\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx\) [754]

Optimal result

Integrand size = 48, antiderivative size = 294 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=-\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {15 \sqrt {c} \sqrt {d} (c d f-a e g)^2 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{4 g^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

-2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/g/(e*x+d)^(5/2)/(g*x+f)^(1/2)+5/2*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(3/2)*(g*x+f)^(1/2)/g^2/(e*x+d)^(3/2)+15/4*(-a*e*g+c*d*f)^2*arctanh(g^(1/2)*(c*d*x+a*e)^(1/2)/c^(1/2)/d^(
1/2)/(g*x+f)^(1/2))*c^(1/2)*d^(1/2)*(c*d*x+a*e)^(1/2)*(e*x+d)^(1/2)/g^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)-15/4*c*d*(-a*e*g+c*d*f)*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {876, 878, 905, 65, 223, 212} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\frac {15 \sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^2 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{4 g^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {15 c d \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(3/2)),x]

[Out]

(-15*c*d*(c*d*f - a*e*g)*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*g^3*Sqrt[d + e*x]) + (5
*c*d*Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(2*g^2*(d + e*x)^(3/2)) - (2*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(5/2))/(g*(d + e*x)^(5/2)*Sqrt[f + g*x]) + (15*Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)^2*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])])/(4*g^(7/2)*S
qrt[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 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 878

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*(m - n - 1))), x] - Dist[m*((c*e*f + c*d*g - b*e
*g)/(e^2*g*(m - n - 1))), Int[(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b,
c, d, e, f, g, 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] &&  !Intege
rQ[p] && EqQ[m + p, 0] && GtQ[p, 0] && NeQ[m - n - 1, 0] &&  !IGtQ[n, 0] &&  !(IntegerQ[n + p] && LtQ[n + p +
2, 0]) && RationalQ[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 {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {(5 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx}{g} \\ & = \frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}-\frac {(15 c d (c d f-a e g)) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} \sqrt {f+g x}} \, dx}{4 g^2} \\ & = -\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {\left (15 c d (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 g^3} \\ & = -\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {\left (15 c d (c d f-a e g)^2 \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}{8 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {\left (15 (c d f-a e g)^2 \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 )}{4 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {\left (15 (c d f-a e g)^2 \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 )}{4 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {15 \sqrt {c} \sqrt {d} (c d f-a e g)^2 \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 )}{4 g^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.68 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {a e+c d x} \left (-8 a^2 e^2 g^2+a c d e g (25 f+9 g x)+c^2 d^2 \left (-15 f^2-5 f g x+2 g^2 x^2\right )\right )+15 \sqrt {c} \sqrt {d} (c d f-a e g)^2 \sqrt {f+g x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{4 g^{7/2} \sqrt {(a e+c d x) (d+e x)} \sqrt {f+g x}} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g*x)^(3/2)),x]

[Out]

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[g]*Sqrt[a*e + c*d*x]*(-8*a^2*e^2*g^2 + a*c*d*e*g*(25*f + 9*g*x) + c^2*d
^2*(-15*f^2 - 5*f*g*x + 2*g^2*x^2)) + 15*Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)^2*Sqrt[f + g*x]*ArcTanh[(Sqrt[c]*Sqrt
[d]*Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])]))/(4*g^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(624\) vs. \(2(248)=496\).

Time = 0.57 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.13

method	result	size
default	\(\frac {\left (15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} g^{3} x -30 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e f \,g^{2} x +15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{3} d^{3} f^{2} g x +15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} f \,g^{2}-30 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e \,f^{2} g +15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{3} d^{3} f^{3}+4 c^{2} d^{2} g^{2} x^{2} \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}+18 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, a c d e \,g^{2} x -10 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, c^{2} d^{2} f g x -16 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a^{2} e^{2} g^{2}+50 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a c d e f g -30 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, c^{2} d^{2} f^{2}\right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{8 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, g^{3} \sqrt {g x +f}\, \sqrt {e x +d}}\)	\(625\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/8*(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^2*c*d*e^2*
g^3*x-30*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*g^2*x+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^2*g*x+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^2*c*d*e
^2*f*g^2-30*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+4*c^2*d^2*g^2*x^2*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)+18*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)*
a*c*d*e*g^2*x-10*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)*c^2*d^2*f*g*x-16*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g
)^(1/2)*a^2*e^2*g^2+50*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a*c*d*e*f*g-30*((g*x+f)*(c*d*x+a*e))^(1/2)*(c
*d*g)^(1/2)*c^2*d^2*f^2)*((c*d*x+a*e)*(e*x+d))^(1/2)/((g*x+f)*(c*d*x+a*e))^(1/2)/(c*d*g)^(1/2)/g^3/(g*x+f)^(1/
2)/(e*x+d)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.82 (sec) , antiderivative size = 915, normalized size of antiderivative = 3.11 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\left [\frac {4 \, {\left (2 \, c^{2} d^{2} g^{2} x^{2} - 15 \, c^{2} d^{2} f^{2} + 25 \, a c d e f g - 8 \, a^{2} e^{2} g^{2} - {\left (5 \, c^{2} d^{2} f g - 9 \, a c d e g^{2}\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^{2} d^{3} f^{3} - 2 \, a c d^{2} e f^{2} g + a^{2} d e^{2} f g^{2} + {\left (c^{2} d^{2} e f^{2} g - 2 \, a c d e^{2} f g^{2} + a^{2} e^{3} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{3} + a^{2} d e^{2} g^{3} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{2} g - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f g^{2}\right )} x\right )} \sqrt {\frac {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 \, {\left (2 \, c d g^{2} x + c d f g + a e g^{2}\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} \sqrt {\frac {c d}{g}} + 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 )}{16 \, {\left (e g^{4} x^{2} + d f g^{3} + {\left (e f g^{3} + d g^{4}\right )} x\right )}}, \frac {2 \, {\left (2 \, c^{2} d^{2} g^{2} x^{2} - 15 \, c^{2} d^{2} f^{2} + 25 \, a c d e f g - 8 \, a^{2} e^{2} g^{2} - {\left (5 \, c^{2} d^{2} f g - 9 \, a c d e g^{2}\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^{2} d^{3} f^{3} - 2 \, a c d^{2} e f^{2} g + a^{2} d e^{2} f g^{2} + {\left (c^{2} d^{2} e f^{2} g - 2 \, a c d e^{2} f g^{2} + a^{2} e^{3} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{3} + a^{2} d e^{2} g^{3} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{2} g - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f g^{2}\right )} x\right )} \sqrt {-\frac {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 {e x + d} \sqrt {g x + f} \sqrt {-\frac {c d}{g}} g}{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 )}{8 \, {\left (e g^{4} x^{2} + d f g^{3} + {\left (e f g^{3} + d g^{4}\right )} x\right )}}\right ] \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(4*(2*c^2*d^2*g^2*x^2 - 15*c^2*d^2*f^2 + 25*a*c*d*e*f*g - 8*a^2*e^2*g^2 - (5*c^2*d^2*f*g - 9*a*c*d*e*g^2
)*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^2*d^3*f^3 - 2*a*c*d^2*e*f
^2*g + a^2*d*e^2*f*g^2 + (c^2*d^2*e*f^2*g - 2*a*c*d*e^2*f*g^2 + a^2*e^3*g^3)*x^2 + (c^2*d^2*e*f^3 + a^2*d*e^2*
g^3 + (c^2*d^3 - 2*a*c*d*e^2)*f^2*g - (2*a*c*d^2*e - a^2*e^3)*f*g^2)*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*(2*c*d*g^2*x + c*d*f*g + a*e*g^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)*sqrt(c*d/g) + 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)))/(e*g^4*x^2
+ d*f*g^3 + (e*f*g^3 + d*g^4)*x), 1/8*(2*(2*c^2*d^2*g^2*x^2 - 15*c^2*d^2*f^2 + 25*a*c*d*e*f*g - 8*a^2*e^2*g^2
- (5*c^2*d^2*f*g - 9*a*c*d*e*g^2)*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^2*d^3*f^3 - 2*a*c*d^2*e*f^2*g + a^2*d*e^2*f*g^2 + (c^2*d^2*e*f^2*g - 2*a*c*d*e^2*f*g^2 + a^2*e^3*g^3)*x
^2 + (c^2*d^2*e*f^3 + a^2*d*e^2*g^3 + (c^2*d^3 - 2*a*c*d*e^2)*f^2*g - (2*a*c*d^2*e - a^2*e^3)*f*g^2)*x)*sqrt(-
c*d/g)*arctan(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)*sqrt(-c*d/g)*g/(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)))/(e*g^4*x^2 + d*f*g^3 + (e*f*g^3 + d*g^4)*x)
]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2)/(g*x+f)**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^(3/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 894 vs. \(2 (248) = 496\).

Time = 0.77 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.04 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} {\left ({\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} {\left (\frac {2 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} {\left | c \right |} {\left | d \right |}}{e^{4} g} - \frac {5 \, {\left (c d e^{2} f g^{3} {\left | c \right |} {\left | d \right |} - a e^{3} g^{4} {\left | c \right |} {\left | d \right |}\right )}}{e^{4} g^{5}}\right )} - \frac {15 \, {\left (c^{2} d^{2} e^{4} f^{2} g^{2} {\left | c \right |} {\left | d \right |} - 2 \, a c d e^{5} f g^{3} {\left | c \right |} {\left | d \right |} + a^{2} e^{6} g^{4} {\left | c \right |} {\left | d \right |}\right )}}{e^{4} g^{5}}\right )}}{4 \, \sqrt {c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g}} - \frac {15 \, {\left (c^{2} d^{2} f^{2} {\left | c \right |} {\left | d \right |} - 2 \, a c d e f g {\left | c \right |} {\left | d \right |} + a^{2} e^{2} g^{2} {\left | c \right |} {\left | d \right |}\right )} \log \left ({\left | -\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} \sqrt {c d g} + \sqrt {c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g} \right |}\right )}{4 \, \sqrt {c d g} g^{3}} + \frac {15 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} c^{2} d^{2} e^{2} f^{2} {\left | c \right |} {\left | d \right |} \log \left ({\left | -\sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} \right |}\right ) - 30 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a c d e^{3} f g {\left | c \right |} {\left | d \right |} \log \left ({\left | -\sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} \right |}\right ) + 15 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} a^{2} e^{4} g^{2} {\left | c \right |} {\left | d \right |} \log \left ({\left | -\sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} + \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} \right |}\right ) + 15 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} c^{2} d^{2} e^{2} f^{2} {\left | c \right |} {\left | d \right |} - 5 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} c^{2} d^{3} e f g {\left | c \right |} {\left | d \right |} - 25 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} a c d e^{3} f g {\left | c \right |} {\left | d \right |} - 2 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} c^{2} d^{4} g^{2} {\left | c \right |} {\left | d \right |} + 9 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} a c d^{2} e^{2} g^{2} {\left | c \right |} {\left | d \right |} + 8 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d g} a^{2} e^{4} g^{2} {\left | c \right |} {\left | d \right |}}{4 \, \sqrt {c^{2} d^{2} e^{2} f - c^{2} d^{3} e g} \sqrt {c d g} e^{2} g^{3}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(3/2),x, algorithm="giac")

[Out]

1/4*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*(((e*x + d)*c*d*e - c*d^2*e + a*e^3)*(2*((e*x + d)*c*d*e - c*d^2*e
+ a*e^3)*abs(c)*abs(d)/(e^4*g) - 5*(c*d*e^2*f*g^3*abs(c)*abs(d) - a*e^3*g^4*abs(c)*abs(d))/(e^4*g^5)) - 15*(c
^2*d^2*e^4*f^2*g^2*abs(c)*abs(d) - 2*a*c*d*e^5*f*g^3*abs(c)*abs(d) + a^2*e^6*g^4*abs(c)*abs(d))/(e^4*g^5))/sqr
t(c^2*d^2*e^2*f - a*c*d*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d*g) - 15/4*(c^2*d^2*f^2*abs(c)*abs(d) -
2*a*c*d*e*f*g*abs(c)*abs(d) + a^2*e^2*g^2*abs(c)*abs(d))*log(abs(-sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*sqr
t(c*d*g) + sqrt(c^2*d^2*e^2*f - a*c*d*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d*g)))/(sqrt(c*d*g)*g^3) +
1/4*(15*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*c^2*d^2*e^2*f^2*abs(c)*abs(d)*log(abs(-sqrt(-c*d^2*e + a*e^3)*sqrt(
c*d*g) + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g))) - 30*sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g)*a*c*d*e^3*f*g*abs(c)*abs(d
)*log(abs(-sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*g) + sqrt(c^2*d^2*e^2*f - c^2*d^3*e*g))) + 15*sqrt(c^2*d^2*e^2*f -
c^2*d^3*e*g)*a^2*e^4*g^2*abs(c)*abs(d)*log(abs(-sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*g) + sqrt(c^2*d^2*e^2*f - c^2*
d^3*e*g))) + 15*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*g)*c^2*d^2*e^2*f^2*abs(c)*abs(d) - 5*sqrt(-c*d^2*e + a*e^3)*sq
rt(c*d*g)*c^2*d^3*e*f*g*abs(c)*abs(d) - 25*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*g)*a*c*d*e^3*f*g*abs(c)*abs(d) - 2*
sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*g)*c^2*d^4*g^2*abs(c)*abs(d) + 9*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*g)*a*c*d^2*e^
2*g^2*abs(c)*abs(d) + 8*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*g)*a^2*e^4*g^2*abs(c)*abs(d))/(sqrt(c^2*d^2*e^2*f - c^
2*d^3*e*g)*sqrt(c*d*g)*e^2*g^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (f+g\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^(3/2)*(d + e*x)^(5/2)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^(3/2)*(d + e*x)^(5/2)), x)