\(\int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [732]

Optimal result

Integrand size = 48, antiderivative size = 260 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 g \sqrt {d+e x}}{(c d f-a e g)^2 (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {32 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \]

[Out]

-2/3*(e*x+d)^(3/2)/(-a*e*g+c*d*f)/(g*x+f)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+4*g*(e*x+d)^(1/2)/(-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)+16/3*g^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)/(-a*e*g+c*d*f)^3/(g*x+f)^(3/2)/(e*x+d)^(1/2)+32/3*c*d*g^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*
g+c*d*f)^4/(e*x+d)^(1/2)/(g*x+f)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {882, 886, 874} \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {32 c d g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^4}+\frac {16 g^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac {4 g \sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac {2 (d+e x)^{3/2}}{3 (f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]

[In]

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

[Out]

(-2*(d + e*x)^(3/2))/(3*(c*d*f - a*e*g)*(f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (4*g*
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]) + (16*g^2*Sqrt[
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(3/2)) + (32*c*d*g^2*Sqrt
[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*Sqrt[f + g*x])

Rule 874

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

Rule 882

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[e^2*(d + e*x)^(m - 1)*(f + g*x)^(n + 1)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(c*e*f + c*d*g - b*e*g))), x]
+ Dist[e^2*g*((m - n - 2)/((p + 1)*(c*e*f + c*d*g - b*e*g))), 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] && LtQ[p, -1] && RationalQ[n]

Rule 886

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {(2 g) \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}{c d f-a e g} \\ & = -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 g \sqrt {d+e x}}{(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}}+\frac {\left (8 g^2\right ) \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}{(c d f-a e g)^2} \\ & = -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 g \sqrt {d+e x}}{(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}}+\frac {16 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {\left (16 c d 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}{3 (c d f-a e g)^3} \\ & = -\frac {2 (d+e x)^{3/2}}{3 (c d f-a e g) (f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {4 g \sqrt {d+e x}}{(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}}+\frac {16 g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {32 c d g^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.58 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^{3/2} \left (-a^3 e^3 g^3+3 a^2 c d e^2 g^2 (3 f+2 g x)+3 a c^2 d^2 e g \left (3 f^2+12 f g x+8 g^2 x^2\right )+c^3 d^3 \left (-f^3+6 f^2 g x+24 f g^2 x^2+16 g^3 x^3\right )\right )}{3 (c d f-a e g)^4 ((a e+c d x) (d+e x))^{3/2} (f+g x)^{3/2}} \]

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(-(a^3*e^3*g^3) + 3*a^2*c*d*e^2*g^2*(3*f + 2*g*x) + 3*a*c^2*d^2*e*g*(3*f^2 + 12*f*g*x + 8*g
^2*x^2) + c^3*d^3*(-f^3 + 6*f^2*g*x + 24*f*g^2*x^2 + 16*g^3*x^3)))/(3*(c*d*f - a*e*g)^4*((a*e + c*d*x)*(d + e*
x))^(3/2)*(f + g*x)^(3/2))

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.73

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-16 g^{3} x^{3} c^{3} d^{3}-24 a \,c^{2} d^{2} e \,g^{3} x^{2}-24 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x -6 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-9 a^{2} c d \,e^{2} f \,g^{2}-9 a \,c^{2} d^{2} e \,f^{2} g +f^{3} c^{3} d^{3}\right )}{3 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {3}{2}} \left (c d x +a e \right )^{2} \left (a e g -c d f \right )^{4}}\)	\(191\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}-24 a \,c^{2} d^{2} e \,g^{3} x^{2}-24 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x -36 a \,c^{2} d^{2} e f \,g^{2} x -6 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-9 a^{2} c d \,e^{2} f \,g^{2}-9 a \,c^{2} d^{2} e \,f^{2} g +f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a^{4} e^{4} g^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(258\)

[In]

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

[Out]

-2/3/(e*x+d)^(1/2)/(g*x+f)^(3/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(-16*c^3*d^3*g^3*x^3-24*a*c^2*d^2*e*g^3*x^2-24*c^
3*d^3*f*g^2*x^2-6*a^2*c*d*e^2*g^3*x-36*a*c^2*d^2*e*f*g^2*x-6*c^3*d^3*f^2*g*x+a^3*e^3*g^3-9*a^2*c*d*e^2*f*g^2-9
*a*c^2*d^2*e*f^2*g+c^3*d^3*f^3)/(c*d*x+a*e)^2/(a*e*g-c*d*f)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1065 vs. \(2 (230) = 460\).

Time = 1.03 (sec) , antiderivative size = 1065, normalized size of antiderivative = 4.10 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (16 \, c^{3} d^{3} g^{3} x^{3} - c^{3} d^{3} f^{3} + 9 \, a c^{2} d^{2} e f^{2} g + 9 \, a^{2} c d e^{2} f g^{2} - a^{3} e^{3} g^{3} + 24 \, {\left (c^{3} d^{3} f g^{2} + a c^{2} d^{2} e g^{3}\right )} x^{2} + 6 \, {\left (c^{3} d^{3} f^{2} g + 6 \, a c^{2} d^{2} e f g^{2} + a^{2} c d e^{2} g^{3}\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}}{3 \, {\left (a^{2} c^{4} d^{5} e^{2} f^{6} - 4 \, a^{3} c^{3} d^{4} e^{3} f^{5} g + 6 \, a^{4} c^{2} d^{3} e^{4} f^{4} g^{2} - 4 \, a^{5} c d^{2} e^{5} f^{3} g^{3} + a^{6} d e^{6} f^{2} g^{4} + {\left (c^{6} d^{6} e f^{4} g^{2} - 4 \, a c^{5} d^{5} e^{2} f^{3} g^{3} + 6 \, a^{2} c^{4} d^{4} e^{3} f^{2} g^{4} - 4 \, a^{3} c^{3} d^{3} e^{4} f g^{5} + a^{4} c^{2} d^{2} e^{5} g^{6}\right )} x^{5} + {\left (2 \, c^{6} d^{6} e f^{5} g + {\left (c^{6} d^{7} - 6 \, a c^{5} d^{5} e^{2}\right )} f^{4} g^{2} - 4 \, {\left (a c^{5} d^{6} e - a^{2} c^{4} d^{4} e^{3}\right )} f^{3} g^{3} + 2 \, {\left (3 \, a^{2} c^{4} d^{5} e^{2} + 2 \, a^{3} c^{3} d^{3} e^{4}\right )} f^{2} g^{4} - 2 \, {\left (2 \, a^{3} c^{3} d^{4} e^{3} + 3 \, a^{4} c^{2} d^{2} e^{5}\right )} f g^{5} + {\left (a^{4} c^{2} d^{3} e^{4} + 2 \, a^{5} c d e^{6}\right )} g^{6}\right )} x^{4} + {\left (c^{6} d^{6} e f^{6} + 2 \, c^{6} d^{7} f^{5} g - 6 \, a^{4} c^{2} d^{3} e^{4} f g^{5} - 3 \, {\left (2 \, a c^{5} d^{6} e + 3 \, a^{2} c^{4} d^{4} e^{3}\right )} f^{4} g^{2} + 4 \, {\left (a^{2} c^{4} d^{5} e^{2} + 4 \, a^{3} c^{3} d^{3} e^{4}\right )} f^{3} g^{3} + {\left (4 \, a^{3} c^{3} d^{4} e^{3} - 9 \, a^{4} c^{2} d^{2} e^{5}\right )} f^{2} g^{4} + {\left (2 \, a^{5} c d^{2} e^{5} + a^{6} e^{7}\right )} g^{6}\right )} x^{3} - {\left (6 \, a^{2} c^{4} d^{4} e^{3} f^{5} g - 2 \, a^{6} e^{7} f g^{5} - a^{6} d e^{6} g^{6} - {\left (c^{6} d^{7} + 2 \, a c^{5} d^{5} e^{2}\right )} f^{6} + {\left (9 \, a^{2} c^{4} d^{5} e^{2} - 4 \, a^{3} c^{3} d^{3} e^{4}\right )} f^{4} g^{2} - 4 \, {\left (4 \, a^{3} c^{3} d^{4} e^{3} + a^{4} c^{2} d^{2} e^{5}\right )} f^{3} g^{3} + 3 \, {\left (3 \, a^{4} c^{2} d^{3} e^{4} + 2 \, a^{5} c d e^{6}\right )} f^{2} g^{4}\right )} x^{2} + {\left (2 \, a^{6} d e^{6} f g^{5} + {\left (2 \, a c^{5} d^{6} e + a^{2} c^{4} d^{4} e^{3}\right )} f^{6} - 2 \, {\left (3 \, a^{2} c^{4} d^{5} e^{2} + 2 \, a^{3} c^{3} d^{3} e^{4}\right )} f^{5} g + 2 \, {\left (2 \, a^{3} c^{3} d^{4} e^{3} + 3 \, a^{4} c^{2} d^{2} e^{5}\right )} f^{4} g^{2} + 4 \, {\left (a^{4} c^{2} d^{3} e^{4} - a^{5} c d e^{6}\right )} f^{3} g^{3} - {\left (6 \, a^{5} c d^{2} e^{5} - a^{6} e^{7}\right )} f^{2} g^{4}\right )} x\right )}} \]

[In]

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

[Out]

2/3*(16*c^3*d^3*g^3*x^3 - c^3*d^3*f^3 + 9*a*c^2*d^2*e*f^2*g + 9*a^2*c*d*e^2*f*g^2 - a^3*e^3*g^3 + 24*(c^3*d^3*
f*g^2 + a*c^2*d^2*e*g^3)*x^2 + 6*(c^3*d^3*f^2*g + 6*a*c^2*d^2*e*f*g^2 + a^2*c*d*e^2*g^3)*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)/(a^2*c^4*d^5*e^2*f^6 - 4*a^3*c^3*d^4*e^3*f^5*g + 6*a^4*c
^2*d^3*e^4*f^4*g^2 - 4*a^5*c*d^2*e^5*f^3*g^3 + a^6*d*e^6*f^2*g^4 + (c^6*d^6*e*f^4*g^2 - 4*a*c^5*d^5*e^2*f^3*g^
3 + 6*a^2*c^4*d^4*e^3*f^2*g^4 - 4*a^3*c^3*d^3*e^4*f*g^5 + a^4*c^2*d^2*e^5*g^6)*x^5 + (2*c^6*d^6*e*f^5*g + (c^6
*d^7 - 6*a*c^5*d^5*e^2)*f^4*g^2 - 4*(a*c^5*d^6*e - a^2*c^4*d^4*e^3)*f^3*g^3 + 2*(3*a^2*c^4*d^5*e^2 + 2*a^3*c^3
*d^3*e^4)*f^2*g^4 - 2*(2*a^3*c^3*d^4*e^3 + 3*a^4*c^2*d^2*e^5)*f*g^5 + (a^4*c^2*d^3*e^4 + 2*a^5*c*d*e^6)*g^6)*x
^4 + (c^6*d^6*e*f^6 + 2*c^6*d^7*f^5*g - 6*a^4*c^2*d^3*e^4*f*g^5 - 3*(2*a*c^5*d^6*e + 3*a^2*c^4*d^4*e^3)*f^4*g^
2 + 4*(a^2*c^4*d^5*e^2 + 4*a^3*c^3*d^3*e^4)*f^3*g^3 + (4*a^3*c^3*d^4*e^3 - 9*a^4*c^2*d^2*e^5)*f^2*g^4 + (2*a^5
*c*d^2*e^5 + a^6*e^7)*g^6)*x^3 - (6*a^2*c^4*d^4*e^3*f^5*g - 2*a^6*e^7*f*g^5 - a^6*d*e^6*g^6 - (c^6*d^7 + 2*a*c
^5*d^5*e^2)*f^6 + (9*a^2*c^4*d^5*e^2 - 4*a^3*c^3*d^3*e^4)*f^4*g^2 - 4*(4*a^3*c^3*d^4*e^3 + a^4*c^2*d^2*e^5)*f^
3*g^3 + 3*(3*a^4*c^2*d^3*e^4 + 2*a^5*c*d*e^6)*f^2*g^4)*x^2 + (2*a^6*d*e^6*f*g^5 + (2*a*c^5*d^6*e + a^2*c^4*d^4
*e^3)*f^6 - 2*(3*a^2*c^4*d^5*e^2 + 2*a^3*c^3*d^3*e^4)*f^5*g + 2*(2*a^3*c^3*d^4*e^3 + 3*a^4*c^2*d^2*e^5)*f^4*g^
2 + 4*(a^4*c^2*d^3*e^4 - a^5*c*d*e^6)*f^3*g^3 - (6*a^5*c*d^2*e^5 - a^6*e^7)*f^2*g^4)*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4668 vs. \(2 (230) = 460\).

Time = 3.08 (sec) , antiderivative size = 4668, normalized size of antiderivative = 17.95 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-2/3*e^4*(sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*(8*(c^7*d^7*e^2*f^3*g^4*abs(g) - 3*a*c^6*d^6*e^3*f^2*g^5*abs(g)
+ 3*a^2*c^5*d^5*e^4*f*g^6*abs(g) - a^3*c^4*d^4*e^5*g^7*abs(g))*(e^2*f + (e*x + d)*e*g - d*e*g)/(c^8*d^8*e^6*f^
7*g^2 - 7*a*c^7*d^7*e^7*f^6*g^3 + 21*a^2*c^6*d^6*e^8*f^5*g^4 - 35*a^3*c^5*d^5*e^9*f^4*g^5 + 35*a^4*c^4*d^4*e^1
0*f^3*g^6 - 21*a^5*c^3*d^3*e^11*f^2*g^7 + 7*a^6*c^2*d^2*e^12*f*g^8 - a^7*c*d*e^13*g^9) - 9*(c^7*d^7*e^4*f^4*g^
4*abs(g) - 4*a*c^6*d^6*e^5*f^3*g^5*abs(g) + 6*a^2*c^5*d^5*e^6*f^2*g^6*abs(g) - 4*a^3*c^4*d^4*e^7*f*g^7*abs(g)
+ a^4*c^3*d^3*e^8*g^8*abs(g))/(c^8*d^8*e^6*f^7*g^2 - 7*a*c^7*d^7*e^7*f^6*g^3 + 21*a^2*c^6*d^6*e^8*f^5*g^4 - 35
*a^3*c^5*d^5*e^9*f^4*g^5 + 35*a^4*c^4*d^4*e^10*f^3*g^6 - 21*a^5*c^3*d^3*e^11*f^2*g^7 + 7*a^6*c^2*d^2*e^12*f*g^
8 - a^7*c*d*e^13*g^9))/((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*e^2*f*g +
a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g)) - 4*(4*sqrt(c*d*g)*c^3*d^3*e^4*f^2*g^5 - 8*sqrt(c*d*g)*a*c
^2*d^2*e^5*f*g^6 + 4*sqrt(c*d*g)*a^2*c*d*e^6*g^7 + 9*sqrt(c*d*g)*(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))^2*c^2*d^2*e^2*f*g^4 - 9*sqrt(c*d
*g)*(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))^2*a*c*d*e^3*g^5 + 3*sqrt(c*d*g)*(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))^4*c*d*g^3)/((c^3*d^3*e^2*f^3*abs(g) - 3*a*c^2*d^
2*e^3*f^2*g*abs(g) + 3*a^2*c*d*e^4*f*g^2*abs(g) - a^3*e^5*g^3*abs(g))*(c*d*e^2*f*g - a*e^3*g^2 + (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))^
2)^3)) - 2/3*(sqrt(e^2*f - d*e*g)*c^5*d^5*e^4*f^4*g^2 + 2*sqrt(e^2*f - d*e*g)*c^5*d^6*e^3*f^3*g^3 - 6*sqrt(e^2
*f - d*e*g)*a*c^4*d^4*e^5*f^3*g^3 - 24*sqrt(e^2*f - d*e*g)*c^5*d^7*e^2*f^2*g^4 + 42*sqrt(e^2*f - d*e*g)*a*c^4*
d^5*e^4*f^2*g^4 - 12*sqrt(e^2*f - d*e*g)*a^2*c^3*d^3*e^6*f^2*g^4 + 56*sqrt(e^2*f - d*e*g)*c^5*d^8*e*f*g^5 - 12
0*sqrt(e^2*f - d*e*g)*a*c^4*d^6*e^3*f*g^5 + 78*sqrt(e^2*f - d*e*g)*a^2*c^3*d^4*e^5*f*g^5 - 18*sqrt(e^2*f - d*e
*g)*a^3*c^2*d^2*e^7*f*g^5 - 32*sqrt(e^2*f - d*e*g)*c^5*d^9*g^6 + 72*sqrt(e^2*f - d*e*g)*a*c^4*d^7*e^2*g^6 - 48
*sqrt(e^2*f - d*e*g)*a^2*c^3*d^5*e^4*g^6 + 6*sqrt(e^2*f - d*e*g)*a^3*c^2*d^3*e^6*g^6 + 3*sqrt(e^2*f - d*e*g)*a
^4*c*d*e^8*g^6 - 3*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^4*d^4*e^4*f^4*g - 6*sqrt(-c*d^2*e*g^2 + a*e^3*
g^2)*sqrt(c*d*g)*c^4*d^5*e^3*f^3*g^2 + 18*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a*c^3*d^3*e^5*f^3*g^2 + 4
8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^4*d^6*e^2*f^2*g^3 - 78*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*
g)*a*c^3*d^4*e^4*f^2*g^3 + 12*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^2*c^2*d^2*e^6*f^2*g^3 - 72*sqrt(-c*
d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^4*d^7*e*f*g^4 + 120*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a*c^3*d^5*
e^3*f*g^4 - 42*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^2*c^2*d^3*e^5*f*g^4 + 6*sqrt(-c*d^2*e*g^2 + a*e^3*
g^2)*sqrt(c*d*g)*a^3*c*d*e^7*f*g^4 + 32*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^4*d^8*g^5 - 56*sqrt(-c*d^
2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a*c^3*d^6*e^2*g^5 + 24*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^2*c^2*d^4
*e^4*g^5 - 2*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^3*c*d^2*e^6*g^5 - sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqr
t(c*d*g)*a^4*e^8*g^5)/(3*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^7*d^9*e^2*f^6*g*abs(g) - 6*sqrt(e^2*f - d*e*g)*sqrt
(c*d*g)*a*c^6*d^7*e^4*f^6*g*abs(g) + 3*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^5*d^5*e^6*f^6*g*abs(g) - 7*sqrt(e
^2*f - d*e*g)*sqrt(c*d*g)*c^7*d^10*e*f^5*g^2*abs(g) + 3*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^6*d^8*e^3*f^5*g^2*
abs(g) + 15*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^5*d^6*e^5*f^5*g^2*abs(g) - 11*sqrt(e^2*f - d*e*g)*sqrt(c*d*g
)*a^3*c^4*d^4*e^7*f^5*g^2*abs(g) + 4*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^7*d^11*f^4*g^3*abs(g) + 19*sqrt(e^2*f -
d*e*g)*sqrt(c*d*g)*a*c^6*d^9*e^2*f^4*g^3*abs(g) - 36*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^5*d^7*e^4*f^4*g^3*
abs(g) - sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*c^4*d^5*e^6*f^4*g^3*abs(g) + 14*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a
^4*c^3*d^3*e^8*f^4*g^3*abs(g) - 16*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^6*d^10*e*f^3*g^4*abs(g) - 6*sqrt(e^2*f
- d*e*g)*sqrt(c*d*g)*a^2*c^5*d^8*e^3*f^3*g^4*abs(g) + 54*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*c^4*d^6*e^5*f^3*g
^4*abs(g) - 26*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*c^3*d^4*e^7*f^3*g^4*abs(g) - 6*sqrt(e^2*f - d*e*g)*sqrt(c*d
*g)*a^5*c^2*d^2*e^9*f^3*g^4*abs(g) + 24*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^5*d^9*e^2*f^2*g^5*abs(g) - 26*sq
rt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*c^4*d^7*e^4*f^2*g^5*abs(g) - 21*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*c^3*d^5*
e^6*f^2*g^5*abs(g) + 24*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^5*c^2*d^3*e^8*f^2*g^5*abs(g) - sqrt(e^2*f - d*e*g)*s
qrt(c*d*g)*a^6*c*d*e^10*f^2*g^5*abs(g) - 16*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*c^4*d^8*e^3*f*g^6*abs(g) + 29*
sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*c^3*d^6*e^5*f*g^6*abs(g) - 9*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^5*c^2*d^4*e
^7*f*g^6*abs(g) - 5*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^6*c*d^2*e^9*f*g^6*abs(g) + sqrt(e^2*f - d*e*g)*sqrt(c*d*
g)*a^7*e^11*f*g^6*abs(g) + 4*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*c^3*d^7*e^4*g^7*abs(g) - 9*sqrt(e^2*f - d*e*g
)*sqrt(c*d*g)*a^5*c^2*d^5*e^6*g^7*abs(g) + 6*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^6*c*d^3*e^8*g^7*abs(g) - sqrt(e
^2*f - d*e*g)*sqrt(c*d*g)*a^7*d*e^10*g^7*abs(g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^7*d^8*e^3*f^7*abs(g) - sqrt
(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^6*d^6*e^5*f^7*abs(g) - 6*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^7*d^9*e^2*f^6*g*abs(g
) + 5*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^6*d^7*e^4*f^6*g*abs(g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^5*d^5*e
^6*f^6*g*abs(g) + 9*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^7*d^10*e*f^5*g^2*abs(g) + 9*sqrt(-c*d^2*e*g^2 + a*e^3*g^2
)*a*c^6*d^8*e^3*f^5*g^2*abs(g) - 24*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^5*d^6*e^5*f^5*g^2*abs(g) + 6*sqrt(-c*
d^2*e*g^2 + a*e^3*g^2)*a^3*c^4*d^4*e^7*f^5*g^2*abs(g) - 4*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^7*d^11*f^4*g^3*abs(
g) - 29*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^6*d^9*e^2*f^4*g^3*abs(g) + 21*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^
5*d^7*e^4*f^4*g^3*abs(g) + 26*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^3*c^4*d^5*e^6*f^4*g^3*abs(g) - 14*sqrt(-c*d^2*e
*g^2 + a*e^3*g^2)*a^4*c^3*d^3*e^8*f^4*g^3*abs(g) + 16*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^6*d^10*e*f^3*g^4*abs(
g) + 26*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^5*d^8*e^3*f^3*g^4*abs(g) - 54*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^3*
c^4*d^6*e^5*f^3*g^4*abs(g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^4*c^3*d^4*e^7*f^3*g^4*abs(g) + 11*sqrt(-c*d^2*e*
g^2 + a*e^3*g^2)*a^5*c^2*d^2*e^9*f^3*g^4*abs(g) - 24*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^5*d^9*e^2*f^2*g^5*ab
s(g) + 6*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^3*c^4*d^7*e^4*f^2*g^5*abs(g) + 36*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^4
*c^3*d^5*e^6*f^2*g^5*abs(g) - 15*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^5*c^2*d^3*e^8*f^2*g^5*abs(g) - 3*sqrt(-c*d^2
*e*g^2 + a*e^3*g^2)*a^6*c*d*e^10*f^2*g^5*abs(g) + 16*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^3*c^4*d^8*e^3*f*g^6*abs(
g) - 19*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^4*c^3*d^6*e^5*f*g^6*abs(g) - 3*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^5*c^2
*d^4*e^7*f*g^6*abs(g) + 6*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^6*c*d^2*e^9*f*g^6*abs(g) - 4*sqrt(-c*d^2*e*g^2 + a*
e^3*g^2)*a^4*c^3*d^7*e^4*g^7*abs(g) + 7*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^5*c^2*d^5*e^6*g^7*abs(g) - 3*sqrt(-c*
d^2*e*g^2 + a*e^3*g^2)*a^6*c*d^3*e^8*g^7*abs(g))

Mupad [B] (verification not implemented)

Time = 14.25 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.60 \[ \int \frac {(d+e x)^{5/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {16\,g\,x^2\,\left (a\,e\,g+c\,d\,f\right )\,\sqrt {d+e\,x}}{e\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {\sqrt {d+e\,x}\,\left (2\,a^3\,e^3\,g^3-18\,a^2\,c\,d\,e^2\,f\,g^2-18\,a\,c^2\,d^2\,e\,f^2\,g+2\,c^3\,d^3\,f^3\right )}{3\,c^2\,d^2\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {32\,c\,d\,g^2\,x^3\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {4\,x\,\sqrt {d+e\,x}\,\left (a^2\,e^2\,g^2+6\,a\,c\,d\,e\,f\,g+c^2\,d^2\,f^2\right )}{c\,d\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}+\frac {x^2\,\sqrt {f+g\,x}\,\left (g\,a^2\,e^3+2\,g\,a\,c\,d^2\,e+2\,f\,a\,c\,d\,e^2+f\,c^2\,d^3\right )}{c^2\,d^2\,e\,g}+\frac {a\,x\,\sqrt {f+g\,x}\,\left (2\,c\,f\,d^2+a\,g\,d\,e+a\,f\,e^2\right )}{c^2\,d^2\,g}+\frac {a^2\,e\,f\,\sqrt {f+g\,x}}{c^2\,d\,g}+\frac {x^3\,\sqrt {f+g\,x}\,\left (c\,g\,d^2+c\,f\,d\,e+2\,a\,g\,e^2\right )}{c\,d\,e\,g}} \]

[In]

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

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((16*g*x^2*(a*e*g + c*d*f)*(d + e*x)^(1/2))/(e*(a*e*g - c*d*f)^
4) - ((d + e*x)^(1/2)*(2*a^3*e^3*g^3 + 2*c^3*d^3*f^3 - 18*a*c^2*d^2*e*f^2*g - 18*a^2*c*d*e^2*f*g^2))/(3*c^2*d^
2*e*g*(a*e*g - c*d*f)^4) + (32*c*d*g^2*x^3*(d + e*x)^(1/2))/(3*e*(a*e*g - c*d*f)^4) + (4*x*(d + e*x)^(1/2)*(a^
2*e^2*g^2 + c^2*d^2*f^2 + 6*a*c*d*e*f*g))/(c*d*e*(a*e*g - c*d*f)^4)))/(x^4*(f + g*x)^(1/2) + (x^2*(f + g*x)^(1
/2)*(a^2*e^3*g + c^2*d^3*f + 2*a*c*d*e^2*f + 2*a*c*d^2*e*g))/(c^2*d^2*e*g) + (a*x*(f + g*x)^(1/2)*(a*e^2*f + 2
*c*d^2*f + a*d*e*g))/(c^2*d^2*g) + (a^2*e*f*(f + g*x)^(1/2))/(c^2*d*g) + (x^3*(f + g*x)^(1/2)*(2*a*e^2*g + c*d
^2*g + c*d*e*f))/(c*d*e*g))