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

Optimal. Leaf size=376 \[ -\frac {5 \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^4 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{64 c^{3/2} d^{3/2} g^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {5 \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt {d+e x}}+\frac {5 (f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{32 g^3 \sqrt {d+e x}}-\frac {5 (f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{24 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}} \]

[Out]

-5/24*(-a*e*g+c*d*f)*(g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g^2/(e*x+d)^(3/2)+1/4*(g*x+f)^(3/2)
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/g/(e*x+d)^(5/2)-5/64*(-a*e*g+c*d*f)^4*arctanh(g^(1/2)*(c*d*x+a*e)^(1/
2)/c^(1/2)/d^(1/2)/(g*x+f)^(1/2))*(c*d*x+a*e)^(1/2)*(e*x+d)^(1/2)/c^(3/2)/d^(3/2)/g^(7/2)/(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)+5/32*(-a*e*g+c*d*f)^2*(g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(e*x+d)^(1
/2)-5/64*(-a*e*g+c*d*f)^3*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/g^3/(e*x+d)^(1/2)

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Rubi [A]  time = 0.68, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {864, 870, 891, 63, 217, 206} \[ -\frac {5 \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^4 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{64 c^{3/2} d^{3/2} g^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {5 \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c d g^3 \sqrt {d+e x}}+\frac {5 (f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{32 g^3 \sqrt {d+e x}}-\frac {5 (f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{24 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*(c*d*f - a*e*g)^3*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*c*d*g^3*Sqrt[d + e*x]) +
(5*(c*d*f - a*e*g)^2*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(32*g^3*Sqrt[d + e*x]) - (5*
(c*d*f - a*e*g)*(f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*g^2*(d + e*x)^(3/2)) + ((f
+ g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(4*g*(d + e*x)^(5/2)) - (5*(c*d*f - a*e*g)^4*Sqrt[
a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(64*c^(3/2)*d
^(3/2)*g^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 864

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] &&  !IntegerQ
[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 870

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

Rule 891

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Dist[(a + b*x + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(d + e*x)^(m + p)*
(f + g*x)^n*(a/d + (c*x)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, 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] &&  !IGtQ[m, 0] &&  !IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {f+g x} \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}} \, dx &=\frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {(5 (c d f-a e g)) \int \frac {\sqrt {f+g x} \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}} \, dx}{8 g}\\ &=-\frac {5 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}+\frac {\left (5 (c d f-a e g)^2\right ) \int \frac {\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{16 g^2}\\ &=\frac {5 (c d f-a e g)^2 (f+g x)^{3/2} \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}}-\frac {5 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {\left (5 (c d f-a e g)^3\right ) \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{64 g^3}\\ &=-\frac {5 (c d f-a e g)^3 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g)^2 (f+g x)^{3/2} \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}}-\frac {5 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {\left (5 (c d f-a e g)^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c d g^3}\\ &=-\frac {5 (c d f-a e g)^3 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g)^2 (f+g x)^{3/2} \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}}-\frac {5 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {\left (5 (c d f-a e g)^4 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}} \, dx}{128 c d g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {5 (c d f-a e g)^3 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g)^2 (f+g x)^{3/2} \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}}-\frac {5 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {\left (5 (c d f-a e g)^4 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {f-\frac {a e g}{c d}+\frac {g x^2}{c d}}} \, dx,x,\sqrt {a e+c d x}\right )}{64 c^2 d^2 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {5 (c d f-a e g)^3 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g)^2 (f+g x)^{3/2} \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}}-\frac {5 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {\left (5 (c d f-a e g)^4 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {g x^2}{c d}} \, dx,x,\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}\right )}{64 c^2 d^2 g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {5 (c d f-a e g)^3 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {5 (c d f-a e g)^2 (f+g x)^{3/2} \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}}-\frac {5 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {5 (c d f-a e g)^4 \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{64 c^{3/2} d^{3/2} g^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}

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Mathematica [A]  time = 1.13, size = 299, normalized size = 0.80 \[ \frac {\sqrt {c} \sqrt {d} \sqrt {d+e x} \left (\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {c d} (f+g x) (a e+c d x) \left (15 a^3 e^3 g^3+a^2 c d e^2 g^2 (73 f+118 g x)+a c^2 d^2 e g \left (-55 f^2+36 f g x+136 g^2 x^2\right )+c^3 d^3 \left (15 f^3-10 f^2 g x+8 f g^2 x^2+48 g^3 x^3\right )\right )-15 \sqrt {a e+c d x} (c d f-a e g)^{9/2} \sqrt {\frac {c d (f+g x)}{c d f-a e g}} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d f-a e g}}\right )\right )}{192 g^{7/2} (c d)^{5/2} \sqrt {f+g x} \sqrt {(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x]*(Sqrt[c]*Sqrt[d]*Sqrt[c*d]*Sqrt[g]*(a*e + c*d*x)*(f + g*x)*(15*a^3*e^3*g^3 + a^
2*c*d*e^2*g^2*(73*f + 118*g*x) + a*c^2*d^2*e*g*(-55*f^2 + 36*f*g*x + 136*g^2*x^2) + c^3*d^3*(15*f^3 - 10*f^2*g
*x + 8*f*g^2*x^2 + 48*g^3*x^3)) - 15*(c*d*f - a*e*g)^(9/2)*Sqrt[a*e + c*d*x]*Sqrt[(c*d*(f + g*x))/(c*d*f - a*e
*g)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/(192*(c*d)^(5/2)*g
^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])

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fricas [A]  time = 5.35, size = 1065, normalized size = 2.83 \[ \left [\frac {4 \, {\left (48 \, c^{4} d^{4} g^{4} x^{3} + 15 \, c^{4} d^{4} f^{3} g - 55 \, a c^{3} d^{3} e f^{2} g^{2} + 73 \, a^{2} c^{2} d^{2} e^{2} f g^{3} + 15 \, a^{3} c d e^{3} g^{4} + 8 \, {\left (c^{4} d^{4} f g^{3} + 17 \, a c^{3} d^{3} e g^{4}\right )} x^{2} - 2 \, {\left (5 \, c^{4} d^{4} f^{2} g^{2} - 18 \, a c^{3} d^{3} e f g^{3} - 59 \, a^{2} c^{2} d^{2} e^{2} g^{4}\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} \sqrt {g x + f} + 15 \, {\left (c^{4} d^{5} f^{4} - 4 \, a c^{3} d^{4} e f^{3} g + 6 \, a^{2} c^{2} d^{3} e^{2} f^{2} g^{2} - 4 \, a^{3} c d^{2} e^{3} f g^{3} + a^{4} d e^{4} g^{4} + {\left (c^{4} d^{4} e f^{4} - 4 \, a c^{3} d^{3} e^{2} f^{3} g + 6 \, a^{2} c^{2} d^{2} e^{3} f^{2} g^{2} - 4 \, a^{3} c d e^{4} f g^{3} + a^{4} e^{5} g^{4}\right )} x\right )} \sqrt {c d g} \log \left (-\frac {8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + c d f + a e g\right )} \sqrt {c d g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (c^{2} d^{2} e f g + {\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, {\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g + {\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right )}{768 \, {\left (c^{2} d^{2} e g^{4} x + c^{2} d^{3} g^{4}\right )}}, \frac {2 \, {\left (48 \, c^{4} d^{4} g^{4} x^{3} + 15 \, c^{4} d^{4} f^{3} g - 55 \, a c^{3} d^{3} e f^{2} g^{2} + 73 \, a^{2} c^{2} d^{2} e^{2} f g^{3} + 15 \, a^{3} c d e^{3} g^{4} + 8 \, {\left (c^{4} d^{4} f g^{3} + 17 \, a c^{3} d^{3} e g^{4}\right )} x^{2} - 2 \, {\left (5 \, c^{4} d^{4} f^{2} g^{2} - 18 \, a c^{3} d^{3} e f g^{3} - 59 \, a^{2} c^{2} d^{2} e^{2} g^{4}\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} \sqrt {g x + f} + 15 \, {\left (c^{4} d^{5} f^{4} - 4 \, a c^{3} d^{4} e f^{3} g + 6 \, a^{2} c^{2} d^{3} e^{2} f^{2} g^{2} - 4 \, a^{3} c d^{2} e^{3} f g^{3} + a^{4} d e^{4} g^{4} + {\left (c^{4} d^{4} e f^{4} - 4 \, a c^{3} d^{3} e^{2} f^{3} g + 6 \, a^{2} c^{2} d^{2} e^{3} f^{2} g^{2} - 4 \, a^{3} c d e^{4} f g^{3} + a^{4} e^{5} g^{4}\right )} x\right )} \sqrt {-c d g} \arctan \left (\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, c d e g x^{2} + c d^{2} f + a d e g + {\left (c d e f + {\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x}\right )}{384 \, {\left (c^{2} d^{2} e g^{4} x + c^{2} d^{3} g^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

[1/768*(4*(48*c^4*d^4*g^4*x^3 + 15*c^4*d^4*f^3*g - 55*a*c^3*d^3*e*f^2*g^2 + 73*a^2*c^2*d^2*e^2*f*g^3 + 15*a^3*
c*d*e^3*g^4 + 8*(c^4*d^4*f*g^3 + 17*a*c^3*d^3*e*g^4)*x^2 - 2*(5*c^4*d^4*f^2*g^2 - 18*a*c^3*d^3*e*f*g^3 - 59*a^
2*c^2*d^2*e^2*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f) + 15*(c^4*d^5*f^
4 - 4*a*c^3*d^4*e*f^3*g + 6*a^2*c^2*d^3*e^2*f^2*g^2 - 4*a^3*c*d^2*e^3*f*g^3 + a^4*d*e^4*g^4 + (c^4*d^4*e*f^4 -
4*a*c^3*d^3*e^2*f^3*g + 6*a^2*c^2*d^2*e^3*f^2*g^2 - 4*a^3*c*d*e^4*f*g^3 + a^4*e^5*g^4)*x)*sqrt(c*d*g)*log(-(8
*c^2*d^2*e*g^2*x^3 + c^2*d^3*f^2 + 6*a*c*d^2*e*f*g + a^2*d*e^2*g^2 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2
)*x)*(2*c*d*g*x + c*d*f + a*e*g)*sqrt(c*d*g)*sqrt(e*x + d)*sqrt(g*x + f) + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d
*e^2)*g^2)*x^2 + (c^2*d^2*e*f^2 + 2*(4*c^2*d^3 + 3*a*c*d*e^2)*f*g + (8*a*c*d^2*e + a^2*e^3)*g^2)*x)/(e*x + d))
)/(c^2*d^2*e*g^4*x + c^2*d^3*g^4), 1/384*(2*(48*c^4*d^4*g^4*x^3 + 15*c^4*d^4*f^3*g - 55*a*c^3*d^3*e*f^2*g^2 +
73*a^2*c^2*d^2*e^2*f*g^3 + 15*a^3*c*d*e^3*g^4 + 8*(c^4*d^4*f*g^3 + 17*a*c^3*d^3*e*g^4)*x^2 - 2*(5*c^4*d^4*f^2*
g^2 - 18*a*c^3*d^3*e*f*g^3 - 59*a^2*c^2*d^2*e^2*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x +
d)*sqrt(g*x + f) + 15*(c^4*d^5*f^4 - 4*a*c^3*d^4*e*f^3*g + 6*a^2*c^2*d^3*e^2*f^2*g^2 - 4*a^3*c*d^2*e^3*f*g^3
+ a^4*d*e^4*g^4 + (c^4*d^4*e*f^4 - 4*a*c^3*d^3*e^2*f^3*g + 6*a^2*c^2*d^2*e^3*f^2*g^2 - 4*a^3*c*d*e^4*f*g^3 + a
^4*e^5*g^4)*x)*sqrt(-c*d*g)*arctan(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*g)*sqrt(e*x + d)*sq
rt(g*x + f)/(2*c*d*e*g*x^2 + c*d^2*f + a*d*e*g + (c*d*e*f + (2*c*d^2 + a*e^2)*g)*x)))/(c^2*d^2*e*g^4*x + c^2*d
^3*g^4)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:inde
x.cc index_m operator + Error: Bad Argument Valueindex.cc index_m operator + Error: Bad Argument Valueindex.cc
index_m operator + Error: Bad Argument ValueEvaluation time: 7.91Done

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maple [B]  time = 0.02, size = 870, normalized size = 2.31 \[ -\frac {\sqrt {g x +f}\, \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (15 a^{4} e^{4} g^{4} \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )-60 a^{3} c d \,e^{3} f \,g^{3} \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )+90 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2} \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )-60 a \,c^{3} d^{3} e \,f^{3} g \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )+15 c^{4} d^{4} f^{4} \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right )-96 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, c^{3} d^{3} g^{3} x^{3}-272 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, a \,c^{2} d^{2} e \,g^{3} x^{2}-16 \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, c^{3} d^{3} f \,g^{2} x^{2}-236 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a^{2} c d \,e^{2} g^{3} x -72 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a \,c^{2} d^{2} e f \,g^{2} x +20 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, c^{3} d^{3} f^{2} g x -30 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a^{3} e^{3} g^{3}-146 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a^{2} c d \,e^{2} f \,g^{2}+110 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, a \,c^{2} d^{2} e \,f^{2} g -30 \sqrt {c d g}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, c^{3} d^{3} f^{3}\right )}{384 \sqrt {e x +d}\, \sqrt {c d g \,x^{2}+a e g x +c d f x +a e f}\, \sqrt {c d g}\, c d \,g^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(g*x+f)^(1/2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(-96*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d
*g)^(1/2)*c^3*d^3*g^3*x^3+15*a^4*e^4*g^4*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/
2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))-60*a^3*c*d*e^3*f*g^3*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g*x+c*d*f
*x+a*e*f)^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))+90*a^2*c^2*d^2*e^2*f^2*g^2*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*(c*d*
g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))-60*a*c^3*d^3*e*f^3*g*ln(1/2*(2*c*d*g*x+a*e*g+
c*d*f+2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))+15*c^4*d^4*f^4*ln(1/2*(2*c*d*g*x
+a*e*g+c*d*f+2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))-272*(c*d*g*x^2+a*e*g*x+c*
d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2)*a*c^2*d^2*e*g^3*x^2-16*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2)*
c^3*d^3*f*g^2*x^2-236*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*a^2*c*d*e^2*g^3*x-72*(c*d*g)^(1/2)
*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*a*c^2*d^2*e*f*g^2*x+20*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f
)^(1/2)*c^3*d^3*f^2*g*x-30*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*a^3*e^3*g^3-146*(c*d*g)^(1/2)
*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*a^2*c*d*e^2*f*g^2+110*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)
^(1/2)*a*c^2*d^2*e*f^2*g-30*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*c^3*d^3*f^3)/(e*x+d)^(1/2)/c
/d/(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)/g^3/(c*d*g)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}} \sqrt {g x + f}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {f+g\,x}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**(1/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

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