\(\int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [714]

Optimal result

Integrand size = 48, antiderivative size = 169 \[ \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d \sqrt {d+e x}}+\frac {(c d f-a e g) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {884, 905, 65, 223, 212} \[ \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}} \]

[In]

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

[Out]

(Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d*Sqrt[d + e*x]) + ((c*d*f - a*e*g)*Sqrt[a*e +
c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(c^(3/2)*d^(3/2)*Sq
rt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 884

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

Rule 905

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

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d \sqrt {d+e x}}+\frac {\left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right ) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 c d e^2} \\ & = \frac {\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d \sqrt {d+e x}}+\frac {\left (\left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right ) \sqrt {a e+c d x} \sqrt {d+e x}\right ) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}} \, dx}{2 c d e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d \sqrt {d+e x}}+\frac {\left (\left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right ) \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {f-\frac {a e g}{c d}+\frac {g x^2}{c d}}} \, dx,x,\sqrt {a e+c d x}\right )}{c^2 d^2 e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d \sqrt {d+e x}}+\frac {\left (\left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right ) \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {g x^2}{c d}} \, dx,x,\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}\right )}{c^2 d^2 e^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d \sqrt {d+e x}}+\frac {(c d f-a e g) \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {d+e x} \left (\sqrt {c} \sqrt {d} \sqrt {g} (a e+c d x) \sqrt {f+g x}+(c d f-a e g) \sqrt {a e+c d x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{c^{3/2} d^{3/2} \sqrt {g} \sqrt {(a e+c d x) (d+e x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.13

method	result	size
default	\(-\frac {\sqrt {g x +f}\, \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (\ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a e g -\ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c d f -2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\right )}{2 \sqrt {e x +d}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, c d \sqrt {c d g}}\)	\(191\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.75 (sec) , antiderivative size = 521, normalized size of antiderivative = 3.08 \[ \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [\frac {4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} c d g - {\left (c d^{2} f - a d e g + {\left (c d e f - a e^{2} g\right )} x\right )} \sqrt {c d g} \log \left (-\frac {8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + c d f + a e g\right )} \sqrt {c d g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (c^{2} d^{2} e f g + {\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, {\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g + {\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right )}{4 \, {\left (c^{2} d^{2} e g x + c^{2} d^{3} g\right )}}, \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} c d g - {\left (c d^{2} f - a d e g + {\left (c d e f - a e^{2} g\right )} x\right )} \sqrt {-c d g} \arctan \left (\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, c d e g x^{2} + c d^{2} f + a d e g + {\left (c d e f + {\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x}\right )}{2 \, {\left (c^{2} d^{2} e g x + c^{2} d^{3} g\right )}}\right ] \]

[In]

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

[Out]

[1/4*(4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)*c*d*g - (c*d^2*f - a*d*e*g + (
c*d*e*f - a*e^2*g)*x)*sqrt(c*d*g)*log(-(8*c^2*d^2*e*g^2*x^3 + c^2*d^3*f^2 + 6*a*c*d^2*e*f*g + a^2*d*e^2*g^2 -
4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*g*x + c*d*f + a*e*g)*sqrt(c*d*g)*sqrt(e*x + d)*sqrt(g*x +
f) + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x^2 + (c^2*d^2*e*f^2 + 2*(4*c^2*d^3 + 3*a*c*d*e^2)*f*g + (
8*a*c*d^2*e + a^2*e^3)*g^2)*x)/(e*x + d)))/(c^2*d^2*e*g*x + c^2*d^3*g), 1/2*(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)*c*d*g - (c*d^2*f - a*d*e*g + (c*d*e*f - a*e^2*g)*x)*sqrt(-c*d*g)*arct
an(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*g)*sqrt(e*x + d)*sqrt(g*x + f)/(2*c*d*e*g*x^2 + c*d
^2*f + a*d*e*g + (c*d*e*f + (2*c*d^2 + a*e^2)*g)*x)))/(c^2*d^2*e*g*x + c^2*d^3*g)]

Sympy [F]

\[ \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {d + e x} \sqrt {f + g x}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(d + e*x)*sqrt(f + g*x)/sqrt((d + e*x)*(a*e + c*d*x)), x)

Maxima [F]

\[ \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {\sqrt {e x + d} \sqrt {g x + f}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (141) = 282\).

Time = 0.43 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {e {\left (\frac {g {\left (\frac {{\left (c d e f {\left | e \right |} - a e^{2} g {\left | e \right |}\right )} \log \left ({\left | -\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} + \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g} \right |}\right )}{\sqrt {c d g} c d} - \frac {\sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g} \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} {\left | e \right |}}{c d e g}\right )}}{e^{2} {\left | g \right |}} - \frac {c d e^{2} f g {\left | e \right |} \log \left ({\left | -\sqrt {e^{2} f - d e g} \sqrt {c d g} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \right |}\right ) - a e^{3} g^{2} {\left | e \right |} \log \left ({\left | -\sqrt {e^{2} f - d e g} \sqrt {c d g} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \right |}\right ) - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {e^{2} f - d e g} \sqrt {c d g} {\left | e \right |}}{\sqrt {c d g} c d e^{3} {\left | g \right |}}\right )}}{{\left | e \right |}} \]

[In]

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

[Out]

-e*(g*((c*d*e*f*abs(e) - a*e^2*g*abs(e))*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*
e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c*d) - sqrt(-c*d*e^2*f*g + a*e^3*g
^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*abs(e)/(c*d*e*g))/(e^2*abs(g))
- (c*d*e^2*f*g*abs(e)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - a*e^3*g^2
*abs(e)*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - sqrt(-c*d^2*e*g^2 + a*e^
3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*abs(e))/(sqrt(c*d*g)*c*d*e^3*abs(g)))/abs(e)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]

[In]

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

[Out]

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