\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx\) [706]

Optimal result

Integrand size = 46, antiderivative size = 235 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx=-\frac {5 c d (c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^3 \sqrt {d+e x}}+\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac {5 c d (c d f-a e g)^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{g^{7/2}} \]

[Out]

5/3*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)-(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*c*d*(-a*e*g+c*d*f)^(3/2)*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)-5*c*d*(-a*e*g+c*d*f)*(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 = 0.23 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {876, 878, 888, 211} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx=\frac {5 c d (c d f-a e g)^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{g^{7/2}}-\frac {5 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{g^3 \sqrt {d+e x}}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}} \]

[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)^2),x]

[Out]

(-5*c*d*(c*d*f - a*e*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g^3*Sqrt[d + e*x]) + (5*c*d*(a*d*e + (c*
d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g^2*(d + e*x)^(3/2)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(g*
(d + e*x)^(5/2)*(f + g*x)) + (5*c*d*(c*d*f - a*e*g)^(3/2)*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])])/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 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 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}}{g (d+e x)^{5/2} (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} (f+g x)} \, dx}{2 g} \\ & = \frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}}-\frac {\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} (f+g x)}-\frac {(5 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} (f+g x)} \, dx}{2 g^2} \\ & = -\frac {5 c d (c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^3 \sqrt {d+e x}}+\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}}-\frac {\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} (f+g x)}+\frac {\left (5 c d (c d f-a e g)^2\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}{2 g^3} \\ & = -\frac {5 c d (c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^3 \sqrt {d+e x}}+\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}}-\frac {\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} (f+g x)}+\frac {\left (5 c d e^2 (c d f-a e g)^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 )}{g^3} \\ & = -\frac {5 c d (c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^3 \sqrt {d+e x}}+\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}}-\frac {\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} (f+g x)}+\frac {5 c d (c d f-a e g)^{3/2} \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 )}{g^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {a e+c d x} \left (-3 a^2 e^2 g^2+2 a c d e g (10 f+7 g x)+c^2 d^2 \left (-15 f^2-10 f g x+2 g^2 x^2\right )\right )+15 c d (c d f-a e g)^{3/2} (f+g x) \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{3 g^{7/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)} \]

[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)^2),x]

[Out]

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[g]*Sqrt[a*e + c*d*x]*(-3*a^2*e^2*g^2 + 2*a*c*d*e*g*(10*f + 7*g*x) + c^2
*d^2*(-15*f^2 - 10*f*g*x + 2*g^2*x^2)) + 15*c*d*(c*d*f - a*e*g)^(3/2)*(f + g*x)*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d
*x])/Sqrt[c*d*f - a*e*g]]))/(3*g^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(512\) vs. \(2(209)=418\).

Time = 0.60 (sec) , antiderivative size = 513, normalized size of antiderivative = 2.18

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a^{2} c d \,e^{2} g^{3} x -30 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e f \,g^{2} x +15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{2} g x +15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a^{2} c d \,e^{2} f \,g^{2}-30 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e \,f^{2} g +15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{3}-2 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} g^{2} x^{2}-14 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d e \,g^{2} x +10 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f g x +3 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} e^{2} g^{2}-20 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d e f g +15 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f^{2}\right )}{3 \sqrt {e x +d}\, \sqrt {c d x +a e}\, g^{3} \left (g x +f \right ) \sqrt {\left (a e g -c d f \right ) g}}\)	\(513\)

[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)^2,x,method=_RETURNVERBOSE)

[Out]

-1/3*((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))*a^2*c*d*e^2*g^3*x-30
*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c^2*d^2*e*f*g^2*x+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e
*g-c*d*f)*g)^(1/2))*c^3*d^3*f^2*g*x+15*arctanh(g*(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-
30*arctanh(g*(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*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e
*g-c*d*f)*g)^(1/2))*c^3*d^3*f^3-2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*g^2*x^2-14*(c*d*x+a*e)^(1/
2)*((a*e*g-c*d*f)*g)^(1/2)*a*c*d*e*g^2*x+10*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*f*g*x+3*(c*d*x+a
*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^2*e^2*g^2-20*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a*c*d*e*f*g+15*(c*d
*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*f^2)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/g^3/(g*x+f)/((a*e*g-c*d*f)*
g)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.86 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx=\left [-\frac {15 \, {\left (c^{2} d^{3} f^{2} - a c d^{2} e f g + {\left (c^{2} d^{2} e f g - a c d e^{2} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} - a c d^{2} e g^{2} + {\left (c^{2} d^{3} - a c d e^{2}\right )} f g\right )} x\right )} \sqrt {-\frac {c d f - a e g}{g}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} g \sqrt {-\frac {c d f - a e g}{g}} - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right ) - 2 \, {\left (2 \, c^{2} d^{2} g^{2} x^{2} - 15 \, c^{2} d^{2} f^{2} + 20 \, a c d e f g - 3 \, a^{2} e^{2} g^{2} - 2 \, {\left (5 \, c^{2} d^{2} f g - 7 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{6 \, {\left (e g^{4} x^{2} + d f g^{3} + {\left (e f g^{3} + d g^{4}\right )} x\right )}}, -\frac {15 \, {\left (c^{2} d^{3} f^{2} - a c d^{2} e f g + {\left (c^{2} d^{2} e f g - a c d e^{2} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} - a c d^{2} e g^{2} + {\left (c^{2} d^{3} - a c d e^{2}\right )} f g\right )} x\right )} \sqrt {\frac {c d f - a e g}{g}} \arctan \left (\frac {\sqrt {e x + d} \sqrt {\frac {c d f - a e g}{g}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\right ) - {\left (2 \, c^{2} d^{2} g^{2} x^{2} - 15 \, c^{2} d^{2} f^{2} + 20 \, a c d e f g - 3 \, a^{2} e^{2} g^{2} - 2 \, {\left (5 \, c^{2} d^{2} f g - 7 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3 \, {\left (e g^{4} x^{2} + d f g^{3} + {\left (e f g^{3} + d g^{4}\right )} x\right )}}\right ] \]

[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)^2,x, algorithm="fricas")

[Out]

[-1/6*(15*(c^2*d^3*f^2 - a*c*d^2*e*f*g + (c^2*d^2*e*f*g - a*c*d*e^2*g^2)*x^2 + (c^2*d^2*e*f^2 - a*c*d^2*e*g^2
+ (c^2*d^3 - a*c*d*e^2)*f*g)*x)*sqrt(-(c*d*f - a*e*g)/g)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - 2*sqrt(c*d*
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*g*sqrt(-(c*d*f - a*e*g)/g) - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*
x)/(e*g*x^2 + d*f + (e*f + d*g)*x)) - 2*(2*c^2*d^2*g^2*x^2 - 15*c^2*d^2*f^2 + 20*a*c*d*e*f*g - 3*a^2*e^2*g^2 -
2*(5*c^2*d^2*f*g - 7*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))/(e*g^4*x^2 +
d*f*g^3 + (e*f*g^3 + d*g^4)*x), -1/3*(15*(c^2*d^3*f^2 - a*c*d^2*e*f*g + (c^2*d^2*e*f*g - a*c*d*e^2*g^2)*x^2 +
(c^2*d^2*e*f^2 - a*c*d^2*e*g^2 + (c^2*d^3 - a*c*d*e^2)*f*g)*x)*sqrt((c*d*f - a*e*g)/g)*arctan(sqrt(e*x + d)*sq
rt((c*d*f - a*e*g)/g)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)) - (2*c^2*d^2*g^2*x^2 - 15*c^2*d^2*f^2 + 20*
a*c*d*e*f*g - 3*a^2*e^2*g^2 - 2*(5*c^2*d^2*f*g - 7*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))/(e*g^4*x^2 + d*f*g^3 + (e*f*g^3 + d*g^4)*x)]

Sympy [F(-1)]

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

[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)**2,x)

[Out]

Timed out

Maxima [F]

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

[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)^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)^2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1025 vs. \(2 (209) = 418\).

Time = 0.60 (sec) , antiderivative size = 1025, normalized size of antiderivative = 4.36 \[ \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)^2} \, dx=-\frac {15 \, c^{3} d^{3} e^{3} f^{3} {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 15 \, c^{3} d^{4} e^{2} f^{2} g {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 30 \, a c^{2} d^{2} e^{4} f^{2} g {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) + 30 \, a c^{2} d^{3} e^{3} f g^{2} {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) + 15 \, a^{2} c d e^{5} f g^{2} {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 15 \, a^{2} c d^{2} e^{4} g^{3} {\left | e \right |} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 15 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c^{2} d^{2} e^{2} f^{2} {\left | e \right |} + 10 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c^{2} d^{3} e f g {\left | e \right |} + 20 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a c d e^{3} f g {\left | e \right |} + 2 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c^{2} d^{4} g^{2} {\left | e \right |} - 14 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a c d^{2} e^{2} g^{2} {\left | e \right |} - 3 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a^{2} e^{4} g^{2} {\left | e \right |}}{3 \, {\left (\sqrt {c d f g - a e g^{2}} e^{4} f g^{3} - \sqrt {c d f g - a e g^{2}} d e^{3} g^{4}\right )}} + \frac {5 \, {\left (c^{3} d^{3} f^{2} {\left | e \right |} - 2 \, a c^{2} d^{2} e f g {\left | e \right |} + a^{2} c d e^{2} g^{2} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{\sqrt {c d f g - a e g^{2}} e g^{3}} - \frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{3} f^{2} {\left | e \right |} - 2 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{2} d^{2} e f g {\left | e \right |} + \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c d e^{2} g^{2} {\left | e \right |}}{{\left (c d e^{2} f - a e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} g\right )} g^{3}} - \frac {2 \, {\left (6 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{2} e^{10} f g^{3} {\left | e \right |} - 6 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c d e^{11} g^{4} {\left | e \right |} - {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d e^{8} g^{4} {\left | e \right |}\right )}}{3 \, e^{12} g^{6}} \]

[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)^2,x, algorithm="giac")

[Out]

-1/3*(15*c^3*d^3*e^3*f^3*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 15*c^3*d^4*e^2*
f^2*g*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 30*a*c^2*d^2*e^4*f^2*g*abs(e)*arct
an(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 30*a*c^2*d^3*e^3*f*g^2*abs(e)*arctan(sqrt(-c*d^2*e
+ a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 15*a^2*c*d*e^5*f*g^2*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*
d*f*g - a*e*g^2)*e)) - 15*a^2*c*d^2*e^4*g^3*abs(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^2*d^2*e^2*f^2*abs(e) + 10*sqrt(-c*d^2*e + a*e^3)*sqrt(
c*d*f*g - a*e*g^2)*c^2*d^3*e*f*g*abs(e) + 20*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c*d*e^3*f*g*abs(
e) + 2*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^2*d^4*g^2*abs(e) - 14*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*
f*g - a*e*g^2)*a*c*d^2*e^2*g^2*abs(e) - 3*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*e^4*g^2*abs(e))/(
sqrt(c*d*f*g - a*e*g^2)*e^4*f*g^3 - sqrt(c*d*f*g - a*e*g^2)*d*e^3*g^4) + 5*(c^3*d^3*f^2*abs(e) - 2*a*c^2*d^2*e
*f*g*abs(e) + a^2*c*d*e^2*g^2*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))/(sqrt(c*d*f*g - a*e*g^2)*e*g^3) - (sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^3*d^3*f^2*abs(e) - 2*sqrt((
e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^2*d^2*e*f*g*abs(e) + sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c*d*e^2
*g^2*abs(e))/((c*d*e^2*f - a*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g)*g^3) - 2/3*(6*sqrt((e*x + d)*c*d*e
- c*d^2*e + a*e^3)*c^2*d^2*e^10*f*g^3*abs(e) - 6*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c*d*e^11*g^4*abs(e
) - ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c*d*e^8*g^4*abs(e))/(e^12*g^6)

Mupad [F(-1)]

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

[In]

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