3.8.51 \(\int \frac {(f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\) [751]

Optimal. Leaf size=448 \[ -\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}+\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 g^2 (d+e x)^{3/2}}+\frac {(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {3 (c d f-a e g)^5 \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

-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)^(3/2)/g^2/(e*x+d)^(3/2)+1/5*(g*x+f)^(5/2)*
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/g/(e*x+d)^(5/2)-3/128*(-a*e*g+c*d*f)^5*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^(7/2)/(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)-1/64*(-a*e*g+c*d*f)^3*(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^3/(e*x+d
)^(1/2)+1/16*(-a*e*g+c*d*f)^2*(g*x+f)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(e*x+d)^(1/2)-3/128*(-
a*e*g+c*d*f)^4*(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^3/(e*x+d)^(1/2)

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Rubi [A]
time = 0.58, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 9, 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)^5 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2} \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)^4}{128 c^2 d^2 g^3 \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)^3}{64 c d g^3 \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)^2}{16 g^3 \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} (c d f-a e g)}{8 g^2 (d+e x)^{3/2}}+\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 )^{5/2}}{5 g (d+e x)^{5/2}} \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)^(5/2))/(d + e*x)^(5/2),x]

[Out]

(-3*(c*d*f - a*e*g)^4*Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*c^2*d^2*g^3*Sqrt[d + e*x
]) - ((c*d*f - a*e*g)^3*(f + g*x)^(3/2)*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]
) + ((c*d*f - a*e*g)^2*(f + g*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(16*g^3*Sqrt[d + e*x]) - (
(c*d*f - a*e*g)*(f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(8*g^2*(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)^(5/2))/(5*g*(d + e*x)^(5/2)) - (3*(c*d*f - a*e*g)^5*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])])/(128*c^(5/2)*d
^(5/2)*g^(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 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 )^{5/2}}{(d+e x)^{5/2}} \, 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 )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g) \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}{2 g}\\ &=-\frac {(c d f-a e g) (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}}{8 g^2 (d+e x)^{3/2}}+\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 )^{5/2}}{5 g (d+e x)^{5/2}}+\frac {\left (3 (c d f-a e g)^2\right ) \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}{16 g^2}\\ &=\frac {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (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}}{8 g^2 (d+e x)^{3/2}}+\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 )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g)^3 \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}{32 g^3}\\ &=-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \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 {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (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}}{8 g^2 (d+e x)^{3/2}}+\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 )^{5/2}}{5 g (d+e x)^{5/2}}-\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 d g^3}\\ &=-\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \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 {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (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}}{8 g^2 (d+e x)^{3/2}}+\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 )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5\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}{256 c^2 d^2 g^3}\\ &=-\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \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 {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (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}}{8 g^2 (d+e x)^{3/2}}+\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 )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5 \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}{256 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 {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \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 {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (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}}{8 g^2 (d+e x)^{3/2}}+\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 )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5 \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 )}{128 c^3 d^3 g^3 \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)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \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 {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (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}}{8 g^2 (d+e x)^{3/2}}+\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 )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {\left (3 (c d f-a e g)^5 \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 )}{128 c^3 d^3 g^3 \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)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 (f+g x)^{3/2} \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 {(c d f-a e g)^2 (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g) (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}}{8 g^2 (d+e x)^{3/2}}+\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 )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {3 (c d f-a e g)^5 \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 )}{128 c^{5/2} d^{5/2} 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 = 0.92, size = 265, normalized size = 0.59 \begin {gather*} \frac {(c d f-a e g)^5 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {g} (f+g x)^{9/2} \left (15 c^4 d^4-\frac {15 g^4 (a e+c d x)^4}{(f+g x)^4}+\frac {70 c d g^3 (a e+c d x)^3}{(f+g x)^3}+\frac {128 c^2 d^2 g^2 (a e+c d x)^2}{(f+g x)^2}-\frac {70 c^3 d^3 g (a e+c d x)}{f+g x}\right )}{(c d f-a e g)^5 (a e+c d x)^2}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{(a e+c d x)^{5/2}}\right )}{640 c^{5/2} d^{5/2} g^{7/2} (d+e x)^{5/2}} \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)^(5/2))/(d + e*x)^(5/2),x]

[Out]

((c*d*f - a*e*g)^5*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[g]*(f + g*x)^(9/2)*(15*c^4*d^4 - (15
*g^4*(a*e + c*d*x)^4)/(f + g*x)^4 + (70*c*d*g^3*(a*e + c*d*x)^3)/(f + g*x)^3 + (128*c^2*d^2*g^2*(a*e + c*d*x)^
2)/(f + g*x)^2 - (70*c^3*d^3*g*(a*e + c*d*x))/(f + g*x)))/((c*d*f - a*e*g)^5*(a*e + c*d*x)^2) - (15*ArcTanh[(S
qrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(a*e + c*d*x)^(5/2)))/(640*c^(5/2)*d^(5/2)*g^(7/2)
*(d + e*x)^(5/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1004\) vs. \(2(384)=768\).
time = 0.14, size = 1005, normalized size = 2.24

method	result	size
default	\(\frac {\sqrt {g x +f}\, \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (256 c^{4} d^{4} g^{4} x^{4} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}+672 a \,c^{3} d^{3} e \,g^{4} x^{3} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}+352 c^{4} d^{4} f \,g^{3} x^{3} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}+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 {d g c}}{2 \sqrt {d g c}}\right ) a^{5} e^{5} g^{5}-75 \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} c d \,e^{4} f \,g^{4}+150 \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^{2} d^{2} e^{3} f^{2} g^{3}-150 \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^{3} d^{3} e^{2} f^{3} g^{2}+75 \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^{4} d^{4} e \,f^{4} g -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 {d g c}}{2 \sqrt {d g c}}\right ) c^{5} d^{5} f^{5}+496 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}+1024 a \,c^{3} d^{3} e f \,g^{3} x^{2} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}+16 c^{4} d^{4} f^{2} g^{2} x^{2} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}+20 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, a^{3} c d \,e^{3} g^{4} x +932 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, a^{2} c^{2} d^{2} e^{2} f \,g^{3} x +92 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, a \,c^{3} d^{3} e \,f^{2} g^{2} x -20 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, c^{4} d^{4} f^{3} g x -30 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, a^{4} e^{4} g^{4}+140 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, a^{3} c d \,e^{3} f \,g^{3}+256 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}-140 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, a \,c^{3} d^{3} e \,f^{3} g +30 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\, c^{4} d^{4} f^{4}\right )}{1280 \sqrt {e x +d}\, c^{2} d^{2} g^{3} \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}}\)	\(1005\)

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)^(5/2)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/1280*(g*x+f)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(256*c^4*d^4*g^4*x^4*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2
)+672*a*c^3*d^3*e*g^4*x^3*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)+352*c^4*d^4*f*g^3*x^3*((g*x+f)*(c*d*x+a*e)
)^(1/2)*(d*g*c)^(1/2)+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)*(d*g*c)^(1/2))/(d*g*c)^(1
/2))*a^5*e^5*g^5-75*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*c*d*e^4*f*g^4+150*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^2*d^2*e^3*f^2*g^3-150*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^3*d^3*e^2*f^3*g^2+75*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^4*d^4*e*f^4*g-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)*(d*g*c)^(1/2
))/(d*g*c)^(1/2))*c^5*d^5*f^5+496*a^2*c^2*d^2*e^2*g^4*x^2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)+1024*a*c^3
*d^3*e*f*g^3*x^2*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)+16*c^4*d^4*f^2*g^2*x^2*((g*x+f)*(c*d*x+a*e))^(1/2)*
(d*g*c)^(1/2)+20*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)*a^3*c*d*e^3*g^4*x+932*((g*x+f)*(c*d*x+a*e))^(1/2)*(
d*g*c)^(1/2)*a^2*c^2*d^2*e^2*f*g^3*x+92*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)*a*c^3*d^3*e*f^2*g^2*x-20*((g
*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)*c^4*d^4*f^3*g*x-30*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)*a^4*e^4*g^
4+140*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)*a^3*c*d*e^3*f*g^3+256*a^2*c^2*d^2*e^2*f^2*g^2*((g*x+f)*(c*d*x+
a*e))^(1/2)*(d*g*c)^(1/2)-140*((g*x+f)*(c*d*x+a*e))^(1/2)*(d*g*c)^(1/2)*a*c^3*d^3*e*f^3*g+30*((g*x+f)*(c*d*x+a
*e))^(1/2)*(d*g*c)^(1/2)*c^4*d^4*f^4)/(e*x+d)^(1/2)/c^2/d^2/g^3/((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)^(5/2)/(e*x+d)^(5/2),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)^(3/2)/(x*e + d)^(5/2), x)

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Fricas [A]
time = 6.33, size = 1331, normalized size = 2.97 \begin {gather*} \left [\frac {4 \, {\left (128 \, c^{5} d^{5} g^{5} x^{4} + 176 \, c^{5} d^{5} f g^{4} x^{3} + 8 \, c^{5} d^{5} f^{2} g^{3} x^{2} - 10 \, c^{5} d^{5} f^{3} g^{2} x + 15 \, c^{5} d^{5} f^{4} g - 15 \, a^{4} c d g^{5} e^{4} + 10 \, {\left (a^{3} c^{2} d^{2} g^{5} x + 7 \, a^{3} c^{2} d^{2} f g^{4}\right )} e^{3} + 2 \, {\left (124 \, a^{2} c^{3} d^{3} g^{5} x^{2} + 233 \, a^{2} c^{3} d^{3} f g^{4} x + 64 \, a^{2} c^{3} d^{3} f^{2} g^{3}\right )} e^{2} + 2 \, {\left (168 \, a c^{4} d^{4} g^{5} x^{3} + 256 \, a c^{4} d^{4} f g^{4} x^{2} + 23 \, a c^{4} d^{4} f^{2} g^{3} x - 35 \, a c^{4} d^{4} f^{3} 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} - 15 \, {\left (c^{5} d^{6} f^{5} - a^{5} g^{5} x e^{6} + {\left (5 \, a^{4} c d f g^{4} x - a^{5} d g^{5}\right )} e^{5} - 5 \, {\left (2 \, a^{3} c^{2} d^{2} f^{2} g^{3} x - a^{4} c d^{2} f g^{4}\right )} e^{4} + 10 \, {\left (a^{2} c^{3} d^{3} f^{3} g^{2} x - a^{3} c^{2} d^{3} f^{2} g^{3}\right )} e^{3} - 5 \, {\left (a c^{4} d^{4} f^{4} g x - 2 \, a^{2} c^{3} d^{4} f^{3} g^{2}\right )} e^{2} + {\left (c^{5} d^{5} f^{5} x - 5 \, a c^{4} d^{5} f^{4} 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 )}{2560 \, {\left (c^{3} d^{3} g^{4} x e + c^{3} d^{4} g^{4}\right )}}, \frac {2 \, {\left (128 \, c^{5} d^{5} g^{5} x^{4} + 176 \, c^{5} d^{5} f g^{4} x^{3} + 8 \, c^{5} d^{5} f^{2} g^{3} x^{2} - 10 \, c^{5} d^{5} f^{3} g^{2} x + 15 \, c^{5} d^{5} f^{4} g - 15 \, a^{4} c d g^{5} e^{4} + 10 \, {\left (a^{3} c^{2} d^{2} g^{5} x + 7 \, a^{3} c^{2} d^{2} f g^{4}\right )} e^{3} + 2 \, {\left (124 \, a^{2} c^{3} d^{3} g^{5} x^{2} + 233 \, a^{2} c^{3} d^{3} f g^{4} x + 64 \, a^{2} c^{3} d^{3} f^{2} g^{3}\right )} e^{2} + 2 \, {\left (168 \, a c^{4} d^{4} g^{5} x^{3} + 256 \, a c^{4} d^{4} f g^{4} x^{2} + 23 \, a c^{4} d^{4} f^{2} g^{3} x - 35 \, a c^{4} d^{4} f^{3} 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} + 15 \, {\left (c^{5} d^{6} f^{5} - a^{5} g^{5} x e^{6} + {\left (5 \, a^{4} c d f g^{4} x - a^{5} d g^{5}\right )} e^{5} - 5 \, {\left (2 \, a^{3} c^{2} d^{2} f^{2} g^{3} x - a^{4} c d^{2} f g^{4}\right )} e^{4} + 10 \, {\left (a^{2} c^{3} d^{3} f^{3} g^{2} x - a^{3} c^{2} d^{3} f^{2} g^{3}\right )} e^{3} - 5 \, {\left (a c^{4} d^{4} f^{4} g x - 2 \, a^{2} c^{3} d^{4} f^{3} g^{2}\right )} e^{2} + {\left (c^{5} d^{5} f^{5} x - 5 \, a c^{4} d^{5} f^{4} 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 )}{1280 \, {\left (c^{3} d^{3} g^{4} x e + c^{3} d^{4} g^{4}\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)^(5/2)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

[1/2560*(4*(128*c^5*d^5*g^5*x^4 + 176*c^5*d^5*f*g^4*x^3 + 8*c^5*d^5*f^2*g^3*x^2 - 10*c^5*d^5*f^3*g^2*x + 15*c^
5*d^5*f^4*g - 15*a^4*c*d*g^5*e^4 + 10*(a^3*c^2*d^2*g^5*x + 7*a^3*c^2*d^2*f*g^4)*e^3 + 2*(124*a^2*c^3*d^3*g^5*x
^2 + 233*a^2*c^3*d^3*f*g^4*x + 64*a^2*c^3*d^3*f^2*g^3)*e^2 + 2*(168*a*c^4*d^4*g^5*x^3 + 256*a*c^4*d^4*f*g^4*x^
2 + 23*a*c^4*d^4*f^2*g^3*x - 35*a*c^4*d^4*f^3*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) - 15*(c^5*d^6*f^5 - a^5*g^5*x*e^6 + (5*a^4*c*d*f*g^4*x - a^5*d*g^5)*e^5 - 5*(2*a^3*c^2*d^2*f^2
*g^3*x - a^4*c*d^2*f*g^4)*e^4 + 10*(a^2*c^3*d^3*f^3*g^2*x - a^3*c^2*d^3*f^2*g^3)*e^3 - 5*(a*c^4*d^4*f^4*g*x -
2*a^2*c^3*d^4*f^3*g^2)*e^2 + (c^5*d^5*f^5*x - 5*a*c^4*d^5*f^4*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^4*
x*e + c^3*d^4*g^4), 1/1280*(2*(128*c^5*d^5*g^5*x^4 + 176*c^5*d^5*f*g^4*x^3 + 8*c^5*d^5*f^2*g^3*x^2 - 10*c^5*d^
5*f^3*g^2*x + 15*c^5*d^5*f^4*g - 15*a^4*c*d*g^5*e^4 + 10*(a^3*c^2*d^2*g^5*x + 7*a^3*c^2*d^2*f*g^4)*e^3 + 2*(12
4*a^2*c^3*d^3*g^5*x^2 + 233*a^2*c^3*d^3*f*g^4*x + 64*a^2*c^3*d^3*f^2*g^3)*e^2 + 2*(168*a*c^4*d^4*g^5*x^3 + 256
*a*c^4*d^4*f*g^4*x^2 + 23*a*c^4*d^4*f^2*g^3*x - 35*a*c^4*d^4*f^3*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) + 15*(c^5*d^6*f^5 - a^5*g^5*x*e^6 + (5*a^4*c*d*f*g^4*x - a^5*d*g^5)*e^5 - 5
*(2*a^3*c^2*d^2*f^2*g^3*x - a^4*c*d^2*f*g^4)*e^4 + 10*(a^2*c^3*d^3*f^3*g^2*x - a^3*c^2*d^3*f^2*g^3)*e^3 - 5*(a
*c^4*d^4*f^4*g*x - 2*a^2*c^3*d^4*f^3*g^2)*e^2 + (c^5*d^5*f^5*x - 5*a*c^4*d^5*f^4*g)*e)*sqrt(-c*d*g)*arctan(2*s
qrt(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^4*x*e + c^3*d^4*g^4)]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8569 deep

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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)^(5/2)/(e*x+d)^(5/2),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 )}^{5/2}}{{\left (d+e\,x\right )}^{5/2}} \,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)^(5/2))/(d + e*x)^(5/2),x)

[Out]

int(((f + g*x)^(3/2)*(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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