\(\int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [726]

Optimal result

Integrand size = 48, antiderivative size = 262 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {12 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {16 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {32 c^2 d^2 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \]

[Out]

-2*(e*x+d)^(1/2)/(-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)-12/5*g*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^2/(g*x+f)^(5/2)/(e*x+d)^(1/2)-16/5*c*d*g*(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)-32/5*c^2*d^2*g*(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 = 262, 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} \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {32 c^2 d^2 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^4}-\frac {16 c d g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^3}-\frac {12 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

[In]

Int[(d + e*x)^(3/2)/((f + g*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-2*Sqrt[d + e*x])/((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]) - (12*g*Sqrt[
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(5/2)) - (16*c*d*g*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(3/2)) - (32*c^2*d^2*g*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*(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 \sqrt {d+e x}}{(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}}-\frac {(6 g) \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \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} \\ & = -\frac {2 \sqrt {d+e x}}{(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}}-\frac {12 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {(24 c d g) \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{5 (c d f-a e g)^2} \\ & = -\frac {2 \sqrt {d+e x}}{(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}}-\frac {12 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {16 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {\left (16 c^2 d^2 g\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}{5 (c d f-a e g)^3} \\ & = -\frac {2 \sqrt {d+e x}}{(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}}-\frac {12 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {16 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {32 c^2 d^2 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.57 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (a^3 e^3 g^3-a^2 c d e^2 g^2 (5 f+2 g x)+a c^2 d^2 e g \left (15 f^2+20 f g x+8 g^2 x^2\right )+c^3 d^3 \left (5 f^3+30 f^2 g x+40 f g^2 x^2+16 g^3 x^3\right )\right )}{5 (c d f-a e g)^4 \sqrt {(a e+c d x) (d+e x)} (f+g x)^{5/2}} \]

[In]

Integrate[(d + e*x)^(3/2)/((f + g*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-2*Sqrt[d + e*x]*(a^3*e^3*g^3 - a^2*c*d*e^2*g^2*(5*f + 2*g*x) + a*c^2*d^2*e*g*(15*f^2 + 20*f*g*x + 8*g^2*x^2)
+ c^3*d^3*(5*f^3 + 30*f^2*g*x + 40*f*g^2*x^2 + 16*g^3*x^3)))/(5*(c*d*f - a*e*g)^4*Sqrt[(a*e + c*d*x)*(d + e*x
)]*(f + g*x)^(5/2))

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.73

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (16 g^{3} x^{3} c^{3} d^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}+40 c^{3} d^{3} f \,g^{2} x^{2}-2 a^{2} c d \,e^{2} g^{3} x +20 a \,c^{2} d^{2} e f \,g^{2} x +30 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-5 a^{2} c d \,e^{2} f \,g^{2}+15 a \,c^{2} d^{2} e \,f^{2} g +5 f^{3} c^{3} d^{3}\right )}{5 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {5}{2}} \left (c d x +a e \right ) \left (a e g -c d f \right )^{4}}\)	\(192\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (16 g^{3} x^{3} c^{3} d^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}+40 c^{3} d^{3} f \,g^{2} x^{2}-2 a^{2} c d \,e^{2} g^{3} x +20 a \,c^{2} d^{2} e f \,g^{2} x +30 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-5 a^{2} c d \,e^{2} f \,g^{2}+15 a \,c^{2} d^{2} e \,f^{2} g +5 f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{5 \left (g x +f \right )^{\frac {5}{2}} \left (a^{4} e^{4} g^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(259\)

[In]

int((e*x+d)^(3/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/5/(e*x+d)^(1/2)/(g*x+f)^(5/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(16*c^3*d^3*g^3*x^3+8*a*c^2*d^2*e*g^3*x^2+40*c^3*
d^3*f*g^2*x^2-2*a^2*c*d*e^2*g^3*x+20*a*c^2*d^2*e*f*g^2*x+30*c^3*d^3*f^2*g*x+a^3*e^3*g^3-5*a^2*c*d*e^2*f*g^2+15
*a*c^2*d^2*e*f^2*g+5*c^3*d^3*f^3)/(c*d*x+a*e)/(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. 1062 vs. \(2 (232) = 464\).

Time = 1.40 (sec) , antiderivative size = 1062, normalized size of antiderivative = 4.05 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (16 \, c^{3} d^{3} g^{3} x^{3} + 5 \, c^{3} d^{3} f^{3} + 15 \, a c^{2} d^{2} e f^{2} g - 5 \, a^{2} c d e^{2} f g^{2} + a^{3} e^{3} g^{3} + 8 \, {\left (5 \, c^{3} d^{3} f g^{2} + a c^{2} d^{2} e g^{3}\right )} x^{2} + 2 \, {\left (15 \, c^{3} d^{3} f^{2} g + 10 \, 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}}{5 \, {\left (a c^{4} d^{5} e f^{7} - 4 \, a^{2} c^{3} d^{4} e^{2} f^{6} g + 6 \, a^{3} c^{2} d^{3} e^{3} f^{5} g^{2} - 4 \, a^{4} c d^{2} e^{4} f^{4} g^{3} + a^{5} d e^{5} f^{3} g^{4} + {\left (c^{5} d^{5} e f^{4} g^{3} - 4 \, a c^{4} d^{4} e^{2} f^{3} g^{4} + 6 \, a^{2} c^{3} d^{3} e^{3} f^{2} g^{5} - 4 \, a^{3} c^{2} d^{2} e^{4} f g^{6} + a^{4} c d e^{5} g^{7}\right )} x^{5} + {\left (3 \, c^{5} d^{5} e f^{5} g^{2} + {\left (c^{5} d^{6} - 11 \, a c^{4} d^{4} e^{2}\right )} f^{4} g^{3} - 2 \, {\left (2 \, a c^{4} d^{5} e - 7 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{3} g^{4} + 6 \, {\left (a^{2} c^{3} d^{4} e^{2} - a^{3} c^{2} d^{2} e^{4}\right )} f^{2} g^{5} - {\left (4 \, a^{3} c^{2} d^{3} e^{3} + a^{4} c d e^{5}\right )} f g^{6} + {\left (a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} g^{7}\right )} x^{4} + {\left (3 \, c^{5} d^{5} e f^{6} g + a^{5} d e^{5} g^{7} + 3 \, {\left (c^{5} d^{6} - 3 \, a c^{4} d^{4} e^{2}\right )} f^{5} g^{2} - {\left (11 \, a c^{4} d^{5} e - 6 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{4} g^{3} + 2 \, {\left (7 \, a^{2} c^{3} d^{4} e^{2} + 3 \, a^{3} c^{2} d^{2} e^{4}\right )} f^{3} g^{4} - 3 \, {\left (2 \, a^{3} c^{2} d^{3} e^{3} + 3 \, a^{4} c d e^{5}\right )} f^{2} g^{5} - {\left (a^{4} c d^{2} e^{4} - 3 \, a^{5} e^{6}\right )} f g^{6}\right )} x^{3} + {\left (c^{5} d^{5} e f^{7} + 3 \, a^{5} d e^{5} f g^{6} + {\left (3 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} f^{6} g - 3 \, {\left (3 \, a c^{4} d^{5} e + 2 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{5} g^{2} + 2 \, {\left (3 \, a^{2} c^{3} d^{4} e^{2} + 7 \, a^{3} c^{2} d^{2} e^{4}\right )} f^{4} g^{3} + {\left (6 \, a^{3} c^{2} d^{3} e^{3} - 11 \, a^{4} c d e^{5}\right )} f^{3} g^{4} - 3 \, {\left (3 \, a^{4} c d^{2} e^{4} - a^{5} e^{6}\right )} f^{2} g^{5}\right )} x^{2} + {\left (3 \, a^{5} d e^{5} f^{2} g^{5} + {\left (c^{5} d^{6} + a c^{4} d^{4} e^{2}\right )} f^{7} - {\left (a c^{4} d^{5} e + 4 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{6} g - 6 \, {\left (a^{2} c^{3} d^{4} e^{2} - a^{3} c^{2} d^{2} e^{4}\right )} f^{5} g^{2} + 2 \, {\left (7 \, a^{3} c^{2} d^{3} e^{3} - 2 \, a^{4} c d e^{5}\right )} f^{4} g^{3} - {\left (11 \, a^{4} c d^{2} e^{4} - a^{5} e^{6}\right )} f^{3} g^{4}\right )} x\right )}} \]

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

-2/5*(16*c^3*d^3*g^3*x^3 + 5*c^3*d^3*f^3 + 15*a*c^2*d^2*e*f^2*g - 5*a^2*c*d*e^2*f*g^2 + a^3*e^3*g^3 + 8*(5*c^3
*d^3*f*g^2 + a*c^2*d^2*e*g^3)*x^2 + 2*(15*c^3*d^3*f^2*g + 10*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*c^4*d^5*e*f^7 - 4*a^2*c^3*d^4*e^2*f^6*g + 6*
a^3*c^2*d^3*e^3*f^5*g^2 - 4*a^4*c*d^2*e^4*f^4*g^3 + a^5*d*e^5*f^3*g^4 + (c^5*d^5*e*f^4*g^3 - 4*a*c^4*d^4*e^2*f
^3*g^4 + 6*a^2*c^3*d^3*e^3*f^2*g^5 - 4*a^3*c^2*d^2*e^4*f*g^6 + a^4*c*d*e^5*g^7)*x^5 + (3*c^5*d^5*e*f^5*g^2 + (
c^5*d^6 - 11*a*c^4*d^4*e^2)*f^4*g^3 - 2*(2*a*c^4*d^5*e - 7*a^2*c^3*d^3*e^3)*f^3*g^4 + 6*(a^2*c^3*d^4*e^2 - a^3
*c^2*d^2*e^4)*f^2*g^5 - (4*a^3*c^2*d^3*e^3 + a^4*c*d*e^5)*f*g^6 + (a^4*c*d^2*e^4 + a^5*e^6)*g^7)*x^4 + (3*c^5*
d^5*e*f^6*g + a^5*d*e^5*g^7 + 3*(c^5*d^6 - 3*a*c^4*d^4*e^2)*f^5*g^2 - (11*a*c^4*d^5*e - 6*a^2*c^3*d^3*e^3)*f^4
*g^3 + 2*(7*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4)*f^3*g^4 - 3*(2*a^3*c^2*d^3*e^3 + 3*a^4*c*d*e^5)*f^2*g^5 - (a^
4*c*d^2*e^4 - 3*a^5*e^6)*f*g^6)*x^3 + (c^5*d^5*e*f^7 + 3*a^5*d*e^5*f*g^6 + (3*c^5*d^6 - a*c^4*d^4*e^2)*f^6*g -
3*(3*a*c^4*d^5*e + 2*a^2*c^3*d^3*e^3)*f^5*g^2 + 2*(3*a^2*c^3*d^4*e^2 + 7*a^3*c^2*d^2*e^4)*f^4*g^3 + (6*a^3*c^
2*d^3*e^3 - 11*a^4*c*d*e^5)*f^3*g^4 - 3*(3*a^4*c*d^2*e^4 - a^5*e^6)*f^2*g^5)*x^2 + (3*a^5*d*e^5*f^2*g^5 + (c^5
*d^6 + a*c^4*d^4*e^2)*f^7 - (a*c^4*d^5*e + 4*a^2*c^3*d^3*e^3)*f^6*g - 6*(a^2*c^3*d^4*e^2 - a^3*c^2*d^2*e^4)*f^
5*g^2 + 2*(7*a^3*c^2*d^3*e^3 - 2*a^4*c*d*e^5)*f^4*g^3 - (11*a^4*c*d^2*e^4 - a^5*e^6)*f^3*g^4)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(3/2)/(g*x+f)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^(7/2)/(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)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^(7/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6137 vs. \(2 (232) = 464\).

Time = 24.65 (sec) , antiderivative size = 6137, normalized size of antiderivative = 23.42 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

-2/5*(5*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*c^3*d^3*g^2/((c^4*d^4*e^4*f^4*abs(g) - 4*a*c^3*d^3*e^5*f^3*g*abs(g
) + 6*a^2*c^2*d^2*e^6*f^2*g^2*abs(g) - 4*a^3*c*d*e^7*f*g^3*abs(g) + a^4*e^8*g^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)) + 2*(11*sqrt(c*d*g)*c^6*d^6*e^8*f^4*g^6 - 44*sqrt(c*d*g)*a*c
^5*d^5*e^9*f^3*g^7 + 66*sqrt(c*d*g)*a^2*c^4*d^4*e^10*f^2*g^8 - 44*sqrt(c*d*g)*a^3*c^3*d^3*e^11*f*g^9 + 11*sqrt
(c*d*g)*a^4*c^2*d^2*e^12*g^10 + 50*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^5*d^5*e^6*f^3*g^5 - 150*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^4*d^4*e^7*f^2*g^6 + 150*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^2*c^3*d^3*e^8*f*g^7 - 50*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^3*c^2*d^2*e^9*g^8 + 80*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^4*d^4*e^4*f^2*g^4 - 160*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*a
*c^3*d^3*e^5*f*g^5 + 80*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*a^2*c^2*d^2*e^6*g^6 + 30*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))^6*c^3*d^3
*e^2*f*g^3 - 30*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))^6*a*c^2*d^2*e^3*g^4 + 5*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))^8*c^2*d^2*g^2)/((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)^5))*e^4 + 2/5*(5*sqrt(e^2*f - d*e*g)*c^6*d^6*e^5*f^5*g^2 - 35*sqrt(e^2*f -
d*e*g)*c^6*d^7*e^4*f^4*g^3 + 10*sqrt(e^2*f - d*e*g)*a*c^5*d^5*e^6*f^4*g^3 + 150*sqrt(e^2*f - d*e*g)*c^6*d^8*e^
3*f^3*g^4 - 160*sqrt(e^2*f - d*e*g)*a*c^5*d^6*e^5*f^3*g^4 + 60*sqrt(e^2*f - d*e*g)*a^2*c^4*d^4*e^7*f^3*g^4 - 3
05*sqrt(e^2*f - d*e*g)*c^6*d^9*e^2*f^2*g^5 + 465*sqrt(e^2*f - d*e*g)*a*c^5*d^7*e^4*f^2*g^5 - 225*sqrt(e^2*f -
d*e*g)*a^2*c^4*d^5*e^6*f^2*g^5 + 15*sqrt(e^2*f - d*e*g)*a^3*c^3*d^3*e^8*f^2*g^5 + 260*sqrt(e^2*f - d*e*g)*c^6*
d^10*e*f*g^6 - 430*sqrt(e^2*f - d*e*g)*a*c^5*d^8*e^3*f*g^6 + 180*sqrt(e^2*f - d*e*g)*a^2*c^4*d^6*e^5*f*g^6 + 3
0*sqrt(e^2*f - d*e*g)*a^3*c^3*d^4*e^7*f*g^6 - 15*sqrt(e^2*f - d*e*g)*a^4*c^2*d^2*e^9*f*g^6 - 80*sqrt(e^2*f - d
*e*g)*c^6*d^11*g^7 + 140*sqrt(e^2*f - d*e*g)*a*c^5*d^9*e^2*g^7 - 65*sqrt(e^2*f - d*e*g)*a^2*c^4*d^7*e^4*g^7 +
5*sqrt(e^2*f - d*e*g)*a^3*c^3*d^5*e^6*g^7 - 10*sqrt(e^2*f - d*e*g)*a^4*c^2*d^3*e^8*g^7 + 5*sqrt(e^2*f - d*e*g)
*a^5*c*d*e^10*g^7 - 14*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^5*d^5*e^5*f^5*g + 85*sqrt(-c*d^2*e*g^2 + a
*e^3*g^2)*sqrt(c*d*g)*c^5*d^6*e^4*f^4*g^2 - 15*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a*c^4*d^4*e^6*f^4*g^
2 - 275*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^5*d^7*e^3*f^3*g^3 + 210*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sq
rt(c*d*g)*a*c^4*d^5*e^5*f^3*g^3 - 75*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^2*c^3*d^3*e^7*f^3*g^3 + 425*
sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^5*d^8*e^2*f^2*g^4 - 450*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g
)*a*c^4*d^6*e^4*f^2*g^4 + 135*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^2*c^3*d^4*e^6*f^2*g^4 + 30*sqrt(-c*
d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^3*c^2*d^2*e^8*f^2*g^4 - 300*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^
5*d^9*e*f*g^5 + 350*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a*c^4*d^7*e^3*f*g^5 - 75*sqrt(-c*d^2*e*g^2 + a*
e^3*g^2)*sqrt(c*d*g)*a^2*c^3*d^5*e^5*f*g^5 - 40*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^3*c^2*d^3*e^7*f*g
^5 - 5*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^4*c*d*e^9*f*g^5 + 80*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c
*d*g)*c^5*d^10*g^6 - 100*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a*c^4*d^8*e^2*g^6 + 25*sqrt(-c*d^2*e*g^2 +
a*e^3*g^2)*sqrt(c*d*g)*a^2*c^3*d^6*e^4*g^6 + 10*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^4*c*d^2*e^8*g^6
- sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^5*e^10*g^6)/(5*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^7*d^8*e^4*f^8*
g*abs(g) - 5*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^6*d^6*e^6*f^8*g*abs(g) - 30*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c
^7*d^9*e^3*f^7*g^2*abs(g) + 20*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^6*d^7*e^5*f^7*g^2*abs(g) + 10*sqrt(e^2*f -
d*e*g)*sqrt(c*d*g)*a^2*c^5*d^5*e^7*f^7*g^2*abs(g) + 61*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^7*d^10*e^2*f^6*g^3*ab
s(g) + 27*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^6*d^8*e^4*f^6*g^3*abs(g) - 97*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^
2*c^5*d^6*e^6*f^6*g^3*abs(g) + 9*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*c^4*d^4*e^8*f^6*g^3*abs(g) - 52*sqrt(e^2*
f - d*e*g)*sqrt(c*d*g)*c^7*d^11*e*f^5*g^4*abs(g) - 158*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^6*d^9*e^3*f^5*g^4*a
bs(g) + 156*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^5*d^7*e^5*f^5*g^4*abs(g) + 90*sqrt(e^2*f - d*e*g)*sqrt(c*d*g
)*a^3*c^4*d^5*e^7*f^5*g^4*abs(g) - 36*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*c^3*d^3*e^9*f^5*g^4*abs(g) + 16*sqrt
(e^2*f - d*e*g)*sqrt(c*d*g)*c^7*d^12*f^4*g^5*abs(g) + 180*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^6*d^10*e^2*f^4*g
^5*abs(g) + 35*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^5*d^8*e^4*f^4*g^5*abs(g) - 295*sqrt(e^2*f - d*e*g)*sqrt(c
*d*g)*a^3*c^4*d^6*e^6*f^4*g^5*abs(g) + 35*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*c^3*d^4*e^8*f^4*g^5*abs(g) + 29*
sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^5*c^2*d^2*e^10*f^4*g^5*abs(g) - 64*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^6*d^1
1*e*f^3*g^6*abs(g) - 200*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^5*d^9*e^3*f^3*g^6*abs(g) + 220*sqrt(e^2*f - d*e
*g)*sqrt(c*d*g)*a^3*c^4*d^7*e^5*f^3*g^6*abs(g) + 130*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*c^3*d^5*e^7*f^3*g^6*a
bs(g) - 80*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^5*c^2*d^3*e^9*f^3*g^6*abs(g) - 6*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*
a^6*c*d*e^11*f^3*g^6*abs(g) + 96*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^5*d^10*e^2*f^2*g^7*abs(g) + 40*sqrt(e^2
*f - d*e*g)*sqrt(c*d*g)*a^3*c^4*d^8*e^4*f^2*g^7*abs(g) - 205*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*c^3*d^6*e^6*f
^2*g^7*abs(g) + 45*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^5*c^2*d^4*e^8*f^2*g^7*abs(g) + 25*sqrt(e^2*f - d*e*g)*sqr
t(c*d*g)*a^6*c*d^2*e^10*f^2*g^7*abs(g) - sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^7*e^12*f^2*g^7*abs(g) - 64*sqrt(e^2
*f - d*e*g)*sqrt(c*d*g)*a^3*c^4*d^9*e^3*f*g^8*abs(g) + 60*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*c^3*d^7*e^5*f*g^
8*abs(g) + 34*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^5*c^2*d^5*e^7*f*g^8*abs(g) - 32*sqrt(e^2*f - d*e*g)*sqrt(c*d*g
)*a^6*c*d^3*e^9*f*g^8*abs(g) + 2*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^7*d*e^11*f*g^8*abs(g) + 16*sqrt(e^2*f - d*e
*g)*sqrt(c*d*g)*a^4*c^3*d^8*e^4*g^9*abs(g) - 28*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^5*c^2*d^6*e^6*g^9*abs(g) + 1
3*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^6*c*d^4*e^8*g^9*abs(g) - sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^7*d^2*e^10*g^9*
abs(g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^7*d^7*e^5*f^9*abs(g) - 15*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^7*d^8*e^4
*f^8*g*abs(g) + 6*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^6*d^6*e^6*f^8*g*abs(g) + 55*sqrt(-c*d^2*e*g^2 + a*e^3*g^2
)*c^7*d^9*e^3*f^7*g^2*abs(g) + 10*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^6*d^7*e^5*f^7*g^2*abs(g) - 29*sqrt(-c*d^2
*e*g^2 + a*e^3*g^2)*a^2*c^5*d^5*e^7*f^7*g^2*abs(g) - 85*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^7*d^10*e^2*f^6*g^3*ab
s(g) - 130*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^6*d^8*e^4*f^6*g^3*abs(g) + 95*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2
*c^5*d^6*e^6*f^6*g^3*abs(g) + 36*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^3*c^4*d^4*e^8*f^6*g^3*abs(g) + 60*sqrt(-c*d^
2*e*g^2 + a*e^3*g^2)*c^7*d^11*e*f^5*g^4*abs(g) + 270*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^6*d^9*e^3*f^5*g^4*abs(
g) - 15*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^5*d^7*e^5*f^5*g^4*abs(g) - 180*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^3
*c^4*d^5*e^7*f^5*g^4*abs(g) - 9*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^4*c^3*d^3*e^9*f^5*g^4*abs(g) - 16*sqrt(-c*d^2
*e*g^2 + a*e^3*g^2)*c^7*d^12*f^4*g^5*abs(g) - 220*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^6*d^10*e^2*f^4*g^5*abs(g)
- 235*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^5*d^8*e^4*f^4*g^5*abs(g) + 260*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^3*
c^4*d^6*e^6*f^4*g^5*abs(g) + 95*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^4*c^3*d^4*e^8*f^4*g^5*abs(g) - 10*sqrt(-c*d^2
*e*g^2 + a*e^3*g^2)*a^5*c^2*d^2*e^10*f^4*g^5*abs(g) + 64*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^6*d^11*e*f^3*g^6*a
bs(g) + 280*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^5*d^9*e^3*f^3*g^6*abs(g) - 60*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*
a^3*c^4*d^7*e^5*f^3*g^6*abs(g) - 215*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^4*c^3*d^5*e^7*f^3*g^6*abs(g) + 10*sqrt(-
c*d^2*e*g^2 + a*e^3*g^2)*a^5*c^2*d^3*e^9*f^3*g^6*abs(g) + 5*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^6*c*d*e^11*f^3*g^
6*abs(g) - 96*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^5*d^10*e^2*f^2*g^7*abs(g) - 120*sqrt(-c*d^2*e*g^2 + a*e^3*g
^2)*a^3*c^4*d^8*e^4*f^2*g^7*abs(g) + 165*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^4*c^3*d^6*e^6*f^2*g^7*abs(g) + 30*sq
rt(-c*d^2*e*g^2 + a*e^3*g^2)*a^5*c^2*d^4*e^8*f^2*g^7*abs(g) - 15*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^6*c*d^2*e^10
*f^2*g^7*abs(g) + 64*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^3*c^4*d^9*e^3*f*g^8*abs(g) - 20*sqrt(-c*d^2*e*g^2 + a*e^
3*g^2)*a^4*c^3*d^7*e^5*f*g^8*abs(g) - 50*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^5*c^2*d^5*e^7*f*g^8*abs(g) + 15*sqrt
(-c*d^2*e*g^2 + a*e^3*g^2)*a^6*c*d^3*e^9*f*g^8*abs(g) - 16*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^4*c^3*d^8*e^4*g^9*
abs(g) + 20*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^5*c^2*d^6*e^6*g^9*abs(g) - 5*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^6*c
*d^4*e^8*g^9*abs(g))

Mupad [B] (verification not implemented)

Time = 14.40 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.58 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,x\,\sqrt {d+e\,x}\,\left (-a^2\,e^2\,g^2+10\,a\,c\,d\,e\,f\,g+15\,c^2\,d^2\,f^2\right )}{5\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {\sqrt {d+e\,x}\,\left (\frac {2\,a^3\,e^3\,g^3}{5}-2\,a^2\,c\,d\,e^2\,f\,g^2+6\,a\,c^2\,d^2\,e\,f^2\,g+2\,c^3\,d^3\,f^3\right )}{c\,d\,e\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {32\,c^2\,d^2\,g\,x^3\,\sqrt {d+e\,x}}{5\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c\,d\,x^2\,\left (a\,e\,g+5\,c\,d\,f\right )\,\sqrt {d+e\,x}}{5\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}+\frac {a\,f^2\,\sqrt {f+g\,x}}{c\,g^2}+\frac {x^2\,\sqrt {f+g\,x}\,\left (2\,c\,d^2\,f\,g+c\,d\,e\,f^2+a\,d\,e\,g^2+2\,a\,e^2\,f\,g\right )}{c\,d\,e\,g^2}+\frac {x^3\,\sqrt {f+g\,x}\,\left (c\,g\,d^2+2\,c\,f\,d\,e+a\,g\,e^2\right )}{c\,d\,e\,g}+\frac {f\,x\,\sqrt {f+g\,x}\,\left (c\,f\,d^2+2\,a\,g\,d\,e+a\,f\,e^2\right )}{c\,d\,e\,g^2}} \]

[In]

int((d + e*x)^(3/2)/((f + g*x)^(7/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)

[Out]

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((4*x*(d + e*x)^(1/2)*(15*c^2*d^2*f^2 - a^2*e^2*g^2 + 10*a*c*d
*e*f*g))/(5*e*g*(a*e*g - c*d*f)^4) + ((d + e*x)^(1/2)*((2*a^3*e^3*g^3)/5 + 2*c^3*d^3*f^3 + 6*a*c^2*d^2*e*f^2*g
- 2*a^2*c*d*e^2*f*g^2))/(c*d*e*g^2*(a*e*g - c*d*f)^4) + (32*c^2*d^2*g*x^3*(d + e*x)^(1/2))/(5*e*(a*e*g - c*d*
f)^4) + (16*c*d*x^2*(a*e*g + 5*c*d*f)*(d + e*x)^(1/2))/(5*e*(a*e*g - c*d*f)^4)))/(x^4*(f + g*x)^(1/2) + (a*f^2
*(f + g*x)^(1/2))/(c*g^2) + (x^2*(f + g*x)^(1/2)*(a*d*e*g^2 + c*d*e*f^2 + 2*a*e^2*f*g + 2*c*d^2*f*g))/(c*d*e*g
^2) + (x^3*(f + g*x)^(1/2)*(a*e^2*g + c*d^2*g + 2*c*d*e*f))/(c*d*e*g) + (f*x*(f + g*x)^(1/2)*(a*e^2*f + c*d^2*
f + 2*a*d*e*g))/(c*d*e*g^2))