\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^6} \, dx\) [710]

Optimal result

Integrand size = 46, antiderivative size = 393 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^6} \, dx=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x} (f+g x)^3}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2} (f+g x)^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5}+\frac {3 c^5 d^5 \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{128 g^{7/2} (c d f-a e g)^{5/2}} \]

[Out]

-1/8*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g^2/(e*x+d)^(3/2)/(g*x+f)^4-1/5*(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)^5+3/128*c^5*d^5*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^(7/2)/(-a*e*g+c*d*f)^(5/2)-1/16*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)/g^3/(g*x+f)^3/(e*x+d)^(1/2)+1/64*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)/(g*x+f
)^2/(e*x+d)^(1/2)+3/128*c^4*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)^2/(g*x+f)/(e*x+d)^(
1/2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {876, 886, 888, 211} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^6} \, dx=\frac {3 c^5 d^5 \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{128 g^{7/2} (c d f-a e g)^{5/2}}+\frac {3 c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 g^3 \sqrt {d+e x} (f+g x) (c d f-a e g)^2}+\frac {c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g^3 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}-\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16 g^3 \sqrt {d+e x} (f+g x)^3}-\frac {c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2} (f+g x)^4}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5} \]

[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)^6),x]

[Out]

-1/16*(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g^3*Sqrt[d + e*x]*(f + g*x)^3) + (c^3*d^3*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*g^3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^2) + (3*c^4*d^4*Sqrt[a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*g^3*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)) - (c*d*(a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*g^2*(d + e*x)^(3/2)*(f + g*x)^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)
^(5/2)/(5*g*(d + e*x)^(5/2)*(f + g*x)^5) + (3*c^5*d^5*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])])/(128*g^(7/2)*(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*} \text {integral}& = -\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5}+\frac {(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} (f+g x)^5} \, dx}{2 g} \\ & = -\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2} (f+g x)^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5}+\frac {\left (3 c^2 d^2\right ) \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}{16 g^2} \\ & = -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x} (f+g x)^3}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2} (f+g x)^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5}+\frac {\left (c^3 d^3\right ) \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}{32 g^3} \\ & = -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x} (f+g x)^3}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2} (f+g x)^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5}+\frac {\left (3 c^4 d^4\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}{128 g^3 (c d f-a e g)} \\ & = -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x} (f+g x)^3}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2} (f+g x)^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5}+\frac {\left (3 c^5 d^5\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}{256 g^3 (c d f-a e g)^2} \\ & = -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x} (f+g x)^3}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2} (f+g x)^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5}+\frac {\left (3 c^5 d^5 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 )}{128 g^3 (c d f-a e g)^2} \\ & = -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x} (f+g x)^3}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2} (f+g x)^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2} (f+g x)^5}+\frac {3 c^5 d^5 \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 )}{128 g^{7/2} (c d f-a e g)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^6} \, dx=\frac {c^5 d^5 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {g} \left (-128 a^4 e^4 g^4+16 a^3 c d e^3 g^3 (11 f-21 g x)-8 a^2 c^2 d^2 e^2 g^2 \left (f^2-64 f g x+31 g^2 x^2\right )-2 a c^3 d^3 e g \left (5 f^3+23 f^2 g x-233 f g^2 x^2+5 g^3 x^3\right )+c^4 d^4 \left (-15 f^4-70 f^3 g x-128 f^2 g^2 x^2+70 f g^3 x^3+15 g^4 x^4\right )\right )}{c^5 d^5 (c d f-a e g)^2 (a e+c d x)^2 (f+g x)^5}+\frac {15 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{5/2} (a e+c d x)^{5/2}}\right )}{640 g^{7/2} (d+e x)^{5/2}} \]

[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)^6),x]

[Out]

(c^5*d^5*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[g]*(-128*a^4*e^4*g^4 + 16*a^3*c*d*e^3*g^3*(11*f - 21*g*x) - 8*
a^2*c^2*d^2*e^2*g^2*(f^2 - 64*f*g*x + 31*g^2*x^2) - 2*a*c^3*d^3*e*g*(5*f^3 + 23*f^2*g*x - 233*f*g^2*x^2 + 5*g^
3*x^3) + c^4*d^4*(-15*f^4 - 70*f^3*g*x - 128*f^2*g^2*x^2 + 70*f*g^3*x^3 + 15*g^4*x^4)))/(c^5*d^5*(c*d*f - a*e*
g)^2*(a*e + c*d*x)^2*(f + g*x)^5) + (15*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/((c*d*f - a*e
*g)^(5/2)*(a*e + c*d*x)^(5/2))))/(640*g^(7/2)*(d + e*x)^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(913\) vs. \(2(349)=698\).

Time = 0.57 (sec) , antiderivative size = 914, normalized size of antiderivative = 2.33

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} g^{5} x^{5}+75 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} f \,g^{4} x^{4}+150 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} f^{2} g^{3} x^{3}+150 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} f^{3} g^{2} x^{2}-15 c^{4} d^{4} g^{4} x^{4} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+75 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} f^{4} g x +10 a \,c^{3} d^{3} e \,g^{4} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-70 c^{4} d^{4} f \,g^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{5} d^{5} f^{5}+248 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-466 a \,c^{3} d^{3} e f \,g^{3} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+128 c^{4} d^{4} f^{2} g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+336 a^{3} c d \,e^{3} g^{4} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-512 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+46 a \,c^{3} d^{3} e \,f^{2} g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+70 c^{4} d^{4} f^{3} g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+128 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{4} e^{4} g^{4}-176 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{3} c d \,e^{3} f \,g^{3}+8 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}+10 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{3} d^{3} e \,f^{3} g +15 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{4} d^{4} f^{4}\right )}{640 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{5} g^{3} \left (a e g -c d f \right )^{2} \sqrt {c d x +a e}}\)	\(914\)

[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)^6,x,method=_RETURNVERBOSE)

[Out]

-1/640*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^5*d^5*g^5*x^5+75
*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^5*d^5*f*g^4*x^4+150*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*
g-c*d*f)*g)^(1/2))*c^5*d^5*f^2*g^3*x^3+150*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^5*d^5*f^3*g^
2*x^2-15*c^4*d^4*g^4*x^4*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+75*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*
f)*g)^(1/2))*c^5*d^5*f^4*g*x+10*a*c^3*d^3*e*g^4*x^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-70*c^4*d^4*f*g^3
*x^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^5*d^5
*f^5+248*a^2*c^2*d^2*e^2*g^4*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-466*a*c^3*d^3*e*f*g^3*x^2*(c*d*x+a*
e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+128*c^4*d^4*f^2*g^2*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+336*a^3*c*d
*e^3*g^4*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-512*a^2*c^2*d^2*e^2*f*g^3*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d
*f)*g)^(1/2)+46*a*c^3*d^3*e*f^2*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+70*c^4*d^4*f^3*g*x*(c*d*x+a*e)
^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+128*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^4*e^4*g^4-176*(c*d*x+a*e)^(1/2)
*((a*e*g-c*d*f)*g)^(1/2)*a^3*c*d*e^3*f*g^3+8*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^2*c^2*d^2*e^2*f^2*g^2
+10*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a*c^3*d^3*e*f^3*g+15*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c
^4*d^4*f^4)/(e*x+d)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)^5/g^3/(a*e*g-c*d*f)^2/(c*d*x+a*e)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1354 vs. \(2 (349) = 698\).

Time = 1.37 (sec) , antiderivative size = 2750, normalized size of antiderivative = 7.00 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^6} \, dx=\text {Too large to display} \]

[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)^6,x, algorithm="fricas")

[Out]

[-1/1280*(15*(c^5*d^5*e*g^5*x^6 + c^5*d^6*f^5 + (5*c^5*d^5*e*f*g^4 + c^5*d^6*g^5)*x^5 + 5*(2*c^5*d^5*e*f^2*g^3
+ c^5*d^6*f*g^4)*x^4 + 10*(c^5*d^5*e*f^3*g^2 + c^5*d^6*f^2*g^3)*x^3 + 5*(c^5*d^5*e*f^4*g + 2*c^5*d^6*f^3*g^2)
*x^2 + (c^5*d^5*e*f^5 + 5*c^5*d^6*f^4*g)*x)*sqrt(-c*d*f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g -
(c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*sq
rt(e*x + d))/(e*g*x^2 + d*f + (e*f + d*g)*x)) + 2*(15*c^5*d^5*f^5*g - 5*a*c^4*d^4*e*f^4*g^2 - 2*a^2*c^3*d^3*e^
2*f^3*g^3 - 184*a^3*c^2*d^2*e^3*f^2*g^4 + 304*a^4*c*d*e^4*f*g^5 - 128*a^5*e^5*g^6 - 15*(c^5*d^5*f*g^5 - a*c^4*
d^4*e*g^6)*x^4 - 10*(7*c^5*d^5*f^2*g^4 - 8*a*c^4*d^4*e*f*g^5 + a^2*c^3*d^3*e^2*g^6)*x^3 + 2*(64*c^5*d^5*f^3*g^
3 - 297*a*c^4*d^4*e*f^2*g^4 + 357*a^2*c^3*d^3*e^2*f*g^5 - 124*a^3*c^2*d^2*e^3*g^6)*x^2 + 2*(35*c^5*d^5*f^4*g^2
- 12*a*c^4*d^4*e*f^3*g^3 - 279*a^2*c^3*d^3*e^2*f^2*g^4 + 424*a^3*c^2*d^2*e^3*f*g^5 - 168*a^4*c*d*e^4*g^6)*x)*
sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^4*f^8*g^4 - 3*a*c^2*d^3*e*f^7*g^5 + 3*a^2*c*
d^2*e^2*f^6*g^6 - a^3*d*e^3*f^5*g^7 + (c^3*d^3*e*f^3*g^9 - 3*a*c^2*d^2*e^2*f^2*g^10 + 3*a^2*c*d*e^3*f*g^11 - a
^3*e^4*g^12)*x^6 + (5*c^3*d^3*e*f^4*g^8 - a^3*d*e^3*g^12 + (c^3*d^4 - 15*a*c^2*d^2*e^2)*f^3*g^9 - 3*(a*c^2*d^3
*e - 5*a^2*c*d*e^3)*f^2*g^10 + (3*a^2*c*d^2*e^2 - 5*a^3*e^4)*f*g^11)*x^5 + 5*(2*c^3*d^3*e*f^5*g^7 - a^3*d*e^3*
f*g^11 + (c^3*d^4 - 6*a*c^2*d^2*e^2)*f^4*g^8 - 3*(a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^3*g^9 + (3*a^2*c*d^2*e^2 - 2*
a^3*e^4)*f^2*g^10)*x^4 + 10*(c^3*d^3*e*f^6*g^6 - a^3*d*e^3*f^2*g^10 + (c^3*d^4 - 3*a*c^2*d^2*e^2)*f^5*g^7 - 3*
(a*c^2*d^3*e - a^2*c*d*e^3)*f^4*g^8 + (3*a^2*c*d^2*e^2 - a^3*e^4)*f^3*g^9)*x^3 + 5*(c^3*d^3*e*f^7*g^5 - 2*a^3*
d*e^3*f^3*g^9 + (2*c^3*d^4 - 3*a*c^2*d^2*e^2)*f^6*g^6 - 3*(2*a*c^2*d^3*e - a^2*c*d*e^3)*f^5*g^7 + (6*a^2*c*d^2
*e^2 - a^3*e^4)*f^4*g^8)*x^2 + (c^3*d^3*e*f^8*g^4 - 5*a^3*d*e^3*f^4*g^8 + (5*c^3*d^4 - 3*a*c^2*d^2*e^2)*f^7*g^
5 - 3*(5*a*c^2*d^3*e - a^2*c*d*e^3)*f^6*g^6 + (15*a^2*c*d^2*e^2 - a^3*e^4)*f^5*g^7)*x), -1/640*(15*(c^5*d^5*e*
g^5*x^6 + c^5*d^6*f^5 + (5*c^5*d^5*e*f*g^4 + c^5*d^6*g^5)*x^5 + 5*(2*c^5*d^5*e*f^2*g^3 + c^5*d^6*f*g^4)*x^4 +
10*(c^5*d^5*e*f^3*g^2 + c^5*d^6*f^2*g^3)*x^3 + 5*(c^5*d^5*e*f^4*g + 2*c^5*d^6*f^3*g^2)*x^2 + (c^5*d^5*e*f^5 +
5*c^5*d^6*f^4*g)*x)*sqrt(c*d*f*g - a*e*g^2)*arctan(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*f*g -
a*e*g^2)*sqrt(e*x + d)/(c*d*e*g*x^2 + a*d*e*g + (c*d^2 + a*e^2)*g*x)) + (15*c^5*d^5*f^5*g - 5*a*c^4*d^4*e*f^4*
g^2 - 2*a^2*c^3*d^3*e^2*f^3*g^3 - 184*a^3*c^2*d^2*e^3*f^2*g^4 + 304*a^4*c*d*e^4*f*g^5 - 128*a^5*e^5*g^6 - 15*(
c^5*d^5*f*g^5 - a*c^4*d^4*e*g^6)*x^4 - 10*(7*c^5*d^5*f^2*g^4 - 8*a*c^4*d^4*e*f*g^5 + a^2*c^3*d^3*e^2*g^6)*x^3
+ 2*(64*c^5*d^5*f^3*g^3 - 297*a*c^4*d^4*e*f^2*g^4 + 357*a^2*c^3*d^3*e^2*f*g^5 - 124*a^3*c^2*d^2*e^3*g^6)*x^2 +
2*(35*c^5*d^5*f^4*g^2 - 12*a*c^4*d^4*e*f^3*g^3 - 279*a^2*c^3*d^3*e^2*f^2*g^4 + 424*a^3*c^2*d^2*e^3*f*g^5 - 16
8*a^4*c*d*e^4*g^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^4*f^8*g^4 - 3*a*c^2*d^
3*e*f^7*g^5 + 3*a^2*c*d^2*e^2*f^6*g^6 - a^3*d*e^3*f^5*g^7 + (c^3*d^3*e*f^3*g^9 - 3*a*c^2*d^2*e^2*f^2*g^10 + 3*
a^2*c*d*e^3*f*g^11 - a^3*e^4*g^12)*x^6 + (5*c^3*d^3*e*f^4*g^8 - a^3*d*e^3*g^12 + (c^3*d^4 - 15*a*c^2*d^2*e^2)*
f^3*g^9 - 3*(a*c^2*d^3*e - 5*a^2*c*d*e^3)*f^2*g^10 + (3*a^2*c*d^2*e^2 - 5*a^3*e^4)*f*g^11)*x^5 + 5*(2*c^3*d^3*
e*f^5*g^7 - a^3*d*e^3*f*g^11 + (c^3*d^4 - 6*a*c^2*d^2*e^2)*f^4*g^8 - 3*(a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^3*g^9 +
(3*a^2*c*d^2*e^2 - 2*a^3*e^4)*f^2*g^10)*x^4 + 10*(c^3*d^3*e*f^6*g^6 - a^3*d*e^3*f^2*g^10 + (c^3*d^4 - 3*a*c^2
*d^2*e^2)*f^5*g^7 - 3*(a*c^2*d^3*e - a^2*c*d*e^3)*f^4*g^8 + (3*a^2*c*d^2*e^2 - a^3*e^4)*f^3*g^9)*x^3 + 5*(c^3*
d^3*e*f^7*g^5 - 2*a^3*d*e^3*f^3*g^9 + (2*c^3*d^4 - 3*a*c^2*d^2*e^2)*f^6*g^6 - 3*(2*a*c^2*d^3*e - a^2*c*d*e^3)*
f^5*g^7 + (6*a^2*c*d^2*e^2 - a^3*e^4)*f^4*g^8)*x^2 + (c^3*d^3*e*f^8*g^4 - 5*a^3*d*e^3*f^4*g^8 + (5*c^3*d^4 - 3
*a*c^2*d^2*e^2)*f^7*g^5 - 3*(5*a*c^2*d^3*e - a^2*c*d*e^3)*f^6*g^6 + (15*a^2*c*d^2*e^2 - a^3*e^4)*f^5*g^7)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^6} \, dx=\text {Timed out} \]

[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)**6,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^6} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{6}} \,d x } \]

[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)^6,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)^6), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2407 vs. \(2 (349) = 698\).

Time = 2.83 (sec) , antiderivative size = 2407, normalized size of antiderivative = 6.12 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^6} \, dx=\text {Too large to display} \]

[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)^6,x, algorithm="giac")

[Out]

3/128*c^5*d^5*abs(e)*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e))/((c^2*d^2*f
^2*g^3 - 2*a*c*d*e*f*g^4 + a^2*e^2*g^5)*sqrt(c*d*f*g - a*e*g^2)*e) - 1/640*(15*c^5*d^5*e^5*f^5*abs(e)*arctan(s
qrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 75*c^5*d^6*e^4*f^4*g*abs(e)*arctan(sqrt(-c*d^2*e + a*e^
3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 150*c^5*d^7*e^3*f^3*g^2*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f
*g - a*e*g^2)*e)) - 150*c^5*d^8*e^2*f^2*g^3*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)
) + 75*c^5*d^9*e*f*g^4*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 15*c^5*d^10*g^5*a
bs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 15*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g -
a*e*g^2)*c^4*d^4*e^4*f^4*abs(e) + 70*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^4*d^5*e^3*f^3*g*abs(e) -
10*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^3*d^3*e^5*f^3*g*abs(e) - 128*sqrt(-c*d^2*e + a*e^3)*sqr
t(c*d*f*g - a*e*g^2)*c^4*d^6*e^2*f^2*g^2*abs(e) + 46*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^3*d^4*
e^4*f^2*g^2*abs(e) - 8*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c^2*d^2*e^6*f^2*g^2*abs(e) - 70*sqrt
(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^4*d^7*e*f*g^3*abs(e) + 466*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g -
a*e*g^2)*a*c^3*d^5*e^3*f*g^3*abs(e) - 512*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c^2*d^3*e^5*f*g^3
*abs(e) + 176*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^3*c*d*e^7*f*g^3*abs(e) + 15*sqrt(-c*d^2*e + a*e
^3)*sqrt(c*d*f*g - a*e*g^2)*c^4*d^8*g^4*abs(e) + 10*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^3*d^6*e
^2*g^4*abs(e) - 248*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c^2*d^4*e^4*g^4*abs(e) + 336*sqrt(-c*d^
2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^3*c*d^2*e^6*g^4*abs(e) - 128*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*
g^2)*a^4*e^8*g^4*abs(e))/(sqrt(c*d*f*g - a*e*g^2)*c^2*d^2*e^6*f^7*g^3 - 5*sqrt(c*d*f*g - a*e*g^2)*c^2*d^3*e^5*
f^6*g^4 - 2*sqrt(c*d*f*g - a*e*g^2)*a*c*d*e^7*f^6*g^4 + 10*sqrt(c*d*f*g - a*e*g^2)*c^2*d^4*e^4*f^5*g^5 + 10*sq
rt(c*d*f*g - a*e*g^2)*a*c*d^2*e^6*f^5*g^5 + sqrt(c*d*f*g - a*e*g^2)*a^2*e^8*f^5*g^5 - 10*sqrt(c*d*f*g - a*e*g^
2)*c^2*d^5*e^3*f^4*g^6 - 20*sqrt(c*d*f*g - a*e*g^2)*a*c*d^3*e^5*f^4*g^6 - 5*sqrt(c*d*f*g - a*e*g^2)*a^2*d*e^7*
f^4*g^6 + 5*sqrt(c*d*f*g - a*e*g^2)*c^2*d^6*e^2*f^3*g^7 + 20*sqrt(c*d*f*g - a*e*g^2)*a*c*d^4*e^4*f^3*g^7 + 10*
sqrt(c*d*f*g - a*e*g^2)*a^2*d^2*e^6*f^3*g^7 - sqrt(c*d*f*g - a*e*g^2)*c^2*d^7*e*f^2*g^8 - 10*sqrt(c*d*f*g - a*
e*g^2)*a*c*d^5*e^3*f^2*g^8 - 10*sqrt(c*d*f*g - a*e*g^2)*a^2*d^3*e^5*f^2*g^8 + 2*sqrt(c*d*f*g - a*e*g^2)*a*c*d^
6*e^2*f*g^9 + 5*sqrt(c*d*f*g - a*e*g^2)*a^2*d^4*e^4*f*g^9 - sqrt(c*d*f*g - a*e*g^2)*a^2*d^5*e^3*g^10) - 1/640*
(15*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^9*d^9*e^8*f^4*abs(e) - 60*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3
)*a*c^8*d^8*e^9*f^3*g*abs(e) + 90*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^7*d^7*e^10*f^2*g^2*abs(e) - 60
*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^6*d^6*e^11*f*g^3*abs(e) + 15*sqrt((e*x + d)*c*d*e - c*d^2*e + a
*e^3)*a^4*c^5*d^5*e^12*g^4*abs(e) + 70*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^8*d^8*e^6*f^3*g*abs(e) - 21
0*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^7*d^7*e^7*f^2*g^2*abs(e) + 210*((e*x + d)*c*d*e - c*d^2*e + a*
e^3)^(3/2)*a^2*c^6*d^6*e^8*f*g^3*abs(e) - 70*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*c^5*d^5*e^9*g^4*abs
(e) + 128*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^7*d^7*e^4*f^2*g^2*abs(e) - 256*((e*x + d)*c*d*e - c*d^2*
e + a*e^3)^(5/2)*a*c^6*d^6*e^5*f*g^3*abs(e) + 128*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*c^5*d^5*e^6*g^
4*abs(e) - 70*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^6*d^6*e^2*f*g^3*abs(e) + 70*((e*x + d)*c*d*e - c*d^2
*e + a*e^3)^(7/2)*a*c^5*d^5*e^3*g^4*abs(e) - 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*c^5*d^5*g^4*abs(e))/
((c^2*d^2*f^2*g^3 - 2*a*c*d*e*f*g^4 + a^2*e^2*g^5)*(c*d*e^2*f - a*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*
g)^5)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^6} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (f+g\,x\right )}^6\,{\left (d+e\,x\right )}^{5/2}} \,d x \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^6*(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)^6*(d + e*x)^(5/2)), x)