\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^5} \, dx\) [688]

Optimal result

Integrand size = 46, antiderivative size = 347 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^5} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {5 c^4 d^4 \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{64 g^{3/2} (c d f-a e g)^{7/2}} \]

[Out]

5/64*c^4*d^4*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)^(7/2)-1/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g/(g*x+f)^4/(e*x+d)^(1/2)+1/24*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)^3/(e*x+d)^(1/2)+5/96*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)/g/(-a*e*g+c*d*f)^2/(g*x+f)^2/(e*x+d)^(1/2)+5/64*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)/g/(-a*e*g+c*d*f)^3/(g*x+f)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {876, 886, 888, 211} \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^5} \, dx=\frac {5 c^4 d^4 \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{64 g^{3/2} (c d f-a e g)^{7/2}}+\frac {5 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g \sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4} \]

[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)^5),x]

[Out]

-1/4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(g*Sqrt[d + e*x]*(f + g*x)^4) + (c*d*Sqrt[a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2])/(24*g*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) + (5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2
)*x + c*d*e*x^2])/(96*g*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^2) + (5*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(64*g*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) + (5*c^4*d^4*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])])/(64*g^(3/2)*(c*d*f - a*e*g)^(7/2))

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 {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {(c d) \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 g} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {\left (5 c^2 d^2\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}{48 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}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {\left (5 c^3 d^3\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}{64 g (c d f-a e g)^2} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {\left (5 c^4 d^4\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}{128 g (c d f-a e g)^3} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {\left (5 c^4 d^4 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 )}{64 g (c d f-a e g)^3} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {5 c^4 d^4 \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 )}{64 g^{3/2} (c d f-a e g)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^5} \, dx=\frac {c^4 d^4 \sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {g} \left (48 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (-17 f+g x)-2 a c^2 d^2 e g \left (-59 f^2+18 f g x+5 g^2 x^2\right )+c^3 d^3 \left (-15 f^3+73 f^2 g x+55 f g^2 x^2+15 g^3 x^3\right )\right )}{c^4 d^4 (c d f-a e g)^3 (f+g x)^4}+\frac {15 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{7/2} \sqrt {a e+c d x}}\right )}{192 g^{3/2} \sqrt {d+e x}} \]

[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)^5),x]

[Out]

(c^4*d^4*Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[g]*(48*a^3*e^3*g^3 + 8*a^2*c*d*e^2*g^2*(-17*f + g*x) - 2*a*c^2*d
^2*e*g*(-59*f^2 + 18*f*g*x + 5*g^2*x^2) + c^3*d^3*(-15*f^3 + 73*f^2*g*x + 55*f*g^2*x^2 + 15*g^3*x^3)))/(c^4*d^
4*(c*d*f - a*e*g)^3*(f + g*x)^4) + (15*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/((c*d*f - a*e*
g)^(7/2)*Sqrt[a*e + c*d*x])))/(192*g^(3/2)*Sqrt[d + e*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(685\) vs. \(2(309)=618\).

Time = 0.54 (sec) , antiderivative size = 686, normalized size of antiderivative = 1.98

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} g^{4} x^{4}+60 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f \,g^{3} x^{3}+90 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{2} g^{2} x^{2}+60 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{3} g x -15 c^{3} d^{3} g^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{4} d^{4} f^{4}+10 a \,c^{2} d^{2} e \,g^{3} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-55 c^{3} d^{3} f \,g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-8 a^{2} c d \,e^{2} g^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+36 a \,c^{2} d^{2} e f \,g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-73 c^{3} d^{3} f^{2} g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-48 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{3} e^{3} g^{3}+136 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c d \,e^{2} f \,g^{2}-118 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e \,f^{2} g +15 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f^{3}\right )}{192 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{4} g \left (a e g -c d f \right ) \left (a^{2} e^{2} g^{2}-2 a c d e f g +c^{2} d^{2} f^{2}\right ) \sqrt {c d x +a e}}\)	\(686\)

[In]

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

[Out]

1/192*((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^4*d^4*g^4*x^4+60*
arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^4*d^4*f*g^3*x^3+90*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-
c*d*f)*g)^(1/2))*c^4*d^4*f^2*g^2*x^2+60*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^4*d^4*f^3*g*x-1
5*c^3*d^3*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^4*d^4*f^4+10*a*c^2*d^2*e*g^3*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-55*c^3*d^3*f*g^2*x^2*(c*d*x
+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-8*a^2*c*d*e^2*g^3*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+36*a*c^2*d^2
*e*f*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-73*c^3*d^3*f^2*g*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1
/2)-48*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^3*e^3*g^3+136*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^2
*c*d*e^2*f*g^2-118*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a*c^2*d^2*e*f^2*g+15*(c*d*x+a*e)^(1/2)*((a*e*g-c*
d*f)*g)^(1/2)*c^3*d^3*f^3)/(e*x+d)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)^4/g/(a*e*g-c*d*f)/(a^2*e^2*g^2-2*a*c*
d*e*f*g+c^2*d^2*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. 1284 vs. \(2 (309) = 618\).

Time = 1.35 (sec) , antiderivative size = 2610, normalized size of antiderivative = 7.52 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^5} \, dx=\text {Too large to display} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^5/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

[1/384*(15*(c^4*d^4*e*g^4*x^5 + c^4*d^5*f^4 + (4*c^4*d^4*e*f*g^3 + c^4*d^5*g^4)*x^4 + 2*(3*c^4*d^4*e*f^2*g^2 +
2*c^4*d^5*f*g^3)*x^3 + 2*(2*c^4*d^4*e*f^3*g + 3*c^4*d^5*f^2*g^2)*x^2 + (c^4*d^4*e*f^4 + 4*c^4*d^5*f^3*g)*x)*s
qrt(-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)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f + d*g)*x))
- 2*(15*c^4*d^4*f^4*g - 133*a*c^3*d^3*e*f^3*g^2 + 254*a^2*c^2*d^2*e^2*f^2*g^3 - 184*a^3*c*d*e^3*f*g^4 + 48*a^
4*e^4*g^5 - 15*(c^4*d^4*f*g^4 - a*c^3*d^3*e*g^5)*x^3 - 5*(11*c^4*d^4*f^2*g^3 - 13*a*c^3*d^3*e*f*g^4 + 2*a^2*c^
2*d^2*e^2*g^5)*x^2 - (73*c^4*d^4*f^3*g^2 - 109*a*c^3*d^3*e*f^2*g^3 + 44*a^2*c^2*d^2*e^2*f*g^4 - 8*a^3*c*d*e^3*
g^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^5*f^8*g^2 - 4*a*c^3*d^4*e*f^7*g^3 +
6*a^2*c^2*d^3*e^2*f^6*g^4 - 4*a^3*c*d^2*e^3*f^5*g^5 + a^4*d*e^4*f^4*g^6 + (c^4*d^4*e*f^4*g^6 - 4*a*c^3*d^3*e^2
*f^3*g^7 + 6*a^2*c^2*d^2*e^3*f^2*g^8 - 4*a^3*c*d*e^4*f*g^9 + a^4*e^5*g^10)*x^5 + (4*c^4*d^4*e*f^5*g^5 + a^4*d*
e^4*g^10 + (c^4*d^5 - 16*a*c^3*d^3*e^2)*f^4*g^6 - 4*(a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*f^3*g^7 + 2*(3*a^2*c^2*d
^3*e^2 - 8*a^3*c*d*e^4)*f^2*g^8 - 4*(a^3*c*d^2*e^3 - a^4*e^5)*f*g^9)*x^4 + 2*(3*c^4*d^4*e*f^6*g^4 + 2*a^4*d*e^
4*f*g^9 + 2*(c^4*d^5 - 6*a*c^3*d^3*e^2)*f^5*g^5 - 2*(4*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^4*g^6 + 12*(a^2*c^2*
d^3*e^2 - a^3*c*d*e^4)*f^3*g^7 - (8*a^3*c*d^2*e^3 - 3*a^4*e^5)*f^2*g^8)*x^3 + 2*(2*c^4*d^4*e*f^7*g^3 + 3*a^4*d
*e^4*f^2*g^8 + (3*c^4*d^5 - 8*a*c^3*d^3*e^2)*f^6*g^4 - 12*(a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^5*g^5 + 2*(9*a^2*c
^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^4*g^6 - 2*(6*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g^7)*x^2 + (c^4*d^4*e*f^8*g^2 + 4*a^4*
d*e^4*f^3*g^7 + 4*(c^4*d^5 - a*c^3*d^3*e^2)*f^7*g^3 - 2*(8*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^6*g^4 + 4*(6*a^2
*c^2*d^3*e^2 - a^3*c*d*e^4)*f^5*g^5 - (16*a^3*c*d^2*e^3 - a^4*e^5)*f^4*g^6)*x), -1/192*(15*(c^4*d^4*e*g^4*x^5
+ c^4*d^5*f^4 + (4*c^4*d^4*e*f*g^3 + c^4*d^5*g^4)*x^4 + 2*(3*c^4*d^4*e*f^2*g^2 + 2*c^4*d^5*f*g^3)*x^3 + 2*(2*c
^4*d^4*e*f^3*g + 3*c^4*d^5*f^2*g^2)*x^2 + (c^4*d^4*e*f^4 + 4*c^4*d^5*f^3*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^4*d^4*f^4*g - 133*a*c^3*d^3*e*f^3*g^2 + 254*a^2*c^2*d^2*e^2*f^2*g^3 - 184*a^3*c*d*e
^3*f*g^4 + 48*a^4*e^4*g^5 - 15*(c^4*d^4*f*g^4 - a*c^3*d^3*e*g^5)*x^3 - 5*(11*c^4*d^4*f^2*g^3 - 13*a*c^3*d^3*e*
f*g^4 + 2*a^2*c^2*d^2*e^2*g^5)*x^2 - (73*c^4*d^4*f^3*g^2 - 109*a*c^3*d^3*e*f^2*g^3 + 44*a^2*c^2*d^2*e^2*f*g^4
- 8*a^3*c*d*e^3*g^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^5*f^8*g^2 - 4*a*c^3*
d^4*e*f^7*g^3 + 6*a^2*c^2*d^3*e^2*f^6*g^4 - 4*a^3*c*d^2*e^3*f^5*g^5 + a^4*d*e^4*f^4*g^6 + (c^4*d^4*e*f^4*g^6 -
4*a*c^3*d^3*e^2*f^3*g^7 + 6*a^2*c^2*d^2*e^3*f^2*g^8 - 4*a^3*c*d*e^4*f*g^9 + a^4*e^5*g^10)*x^5 + (4*c^4*d^4*e*
f^5*g^5 + a^4*d*e^4*g^10 + (c^4*d^5 - 16*a*c^3*d^3*e^2)*f^4*g^6 - 4*(a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*f^3*g^7
+ 2*(3*a^2*c^2*d^3*e^2 - 8*a^3*c*d*e^4)*f^2*g^8 - 4*(a^3*c*d^2*e^3 - a^4*e^5)*f*g^9)*x^4 + 2*(3*c^4*d^4*e*f^6*
g^4 + 2*a^4*d*e^4*f*g^9 + 2*(c^4*d^5 - 6*a*c^3*d^3*e^2)*f^5*g^5 - 2*(4*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^4*g^
6 + 12*(a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^3*g^7 - (8*a^3*c*d^2*e^3 - 3*a^4*e^5)*f^2*g^8)*x^3 + 2*(2*c^4*d^4*e*f
^7*g^3 + 3*a^4*d*e^4*f^2*g^8 + (3*c^4*d^5 - 8*a*c^3*d^3*e^2)*f^6*g^4 - 12*(a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^5*
g^5 + 2*(9*a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^4*g^6 - 2*(6*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g^7)*x^2 + (c^4*d^4*e*
f^8*g^2 + 4*a^4*d*e^4*f^3*g^7 + 4*(c^4*d^5 - a*c^3*d^3*e^2)*f^7*g^3 - 2*(8*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^
6*g^4 + 4*(6*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^5*g^5 - (16*a^3*c*d^2*e^3 - a^4*e^5)*f^4*g^6)*x)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^5/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1955 vs. \(2 (309) = 618\).

Time = 1.73 (sec) , antiderivative size = 1955, normalized size of antiderivative = 5.63 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^5} \, dx=\text {Too large to display} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(g*x+f)^5/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

1/192*(15*c^4*d^4*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^3*d^3*f^
3*g - 3*a*c^2*d^2*e*f^2*g^2 + 3*a^2*c*d*e^2*f*g^3 - a^3*e^3*g^4)*sqrt(c*d*f*g - a*e*g^2)) - (15*c^4*d^4*e^5*f^
4*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 60*c^4*d^5*e^4*f^3*g*arctan(sqrt(-c*d^2*e + a
*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 90*c^4*d^6*e^3*f^2*g^2*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g -
a*e*g^2)*e)) - 60*c^4*d^7*e^2*f*g^3*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 15*c^4*d^8*
e*g^4*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^3*d^3*e^4*f^3 - 73*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^3*d^4*e^3*f^2*g + 118*sqrt(-c*d
^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^2*e^5*f^2*g + 55*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*
c^3*d^5*e^2*f*g^2 + 36*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^3*e^4*f*g^2 - 136*sqrt(-c*d^2*e
+ a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d*e^6*f*g^2 - 15*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^3*d^6
*e*g^3 - 10*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^4*e^3*g^3 - 8*sqrt(-c*d^2*e + a*e^3)*sqrt(c
*d*f*g - a*e*g^2)*a^2*c*d^2*e^5*g^3 + 48*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^3*e^7*g^3)/(sqrt(c*d
*f*g - a*e*g^2)*c^3*d^3*e^4*f^7*g - 4*sqrt(c*d*f*g - a*e*g^2)*c^3*d^4*e^3*f^6*g^2 - 3*sqrt(c*d*f*g - a*e*g^2)*
a*c^2*d^2*e^5*f^6*g^2 + 6*sqrt(c*d*f*g - a*e*g^2)*c^3*d^5*e^2*f^5*g^3 + 12*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^3*e
^4*f^5*g^3 + 3*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d*e^6*f^5*g^3 - 4*sqrt(c*d*f*g - a*e*g^2)*c^3*d^6*e*f^4*g^4 - 18*
sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^4*e^3*f^4*g^4 - 12*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^2*e^5*f^4*g^4 - sqrt(c*d*f*
g - a*e*g^2)*a^3*e^7*f^4*g^4 + sqrt(c*d*f*g - a*e*g^2)*c^3*d^7*f^3*g^5 + 12*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^5*
e^2*f^3*g^5 + 18*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^3*e^4*f^3*g^5 + 4*sqrt(c*d*f*g - a*e*g^2)*a^3*d*e^6*f^3*g^5 -
3*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^6*e*f^2*g^6 - 12*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^4*e^3*f^2*g^6 - 6*sqrt(c*d
*f*g - a*e*g^2)*a^3*d^2*e^5*f^2*g^6 + 3*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^5*e^2*f*g^7 + 4*sqrt(c*d*f*g - a*e*g^2
)*a^3*d^3*e^4*f*g^7 - sqrt(c*d*f*g - a*e*g^2)*a^3*d^4*e^3*g^8) - (15*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c
^7*d^7*e^8*f^3 - 45*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^6*d^6*e^9*f^2*g + 45*sqrt((e*x + d)*c*d*e - c*
d^2*e + a*e^3)*a^2*c^5*d^5*e^10*f*g^2 - 15*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^4*d^4*e^11*g^3 - 73*(
(e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^6*d^6*e^6*f^2*g + 146*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c
^5*d^5*e^7*f*g^2 - 73*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^4*d^4*e^8*g^3 - 55*((e*x + d)*c*d*e - c*
d^2*e + a*e^3)^(5/2)*c^5*d^5*e^4*f*g^2 + 55*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^4*d^4*e^5*g^3 - 15*(
(e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^4*d^4*e^2*g^3)/((c^3*d^3*f^3*g - 3*a*c^2*d^2*e*f^2*g^2 + 3*a^2*c*d*
e^2*f*g^3 - a^3*e^3*g^4)*(c*d*e^2*f - a*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g)^4))*abs(e)/e^2

Mupad [F(-1)]

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

[In]

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