\(\int \frac {(d+e x)^{5/2}}{(f+g x) (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [676]

Optimal result

Integrand size = 46, antiderivative size = 188 \[ \int \frac {(d+e x)^{5/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 g \sqrt {d+e x}}{(c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 g^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{(c d f-a e g)^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {882, 888, 211} \[ \int \frac {(d+e x)^{5/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 g^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{5/2}}+\frac {2 g \sqrt {d+e x}}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac {2 (d+e x)^{3/2}}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]

[In]

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

[Out]

(-2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (2*g*Sqrt[d + e*x])/(
(c*d*f - a*e*g)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (2*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])])/(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 882

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)/((p + 1)*(c*e*f + c*d*g - b*e*g))), x]
+ Dist[e^2*g*((m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g))), 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] &&  !IntegerQ[p] && EqQ[m + p, 0] && LtQ[p, -1] && 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 {2 (d+e x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {g \int \frac {(d+e x)^{3/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{c d f-a e g} \\ & = -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 g \sqrt {d+e x}}{(c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {g^2 \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}{(c d f-a e g)^2} \\ & = -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 g \sqrt {d+e x}}{(c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (2 e^2 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 )}{(c d f-a e g)^2} \\ & = -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 g \sqrt {d+e x}}{(c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 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 )}{(c d f-a e g)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.69 \[ \int \frac {(d+e x)^{5/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^{3/2} \left (\sqrt {c d f-a e g} (4 a e g-c d (f-3 g x))+3 g^{3/2} (a e+c d x)^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{3 (c d f-a e g)^{5/2} ((a e+c d x) (d+e x))^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \sqrt {c d x +a e}\, \operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c d \,g^{2} x +3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) a e \,g^{2} \sqrt {c d x +a e}-3 \sqrt {\left (a e g -c d f \right ) g}\, c d g x -4 \sqrt {\left (a e g -c d f \right ) g}\, a e g +\sqrt {\left (a e g -c d f \right ) g}\, c d f \right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} \left (a e g -c d f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\)	\(209\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (166) = 332\).

Time = 0.39 (sec) , antiderivative size = 1015, normalized size of antiderivative = 5.40 \[ \int \frac {(d+e x)^{5/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (c^{2} d^{2} e g x^{3} + a^{2} d e^{2} g + {\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} g x^{2} + {\left (2 \, a c d^{2} e + a^{2} e^{3}\right )} g x\right )} \sqrt {-\frac {g}{c d f - a e 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} {\left (c d f - a e g\right )} \sqrt {e x + d} \sqrt {-\frac {g}{c d f - a e 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 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d g x - c d f + 4 \, a e g\right )} \sqrt {e x + d}}{3 \, {\left (a^{2} c^{2} d^{3} e^{2} f^{2} - 2 \, a^{3} c d^{2} e^{3} f g + a^{4} d e^{4} g^{2} + {\left (c^{4} d^{4} e f^{2} - 2 \, a c^{3} d^{3} e^{2} f g + a^{2} c^{2} d^{2} e^{3} g^{2}\right )} x^{3} + {\left ({\left (c^{4} d^{5} + 2 \, a c^{3} d^{3} e^{2}\right )} f^{2} - 2 \, {\left (a c^{3} d^{4} e + 2 \, a^{2} c^{2} d^{2} e^{3}\right )} f g + {\left (a^{2} c^{2} d^{3} e^{2} + 2 \, a^{3} c d e^{4}\right )} g^{2}\right )} x^{2} + {\left ({\left (2 \, a c^{3} d^{4} e + a^{2} c^{2} d^{2} e^{3}\right )} f^{2} - 2 \, {\left (2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} f g + {\left (2 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} g^{2}\right )} x\right )}}, \frac {2 \, {\left (3 \, {\left (c^{2} d^{2} e g x^{3} + a^{2} d e^{2} g + {\left (c^{2} d^{3} + 2 \, a c d e^{2}\right )} g x^{2} + {\left (2 \, a c d^{2} e + a^{2} e^{3}\right )} g x\right )} \sqrt {\frac {g}{c d f - a e g}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d f - a e g\right )} \sqrt {e x + d} \sqrt {\frac {g}{c d f - a e g}}}{c d e g x^{2} + a d e g + {\left (c d^{2} + a e^{2}\right )} g x}\right ) + \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d g x - c d f + 4 \, a e g\right )} \sqrt {e x + d}\right )}}{3 \, {\left (a^{2} c^{2} d^{3} e^{2} f^{2} - 2 \, a^{3} c d^{2} e^{3} f g + a^{4} d e^{4} g^{2} + {\left (c^{4} d^{4} e f^{2} - 2 \, a c^{3} d^{3} e^{2} f g + a^{2} c^{2} d^{2} e^{3} g^{2}\right )} x^{3} + {\left ({\left (c^{4} d^{5} + 2 \, a c^{3} d^{3} e^{2}\right )} f^{2} - 2 \, {\left (a c^{3} d^{4} e + 2 \, a^{2} c^{2} d^{2} e^{3}\right )} f g + {\left (a^{2} c^{2} d^{3} e^{2} + 2 \, a^{3} c d e^{4}\right )} g^{2}\right )} x^{2} + {\left ({\left (2 \, a c^{3} d^{4} e + a^{2} c^{2} d^{2} e^{3}\right )} f^{2} - 2 \, {\left (2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} f g + {\left (2 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} g^{2}\right )} x\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (166) = 332\).

Time = 0.44 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.63 \[ \int \frac {(d+e x)^{5/2}}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2}{3} \, e^{3} {\left (\frac {3 \, g^{2} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{{\left (c^{2} d^{2} e f^{2} {\left | e \right |} - 2 \, a c d e^{2} f g {\left | e \right |} + a^{2} e^{3} g^{2} {\left | e \right |}\right )} \sqrt {c d f g - a e g^{2}} e} - \frac {c d e^{2} f - a e^{3} g - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} g}{{\left (c^{2} d^{2} e f^{2} {\left | e \right |} - 2 \, a c d e^{2} f g {\left | e \right |} + a^{2} e^{3} g^{2} {\left | e \right |}\right )} {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}}\right )} - \frac {2 \, {\left (3 \, \sqrt {-c d^{2} e + a e^{3}} c d^{2} e g^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 3 \, \sqrt {-c d^{2} e + a e^{3}} a e^{3} g^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) + \sqrt {c d f g - a e g^{2}} c d e^{3} f + 3 \, \sqrt {c d f g - a e g^{2}} c d^{2} e^{2} g - 4 \, \sqrt {c d f g - a e g^{2}} a e^{4} g\right )}}{3 \, {\left (\sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c^{3} d^{4} f^{2} {\left | e \right |} - \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a c^{2} d^{2} e^{2} f^{2} {\left | e \right |} - 2 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a c^{2} d^{3} e f g {\left | e \right |} + 2 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a^{2} c d e^{3} f g {\left | e \right |} + \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a^{2} c d^{2} e^{2} g^{2} {\left | e \right |} - \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a^{3} e^{4} g^{2} {\left | e \right |}\right )}} \]

[In]

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

[Out]

2/3*e^3*(3*g^2*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*e*f^2*a
bs(e) - 2*a*c*d*e^2*f*g*abs(e) + a^2*e^3*g^2*abs(e))*sqrt(c*d*f*g - a*e*g^2)*e) - (c*d*e^2*f - a*e^3*g - 3*((e
*x + d)*c*d*e - c*d^2*e + a*e^3)*g)/((c^2*d^2*e*f^2*abs(e) - 2*a*c*d*e^2*f*g*abs(e) + a^2*e^3*g^2*abs(e))*((e*
x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2))) - 2/3*(3*sqrt(-c*d^2*e + a*e^3)*c*d^2*e*g^2*arctan(sqrt(-c*d^2*e + a*e
^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 3*sqrt(-c*d^2*e + a*e^3)*a*e^3*g^2*arctan(sqrt(-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)*c*d*e^3*f + 3*sqrt(c*d*f*g - a*e*g^2)*c*d^2*e^2*g - 4*sqrt(c*
d*f*g - a*e*g^2)*a*e^4*g)/(sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^3*d^4*f^2*abs(e) - sqrt(-c*d^2*e +
a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^2*e^2*f^2*abs(e) - 2*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*
c^2*d^3*e*f*g*abs(e) + 2*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d*e^3*f*g*abs(e) + sqrt(-c*d^2*e
+ a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^2*e^2*g^2*abs(e) - sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^
3*e^4*g^2*abs(e))

Mupad [F(-1)]

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

[In]

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

[Out]

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