\(\int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [790]

Optimal result

Integrand size = 46, antiderivative size = 261 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {c d \left (4 a e^2 g-c d (e f+3 d g)\right ) \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 g^{3/2} (c d f-a e g)^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {892, 886, 888, 211} \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {c d \left (4 a e^2 g-c d (3 d g+e f)\right ) \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{4 g^{3/2} (c d f-a e g)^{5/2}}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (4 a e^2 g-c d (3 d g+e f)\right )}{4 g \sqrt {d+e x} (f+g x) (c d f-a e g)^2} \]

[In]

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

[Out]

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

Rule 892

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[e^2*(e*f - d*g)*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/(g*(n + 1)*(c*e*f + c*d*g
- b*e*g))), x] - Dist[e*((b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3))/(g*(n + 1)*(c*e*f + c*d*g - b*e
*g))), Int[(d + e*x)^(m - 1)*(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 - 1, 0] && LtQ[n, -1] && IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {\left (e \left (\frac {1}{2} c d e^2 f+\frac {7}{2} c d^2 e g-2 e \left (c d^2+a e^2\right ) g\right )\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}{2 g \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )} \\ & = -\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {\left (c d \left (4 a e^2 g-c d (e f+3 d g)\right )\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 (c d f-a e g)^2} \\ & = -\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {\left (c d e^2 \left (4 a e^2 g-c d (e f+3 d g)\right )\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 (c d f-a e g)^2} \\ & = -\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {c d \left (4 a e^2 g-c d (e f+3 d g)\right ) \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^{3/2} (c d f-a e g)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {c d \sqrt {d+e x} \left (\frac {\sqrt {g} (a e+c d x) (-2 a e g (d g+e (f+2 g x))+c d (e f (-f+g x)+d g (5 f+3 g x)))}{c d (c d f-a e g)^2 (f+g x)^2}+\frac {\left (-4 a e^2 g+c d (e f+3 d g)\right ) \sqrt {a e+c d x} \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{5/2}}\right )}{4 g^{3/2} \sqrt {(a e+c d x) (d+e x)}} \]

[In]

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

[Out]

(c*d*Sqrt[d + e*x]*((Sqrt[g]*(a*e + c*d*x)*(-2*a*e*g*(d*g + e*(f + 2*g*x)) + c*d*(e*f*(-f + g*x) + d*g*(5*f +
3*g*x))))/(c*d*(c*d*f - a*e*g)^2*(f + g*x)^2) + ((-4*a*e^2*g + c*d*(e*f + 3*d*g))*Sqrt[a*e + c*d*x]*ArcTan[(Sq
rt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(c*d*f - a*e*g)^(5/2)))/(4*g^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)
])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(662\) vs. \(2(235)=470\).

Time = 0.58 (sec) , antiderivative size = 663, normalized size of antiderivative = 2.54

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (4 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a c d \,e^{2} g^{3} x^{2}-3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{3} g^{3} x^{2}-\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} e f \,g^{2} x^{2}+8 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a c d \,e^{2} f \,g^{2} x -6 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{3} f \,g^{2} x -2 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} e \,f^{2} g x +4 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a c d \,e^{2} f^{2} g -3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{3} f^{2} g -\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{2} d^{2} e \,f^{3}-4 a \,e^{2} g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+3 c \,d^{2} g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+c d e f g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-2 a d e \,g^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-2 a \,e^{2} f g \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}+5 c \,d^{2} f g \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}-c d e \,f^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\right )}{4 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{2} g \left (a e g -c d f \right )^{2} \sqrt {c d x +a e}}\)	\(663\)

[In]

int((e*x+d)^(3/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*((c*d*x+a*e)*(e*x+d))^(1/2)*(4*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c*d*e^2*g^3*x^2-3*ar
ctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^3*g^3*x^2-arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*
g)^(1/2))*c^2*d^2*e*f*g^2*x^2+8*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c*d*e^2*f*g^2*x-6*arcta
nh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^3*f*g^2*x-2*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g
)^(1/2))*c^2*d^2*e*f^2*g*x+4*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*a*c*d*e^2*f^2*g-3*arctanh(g*
(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^3*f^2*g-arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*
c^2*d^2*e*f^3-4*a*e^2*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+3*c*d^2*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-
c*d*f)*g)^(1/2)+c*d*e*f*g*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-2*a*d*e*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*
d*f)*g)^(1/2)-2*a*e^2*f*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+5*c*d^2*f*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*
f)*g)^(1/2)-c*d*e*f^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2))/(e*x+d)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)
^2/g/(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. 831 vs. \(2 (235) = 470\).

Time = 0.35 (sec) , antiderivative size = 1704, normalized size of antiderivative = 6.53 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

[1/8*((c^2*d^3*e*f^3 + (3*c^2*d^4 - 4*a*c*d^2*e^2)*f^2*g + (c^2*d^2*e^2*f*g^2 + (3*c^2*d^3*e - 4*a*c*d*e^3)*g^
3)*x^3 + (2*c^2*d^2*e^2*f^2*g + (7*c^2*d^3*e - 8*a*c*d*e^3)*f*g^2 + (3*c^2*d^4 - 4*a*c*d^2*e^2)*g^3)*x^2 + (c^
2*d^2*e^2*f^3 + (5*c^2*d^3*e - 4*a*c*d*e^3)*f^2*g + 2*(3*c^2*d^4 - 4*a*c*d^2*e^2)*f*g^2)*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)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f + d*g)*x)) - 2*(c^2*d^2*e*f
^3*g - 2*a^2*d*e^2*g^4 - (5*c^2*d^3 - a*c*d*e^2)*f^2*g^2 + (7*a*c*d^2*e - 2*a^2*e^3)*f*g^3 - (c^2*d^2*e*f^2*g^
2 + (3*c^2*d^3 - 5*a*c*d*e^2)*f*g^3 - (3*a*c*d^2*e - 4*a^2*e^3)*g^4)*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^5*g^2 - 3*a*c^2*d^3*e*f^4*g^3 + 3*a^2*c*d^2*e^2*f^3*g^4 - a^3*d*e^3*f^2*g^5 +
(c^3*d^3*e*f^3*g^4 - 3*a*c^2*d^2*e^2*f^2*g^5 + 3*a^2*c*d*e^3*f*g^6 - a^3*e^4*g^7)*x^3 + (2*c^3*d^3*e*f^4*g^3 -
a^3*d*e^3*g^7 + (c^3*d^4 - 6*a*c^2*d^2*e^2)*f^3*g^4 - 3*(a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^2*g^5 + (3*a^2*c*d^2*
e^2 - 2*a^3*e^4)*f*g^6)*x^2 + (c^3*d^3*e*f^5*g^2 - 2*a^3*d*e^3*f*g^6 + (2*c^3*d^4 - 3*a*c^2*d^2*e^2)*f^4*g^3 -
3*(2*a*c^2*d^3*e - a^2*c*d*e^3)*f^3*g^4 + (6*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^5)*x), -1/4*((c^2*d^3*e*f^3 + (3*
c^2*d^4 - 4*a*c*d^2*e^2)*f^2*g + (c^2*d^2*e^2*f*g^2 + (3*c^2*d^3*e - 4*a*c*d*e^3)*g^3)*x^3 + (2*c^2*d^2*e^2*f^
2*g + (7*c^2*d^3*e - 8*a*c*d*e^3)*f*g^2 + (3*c^2*d^4 - 4*a*c*d^2*e^2)*g^3)*x^2 + (c^2*d^2*e^2*f^3 + (5*c^2*d^3
*e - 4*a*c*d*e^3)*f^2*g + 2*(3*c^2*d^4 - 4*a*c*d^2*e^2)*f*g^2)*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)) + (c^2*d^2*e*f^3*g - 2*a^2*d*e^2*g^4 - (5*c^2*d^3 - a*c*d*e^2)*f^2*g^2 + (7*a*c*d^2*e - 2*a^2*e^3)*f*g^3
- (c^2*d^2*e*f^2*g^2 + (3*c^2*d^3 - 5*a*c*d*e^2)*f*g^3 - (3*a*c*d^2*e - 4*a^2*e^3)*g^4)*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^5*g^2 - 3*a*c^2*d^3*e*f^4*g^3 + 3*a^2*c*d^2*e^2*f^3*g^4 -
a^3*d*e^3*f^2*g^5 + (c^3*d^3*e*f^3*g^4 - 3*a*c^2*d^2*e^2*f^2*g^5 + 3*a^2*c*d*e^3*f*g^6 - a^3*e^4*g^7)*x^3 + (2
*c^3*d^3*e*f^4*g^3 - a^3*d*e^3*g^7 + (c^3*d^4 - 6*a*c^2*d^2*e^2)*f^3*g^4 - 3*(a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^2
*g^5 + (3*a^2*c*d^2*e^2 - 2*a^3*e^4)*f*g^6)*x^2 + (c^3*d^3*e*f^5*g^2 - 2*a^3*d*e^3*f*g^6 + (2*c^3*d^4 - 3*a*c^
2*d^2*e^2)*f^4*g^3 - 3*(2*a*c^2*d^3*e - a^2*c*d*e^3)*f^3*g^4 + (6*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^5)*x)]

Sympy [F]

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

[In]

integrate((e*x+d)**(3/2)/(g*x+f)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Integral((d + e*x)**(3/2)/(sqrt((d + e*x)*(a*e + c*d*x))*(f + g*x)**3), x)

Maxima [F]

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

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (235) = 470\).

Time = 0.53 (sec) , antiderivative size = 1071, normalized size of antiderivative = 4.10 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {e^{2} {\left (\frac {{\left (c^{3} d^{3} e f + 3 \, c^{3} d^{4} g - 4 \, a c^{2} d^{2} e^{2} g\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 )}{{\left (c^{2} d^{2} f^{2} g {\left | e \right |} - 2 \, a c d e f g^{2} {\left | e \right |} + a^{2} e^{2} g^{3} {\left | e \right |}\right )} \sqrt {c d f g - a e g^{2}} e} - \frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{4} d^{4} e^{3} f^{2} - 5 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{4} d^{5} e^{2} f g + 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{3} d^{3} e^{4} f g + 5 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{3} d^{4} e^{3} g^{2} - 4 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{2} d^{2} e^{5} g^{2} - {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{3} e f g - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{3} d^{4} g^{2} + 4 \, {\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^{2} g^{2}}{{\left (c^{2} d^{2} f^{2} g {\left | e \right |} - 2 \, a c d e f g^{2} {\left | e \right |} + a^{2} e^{2} g^{3} {\left | e \right |}\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 )}^{2}}\right )}}{4 \, c d} - \frac {c^{2} d^{2} e^{3} f^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) + 2 \, c^{2} d^{3} e^{2} f g \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 4 \, a c d e^{4} f g \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 3 \, c^{2} d^{4} e g^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) + 4 \, a c d^{2} e^{3} g^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c d e^{2} f + 3 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c d^{2} e g - 2 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a e^{3} g}{4 \, {\left (\sqrt {c d f g - a e g^{2}} c^{2} d^{2} e f^{3} g {\left | e \right |} - \sqrt {c d f g - a e g^{2}} c^{2} d^{3} f^{2} g^{2} {\left | e \right |} - 2 \, \sqrt {c d f g - a e g^{2}} a c d e^{2} f^{2} g^{2} {\left | e \right |} + 2 \, \sqrt {c d f g - a e g^{2}} a c d^{2} e f g^{3} {\left | e \right |} + \sqrt {c d f g - a e g^{2}} a^{2} e^{3} f g^{3} {\left | e \right |} - \sqrt {c d f g - a e g^{2}} a^{2} d e^{2} g^{4} {\left | e \right |}\right )}} \]

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

1/4*e^2*((c^3*d^3*e*f + 3*c^3*d^4*g - 4*a*c^2*d^2*e^2*g)*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g/(sqr
t(c*d*f*g - a*e*g^2)*e))/((c^2*d^2*f^2*g*abs(e) - 2*a*c*d*e*f*g^2*abs(e) + a^2*e^2*g^3*abs(e))*sqrt(c*d*f*g -
a*e*g^2)*e) - (sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^4*d^4*e^3*f^2 - 5*sqrt((e*x + d)*c*d*e - c*d^2*e + a*
e^3)*c^4*d^5*e^2*f*g + 3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^3*d^3*e^4*f*g + 5*sqrt((e*x + d)*c*d*e -
c*d^2*e + a*e^3)*a*c^3*d^4*e^3*g^2 - 4*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^2*d^2*e^5*g^2 - ((e*x + d
)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^3*d^3*e*f*g - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^3*d^4*g^2 + 4*(
(e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^2*d^2*e^2*g^2)/((c^2*d^2*f^2*g*abs(e) - 2*a*c*d*e*f*g^2*abs(e) +
a^2*e^2*g^3*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))/(c*d) - 1/4*(c^2*d^2*e^3
*f^2*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 2*c^2*d^3*e^2*f*g*arctan(sqrt(-c*d^2*e + a
*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 4*a*c*d*e^4*f*g*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2
)*e)) - 3*c^2*d^4*e*g^2*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 4*a*c*d^2*e^3*g^2*arcta
n(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c*d*e
^2*f + 3*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c*d^2*e*g - 2*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*
e*g^2)*a*e^3*g)/(sqrt(c*d*f*g - a*e*g^2)*c^2*d^2*e*f^3*g*abs(e) - sqrt(c*d*f*g - a*e*g^2)*c^2*d^3*f^2*g^2*abs(
e) - 2*sqrt(c*d*f*g - a*e*g^2)*a*c*d*e^2*f^2*g^2*abs(e) + 2*sqrt(c*d*f*g - a*e*g^2)*a*c*d^2*e*f*g^3*abs(e) + s
qrt(c*d*f*g - a*e*g^2)*a^2*e^3*f*g^3*abs(e) - sqrt(c*d*f*g - a*e*g^2)*a^2*d*e^2*g^4*abs(e))

Mupad [F(-1)]

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

[In]

int((d + e*x)^(3/2)/((f + g*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)),x)

[Out]

int((d + e*x)^(3/2)/((f + g*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)), x)