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

Optimal. Leaf size=385 \[ -\frac {5 \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^4 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{64 c^{7/2} d^{7/2} g^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {5 \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}{64 c^3 d^3 g \sqrt {d+e x}}-\frac {5 (f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {a e}{c d}-\frac {f}{g}\right )}{24 \sqrt {d+e x}} \]

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Rubi [A]  time = 0.72, antiderivative size = 385, 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} \begin {gather*} -\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^{7/2} d^{7/2} g^{3/2} \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^3 d^3 g \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}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (\frac {a e}{c d}-\frac {f}{g}\right )}{24 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x],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^3*d^3*g*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])/(96*c^2*d^2*g*Sqrt[d + e*x
]) + (((a*e)/(c*d) - f/g)*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*Sqrt[d + e*x]) + ((
f + g*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*g*Sqrt[d + e*x]) - (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^(7/2)*d^(
7/2)*g^(3/2)*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 {(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx &=\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\frac {(c d f-a e g) \int \frac {\sqrt {d+e x} (f+g x)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 g}\\ &=\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\frac {\left (5 (c d f-a e g)^2\right ) \int \frac {\sqrt {d+e x} (f+g x)^{3/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{48 c d g}\\ &=-\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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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 c^2 d^2 g}\\ &=-\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^3 d^3 g \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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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^3 d^3 g}\\ &=-\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^3 d^3 g \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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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^3 d^3 g \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^3 d^3 g \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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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^4 d^4 g \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^3 d^3 g \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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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^4 d^4 g \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^3 d^3 g \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}}{96 c^2 d^2 g \sqrt {d+e x}}+\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 \sqrt {d+e x}}+\frac {(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x}}-\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^{7/2} d^{7/2} g^{3/2} \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 = 1.16, size = 300, normalized size = 0.78 \begin {gather*} \frac {\sqrt {c 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^3-5 a^2 c d e^2 g^2 (11 f+2 g x)+a c^2 d^2 e g \left (73 f^2+36 f g x+8 g^2 x^2\right )+c^3 d^3 \left (15 f^3+118 f^2 g x+136 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 c^{9/2} d^{9/2} g^{3/2} \sqrt {f+g x} \sqrt {(d+e x) (a e+c d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c*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 - 5*a^2*c*
d*e^2*g^2*(11*f + 2*g*x) + a*c^2*d^2*e*g*(73*f^2 + 36*f*g*x + 8*g^2*x^2) + c^3*d^3*(15*f^3 + 118*f^2*g*x + 136
*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)]*A
rcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d*f - a*e*g])]))/(192*c^(9/2)*d^(9/2)*g^(
3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])

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IntegrateAlgebraic [A]  time = 8.58, size = 259, normalized size = 0.67 \begin {gather*} \frac {\sqrt {d+e x} \sqrt {a e+c d x} \left (\frac {\sqrt {a e+c d x} (c d f-a e g)^4 \left (\frac {73 c^2 d^2 g (a e+c d x)}{f+g x}+\frac {15 g^3 (a e+c d x)^3}{(f+g x)^3}-\frac {55 c d g^2 (a e+c d x)^2}{(f+g x)^2}+15 c^3 d^3\right )}{192 c^3 d^3 g \sqrt {f+g x} \left (c d-\frac {g (a e+c d x)}{f+g x}\right )^4}-\frac {5 (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^{7/2} d^{7/2} g^{3/2}}\right )}{\sqrt {(d+e x) (a e+c d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(((c*d*f - a*e*g)^4*Sqrt[a*e + c*d*x]*(15*c^3*d^3 + (15*g^3*(a*e + c*d*x)^3)/
(f + g*x)^3 - (55*c*d*g^2*(a*e + c*d*x)^2)/(f + g*x)^2 + (73*c^2*d^2*g*(a*e + c*d*x))/(f + g*x)))/(192*c^3*d^3
*g*Sqrt[f + g*x]*(c*d - (g*(a*e + c*d*x))/(f + g*x))^4) - (5*(c*d*f - a*e*g)^4*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d
*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(64*c^(7/2)*d^(7/2)*g^(3/2))))/Sqrt[(a*e + c*d*x)*(d + e*x)]

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fricas [A]  time = 2.69, size = 1065, normalized size = 2.77 \begin {gather*} \left [\frac {4 \, {\left (48 \, c^{4} d^{4} g^{4} x^{3} + 15 \, c^{4} d^{4} f^{3} g + 73 \, a c^{3} d^{3} e f^{2} g^{2} - 55 \, a^{2} c^{2} d^{2} e^{2} f g^{3} + 15 \, a^{3} c d e^{3} g^{4} + 8 \, {\left (17 \, c^{4} d^{4} f g^{3} + a c^{3} d^{3} e g^{4}\right )} x^{2} + 2 \, {\left (59 \, c^{4} d^{4} f^{2} g^{2} + 18 \, a c^{3} d^{3} e f g^{3} - 5 \, 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^{4} d^{4} e g^{2} x + c^{4} d^{5} g^{2}\right )}}, \frac {2 \, {\left (48 \, c^{4} d^{4} g^{4} x^{3} + 15 \, c^{4} d^{4} f^{3} g + 73 \, a c^{3} d^{3} e f^{2} g^{2} - 55 \, a^{2} c^{2} d^{2} e^{2} f g^{3} + 15 \, a^{3} c d e^{3} g^{4} + 8 \, {\left (17 \, c^{4} d^{4} f g^{3} + a c^{3} d^{3} e g^{4}\right )} x^{2} + 2 \, {\left (59 \, c^{4} d^{4} f^{2} g^{2} + 18 \, a c^{3} d^{3} e f g^{3} - 5 \, 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^{4} d^{4} e g^{2} x + c^{4} d^{5} g^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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maple [B]  time = 0.03, size = 870, normalized size = 2.26 \begin {gather*} -\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}-16 \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}-272 \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}+20 \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 -236 \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}+110 \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}-146 \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^{3} d^{3} g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^(5/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)/(e*x+d)^(1/2),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*x^3*c^3*d^3*g^3*(c*d*g*x^2+a*e*g*x+c*d*f*x+a
*e*f)^(1/2)*(c*d*g)^(1/2)+15*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))*a^4*e^4*g^4-60*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))*a^3*c*d*e^3*f*g^3+90*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))*a^2*c^2*d^2*e^2*f^2*g^2-60*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))*a*c^3*d^3*e*f^3*g+15*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))*c^4*d^4*f^4-16*x^2*a*c^2*d^2*e*g^3*(c
*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*(c*d*g)^(1/2)-272*x^2*c^3*d^3*f*g^2*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1
/2)*(c*d*g)^(1/2)+20*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*x*a^2*c*d*e^2*g^3-72*(c*d*g)^(1/2)*
(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)*x*a*c^2*d^2*e*f*g^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)*x*c^3*d^3*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)*a^3*e^3*g^3+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^2*c*d*e^2*f*g^2-146*(c*d*g)^(1/2)*(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)
^(1/2)*a*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)/g
/(c*d*g*x^2+a*e*g*x+c*d*f*x+a*e*f)^(1/2)/c^3/d^3/(c*d*g)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  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((g*x+f)**(5/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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