\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^3} \, dx\) [707]

Optimal result

Integrand size = 46, antiderivative size = 246 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^3} \, dx=\frac {15 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g^2 (d+e x)^{3/2} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 g (d+e x)^{5/2} (f+g x)^2}-\frac {15 c^2 d^2 \sqrt {c d f-a e g} \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 g^{7/2}} \]

[Out]

-5/4*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)-1/2*(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)^2-15/4*c^2*d^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))*(-a*e*g+c*d*f)^(1/2)/g^(7/2)+15/4*c^2*d^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 = 0.21 (sec) , antiderivative size = 246, 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} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^3} \, dx=-\frac {15 c^2 d^2 \sqrt {c d f-a e g} \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{4 g^{7/2}}+\frac {15 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g^3 \sqrt {d+e x}}-\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g^2 (d+e x)^{3/2} (f+g x)}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{2 g (d+e x)^{5/2} (f+g x)^2} \]

[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)^3),x]

[Out]

(15*c^2*d^2*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*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(3/2))/(4*g^2*(d + e*x)^(3/2)*(f + g*x)) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(2*
g*(d + e*x)^(5/2)*(f + g*x)^2) - (15*c^2*d^2*Sqrt[c*d*f - a*e*g]*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])])/(4*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}}{2 g (d+e x)^{5/2} (f+g x)^2}+\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)^2} \, dx}{4 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}}{4 g^2 (d+e x)^{3/2} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 g (d+e x)^{5/2} (f+g x)^2}+\frac {\left (15 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)} \, dx}{8 g^2} \\ & = \frac {15 c^2 d^2 \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 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g^2 (d+e x)^{3/2} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 g (d+e x)^{5/2} (f+g x)^2}-\frac {\left (15 c^2 d^2 (c d f-a e g)\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}{8 g^3} \\ & = \frac {15 c^2 d^2 \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 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g^2 (d+e x)^{3/2} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 g (d+e x)^{5/2} (f+g x)^2}-\frac {\left (15 c^2 d^2 e^2 (c d f-a e g)\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 )}{4 g^3} \\ & = \frac {15 c^2 d^2 \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 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g^2 (d+e x)^{3/2} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 g (d+e x)^{5/2} (f+g x)^2}-\frac {15 c^2 d^2 \sqrt {c d f-a e g} \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 )}{4 g^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^3} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {a e+c d x} \left (-2 a^2 e^2 g^2-a c d e g (5 f+9 g x)+c^2 d^2 \left (15 f^2+25 f g x+8 g^2 x^2\right )\right )-15 c^2 d^2 \sqrt {c d f-a e g} (f+g x)^2 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{4 g^{7/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)^2} \]

[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)^3),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(515\) vs. \(2(214)=428\).

Time = 0.56 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.10

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e \,g^{3} x^{2}-15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f \,g^{2} x^{2}+30 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e f \,g^{2} x -30 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{2} g x +15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a \,c^{2} d^{2} e \,f^{2} g -15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{3}-8 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} g^{2} x^{2}+9 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d e \,g^{2} x -25 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f g x +2 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} e^{2} g^{2}+5 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d e f g -15 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f^{2}\right )}{4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, g^{3} \left (g x +f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\)	\(516\)

[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,x,method=_RETURNVERBOSE)

[Out]

-1/4*((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*c^2*d^2*e*g^3*x^2-
15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*f*g^2*x^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*g^2*x-30*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*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^3-8*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*g^2*x^2+9*(c*d*x+a*e)^(1
/2)*((a*e*g-c*d*f)*g)^(1/2)*a*c*d*e*g^2*x-25*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*f*g*x+2*(c*d*x+
a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^2*e^2*g^2+5*(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)^2/((a*e*g-c*d*f
)*g)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.55 (sec) , antiderivative size = 683, normalized size of antiderivative = 2.78 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^3} \, dx=\left [\frac {15 \, {\left (c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + {\left (2 \, c^{2} d^{2} e f g + c^{2} d^{3} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, c^{2} d^{3} f g\right )} x\right )} \sqrt {-\frac {c d f - a e g}{g}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} g \sqrt {-\frac {c d f - a e g}{g}} - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right ) + 2 \, {\left (8 \, c^{2} d^{2} g^{2} x^{2} + 15 \, c^{2} d^{2} f^{2} - 5 \, a c d e f g - 2 \, a^{2} e^{2} g^{2} + {\left (25 \, c^{2} d^{2} f g - 9 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{8 \, {\left (e g^{5} x^{3} + d f^{2} g^{3} + {\left (2 \, e f g^{4} + d g^{5}\right )} x^{2} + {\left (e f^{2} g^{3} + 2 \, d f g^{4}\right )} x\right )}}, \frac {15 \, {\left (c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + {\left (2 \, c^{2} d^{2} e f g + c^{2} d^{3} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, c^{2} d^{3} f g\right )} x\right )} \sqrt {\frac {c d f - a e g}{g}} \arctan \left (\frac {\sqrt {e x + d} \sqrt {\frac {c d f - a e g}{g}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\right ) + {\left (8 \, c^{2} d^{2} g^{2} x^{2} + 15 \, c^{2} d^{2} f^{2} - 5 \, a c d e f g - 2 \, a^{2} e^{2} g^{2} + {\left (25 \, c^{2} d^{2} f g - 9 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{4 \, {\left (e g^{5} x^{3} + d f^{2} g^{3} + {\left (2 \, e f g^{4} + d g^{5}\right )} x^{2} + {\left (e f^{2} g^{3} + 2 \, d f g^{4}\right )} x\right )}}\right ] \]

[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,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[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,x)

[Out]

Timed out

Maxima [F]

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

[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,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), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1038 vs. \(2 (214) = 428\).

Time = 0.58 (sec) , antiderivative size = 1038, normalized size of antiderivative = 4.22 \[ \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} \, dx=\frac {2 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{2} {\left | e \right |}}{e^{2} g^{3}} + \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 ) - 30 \, 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 ) - 15 \, 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 ) + 15 \, c^{3} d^{5} e 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 ) + 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 c^{2} d^{4} e^{2} 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 |} + 25 \, \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 |} + 5 \, \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 |} - 8 \, \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 |} - 9 \, \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 |} + 2 \, \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 |}}{4 \, {\left (\sqrt {c d f g - a e g^{2}} e^{4} f^{2} g^{3} - 2 \, \sqrt {c d f g - a e g^{2}} d e^{3} f g^{4} + \sqrt {c d f g - a e g^{2}} d^{2} e^{2} g^{5}\right )}} - \frac {15 \, {\left (c^{3} d^{3} f {\left | e \right |} - a c^{2} d^{2} e g {\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 )}{4 \, \sqrt {c d f g - a e g^{2}} e g^{3}} + \frac {7 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{4} d^{4} e^{2} f^{2} {\left | e \right |} - 14 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{3} d^{3} e^{3} f g {\left | e \right |} + 7 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{2} d^{2} e^{4} g^{2} {\left | e \right |} + 9 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{3} f g {\left | e \right |} - 9 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{2} d^{2} e g^{2} {\left | e \right |}}{4 \, {\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 )}^{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,x, algorithm="giac")

[Out]

2*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^2*d^2*abs(e)/(e^2*g^3) + 1/4*(15*c^3*d^3*e^3*f^3*abs(e)*arctan(sqr
t(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 30*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)) - 15*a*c^2*d^2*e^4*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)) + 15*c^3*d^5*e*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)) + 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*c^2*d^4*e^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)) - 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) + 25*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) + 5*
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) - 8*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) - 9*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) + 2*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^2*g^3 - 2*sqrt(c
*d*f*g - a*e*g^2)*d*e^3*f*g^4 + sqrt(c*d*f*g - a*e*g^2)*d^2*e^2*g^5) - 15/4*(c^3*d^3*f*abs(e) - a*c^2*d^2*e*g*
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) + 1/4*(7*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^4*d^4*e^2*f^2*abs(e) - 14*sqrt((e*x + d)*c*d*e - c*
d^2*e + a*e^3)*a*c^3*d^3*e^3*f*g*abs(e) + 7*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^2*d^2*e^4*g^2*abs(e)
+ 9*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^3*d^3*f*g*abs(e) - 9*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2
)*a*c^2*d^2*e*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)^2*g^3)

Mupad [F(-1)]

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

[In]

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