3.8.8 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^4} \, dx\) [708]

Optimal. Leaf size=253 \[ -\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^3 \sqrt {d+e x} (f+g x)}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}+\frac {5 c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{8 g^{7/2} \sqrt {c d f-a e g}} \]

[Out]

-5/12*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)^2-1/3*(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)^3+5/8*c^3*d^3*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)^(1/2)-5/8*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)/(e*x+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.22, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {876, 888, 211} \begin {gather*} \frac {5 c^3 d^3 \text {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 )}{8 g^{7/2} \sqrt {c d f-a e g}}-\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 g^3 \sqrt {d+e x} (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}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3} \end {gather*}

Antiderivative was successfully verified.

[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)^4),x]

[Out]

(-5*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*g^3*Sqrt[d + e*x]*(f + g*x)) - (5*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)^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(5/2)/(3*g*(d + e*x)^(5/2)*(f + g*x)^3) + (5*c^3*d^3*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])])/(8*g^(7/2)*Sqrt[c*d*f - a*e*g])

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 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*} \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)^4} \, dx &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}+\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)^3} \, dx}{6 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}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}+\frac {\left (5 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)^2} \, dx}{8 g^2}\\ &=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^3 \sqrt {d+e x} (f+g 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}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}+\frac {\left (5 c^3 d^3\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}{16 g^3}\\ &=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^3 \sqrt {d+e x} (f+g 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}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}+\frac {\left (5 c^3 d^3 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 )}{8 g^3}\\ &=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^3 \sqrt {d+e x} (f+g 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}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}+\frac {5 c^3 d^3 \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 )}{8 g^{7/2} \sqrt {c d f-a e g}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.82, size = 171, normalized size = 0.68 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {g} \left (8 a^2 e^2 g^2+2 a c d e g (5 f+13 g x)+c^2 d^2 \left (15 f^2+40 f g x+33 g^2 x^2\right )\right )}{(f+g x)^3}+\frac {15 c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{\sqrt {c d f-a e g} \sqrt {a e+c d x}}\right )}{24 g^{7/2} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[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)^4),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[g]*(8*a^2*e^2*g^2 + 2*a*c*d*e*g*(5*f + 13*g*x) + c^2*d^2*(15*f^2 + 40*
f*g*x + 33*g^2*x^2)))/(f + g*x)^3) + (15*c^3*d^3*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]])/(Sqr
t[c*d*f - a*e*g]*Sqrt[a*e + c*d*x])))/(24*g^(7/2)*Sqrt[d + e*x])

________________________________________________________________________________________

Maple [A]
time = 0.15, size = 431, normalized size = 1.70

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} g^{3} x^{3}+45 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f \,g^{2} x^{2}+45 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{2} g x +15 \arctanh \left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c^{3} d^{3} f^{3}+33 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{2} x^{2}+26 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e \,g^{2} x +40 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f g x +8 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{2}+10 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e f g +15 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2}\right )}{24 \sqrt {e x +d}\, \sqrt {c d x +a e}\, g^{3} \left (g x +f \right )^{3} \sqrt {\left (a e g -c d f \right ) g}}\)	\(431\)

Verification of antiderivative is not currently implemented for this CAS.

[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)^4,x,method=_RETURNVERBOSE)

[Out]

-1/24*((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))*c^3*d^3*g^3*x^3+45*
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+45*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))*c^3*d^3*f^3+33*((a*e*
g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^2*g^2*x^2+26*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*g^2*x
+40*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^2*f*g*x+8*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^
2*g^2+10*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d*e*f*g+15*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(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)^3/((a*e*g-c*d*f)*g)^(1/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)^4,x, algorithm="maxima")

[Out]

integrate((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((g*x + f)^4*(x*e + d)^(5/2)), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (231) = 462\).
time = 1.60, size = 1176, normalized size = 4.65 \begin {gather*} \left [-\frac {15 \, {\left (c^{3} d^{4} g^{3} x^{3} + 3 \, c^{3} d^{4} f g^{2} x^{2} + 3 \, c^{3} d^{4} f^{2} g x + c^{3} d^{4} f^{3} + {\left (c^{3} d^{3} g^{3} x^{4} + 3 \, c^{3} d^{3} f g^{2} x^{3} + 3 \, c^{3} d^{3} f^{2} g x^{2} + c^{3} d^{3} f^{3} x\right )} e\right )} \sqrt {-c d f g + a g^{2} e} \log \left (-\frac {c d^{2} g x - c d^{2} f + 2 \, a g x e^{2} + {\left (c d g x^{2} - c d f x + 2 \, a d g\right )} e - 2 \, \sqrt {-c d f g + a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{d g x + d f + {\left (g x^{2} + f x\right )} e}\right ) + 2 \, {\left (33 \, c^{3} d^{3} f g^{3} x^{2} + 40 \, c^{3} d^{3} f^{2} g^{2} x + 15 \, c^{3} d^{3} f^{3} g - 8 \, a^{3} g^{4} e^{3} - 2 \, {\left (13 \, a^{2} c d g^{4} x + a^{2} c d f g^{3}\right )} e^{2} - {\left (33 \, a c^{2} d^{2} g^{4} x^{2} + 14 \, a c^{2} d^{2} f g^{3} x + 5 \, a c^{2} d^{2} f^{2} g^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{48 \, {\left (c d^{2} f g^{7} x^{3} + 3 \, c d^{2} f^{2} g^{6} x^{2} + 3 \, c d^{2} f^{3} g^{5} x + c d^{2} f^{4} g^{4} - {\left (a g^{8} x^{4} + 3 \, a f g^{7} x^{3} + 3 \, a f^{2} g^{6} x^{2} + a f^{3} g^{5} x\right )} e^{2} + {\left (c d f g^{7} x^{4} - a d f^{3} g^{5} + {\left (3 \, c d f^{2} g^{6} - a d g^{8}\right )} x^{3} + 3 \, {\left (c d f^{3} g^{5} - a d f g^{7}\right )} x^{2} + {\left (c d f^{4} g^{4} - 3 \, a d f^{2} g^{6}\right )} x\right )} e\right )}}, -\frac {15 \, {\left (c^{3} d^{4} g^{3} x^{3} + 3 \, c^{3} d^{4} f g^{2} x^{2} + 3 \, c^{3} d^{4} f^{2} g x + c^{3} d^{4} f^{3} + {\left (c^{3} d^{3} g^{3} x^{4} + 3 \, c^{3} d^{3} f g^{2} x^{3} + 3 \, c^{3} d^{3} f^{2} g x^{2} + c^{3} d^{3} f^{3} x\right )} e\right )} \sqrt {c d f g - a g^{2} e} \arctan \left (\frac {\sqrt {c d f g - a g^{2} e} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{c d^{2} g x + a g x e^{2} + {\left (c d g x^{2} + a d g\right )} e}\right ) + {\left (33 \, c^{3} d^{3} f g^{3} x^{2} + 40 \, c^{3} d^{3} f^{2} g^{2} x + 15 \, c^{3} d^{3} f^{3} g - 8 \, a^{3} g^{4} e^{3} - 2 \, {\left (13 \, a^{2} c d g^{4} x + a^{2} c d f g^{3}\right )} e^{2} - {\left (33 \, a c^{2} d^{2} g^{4} x^{2} + 14 \, a c^{2} d^{2} f g^{3} x + 5 \, a c^{2} d^{2} f^{2} g^{2}\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{24 \, {\left (c d^{2} f g^{7} x^{3} + 3 \, c d^{2} f^{2} g^{6} x^{2} + 3 \, c d^{2} f^{3} g^{5} x + c d^{2} f^{4} g^{4} - {\left (a g^{8} x^{4} + 3 \, a f g^{7} x^{3} + 3 \, a f^{2} g^{6} x^{2} + a f^{3} g^{5} x\right )} e^{2} + {\left (c d f g^{7} x^{4} - a d f^{3} g^{5} + {\left (3 \, c d f^{2} g^{6} - a d g^{8}\right )} x^{3} + 3 \, {\left (c d f^{3} g^{5} - a d f g^{7}\right )} x^{2} + {\left (c d f^{4} g^{4} - 3 \, a d f^{2} g^{6}\right )} x\right )} e\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)^4,x, algorithm="fricas")

[Out]

[-1/48*(15*(c^3*d^4*g^3*x^3 + 3*c^3*d^4*f*g^2*x^2 + 3*c^3*d^4*f^2*g*x + c^3*d^4*f^3 + (c^3*d^3*g^3*x^4 + 3*c^3
*d^3*f*g^2*x^3 + 3*c^3*d^3*f^2*g*x^2 + c^3*d^3*f^3*x)*e)*sqrt(-c*d*f*g + a*g^2*e)*log(-(c*d^2*g*x - c*d^2*f +
2*a*g*x*e^2 + (c*d*g*x^2 - c*d*f*x + 2*a*d*g)*e - 2*sqrt(-c*d*f*g + a*g^2*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2
+ a*d)*e)*sqrt(x*e + d))/(d*g*x + d*f + (g*x^2 + f*x)*e)) + 2*(33*c^3*d^3*f*g^3*x^2 + 40*c^3*d^3*f^2*g^2*x +
15*c^3*d^3*f^3*g - 8*a^3*g^4*e^3 - 2*(13*a^2*c*d*g^4*x + a^2*c*d*f*g^3)*e^2 - (33*a*c^2*d^2*g^4*x^2 + 14*a*c^2
*d^2*f*g^3*x + 5*a*c^2*d^2*f^2*g^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c*d^2*f*g^7
*x^3 + 3*c*d^2*f^2*g^6*x^2 + 3*c*d^2*f^3*g^5*x + c*d^2*f^4*g^4 - (a*g^8*x^4 + 3*a*f*g^7*x^3 + 3*a*f^2*g^6*x^2
+ a*f^3*g^5*x)*e^2 + (c*d*f*g^7*x^4 - a*d*f^3*g^5 + (3*c*d*f^2*g^6 - a*d*g^8)*x^3 + 3*(c*d*f^3*g^5 - a*d*f*g^7
)*x^2 + (c*d*f^4*g^4 - 3*a*d*f^2*g^6)*x)*e), -1/24*(15*(c^3*d^4*g^3*x^3 + 3*c^3*d^4*f*g^2*x^2 + 3*c^3*d^4*f^2*
g*x + c^3*d^4*f^3 + (c^3*d^3*g^3*x^4 + 3*c^3*d^3*f*g^2*x^3 + 3*c^3*d^3*f^2*g*x^2 + c^3*d^3*f^3*x)*e)*sqrt(c*d*
f*g - a*g^2*e)*arctan(sqrt(c*d*f*g - a*g^2*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d)/(c*d^2
*g*x + a*g*x*e^2 + (c*d*g*x^2 + a*d*g)*e)) + (33*c^3*d^3*f*g^3*x^2 + 40*c^3*d^3*f^2*g^2*x + 15*c^3*d^3*f^3*g -
8*a^3*g^4*e^3 - 2*(13*a^2*c*d*g^4*x + a^2*c*d*f*g^3)*e^2 - (33*a*c^2*d^2*g^4*x^2 + 14*a*c^2*d^2*f*g^3*x + 5*a
*c^2*d^2*f^2*g^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(c*d^2*f*g^7*x^3 + 3*c*d^2*f^2
*g^6*x^2 + 3*c*d^2*f^3*g^5*x + c*d^2*f^4*g^4 - (a*g^8*x^4 + 3*a*f*g^7*x^3 + 3*a*f^2*g^6*x^2 + a*f^3*g^5*x)*e^2
+ (c*d*f*g^7*x^4 - a*d*f^3*g^5 + (3*c*d*f^2*g^6 - a*d*g^8)*x^3 + 3*(c*d*f^3*g^5 - a*d*f*g^7)*x^2 + (c*d*f^4*g
^4 - 3*a*d*f^2*g^6)*x)*e)]

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)**4,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3062 deep

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)^4,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^4\,{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^4*(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)^4*(d + e*x)^(5/2)), x)

________________________________________________________________________________________