3.8.42 \(\int \frac {(f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\) [742]

Optimal. Leaf size=382 \[ \frac {3 (c d f-a e g)^3 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c^2 d^2 g^2 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 c d g^2 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}+\frac {3 (c d f-a e g)^4 \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{64 c^{5/2} d^{5/2} g^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

1/4*(g*x+f)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g/(e*x+d)^(3/2)+3/64*(-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^(5/2)/d^(5/2)/g^(5/2)/(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/32*(-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)/c/d/g^2/(e*x+d)^(1/2)-1/8*(-a*e*g+c*d*f)*(g*x+f)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^2/(e*x+d)^
(1/2)+3/64*(-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^2/d^2/g^2/(e*x+d)^(1/2)

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Rubi [A]
time = 0.45, antiderivative size = 382, 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 = {878, 884, 905, 65, 223, 212} \begin {gather*} \frac {3 \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^{5/2} d^{5/2} g^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {3 \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^2 d^2 g^2 \sqrt {d+e x}}+\frac {(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 c d g^2 \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} (c d f-a e g)}{8 g^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

(3*(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^2*d^2*g^2*Sqrt[d + e*x])
+ ((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*c*d*g^2*Sqrt[d + e*x])
- ((c*d*f - a*e*g)*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*g^2*Sqrt[d + e*x]) + ((f +
g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(4*g*(d + e*x)^(3/2)) + (3*(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^(5/2)*d^(
5/2)*g^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

Rule 65

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 212

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

Rule 223

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 878

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] &&  !Intege
rQ[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 884

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] &&  !Int
egerQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])

Rule 905

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/e)*x)^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)^{3/2} \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 &=\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}-\frac {(3 (c d f-a e g)) \int \frac {(f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{8 g}\\ &=-\frac {(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}+\frac {(c d f-a e g)^2 \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}{16 g^2}\\ &=\frac {(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 c d g^2 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}+\frac {\left (3 (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 d g^2}\\ &=\frac {3 (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^2 d^2 g^2 \sqrt {d+e x}}+\frac {(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 c d g^2 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}+\frac {\left (3 (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^2 d^2 g^2}\\ &=\frac {3 (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^2 d^2 g^2 \sqrt {d+e x}}+\frac {(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 c d g^2 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}+\frac {\left (3 (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^2 d^2 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac {3 (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^2 d^2 g^2 \sqrt {d+e x}}+\frac {(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 c d g^2 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}+\frac {\left (3 (c d f-a e g)^4 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {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^3 d^3 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac {3 (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^2 d^2 g^2 \sqrt {d+e x}}+\frac {(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 c d g^2 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}+\frac {\left (3 (c d f-a e g)^4 \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {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^3 d^3 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=\frac {3 (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^2 d^2 g^2 \sqrt {d+e x}}+\frac {(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 c d g^2 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^2 \sqrt {d+e x}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}+\frac {3 (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^{5/2} d^{5/2} g^{5/2} \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 = 0.63, size = 238, normalized size = 0.62 \begin {gather*} \frac {(c d f-a e g)^4 ((a e+c d x) (d+e x))^{3/2} \left (-\frac {\sqrt {c} \sqrt {d} \sqrt {g} (a e+c d x)^2 \sqrt {f+g x} \left (3 g^3-\frac {11 c d g^2 (f+g x)}{a e+c d x}-\frac {11 c^2 d^2 g (f+g x)^2}{(a e+c d x)^2}+\frac {3 c^3 d^3 (f+g x)^3}{(a e+c d x)^3}\right )}{(c d f-a e g)^4}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{3/2}}\right )}{64 c^{5/2} d^{5/2} g^{5/2} (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]

[Out]

((c*d*f - a*e*g)^4*((a*e + c*d*x)*(d + e*x))^(3/2)*(-((Sqrt[c]*Sqrt[d]*Sqrt[g]*(a*e + c*d*x)^2*Sqrt[f + g*x]*(
3*g^3 - (11*c*d*g^2*(f + g*x))/(a*e + c*d*x) - (11*c^2*d^2*g*(f + g*x)^2)/(a*e + c*d*x)^2 + (3*c^3*d^3*(f + g*
x)^3)/(a*e + c*d*x)^3))/(c*d*f - a*e*g)^4) + (3*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*
d*x])])/(a*e + c*d*x)^(3/2)))/(64*c^(5/2)*d^(5/2)*g^(5/2)*(d + e*x)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(731\) vs. \(2(326)=652\).
time = 0.15, size = 732, normalized size = 1.92

method	result	size
default	\(\frac {\sqrt {g x +f}\, \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (32 c^{3} d^{3} g^{3} x^{3} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}+3 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}}{2 \sqrt {d g c}}\right ) a^{4} e^{4} g^{4}-12 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}}{2 \sqrt {d g c}}\right ) a^{3} c d \,e^{3} f \,g^{3}+18 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}}{2 \sqrt {d g c}}\right ) a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-12 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}}{2 \sqrt {d g c}}\right ) a \,c^{3} d^{3} e \,f^{3} g +3 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}}{2 \sqrt {d g c}}\right ) c^{4} d^{4} f^{4}+48 a \,c^{2} d^{2} e \,g^{3} x^{2} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}+48 c^{3} d^{3} f \,g^{2} x^{2} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}+4 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, a^{2} c d \,e^{2} g^{3} x +88 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, a \,c^{2} d^{2} e f \,g^{2} x +4 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, c^{3} d^{3} f^{2} g x -6 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, a^{3} e^{3} g^{3}+22 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, a^{2} c d \,e^{2} f \,g^{2}+22 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, a \,c^{2} d^{2} e \,f^{2} g -6 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, c^{3} d^{3} f^{3}\right )}{128 \sqrt {e x +d}\, d^{2} g^{2} c^{2} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}}\)	\(732\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/128*(g*x+f)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(32*c^3*d^3*g^3*x^3*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)+
3*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*a^4*e^4*g^4-12*ln(
1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*a^3*c*d*e^3*f*g^3+18*ln
(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*a^2*c^2*d^2*e^2*f^2*g^
2-12*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*a*c^3*d^3*e*f^3
*g+3*ln(1/2*(2*c*d*g*x+a*e*g+c*d*f+2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2))/(d*g*c)^(1/2))*c^4*d^4*f^4+48*
a*c^2*d^2*e*g^3*x^2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)+48*c^3*d^3*f*g^2*x^2*((g*x+f)*(c*d*x+a*e))^(1/2)
*(d*g*c)^(1/2)+4*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)*a^2*c*d*e^2*g^3*x+88*((g*x+f)*(c*d*x+a*e))^(1/2)*(d
*g*c)^(1/2)*a*c^2*d^2*e*f*g^2*x+4*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)*c^3*d^3*f^2*g*x-6*((g*x+f)*(c*d*x+
a*e))^(1/2)*(d*g*c)^(1/2)*a^3*e^3*g^3+22*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)*a^2*c*d*e^2*f*g^2+22*((g*x+
f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)*a*c^2*d^2*e*f^2*g-6*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)*c^3*d^3*f^3)
/(e*x+d)^(1/2)/d^2/g^2/c^2/((g*x+f)*(c*d*x+a*e))^(1/2)/(d*g*c)^(1/2)

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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((g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^(3/2)/(x*e + d)^(3/2), x)

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Fricas [A]
time = 6.10, size = 1073, normalized size = 2.81 \begin {gather*} \left [\frac {4 \, {\left (16 \, c^{4} d^{4} g^{4} x^{3} + 24 \, c^{4} d^{4} f g^{3} x^{2} + 2 \, c^{4} d^{4} f^{2} g^{2} x - 3 \, c^{4} d^{4} f^{3} g - 3 \, a^{3} c d g^{4} e^{3} + {\left (2 \, a^{2} c^{2} d^{2} g^{4} x + 11 \, a^{2} c^{2} d^{2} f g^{3}\right )} e^{2} + {\left (24 \, a c^{3} d^{3} g^{4} x^{2} + 44 \, a c^{3} d^{3} f g^{3} x + 11 \, a c^{3} d^{3} 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 {g x + f} \sqrt {x e + d} + 3 \, {\left (c^{4} d^{5} f^{4} + a^{4} g^{4} x e^{5} - {\left (4 \, a^{3} c d f g^{3} x - a^{4} d g^{4}\right )} e^{4} + 2 \, {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{2} x - 2 \, a^{3} c d^{2} f g^{3}\right )} e^{3} - 2 \, {\left (2 \, a c^{3} d^{3} f^{3} g x - 3 \, a^{2} c^{2} d^{3} f^{2} g^{2}\right )} e^{2} + {\left (c^{4} d^{4} f^{4} x - 4 \, a c^{3} d^{4} f^{3} g\right )} e\right )} \sqrt {c d g} \log \left (-\frac {8 \, c^{2} d^{3} g^{2} x^{2} + 8 \, c^{2} d^{3} f g x + c^{2} d^{3} f^{2} + a^{2} g^{2} x e^{3} + 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d g x + c d f + a g e\right )} \sqrt {c d g} \sqrt {g x + f} \sqrt {x e + d} + {\left (8 \, a c d g^{2} x^{2} + 6 \, a c d f g x + a^{2} d g^{2}\right )} e^{2} + {\left (8 \, c^{2} d^{2} g^{2} x^{3} + 8 \, c^{2} d^{2} f g x^{2} + 6 \, a c d^{2} f g + {\left (c^{2} d^{2} f^{2} + 8 \, a c d^{2} g^{2}\right )} x\right )} e}{x e + d}\right )}{256 \, {\left (c^{3} d^{3} g^{3} x e + c^{3} d^{4} g^{3}\right )}}, \frac {2 \, {\left (16 \, c^{4} d^{4} g^{4} x^{3} + 24 \, c^{4} d^{4} f g^{3} x^{2} + 2 \, c^{4} d^{4} f^{2} g^{2} x - 3 \, c^{4} d^{4} f^{3} g - 3 \, a^{3} c d g^{4} e^{3} + {\left (2 \, a^{2} c^{2} d^{2} g^{4} x + 11 \, a^{2} c^{2} d^{2} f g^{3}\right )} e^{2} + {\left (24 \, a c^{3} d^{3} g^{4} x^{2} + 44 \, a c^{3} d^{3} f g^{3} x + 11 \, a c^{3} d^{3} 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 {g x + f} \sqrt {x e + d} - 3 \, {\left (c^{4} d^{5} f^{4} + a^{4} g^{4} x e^{5} - {\left (4 \, a^{3} c d f g^{3} x - a^{4} d g^{4}\right )} e^{4} + 2 \, {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{2} x - 2 \, a^{3} c d^{2} f g^{3}\right )} e^{3} - 2 \, {\left (2 \, a c^{3} d^{3} f^{3} g x - 3 \, a^{2} c^{2} d^{3} f^{2} g^{2}\right )} e^{2} + {\left (c^{4} d^{4} f^{4} x - 4 \, a c^{3} d^{4} f^{3} g\right )} e\right )} \sqrt {-c d g} \arctan \left (\frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-c d g} \sqrt {g x + f} \sqrt {x e + d}}{2 \, c d^{2} g x + c d^{2} f + a g x e^{2} + {\left (2 \, c d g x^{2} + c d f x + a d g\right )} e}\right )}{128 \, {\left (c^{3} d^{3} g^{3} x e + c^{3} d^{4} g^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

[1/256*(4*(16*c^4*d^4*g^4*x^3 + 24*c^4*d^4*f*g^3*x^2 + 2*c^4*d^4*f^2*g^2*x - 3*c^4*d^4*f^3*g - 3*a^3*c*d*g^4*e
^3 + (2*a^2*c^2*d^2*g^4*x + 11*a^2*c^2*d^2*f*g^3)*e^2 + (24*a*c^3*d^3*g^4*x^2 + 44*a*c^3*d^3*f*g^3*x + 11*a*c^
3*d^3*f^2*g^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(g*x + f)*sqrt(x*e + d) + 3*(c^4*d^5*f^4 + a
^4*g^4*x*e^5 - (4*a^3*c*d*f*g^3*x - a^4*d*g^4)*e^4 + 2*(3*a^2*c^2*d^2*f^2*g^2*x - 2*a^3*c*d^2*f*g^3)*e^3 - 2*(
2*a*c^3*d^3*f^3*g*x - 3*a^2*c^2*d^3*f^2*g^2)*e^2 + (c^4*d^4*f^4*x - 4*a*c^3*d^4*f^3*g)*e)*sqrt(c*d*g)*log(-(8*
c^2*d^3*g^2*x^2 + 8*c^2*d^3*f*g*x + c^2*d^3*f^2 + a^2*g^2*x*e^3 + 4*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e
)*(2*c*d*g*x + c*d*f + a*g*e)*sqrt(c*d*g)*sqrt(g*x + f)*sqrt(x*e + d) + (8*a*c*d*g^2*x^2 + 6*a*c*d*f*g*x + a^2
*d*g^2)*e^2 + (8*c^2*d^2*g^2*x^3 + 8*c^2*d^2*f*g*x^2 + 6*a*c*d^2*f*g + (c^2*d^2*f^2 + 8*a*c*d^2*g^2)*x)*e)/(x*
e + d)))/(c^3*d^3*g^3*x*e + c^3*d^4*g^3), 1/128*(2*(16*c^4*d^4*g^4*x^3 + 24*c^4*d^4*f*g^3*x^2 + 2*c^4*d^4*f^2*
g^2*x - 3*c^4*d^4*f^3*g - 3*a^3*c*d*g^4*e^3 + (2*a^2*c^2*d^2*g^4*x + 11*a^2*c^2*d^2*f*g^3)*e^2 + (24*a*c^3*d^3
*g^4*x^2 + 44*a*c^3*d^3*f*g^3*x + 11*a*c^3*d^3*f^2*g^2)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(g*
x + f)*sqrt(x*e + d) - 3*(c^4*d^5*f^4 + a^4*g^4*x*e^5 - (4*a^3*c*d*f*g^3*x - a^4*d*g^4)*e^4 + 2*(3*a^2*c^2*d^2
*f^2*g^2*x - 2*a^3*c*d^2*f*g^3)*e^3 - 2*(2*a*c^3*d^3*f^3*g*x - 3*a^2*c^2*d^3*f^2*g^2)*e^2 + (c^4*d^4*f^4*x - 4
*a*c^3*d^4*f^3*g)*e)*sqrt(-c*d*g)*arctan(2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(-c*d*g)*sqrt(g*x +
f)*sqrt(x*e + d)/(2*c*d^2*g*x + c*d^2*f + a*g*x*e^2 + (2*c*d*g*x^2 + c*d*f*x + a*d*g)*e)))/(c^3*d^3*g^3*x*e +
c^3*d^4*g^3)]

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Sympy [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((g*x+f)**(3/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

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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((g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f+g\,x\right )}^{3/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x)

[Out]

int(((f + g*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2), x)

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