\(\int \frac {(d+e x)^{3/2} (f+g x)^{5/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [720]

Optimal result

Integrand size = 48, antiderivative size = 301 \[ \int \frac {(d+e x)^{3/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} (f+g x)^{5/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {15 g (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^3 d^3 \sqrt {d+e x}}+\frac {5 g (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^2 d^2 \sqrt {d+e x}}+\frac {15 \sqrt {g} (c d f-a e g)^2 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{4 c^{7/2} d^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

-2*(g*x+f)^(5/2)*(e*x+d)^(1/2)/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+15/4*(-a*e*g+c*d*f)^2*arctanh(g^(1/
2)*(c*d*x+a*e)^(1/2)/c^(1/2)/d^(1/2)/(g*x+f)^(1/2))*g^(1/2)*(c*d*x+a*e)^(1/2)*(e*x+d)^(1/2)/c^(7/2)/d^(7/2)/(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+5/2*g*(g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/(e*x
+d)^(1/2)+15/4*g*(-a*e*g+c*d*f)*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {880, 884, 905, 65, 223, 212} \[ \int \frac {(d+e x)^{3/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {15 \sqrt {g} \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g)^2 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{4 c^{7/2} d^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {15 g \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{4 c^3 d^3 \sqrt {d+e x}}+\frac {5 g (f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c^2 d^2 \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} (f+g x)^{5/2}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[In]

Int[((d + e*x)^(3/2)*(f + g*x)^(5/2))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(f + g*x)^(5/2))/(c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (15*g*(c*d*f - a*e*g)*S
qrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*c^3*d^3*Sqrt[d + e*x]) + (5*g*(f + g*x)^(3/2)*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*c^2*d^2*Sqrt[d + e*x]) + (15*Sqrt[g]*(c*d*f - a*e*g)^2*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])])/(4*c^(7/2)*d^(7/2)
*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 880

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*(p + 1))), x] - Dist[e*g*(n/(c*(p + 1))), I
nt[(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] &
& LtQ[p, -1] && GtQ[n, 0]

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*} \text {integral}& = -\frac {2 \sqrt {d+e x} (f+g x)^{5/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(5 g) \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}{c d} \\ & = -\frac {2 \sqrt {d+e x} (f+g x)^{5/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^2 d^2 \sqrt {d+e x}}+\frac {(15 g (c d f-a e g)) \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}{4 c^2 d^2} \\ & = -\frac {2 \sqrt {d+e x} (f+g x)^{5/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {15 g (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^3 d^3 \sqrt {d+e x}}+\frac {5 g (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^2 d^2 \sqrt {d+e x}}+\frac {\left (15 g (c d f-a e g)^2\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}{8 c^3 d^3} \\ & = -\frac {2 \sqrt {d+e x} (f+g x)^{5/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {15 g (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^3 d^3 \sqrt {d+e x}}+\frac {5 g (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^2 d^2 \sqrt {d+e x}}+\frac {\left (15 g (c d f-a e g)^2 \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}{8 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {2 \sqrt {d+e x} (f+g x)^{5/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {15 g (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^3 d^3 \sqrt {d+e x}}+\frac {5 g (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^2 d^2 \sqrt {d+e x}}+\frac {\left (15 g (c d f-a e g)^2 \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 )}{4 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {2 \sqrt {d+e x} (f+g x)^{5/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {15 g (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^3 d^3 \sqrt {d+e x}}+\frac {5 g (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^2 d^2 \sqrt {d+e x}}+\frac {\left (15 g (c d f-a e g)^2 \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 )}{4 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {2 \sqrt {d+e x} (f+g x)^{5/2}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {15 g (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^3 d^3 \sqrt {d+e x}}+\frac {5 g (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^2 d^2 \sqrt {d+e x}}+\frac {15 \sqrt {g} (c d f-a e g)^2 \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 )}{4 c^{7/2} d^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.61 \[ \int \frac {(d+e x)^{3/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (\sqrt {c} \sqrt {d} \sqrt {f+g x} \left (-15 a^2 e^2 g^2-5 a c d e g (-5 f+g x)+c^2 d^2 \left (-8 f^2+9 f g x+2 g^2 x^2\right )\right )+15 \sqrt {g} (c d f-a e g)^2 \sqrt {a e+c d x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{4 c^{7/2} d^{7/2} \sqrt {(a e+c d x) (d+e x)}} \]

[In]

Integrate[((d + e*x)^(3/2)*(f + g*x)^(5/2))/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(Sqrt[d + e*x]*(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x]*(-15*a^2*e^2*g^2 - 5*a*c*d*e*g*(-5*f + g*x) + c^2*d^2*(-8*f^2 +
9*f*g*x + 2*g^2*x^2)) + 15*Sqrt[g]*(c*d*f - a*e*g)^2*Sqrt[a*e + c*d*x]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])
/(Sqrt[g]*Sqrt[a*e + c*d*x])]))/(4*c^(7/2)*d^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(637\) vs. \(2(255)=510\).

Time = 0.55 (sec) , antiderivative size = 638, normalized size of antiderivative = 2.12

method	result	size
default	\(\frac {\left (15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} g^{3} x -30 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e f \,g^{2} x +15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{3} d^{3} f^{2} g x +15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{3} e^{3} g^{3}-30 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} f \,g^{2}+15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e \,f^{2} g +4 c^{2} d^{2} g^{2} x^{2} \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}-10 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, a c d e \,g^{2} x +18 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, c^{2} d^{2} f g x -30 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a^{2} e^{2} g^{2}+50 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a c d e f g -16 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, c^{2} d^{2} f^{2}\right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \sqrt {g x +f}}{8 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, \left (c d x +a e \right ) c^{3} d^{3} \sqrt {e x +d}}\)	\(638\)

[In]

int((e*x+d)^(3/2)*(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/8*(15*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)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^2*c*d*e^2*
g^3*x-30*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)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a*c^2*d^2*e
*f*g^2*x+15*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)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*c^3*d^3*
f^2*g*x+15*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)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^3*e^3*g
^3-30*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)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a^2*c*d*e^2*f*
g^2+15*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)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a*c^2*d^2*e*f
^2*g+4*c^2*d^2*g^2*x^2*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)-10*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)*
a*c*d*e*g^2*x+18*(c*d*g)^(1/2)*((g*x+f)*(c*d*x+a*e))^(1/2)*c^2*d^2*f*g*x-30*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g
)^(1/2)*a^2*e^2*g^2+50*((g*x+f)*(c*d*x+a*e))^(1/2)*(c*d*g)^(1/2)*a*c*d*e*f*g-16*((g*x+f)*(c*d*x+a*e))^(1/2)*(c
*d*g)^(1/2)*c^2*d^2*f^2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(g*x+f)^(1/2)/((g*x+f)*(c*d*x+a*e))^(1/2)/(c*d*g)^(1/2)/(
c*d*x+a*e)/c^3/d^3/(e*x+d)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 1.02 (sec) , antiderivative size = 971, normalized size of antiderivative = 3.23 \[ \int \frac {(d+e x)^{3/2} (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}} \, dx=\left [\frac {4 \, {\left (2 \, c^{2} d^{2} g^{2} x^{2} - 8 \, c^{2} d^{2} f^{2} + 25 \, a c d e f g - 15 \, a^{2} e^{2} g^{2} + {\left (9 \, c^{2} d^{2} f g - 5 \, a c d e g^{2}\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 (a c^{2} d^{3} e f^{2} - 2 \, a^{2} c d^{2} e^{2} f g + a^{3} d e^{3} g^{2} + {\left (c^{3} d^{3} e f^{2} - 2 \, a c^{2} d^{2} e^{2} f g + a^{2} c d e^{3} g^{2}\right )} x^{2} + {\left ({\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} f^{2} - 2 \, {\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f g + {\left (a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g^{2}\right )} x\right )} \sqrt {\frac {g}{c d}} \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} + 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} + 4 \, {\left (2 \, c^{2} d^{2} g x + c^{2} d^{2} f + a c d e g\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} \sqrt {\frac {g}{c d}} + {\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 )}{16 \, {\left (c^{4} d^{4} e x^{2} + a c^{3} d^{4} e + {\left (c^{4} d^{5} + a c^{3} d^{3} e^{2}\right )} x\right )}}, \frac {2 \, {\left (2 \, c^{2} d^{2} g^{2} x^{2} - 8 \, c^{2} d^{2} f^{2} + 25 \, a c d e f g - 15 \, a^{2} e^{2} g^{2} + {\left (9 \, c^{2} d^{2} f g - 5 \, a c d e g^{2}\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 (a c^{2} d^{3} e f^{2} - 2 \, a^{2} c d^{2} e^{2} f g + a^{3} d e^{3} g^{2} + {\left (c^{3} d^{3} e f^{2} - 2 \, a c^{2} d^{2} e^{2} f g + a^{2} c d e^{3} g^{2}\right )} x^{2} + {\left ({\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} f^{2} - 2 \, {\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f g + {\left (a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g^{2}\right )} x\right )} \sqrt {-\frac {g}{c d}} \arctan \left (\frac {2 \, \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} c d \sqrt {-\frac {g}{c d}}}{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 )}{8 \, {\left (c^{4} d^{4} e x^{2} + a c^{3} d^{4} e + {\left (c^{4} d^{5} + a c^{3} d^{3} e^{2}\right )} x\right )}}\right ] \]

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(4*(2*c^2*d^2*g^2*x^2 - 8*c^2*d^2*f^2 + 25*a*c*d*e*f*g - 15*a^2*e^2*g^2 + (9*c^2*d^2*f*g - 5*a*c*d*e*g^2
)*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*(a*c^2*d^3*e*f^2 - 2*a^2*c*d
^2*e^2*f*g + a^3*d*e^3*g^2 + (c^3*d^3*e*f^2 - 2*a*c^2*d^2*e^2*f*g + a^2*c*d*e^3*g^2)*x^2 + ((c^3*d^4 + a*c^2*d
^2*e^2)*f^2 - 2*(a*c^2*d^3*e + a^2*c*d*e^3)*f*g + (a^2*c*d^2*e^2 + a^3*e^4)*g^2)*x)*sqrt(g/(c*d))*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 + 8*(c^2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*
x^2 + 4*(2*c^2*d^2*g*x + c^2*d^2*f + a*c*d*e*g)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt
(g*x + f)*sqrt(g/(c*d)) + (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*x^2 + a*c^3*d^4*e + (c^4*d^5 + a*c^3*d^3*e^2)*x), 1/8*(2*(2*c^2*d^2*g^2*x^2 - 8*c^2*d^2*
f^2 + 25*a*c*d*e*f*g - 15*a^2*e^2*g^2 + (9*c^2*d^2*f*g - 5*a*c*d*e*g^2)*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*(a*c^2*d^3*e*f^2 - 2*a^2*c*d^2*e^2*f*g + a^3*d*e^3*g^2 + (c^3*d^3*e*
f^2 - 2*a*c^2*d^2*e^2*f*g + a^2*c*d*e^3*g^2)*x^2 + ((c^3*d^4 + a*c^2*d^2*e^2)*f^2 - 2*(a*c^2*d^3*e + a^2*c*d*e
^3)*f*g + (a^2*c*d^2*e^2 + a^3*e^4)*g^2)*x)*sqrt(-g/(c*d))*arctan(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
)*sqrt(e*x + d)*sqrt(g*x + f)*c*d*sqrt(-g/(c*d))/(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*x^2 + a*c^3*d^4*e + (c^4*d^5 + a*c^3*d^3*e^2)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(3/2)*(g*x+f)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{3/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {5}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)*(g*x + f)^(5/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 834 vs. \(2 (255) = 510\).

Time = 0.63 (sec) , antiderivative size = 834, normalized size of antiderivative = 2.77 \[ \int \frac {(d+e x)^{3/2} (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}} \, dx=\frac {\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} {\left ({\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} {\left (\frac {2 \, {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} g^{2}}{c d e^{4} {\left | g \right |}} + \frac {5 \, {\left (c^{4} d^{4} e^{2} f g^{2} - a c^{3} d^{3} e^{3} g^{3}\right )}}{c^{5} d^{5} e^{4} {\left | g \right |}}\right )} - \frac {15 \, {\left (c^{4} d^{4} e^{4} f^{2} g^{2} - 2 \, a c^{3} d^{3} e^{5} f g^{3} + a^{2} c^{2} d^{2} e^{6} g^{4}\right )}}{c^{5} d^{5} e^{4} {\left | g \right |}}\right )}}{4 \, \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}} - \frac {15 \, {\left (c^{2} d^{2} f^{2} g^{2} - 2 \, a c d e f g^{3} + a^{2} e^{2} g^{4}\right )} \log \left ({\left | -\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} + \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g} \right |}\right )}{4 \, \sqrt {c d g} c^{3} d^{3} {\left | g \right |}} + \frac {15 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} c^{2} d^{2} e^{2} f^{2} g^{2} \log \left ({\left | -\sqrt {e^{2} f - d e g} \sqrt {c d g} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \right |}\right ) - 30 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a c d e^{3} f g^{3} \log \left ({\left | -\sqrt {e^{2} f - d e g} \sqrt {c d g} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \right |}\right ) + 15 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a^{2} e^{4} g^{4} \log \left ({\left | -\sqrt {e^{2} f - d e g} \sqrt {c d g} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \right |}\right ) + 8 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} c^{2} d^{2} e^{2} f^{2} g^{2} + 9 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} c^{2} d^{3} e f g^{3} - 25 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} a c d e^{3} f g^{3} - 2 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} c^{2} d^{4} g^{4} - 5 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} a c d^{2} e^{2} g^{4} + 15 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} a^{2} e^{4} g^{4}}{4 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {c d g} c^{3} d^{3} e^{2} {\left | g \right |}} \]

[In]

integrate((e*x+d)^(3/2)*(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

1/4*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*((e^2*f + (e*x + d)*e*g - d*e*g)*(2*(e^2*f + (e*x + d)*e*g - d*e*g)*g^
2/(c*d*e^4*abs(g)) + 5*(c^4*d^4*e^2*f*g^2 - a*c^3*d^3*e^3*g^3)/(c^5*d^5*e^4*abs(g))) - 15*(c^4*d^4*e^4*f^2*g^2
- 2*a*c^3*d^3*e^5*f*g^3 + a^2*c^2*d^2*e^6*g^4)/(c^5*d^5*e^4*abs(g)))/sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f +
(e*x + d)*e*g - d*e*g)*c*d*g) - 15/4*(c^2*d^2*f^2*g^2 - 2*a*c*d*e*f*g^3 + a^2*e^2*g^4)*log(abs(-sqrt(e^2*f +
(e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/
(sqrt(c*d*g)*c^3*d^3*abs(g)) + 1/4*(15*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^2*d^2*e^2*f^2*g^2*log(abs(-sqrt(e^2*f
- d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) - 30*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c*d*e^3*f*g^3*lo
g(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 15*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*
a^2*e^4*g^4*log(abs(-sqrt(e^2*f - d*e*g)*sqrt(c*d*g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2))) + 8*sqrt(e^2*f - d*e*g
)*sqrt(c*d*g)*c^2*d^2*e^2*f^2*g^2 + 9*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^3*e*f*g^3 - 25*sqrt(e^2*f - d*e*g)
*sqrt(c*d*g)*a*c*d*e^3*f*g^3 - 2*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^2*d^4*g^4 - 5*sqrt(e^2*f - d*e*g)*sqrt(c*d*
g)*a*c*d^2*e^2*g^4 + 15*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*e^4*g^4)/(sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*
g)*c^3*d^3*e^2*abs(g))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \]

[In]

int(((f + g*x)^(5/2)*(d + e*x)^(3/2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2),x)

[Out]

int(((f + g*x)^(5/2)*(d + e*x)^(3/2))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2), x)