\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx\) [711]

Optimal result

Integrand size = 46, antiderivative size = 463 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^5 d^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 g^3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {5 c^6 d^6 \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{512 g^{7/2} (c d f-a e g)^{7/2}} \]

[Out]

-1/12*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)^5-1/6*(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)^6+5/512*c^6*d^6*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)/(-a*e*g+c*d*f)^(7/2)-1/32*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)/g^3/(g*x+f)^4/(e*x+d)^(1/2)+1/192*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)/(g*x
+f)^3/(e*x+d)^(1/2)+5/768*c^4*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)^2/(g*x+f)^2/(e*x+
d)^(1/2)+5/512*c^5*d^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)^3/(g*x+f)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {876, 886, 888, 211} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\frac {5 c^6 d^6 \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{512 g^{7/2} (c d f-a e g)^{7/2}}+\frac {5 c^5 d^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 g^3 \sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {5 c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{768 g^3 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac {c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 g^3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}-\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}-\frac {c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6} \]

[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)^7),x]

[Out]

-1/32*(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(g^3*Sqrt[d + e*x]*(f + g*x)^4) + (c^3*d^3*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(192*g^3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) + (5*c^4*d^4*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(768*g^3*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^2) + (5*c^5*d^5*Sqrt[
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(512*g^3*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) - (c*d*(a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(12*g^2*(d + e*x)^(3/2)*(f + g*x)^5) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2)^(5/2)/(6*g*(d + e*x)^(5/2)*(f + g*x)^6) + (5*c^6*d^6*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])])/(512*g^(7/2)*(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 {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\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)^6} \, dx}{12 g} \\ & = -\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (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)^5} \, dx}{8 g^2} \\ & = -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (c^3 d^3\right ) \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}{64 g^3} \\ & = -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (5 c^4 d^4\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}{384 g^3 (c d f-a e g)} \\ & = -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (5 c^5 d^5\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}{512 g^3 (c d f-a e g)^2} \\ & = -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^5 d^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 g^3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (5 c^6 d^6\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}{1024 g^3 (c d f-a e g)^3} \\ & = -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^5 d^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 g^3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {\left (5 c^6 d^6 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 )}{512 g^3 (c d f-a e g)^3} \\ & = -\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^5 d^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 g^3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {5 c^6 d^6 \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 )}{512 g^{7/2} (c d f-a e g)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.72 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\frac {c^6 d^6 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {g} \left (256 a^5 e^5 g^5+640 a^4 c d e^4 g^4 (-f+g x)+16 a^3 c^2 d^2 e^3 g^3 \left (27 f^2-106 f g x+27 g^2 x^2\right )+8 a^2 c^3 d^3 e^2 g^2 \left (-f^3+159 f^2 g x-159 f g^2 x^2+g^3 x^3\right )-2 a c^4 d^4 e g \left (5 f^4+28 f^3 g x-594 f^2 g^2 x^2+28 f g^3 x^3+5 g^4 x^4\right )+c^5 d^5 \left (-15 f^5-85 f^4 g x-198 f^3 g^2 x^2+198 f^2 g^3 x^3+85 f g^4 x^4+15 g^5 x^5\right )\right )}{c^6 d^6 (c d f-a e g)^3 (a e+c d x)^2 (f+g x)^6}+\frac {15 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{7/2} (a e+c d x)^{5/2}}\right )}{1536 g^{7/2} (d+e x)^{5/2}} \]

[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)^7),x]

[Out]

(c^6*d^6*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[g]*(256*a^5*e^5*g^5 + 640*a^4*c*d*e^4*g^4*(-f + g*x) + 16*a^3*
c^2*d^2*e^3*g^3*(27*f^2 - 106*f*g*x + 27*g^2*x^2) + 8*a^2*c^3*d^3*e^2*g^2*(-f^3 + 159*f^2*g*x - 159*f*g^2*x^2
+ g^3*x^3) - 2*a*c^4*d^4*e*g*(5*f^4 + 28*f^3*g*x - 594*f^2*g^2*x^2 + 28*f*g^3*x^3 + 5*g^4*x^4) + c^5*d^5*(-15*
f^5 - 85*f^4*g*x - 198*f^3*g^2*x^2 + 198*f^2*g^3*x^3 + 85*f*g^4*x^4 + 15*g^5*x^5)))/(c^6*d^6*(c*d*f - a*e*g)^3
*(a*e + c*d*x)^2*(f + g*x)^6) + (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)*(a*e + c*d*x)^(5/2))))/(1536*g^(7/2)*(d + e*x)^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1250\) vs. \(2(413)=826\).

Time = 0.58 (sec) , antiderivative size = 1251, normalized size of antiderivative = 2.70

method	result	size
default	\(\text {Expression too large to display}\)	\(1251\)

[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)^7,x,method=_RETURNVERBOSE)

[Out]

1/1536*((c*d*x+a*e)*(e*x+d))^(1/2)*(-1188*a*c^4*d^4*e*f^2*g^3*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-15
*c^5*d^5*g^5*x^5*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+90*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1
/2))*c^6*d^6*f^5*g*x+225*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*f^4*g^2*x^2+225*arctanh(
g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*f^2*g^4*x^4+300*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f
)*g)^(1/2))*c^6*d^6*f^3*g^3*x^3+90*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*f*g^5*x^5+15*a
rctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*f^6-256*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*
a^5*e^5*g^5+15*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^5*d^5*f^5+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*
d*f)*g)^(1/2))*c^6*d^6*g^6*x^6-432*a^3*c^2*d^2*e^3*g^5*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-640*a^4*c
*d*e^4*g^5*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+10*a*c^4*d^4*e*g^5*x^4*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)
*g)^(1/2)+640*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^4*c*d*e^4*f*g^4-432*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a
*e)^(1/2)*a^3*c^2*d^2*e^3*f^2*g^3+8*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c^3*d^3*e^2*f^3*g^2+10*((a*e
*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c^4*d^4*e*f^4*g-85*c^5*d^5*f*g^4*x^4*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g
)^(1/2)-198*c^5*d^5*f^2*g^3*x^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+198*c^5*d^5*f^3*g^2*x^2*(c*d*x+a*e)^
(1/2)*((a*e*g-c*d*f)*g)^(1/2)+85*c^5*d^5*f^4*g*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-8*a^2*c^3*d^3*e^2*g
^5*x^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+1272*a^2*c^3*d^3*e^2*f*g^4*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*
f)*g)^(1/2)+56*a*c^4*d^4*e*f*g^4*x^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-1272*a^2*c^3*d^3*e^2*f^2*g^3*x*
(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+56*a*c^4*d^4*e*f^3*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+1
696*a^3*c^2*d^2*e^3*f*g^4*x*(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)^6/g^3/(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. 1915 vs. \(2 (413) = 826\).

Time = 3.97 (sec) , antiderivative size = 3872, normalized size of antiderivative = 8.36 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\text {Too large to display} \]

[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)^7,x, algorithm="fricas")

[Out]

[1/3072*(15*(c^6*d^6*e*g^6*x^7 + c^6*d^7*f^6 + (6*c^6*d^6*e*f*g^5 + c^6*d^7*g^6)*x^6 + 3*(5*c^6*d^6*e*f^2*g^4
+ 2*c^6*d^7*f*g^5)*x^5 + 5*(4*c^6*d^6*e*f^3*g^3 + 3*c^6*d^7*f^2*g^4)*x^4 + 5*(3*c^6*d^6*e*f^4*g^2 + 4*c^6*d^7*
f^3*g^3)*x^3 + 3*(2*c^6*d^6*e*f^5*g + 5*c^6*d^7*f^4*g^2)*x^2 + (c^6*d^6*e*f^6 + 6*c^6*d^7*f^5*g)*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*(15*
c^6*d^6*f^6*g - 5*a*c^5*d^5*e*f^5*g^2 - 2*a^2*c^4*d^4*e^2*f^4*g^3 - 440*a^3*c^3*d^3*e^3*f^3*g^4 + 1072*a^4*c^2
*d^2*e^4*f^2*g^5 - 896*a^5*c*d*e^5*f*g^6 + 256*a^6*e^6*g^7 - 15*(c^6*d^6*f*g^6 - a*c^5*d^5*e*g^7)*x^5 - 5*(17*
c^6*d^6*f^2*g^5 - 19*a*c^5*d^5*e*f*g^6 + 2*a^2*c^4*d^4*e^2*g^7)*x^4 - 2*(99*c^6*d^6*f^3*g^4 - 127*a*c^5*d^5*e*
f^2*g^5 + 32*a^2*c^4*d^4*e^2*f*g^6 - 4*a^3*c^3*d^3*e^3*g^7)*x^3 + 6*(33*c^6*d^6*f^4*g^3 - 231*a*c^5*d^5*e*f^3*
g^4 + 410*a^2*c^4*d^4*e^2*f^2*g^5 - 284*a^3*c^3*d^3*e^3*f*g^6 + 72*a^4*c^2*d^2*e^4*g^7)*x^2 + (85*c^6*d^6*f^5*
g^2 - 29*a*c^5*d^5*e*f^4*g^3 - 1328*a^2*c^4*d^4*e^2*f^3*g^4 + 2968*a^3*c^3*d^3*e^3*f^2*g^5 - 2336*a^4*c^2*d^2*
e^4*f*g^6 + 640*a^5*c*d*e^5*g^7)*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^10*g
^4 - 4*a*c^3*d^4*e*f^9*g^5 + 6*a^2*c^2*d^3*e^2*f^8*g^6 - 4*a^3*c*d^2*e^3*f^7*g^7 + a^4*d*e^4*f^6*g^8 + (c^4*d^
4*e*f^4*g^10 - 4*a*c^3*d^3*e^2*f^3*g^11 + 6*a^2*c^2*d^2*e^3*f^2*g^12 - 4*a^3*c*d*e^4*f*g^13 + a^4*e^5*g^14)*x^
7 + (6*c^4*d^4*e*f^5*g^9 + a^4*d*e^4*g^14 + (c^4*d^5 - 24*a*c^3*d^3*e^2)*f^4*g^10 - 4*(a*c^3*d^4*e - 9*a^2*c^2
*d^2*e^3)*f^3*g^11 + 6*(a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^2*g^12 - 2*(2*a^3*c*d^2*e^3 - 3*a^4*e^5)*f*g^13)*x^
6 + 3*(5*c^4*d^4*e*f^6*g^8 + 2*a^4*d*e^4*f*g^13 + 2*(c^4*d^5 - 10*a*c^3*d^3*e^2)*f^5*g^9 - 2*(4*a*c^3*d^4*e -
15*a^2*c^2*d^2*e^3)*f^4*g^10 + 4*(3*a^2*c^2*d^3*e^2 - 5*a^3*c*d*e^4)*f^3*g^11 - (8*a^3*c*d^2*e^3 - 5*a^4*e^5)*
f^2*g^12)*x^5 + 5*(4*c^4*d^4*e*f^7*g^7 + 3*a^4*d*e^4*f^2*g^12 + (3*c^4*d^5 - 16*a*c^3*d^3*e^2)*f^6*g^8 - 12*(a
*c^3*d^4*e - 2*a^2*c^2*d^2*e^3)*f^5*g^9 + 2*(9*a^2*c^2*d^3*e^2 - 8*a^3*c*d*e^4)*f^4*g^10 - 4*(3*a^3*c*d^2*e^3
- a^4*e^5)*f^3*g^11)*x^4 + 5*(3*c^4*d^4*e*f^8*g^6 + 4*a^4*d*e^4*f^3*g^11 + 4*(c^4*d^5 - 3*a*c^3*d^3*e^2)*f^7*g
^7 - 2*(8*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^6*g^8 + 12*(2*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^5*g^9 - (16*a^3*c*
d^2*e^3 - 3*a^4*e^5)*f^4*g^10)*x^3 + 3*(2*c^4*d^4*e*f^9*g^5 + 5*a^4*d*e^4*f^4*g^10 + (5*c^4*d^5 - 8*a*c^3*d^3*
e^2)*f^8*g^6 - 4*(5*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^7*g^7 + 2*(15*a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^6*g^8
- 2*(10*a^3*c*d^2*e^3 - a^4*e^5)*f^5*g^9)*x^2 + (c^4*d^4*e*f^10*g^4 + 6*a^4*d*e^4*f^5*g^9 + 2*(3*c^4*d^5 - 2*a
*c^3*d^3*e^2)*f^9*g^5 - 6*(4*a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^8*g^6 + 4*(9*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^7*
g^7 - (24*a^3*c*d^2*e^3 - a^4*e^5)*f^6*g^8)*x), -1/1536*(15*(c^6*d^6*e*g^6*x^7 + c^6*d^7*f^6 + (6*c^6*d^6*e*f*
g^5 + c^6*d^7*g^6)*x^6 + 3*(5*c^6*d^6*e*f^2*g^4 + 2*c^6*d^7*f*g^5)*x^5 + 5*(4*c^6*d^6*e*f^3*g^3 + 3*c^6*d^7*f^
2*g^4)*x^4 + 5*(3*c^6*d^6*e*f^4*g^2 + 4*c^6*d^7*f^3*g^3)*x^3 + 3*(2*c^6*d^6*e*f^5*g + 5*c^6*d^7*f^4*g^2)*x^2 +
(c^6*d^6*e*f^6 + 6*c^6*d^7*f^5*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^6*d^6*f^6*g -
5*a*c^5*d^5*e*f^5*g^2 - 2*a^2*c^4*d^4*e^2*f^4*g^3 - 440*a^3*c^3*d^3*e^3*f^3*g^4 + 1072*a^4*c^2*d^2*e^4*f^2*g^5
- 896*a^5*c*d*e^5*f*g^6 + 256*a^6*e^6*g^7 - 15*(c^6*d^6*f*g^6 - a*c^5*d^5*e*g^7)*x^5 - 5*(17*c^6*d^6*f^2*g^5
- 19*a*c^5*d^5*e*f*g^6 + 2*a^2*c^4*d^4*e^2*g^7)*x^4 - 2*(99*c^6*d^6*f^3*g^4 - 127*a*c^5*d^5*e*f^2*g^5 + 32*a^2
*c^4*d^4*e^2*f*g^6 - 4*a^3*c^3*d^3*e^3*g^7)*x^3 + 6*(33*c^6*d^6*f^4*g^3 - 231*a*c^5*d^5*e*f^3*g^4 + 410*a^2*c^
4*d^4*e^2*f^2*g^5 - 284*a^3*c^3*d^3*e^3*f*g^6 + 72*a^4*c^2*d^2*e^4*g^7)*x^2 + (85*c^6*d^6*f^5*g^2 - 29*a*c^5*d
^5*e*f^4*g^3 - 1328*a^2*c^4*d^4*e^2*f^3*g^4 + 2968*a^3*c^3*d^3*e^3*f^2*g^5 - 2336*a^4*c^2*d^2*e^4*f*g^6 + 640*
a^5*c*d*e^5*g^7)*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^10*g^4 - 4*a*c^3*d^4
*e*f^9*g^5 + 6*a^2*c^2*d^3*e^2*f^8*g^6 - 4*a^3*c*d^2*e^3*f^7*g^7 + a^4*d*e^4*f^6*g^8 + (c^4*d^4*e*f^4*g^10 - 4
*a*c^3*d^3*e^2*f^3*g^11 + 6*a^2*c^2*d^2*e^3*f^2*g^12 - 4*a^3*c*d*e^4*f*g^13 + a^4*e^5*g^14)*x^7 + (6*c^4*d^4*e
*f^5*g^9 + a^4*d*e^4*g^14 + (c^4*d^5 - 24*a*c^3*d^3*e^2)*f^4*g^10 - 4*(a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^3*g^
11 + 6*(a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^2*g^12 - 2*(2*a^3*c*d^2*e^3 - 3*a^4*e^5)*f*g^13)*x^6 + 3*(5*c^4*d^4
*e*f^6*g^8 + 2*a^4*d*e^4*f*g^13 + 2*(c^4*d^5 - 10*a*c^3*d^3*e^2)*f^5*g^9 - 2*(4*a*c^3*d^4*e - 15*a^2*c^2*d^2*e
^3)*f^4*g^10 + 4*(3*a^2*c^2*d^3*e^2 - 5*a^3*c*d*e^4)*f^3*g^11 - (8*a^3*c*d^2*e^3 - 5*a^4*e^5)*f^2*g^12)*x^5 +
5*(4*c^4*d^4*e*f^7*g^7 + 3*a^4*d*e^4*f^2*g^12 + (3*c^4*d^5 - 16*a*c^3*d^3*e^2)*f^6*g^8 - 12*(a*c^3*d^4*e - 2*a
^2*c^2*d^2*e^3)*f^5*g^9 + 2*(9*a^2*c^2*d^3*e^2 - 8*a^3*c*d*e^4)*f^4*g^10 - 4*(3*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g
^11)*x^4 + 5*(3*c^4*d^4*e*f^8*g^6 + 4*a^4*d*e^4*f^3*g^11 + 4*(c^4*d^5 - 3*a*c^3*d^3*e^2)*f^7*g^7 - 2*(8*a*c^3*
d^4*e - 9*a^2*c^2*d^2*e^3)*f^6*g^8 + 12*(2*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^5*g^9 - (16*a^3*c*d^2*e^3 - 3*a^4*
e^5)*f^4*g^10)*x^3 + 3*(2*c^4*d^4*e*f^9*g^5 + 5*a^4*d*e^4*f^4*g^10 + (5*c^4*d^5 - 8*a*c^3*d^3*e^2)*f^8*g^6 - 4
*(5*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^7*g^7 + 2*(15*a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^6*g^8 - 2*(10*a^3*c*d^
2*e^3 - a^4*e^5)*f^5*g^9)*x^2 + (c^4*d^4*e*f^10*g^4 + 6*a^4*d*e^4*f^5*g^9 + 2*(3*c^4*d^5 - 2*a*c^3*d^3*e^2)*f^
9*g^5 - 6*(4*a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^8*g^6 + 4*(9*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^7*g^7 - (24*a^3*c*
d^2*e^3 - a^4*e^5)*f^6*g^8)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\text {Timed out} \]

[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)**7,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{7}} \,d x } \]

[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)^7,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)^7), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3412 vs. \(2 (413) = 826\).

Time = 7.62 (sec) , antiderivative size = 3412, normalized size of antiderivative = 7.37 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\text {Too large to display} \]

[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)^7,x, algorithm="giac")

[Out]

5/512*c^6*d^6*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))/((c^3*d^3*f
^3*g^3 - 3*a*c^2*d^2*e*f^2*g^4 + 3*a^2*c*d*e^2*f*g^5 - a^3*e^3*g^6)*sqrt(c*d*f*g - a*e*g^2)*e) - 1/1536*(15*c^
6*d^6*e^6*f^6*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 90*c^6*d^7*e^5*f^5*g*abs(e
)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 225*c^6*d^8*e^4*f^4*g^2*abs(e)*arctan(sqrt(-c
*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 300*c^6*d^9*e^3*f^3*g^3*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*
g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 225*c^6*d^10*e^2*f^2*g^4*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g
- a*e*g^2)*e)) - 90*c^6*d^11*e*f*g^5*abs(e)*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 15
*c^6*d^12*g^6*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^5*d^5*e^5*f^5*abs(e) + 85*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^5*d^6*e^4
*f^4*g*abs(e) - 10*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^4*d^4*e^6*f^4*g*abs(e) - 198*sqrt(-c*d^2
*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^5*d^7*e^3*f^3*g^2*abs(e) + 56*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*
g^2)*a*c^4*d^5*e^5*f^3*g^2*abs(e) - 8*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c^3*d^3*e^7*f^3*g^2*a
bs(e) - 198*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^5*d^8*e^2*f^2*g^3*abs(e) + 1188*sqrt(-c*d^2*e + a
*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^4*d^6*e^4*f^2*g^3*abs(e) - 1272*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^
2)*a^2*c^3*d^4*e^6*f^2*g^3*abs(e) + 432*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^3*c^2*d^2*e^8*f^2*g^3
*abs(e) + 85*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^5*d^9*e*f*g^4*abs(e) + 56*sqrt(-c*d^2*e + a*e^3)
*sqrt(c*d*f*g - a*e*g^2)*a*c^4*d^7*e^3*f*g^4*abs(e) - 1272*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*
c^3*d^5*e^5*f*g^4*abs(e) + 1696*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^3*c^2*d^3*e^7*f*g^4*abs(e) -
640*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^4*c*d*e^9*f*g^4*abs(e) - 15*sqrt(-c*d^2*e + a*e^3)*sqrt(c
*d*f*g - a*e*g^2)*c^5*d^10*g^5*abs(e) - 10*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^4*d^8*e^2*g^5*ab
s(e) - 8*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c^3*d^6*e^4*g^5*abs(e) + 432*sqrt(-c*d^2*e + a*e^3
)*sqrt(c*d*f*g - a*e*g^2)*a^3*c^2*d^4*e^6*g^5*abs(e) - 640*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^4*
c*d^2*e^8*g^5*abs(e) + 256*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^5*e^10*g^5*abs(e))/(sqrt(c*d*f*g -
a*e*g^2)*c^3*d^3*e^7*f^9*g^3 - 6*sqrt(c*d*f*g - a*e*g^2)*c^3*d^4*e^6*f^8*g^4 - 3*sqrt(c*d*f*g - a*e*g^2)*a*c^
2*d^2*e^8*f^8*g^4 + 15*sqrt(c*d*f*g - a*e*g^2)*c^3*d^5*e^5*f^7*g^5 + 18*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^3*e^7*
f^7*g^5 + 3*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d*e^9*f^7*g^5 - 20*sqrt(c*d*f*g - a*e*g^2)*c^3*d^6*e^4*f^6*g^6 - 45*
sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^4*e^6*f^6*g^6 - 18*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^2*e^8*f^6*g^6 - sqrt(c*d*f*
g - a*e*g^2)*a^3*e^10*f^6*g^6 + 15*sqrt(c*d*f*g - a*e*g^2)*c^3*d^7*e^3*f^5*g^7 + 60*sqrt(c*d*f*g - a*e*g^2)*a*
c^2*d^5*e^5*f^5*g^7 + 45*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^3*e^7*f^5*g^7 + 6*sqrt(c*d*f*g - a*e*g^2)*a^3*d*e^9*f
^5*g^7 - 6*sqrt(c*d*f*g - a*e*g^2)*c^3*d^8*e^2*f^4*g^8 - 45*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^6*e^4*f^4*g^8 - 60
*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^4*e^6*f^4*g^8 - 15*sqrt(c*d*f*g - a*e*g^2)*a^3*d^2*e^8*f^4*g^8 + sqrt(c*d*f*g
- a*e*g^2)*c^3*d^9*e*f^3*g^9 + 18*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^7*e^3*f^3*g^9 + 45*sqrt(c*d*f*g - a*e*g^2)*
a^2*c*d^5*e^5*f^3*g^9 + 20*sqrt(c*d*f*g - a*e*g^2)*a^3*d^3*e^7*f^3*g^9 - 3*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^8*e
^2*f^2*g^10 - 18*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^6*e^4*f^2*g^10 - 15*sqrt(c*d*f*g - a*e*g^2)*a^3*d^4*e^6*f^2*g
^10 + 3*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^7*e^3*f*g^11 + 6*sqrt(c*d*f*g - a*e*g^2)*a^3*d^5*e^5*f*g^11 - sqrt(c*d
*f*g - a*e*g^2)*a^3*d^6*e^4*g^12) - 1/1536*(15*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^11*d^11*e^10*f^5*abs(
e) - 75*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^10*d^10*e^11*f^4*g*abs(e) + 150*sqrt((e*x + d)*c*d*e - c*d
^2*e + a*e^3)*a^2*c^9*d^9*e^12*f^3*g^2*abs(e) - 150*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^8*d^8*e^13*f
^2*g^3*abs(e) + 75*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^4*c^7*d^7*e^14*f*g^4*abs(e) - 15*sqrt((e*x + d)*c
*d*e - c*d^2*e + a*e^3)*a^5*c^6*d^6*e^15*g^5*abs(e) + 85*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^10*d^10*e
^8*f^4*g*abs(e) - 340*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^9*d^9*e^9*f^3*g^2*abs(e) + 510*((e*x + d)*
c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^8*d^8*e^10*f^2*g^3*abs(e) - 340*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)
*a^3*c^7*d^7*e^11*f*g^4*abs(e) + 85*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*c^6*d^6*e^12*g^5*abs(e) + 19
8*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^9*d^9*e^6*f^3*g^2*abs(e) - 594*((e*x + d)*c*d*e - c*d^2*e + a*e^
3)^(5/2)*a*c^8*d^8*e^7*f^2*g^3*abs(e) + 594*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*c^7*d^7*e^8*f*g^4*ab
s(e) - 198*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^3*c^6*d^6*e^9*g^5*abs(e) - 198*((e*x + d)*c*d*e - c*d^2
*e + a*e^3)^(7/2)*c^8*d^8*e^4*f^2*g^3*abs(e) + 396*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*c^7*d^7*e^5*f*g
^4*abs(e) - 198*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a^2*c^6*d^6*e^6*g^5*abs(e) - 85*((e*x + d)*c*d*e - c
*d^2*e + a*e^3)^(9/2)*c^7*d^7*e^2*f*g^4*abs(e) + 85*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2)*a*c^6*d^6*e^3*g^
5*abs(e) - 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(11/2)*c^6*d^6*g^5*abs(e))/((c^3*d^3*f^3*g^3 - 3*a*c^2*d^2*e
*f^2*g^4 + 3*a^2*c*d*e^2*f*g^5 - a^3*e^3*g^6)*(c*d*e^2*f - a*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g)^6)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (f+g\,x\right )}^7\,{\left (d+e\,x\right )}^{5/2}} \,d x \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^7*(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)^7*(d + e*x)^(5/2)), x)