*a*c*d^2*f^3*g^3 + 2
*(c^2*d^2*f^3*g^3 - a*c*d^2*f*g^5)*x^2 + (c^2*d^2*f^4*g^2 - 4*a*c*d^2*f^2*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)**3/(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)^3/(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 )}^3\,\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)^3*(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)^3*(d + e*x)^(1/2)), x)

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