\(\int \frac {\sqrt {d+e x} (f+g x)^{5/2}}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [712]

Optimal result

Integrand size = 48, antiderivative size = 313 \[ \int \frac {\sqrt {d+e x} (f+g x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {5 (c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d \sqrt {d+e x}}+\frac {5 (c d f-a e g)^3 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{8 c^{7/2} d^{7/2} \sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {884, 905, 65, 223, 212} \[ \int \frac {\sqrt {d+e x} (f+g x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {5 \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^3 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{8 c^{7/2} d^{7/2} \sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {5 \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{8 c^3 d^3 \sqrt {d+e x}}+\frac {5 (f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{12 c^2 d^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d \sqrt {d+e x}} \]

[In]

Int[(Sqrt[d + e*x]*(f + g*x)^(5/2))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(5*(c*d*f - a*e*g)^2*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*c^3*d^3*Sqrt[d + e*x]) + (5
*(c*d*f - a*e*g)*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*c^2*d^2*Sqrt[d + e*x]) + ((f
+ g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c*d*Sqrt[d + e*x]) + (5*(c*d*f - a*e*g)^3*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])])/(8*c^(7/2)*d^(
7/2)*Sqrt[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 {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d \sqrt {d+e x}}+\frac {\left (5 \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )\right ) \int \frac {\sqrt {d+e x} (f+g x)^{3/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 c d e^2} \\ & = \frac {5 (c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d \sqrt {d+e x}}+\frac {\left (5 (c d f-a e g)^2\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}{8 c^2 d^2} \\ & = \frac {5 (c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d \sqrt {d+e x}}+\frac {\left (5 (c d f-a e g)^3\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}{16 c^3 d^3} \\ & = \frac {5 (c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d \sqrt {d+e x}}+\frac {\left (5 (c d f-a e g)^3 \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}{16 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {5 (c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d \sqrt {d+e x}}+\frac {\left (5 (c d f-a e g)^3 \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 )}{8 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {5 (c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d \sqrt {d+e x}}+\frac {\left (5 (c d f-a e g)^3 \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 )}{8 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {5 (c d f-a e g)^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^3 d^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^2 d^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c d \sqrt {d+e x}}+\frac {5 (c d f-a e g)^3 \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 )}{8 c^{7/2} d^{7/2} \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 = 0.40 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {d+e x} (f+g x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {d+e x} \left (\sqrt {c} \sqrt {d} (a e+c d x) \sqrt {f+g x} \left (15 a^2 e^2 g^2-10 a c d e g (4 f+g x)+c^2 d^2 \left (33 f^2+26 f g x+8 g^2 x^2\right )\right )+\frac {15 (c d f-a e g)^3 \sqrt {a e+c d x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )}{\sqrt {g}}\right )}{24 c^{7/2} d^{7/2} \sqrt {(a e+c d x) (d+e x)}} \]

[In]

Integrate[(Sqrt[d + e*x]*(f + g*x)^(5/2))/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]*(a*e + c*d*x)*Sqrt[f + g*x]*(15*a^2*e^2*g^2 - 10*a*c*d*e*g*(4*f + g*x) + c^2*d
^2*(33*f^2 + 26*f*g*x + 8*g^2*x^2)) + (15*(c*d*f - a*e*g)^3*Sqrt[a*e + c*d*x]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[f
+ g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])])/Sqrt[g]))/(24*c^(7/2)*d^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.60

method	result	size
default	\(-\frac {\sqrt {g x +f}\, \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{3} e^{3} g^{3}-45 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} f \,g^{2}+45 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e \,f^{2} g -15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{3} d^{3} f^{3}-16 c^{2} d^{2} g^{2} x^{2} \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}+20 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, a c d e \,g^{2} x -52 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, c^{2} d^{2} f g x -30 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a^{2} e^{2} g^{2}+80 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a c d e f g -66 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, c^{2} d^{2} f^{2}\right )}{48 \sqrt {e x +d}\, c^{3} d^{3} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}\)	\(501\)

[In]

int((g*x+f)^(5/2)*(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]

-1/48*(g*x+f)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*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^3*e^3*g^3-45*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^2*c*d*e^2*f*g^2+45*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*c^2*d^2*e*f^2*g-15*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^3*d^3*f^3-16*c^2*d^2*g^2*x^2*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)+20*(c*d*
g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)*a*c*d*e*g^2*x-52*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)*c^2*d^2*f*g*x-
30*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a^2*e^2*g^2+80*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a*c*d*e*
f*g-66*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*c^2*d^2*f^2)/(e*x+d)^(1/2)/c^3/d^3/((g*x+f)*(c*d*x+a*e))^(1/2
)/(c*d*g)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 1.02 (sec) , antiderivative size = 841, normalized size of antiderivative = 2.69 \[ \int \frac {\sqrt {d+e x} (f+g x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [\frac {4 \, {\left (8 \, c^{3} d^{3} g^{3} x^{2} + 33 \, c^{3} d^{3} f^{2} g - 40 \, a c^{2} d^{2} e f g^{2} + 15 \, a^{2} c d e^{2} g^{3} + 2 \, {\left (13 \, c^{3} d^{3} f g^{2} - 5 \, a c^{2} d^{2} e g^{3}\right )} x\right )} \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} - 15 \, {\left (c^{3} d^{4} f^{3} - 3 \, a c^{2} d^{3} e f^{2} g + 3 \, a^{2} c d^{2} e^{2} f g^{2} - a^{3} d e^{3} g^{3} + {\left (c^{3} d^{3} e f^{3} - 3 \, a c^{2} d^{2} e^{2} f^{2} g + 3 \, a^{2} c d e^{3} f g^{2} - a^{3} e^{4} g^{3}\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 )}{96 \, {\left (c^{4} d^{4} e g x + c^{4} d^{5} g\right )}}, \frac {2 \, {\left (8 \, c^{3} d^{3} g^{3} x^{2} + 33 \, c^{3} d^{3} f^{2} g - 40 \, a c^{2} d^{2} e f g^{2} + 15 \, a^{2} c d e^{2} g^{3} + 2 \, {\left (13 \, c^{3} d^{3} f g^{2} - 5 \, a c^{2} d^{2} e g^{3}\right )} x\right )} \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} - 15 \, {\left (c^{3} d^{4} f^{3} - 3 \, a c^{2} d^{3} e f^{2} g + 3 \, a^{2} c d^{2} e^{2} f g^{2} - a^{3} d e^{3} g^{3} + {\left (c^{3} d^{3} e f^{3} - 3 \, a c^{2} d^{2} e^{2} f^{2} g + 3 \, a^{2} c d e^{3} f g^{2} - a^{3} e^{4} g^{3}\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 )}{48 \, {\left (c^{4} d^{4} e g x + c^{4} d^{5} g\right )}}\right ] \]

[In]

integrate((g*x+f)^(5/2)*(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]

[1/96*(4*(8*c^3*d^3*g^3*x^2 + 33*c^3*d^3*f^2*g - 40*a*c^2*d^2*e*f*g^2 + 15*a^2*c*d*e^2*g^3 + 2*(13*c^3*d^3*f*g
^2 - 5*a*c^2*d^2*e*g^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f) - 15*(c^3*d
^4*f^3 - 3*a*c^2*d^3*e*f^2*g + 3*a^2*c*d^2*e^2*f*g^2 - a^3*d*e^3*g^3 + (c^3*d^3*e*f^3 - 3*a*c^2*d^2*e^2*f^2*g
+ 3*a^2*c*d*e^3*f*g^2 - a^3*e^4*g^3)*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^4*d^4*e*g*x + c^4*d^5*g), 1/48*(2*(8*c^3*d^3*g
^3*x^2 + 33*c^3*d^3*f^2*g - 40*a*c^2*d^2*e*f*g^2 + 15*a^2*c*d*e^2*g^3 + 2*(13*c^3*d^3*f*g^2 - 5*a*c^2*d^2*e*g^
3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f) - 15*(c^3*d^4*f^3 - 3*a*c^2*d^3*
e*f^2*g + 3*a^2*c*d^2*e^2*f*g^2 - a^3*d*e^3*g^3 + (c^3*d^3*e*f^3 - 3*a*c^2*d^2*e^2*f^2*g + 3*a^2*c*d*e^3*f*g^2
- a^3*e^4*g^3)*x)*sqrt(-c*d*g)*arctan(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^4*d^4*e*g*x + c^4
*d^5*g)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x} (f+g x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Timed out} \]

[In]

integrate((g*x+f)**(5/2)*(e*x+d)**(1/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {d+e x} (f+g x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {\sqrt {e x + d} {\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \]

[In]

integrate((g*x+f)^(5/2)*(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)*(g*x + f)^(5/2)/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. 989 vs. \(2 (265) = 530\).

Time = 0.64 (sec) , antiderivative size = 989, normalized size of antiderivative = 3.16 \[ \int \frac {\sqrt {d+e x} (f+g x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {e {\left (\frac {{\left (\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 (2 \, {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} {\left (\frac {4 \, {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} {\left | e \right |}}{c d e^{3} g} + \frac {5 \, {\left (c^{4} d^{4} e^{2} f {\left | e \right |} - a c^{3} d^{3} e^{3} g {\left | e \right |}\right )}}{c^{5} d^{5} e^{3} g}\right )} + \frac {15 \, {\left (c^{4} d^{4} e^{4} f^{2} {\left | e \right |} - 2 \, a c^{3} d^{3} e^{5} f g {\left | e \right |} + a^{2} c^{2} d^{2} e^{6} g^{2} {\left | e \right |}\right )}}{c^{5} d^{5} e^{3} g}\right )} - \frac {15 \, {\left (c^{3} d^{3} e^{3} f^{3} {\left | e \right |} - 3 \, a c^{2} d^{2} e^{4} f^{2} g {\left | e \right |} + 3 \, a^{2} c d e^{5} f g^{2} {\left | e \right |} - a^{3} e^{6} g^{3} {\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^{3} d^{3}}\right )} g}{e^{4} {\left | g \right |}} + \frac {15 \, c^{3} d^{3} e^{4} f^{3} 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 ) - 45 \, a c^{2} d^{2} e^{5} f^{2} 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 ) + 45 \, a^{2} c d e^{6} f g^{3} {\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 ) - 15 \, a^{3} e^{7} g^{4} {\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 ) - 33 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {e^{2} f - d e g} \sqrt {c d g} c^{2} d^{2} e^{2} f^{2} {\left | e \right |} + 26 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {e^{2} f - d e g} \sqrt {c d g} c^{2} d^{3} e f g {\left | e \right |} + 40 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {e^{2} f - d e g} \sqrt {c d g} a c d e^{3} f g {\left | e \right |} - 8 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {e^{2} f - d e g} \sqrt {c d g} c^{2} d^{4} g^{2} {\left | e \right |} - 10 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {e^{2} f - d e g} \sqrt {c d g} a c d^{2} e^{2} g^{2} {\left | e \right |} - 15 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {e^{2} f - d e g} \sqrt {c d g} a^{2} e^{4} g^{2} {\left | e \right |}}{\sqrt {c d g} c^{3} d^{3} e^{5} {\left | g \right |}}\right )}}{24 \, {\left | e \right |}} \]

[In]

integrate((g*x+f)^(5/2)*(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]

1/24*e*((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)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*(4*(e^2*f + (e*x + d)*e*g - d*e*g)*abs(e)/(c*d*e^3*g) + 5*(c^4*d^4*e^
2*f*abs(e) - a*c^3*d^3*e^3*g*abs(e))/(c^5*d^5*e^3*g)) + 15*(c^4*d^4*e^4*f^2*abs(e) - 2*a*c^3*d^3*e^5*f*g*abs(e
) + a^2*c^2*d^2*e^6*g^2*abs(e))/(c^5*d^5*e^3*g)) - 15*(c^3*d^3*e^3*f^3*abs(e) - 3*a*c^2*d^2*e^4*f^2*g*abs(e) +
3*a^2*c*d*e^5*f*g^2*abs(e) - a^3*e^6*g^3*abs(e))*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + s
qrt(-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^3*d^3))*g/(e^4*abs(g))
+ (15*c^3*d^3*e^4*f^3*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))) - 4
5*a*c^2*d^2*e^5*f^2*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))) + 4
5*a^2*c*d*e^6*f*g^3*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))) - 15*a^
3*e^7*g^4*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))) - 33*sqrt(-c*d^2*
e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^2*e^2*f^2*abs(e) + 26*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*
sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^3*e*f*g*abs(e) + 40*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*s
qrt(c*d*g)*a*c*d*e^3*f*g*abs(e) - 8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^4*g^2
*abs(e) - 10*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c*d^2*e^2*g^2*abs(e) - 15*sqrt(-
c*d^2*e*g^2 + a*e^3*g^2)*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*e^4*g^2*abs(e))/(sqrt(c*d*g)*c^3*d^3*e^5*abs(g)))
/abs(e)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x} (f+g x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^{5/2}\,\sqrt {d+e\,x}}{\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]

[In]

int(((f + g*x)^(5/2)*(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)^(5/2)*(d + e*x)^(1/2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2), x)