\(\int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [725]

Optimal result

Integrand size = 48, antiderivative size = 192 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {16 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^3 \sqrt {d+e x} \sqrt {f+g x}} \]

[Out]

-2*(e*x+d)^(1/2)/(-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)^(1/2)-8/3*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)^(3/2)/(e*x+d)^(1/2)-16/3*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/(e*x+d)^(1/2)/(g*x+f)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {882, 886, 874} \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {16 c d g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^3}-\frac {8 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

[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])/((c*d*f - a*e*g)*(f + g*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (8*g*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^(3/2)) - (16*c*d*g*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]*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)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(4 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}{c d f-a e g} \\ & = -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 g \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)^2 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {(8 c d 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}{3 (c d f-a e g)^2} \\ & = -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {8 g \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)^2 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {16 c d g \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} \sqrt {f+g x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.55 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (-a^2 e^2 g^2+2 a c d e g (3 f+2 g x)+c^2 d^2 \left (3 f^2+12 f g x+8 g^2 x^2\right )\right )}{3 (c d f-a e g)^3 \sqrt {(a e+c d x) (d+e x)} (f+g x)^{3/2}} \]

[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]

(-2*Sqrt[d + e*x]*(-(a^2*e^2*g^2) + 2*a*c*d*e*g*(3*f + 2*g*x) + c^2*d^2*(3*f^2 + 12*f*g*x + 8*g^2*x^2)))/(3*(c
*d*f - a*e*g)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^(3/2))

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.62

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-8 g^{2} x^{2} c^{2} d^{2}-4 a c d e \,g^{2} x -12 c^{2} d^{2} f g x +a^{2} e^{2} g^{2}-6 a c d e f g -3 c^{2} d^{2} f^{2}\right )}{3 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {3}{2}} \left (c d x +a e \right ) \left (a e g -c d f \right )^{3}}\)	\(120\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-8 g^{2} x^{2} c^{2} d^{2}-4 a c d e \,g^{2} x -12 c^{2} d^{2} f g x +a^{2} e^{2} g^{2}-6 a c d e f g -3 c^{2} d^{2} f^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(168\)

[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]

-2/3/(e*x+d)^(1/2)/(g*x+f)^(3/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(-8*c^2*d^2*g^2*x^2-4*a*c*d*e*g^2*x-12*c^2*d^2*f*
g*x+a^2*e^2*g^2-6*a*c*d*e*f*g-3*c^2*d^2*f^2)/(c*d*x+a*e)/(a*e*g-c*d*f)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (170) = 340\).

Time = 0.42 (sec) , antiderivative size = 649, normalized size of antiderivative = 3.38 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (8 \, c^{2} d^{2} g^{2} x^{2} + 3 \, c^{2} d^{2} f^{2} + 6 \, a c d e f g - a^{2} e^{2} g^{2} + 4 \, {\left (3 \, c^{2} d^{2} f g + a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (a c^{3} d^{4} e f^{5} - 3 \, a^{2} c^{2} d^{3} e^{2} f^{4} g + 3 \, a^{3} c d^{2} e^{3} f^{3} g^{2} - a^{4} d e^{4} f^{2} g^{3} + {\left (c^{4} d^{4} e f^{3} g^{2} - 3 \, a c^{3} d^{3} e^{2} f^{2} g^{3} + 3 \, a^{2} c^{2} d^{2} e^{3} f g^{4} - a^{3} c d e^{4} g^{5}\right )} x^{4} + {\left (2 \, c^{4} d^{4} e f^{4} g + {\left (c^{4} d^{5} - 5 \, a c^{3} d^{3} e^{2}\right )} f^{3} g^{2} - 3 \, {\left (a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3}\right )} f^{2} g^{3} + {\left (3 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} f g^{4} - {\left (a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} g^{5}\right )} x^{3} + {\left (c^{4} d^{4} e f^{5} - a^{4} d e^{4} g^{5} + {\left (2 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} f^{4} g - {\left (5 \, a c^{3} d^{4} e + 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{3} g^{2} + {\left (3 \, a^{2} c^{2} d^{3} e^{2} + 5 \, a^{3} c d e^{4}\right )} f^{2} g^{3} + {\left (a^{3} c d^{2} e^{3} - 2 \, a^{4} e^{5}\right )} f g^{4}\right )} x^{2} - {\left (2 \, a^{4} d e^{4} f g^{4} - {\left (c^{4} d^{5} + a c^{3} d^{3} e^{2}\right )} f^{5} + {\left (a c^{3} d^{4} e + 3 \, a^{2} c^{2} d^{2} e^{3}\right )} f^{4} g + 3 \, {\left (a^{2} c^{2} d^{3} e^{2} - a^{3} c d e^{4}\right )} f^{3} g^{2} - {\left (5 \, a^{3} c d^{2} e^{3} - a^{4} e^{5}\right )} f^{2} g^{3}\right )} x\right )}} \]

[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]

-2/3*(8*c^2*d^2*g^2*x^2 + 3*c^2*d^2*f^2 + 6*a*c*d*e*f*g - a^2*e^2*g^2 + 4*(3*c^2*d^2*f*g + a*c*d*e*g^2)*x)*sqr
t(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^3*d^4*e*f^5 - 3*a^2*c^2*d^3*e^2*f^4*
g + 3*a^3*c*d^2*e^3*f^3*g^2 - a^4*d*e^4*f^2*g^3 + (c^4*d^4*e*f^3*g^2 - 3*a*c^3*d^3*e^2*f^2*g^3 + 3*a^2*c^2*d^2
*e^3*f*g^4 - a^3*c*d*e^4*g^5)*x^4 + (2*c^4*d^4*e*f^4*g + (c^4*d^5 - 5*a*c^3*d^3*e^2)*f^3*g^2 - 3*(a*c^3*d^4*e
- a^2*c^2*d^2*e^3)*f^2*g^3 + (3*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*f*g^4 - (a^3*c*d^2*e^3 + a^4*e^5)*g^5)*x^3 + (c
^4*d^4*e*f^5 - a^4*d*e^4*g^5 + (2*c^4*d^5 - a*c^3*d^3*e^2)*f^4*g - (5*a*c^3*d^4*e + 3*a^2*c^2*d^2*e^3)*f^3*g^2
+ (3*a^2*c^2*d^3*e^2 + 5*a^3*c*d*e^4)*f^2*g^3 + (a^3*c*d^2*e^3 - 2*a^4*e^5)*f*g^4)*x^2 - (2*a^4*d*e^4*f*g^4 -
(c^4*d^5 + a*c^3*d^3*e^2)*f^5 + (a*c^3*d^4*e + 3*a^2*c^2*d^2*e^3)*f^4*g + 3*(a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f
^3*g^2 - (5*a^3*c*d^2*e^3 - a^4*e^5)*f^2*g^3)*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]

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

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2618 vs. \(2 (170) = 340\).

Time = 1.84 (sec) , antiderivative size = 2618, normalized size of antiderivative = 13.64 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[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]

-2/3*(3*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*c^2*d^2*g^2/((c^3*d^3*e^3*f^3*abs(g) - 3*a*c^2*d^2*e^4*f^2*g*abs(g
) + 3*a^2*c*d*e^5*f*g^2*abs(g) - a^3*e^6*g^3*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*(5*sqrt(c*d*g)*c^3*d^3*e^4*f^2*g^4 - 10*sqrt(c*d*g)*a*c^2*d^2*e^5*f*g^5 + 5*sqrt(c*d*g)*a^2
*c*d*e^6*g^6 + 12*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^3 - 12*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^4 +
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^2)/((c^2*d^2*e*f^2*abs(g) - 2*a*c*d*e^2*f*g*abs(g) + a^2*e^3*g^2*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))*e^3 + 2/3*(3*sqrt(e^2*f - d*e*g)*c^4*d^4*e^3*f^3*g^2 - 9*sqrt(e^
2*f - d*e*g)*c^4*d^5*e^2*f^2*g^3 + 21*sqrt(e^2*f - d*e*g)*c^4*d^6*e*f*g^4 - 24*sqrt(e^2*f - d*e*g)*a*c^3*d^4*e
^3*f*g^4 + 12*sqrt(e^2*f - d*e*g)*a^2*c^2*d^2*e^5*f*g^4 - 12*sqrt(e^2*f - d*e*g)*c^4*d^7*g^5 + 15*sqrt(e^2*f -
d*e*g)*a*c^3*d^5*e^2*g^5 - 3*sqrt(e^2*f - d*e*g)*a^2*c^2*d^3*e^4*g^5 - 3*sqrt(e^2*f - d*e*g)*a^3*c*d*e^6*g^5
- 4*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^3*d^3*e^3*f^3*g + 18*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*
g)*c^3*d^4*e^2*f^2*g^2 - 6*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a*c^2*d^2*e^4*f^2*g^2 - 27*sqrt(-c*d^2*e
*g^2 + a*e^3*g^2)*sqrt(c*d*g)*c^3*d^5*e*f*g^3 + 18*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a*c^2*d^3*e^3*f*
g^3 - 3*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a^2*c*d*e^5*f*g^3 + 12*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(
c*d*g)*c^3*d^6*g^4 - 9*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*sqrt(c*d*g)*a*c^2*d^4*e^2*g^4 + sqrt(-c*d^2*e*g^2 + a*e^
3*g^2)*sqrt(c*d*g)*a^3*e^6*g^4)/(3*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^5*d^6*e^2*f^5*g*abs(g) - 3*sqrt(e^2*f - d
*e*g)*sqrt(c*d*g)*a*c^4*d^4*e^4*f^5*g*abs(g) - 7*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^5*d^7*e*f^4*g^2*abs(g) - sq
rt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^4*d^5*e^3*f^4*g^2*abs(g) + 8*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^3*d^3*e^5
*f^4*g^2*abs(g) + 4*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*c^5*d^8*f^3*g^3*abs(g) + 16*sqrt(e^2*f - d*e*g)*sqrt(c*d*g
)*a*c^4*d^6*e^2*f^3*g^3*abs(g) - 14*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^3*d^4*e^4*f^3*g^3*abs(g) - 6*sqrt(e^
2*f - d*e*g)*sqrt(c*d*g)*a^3*c^2*d^2*e^6*f^3*g^3*abs(g) - 12*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a*c^4*d^7*e*f^2*g
^4*abs(g) - 6*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^3*d^5*e^3*f^2*g^4*abs(g) + 18*sqrt(e^2*f - d*e*g)*sqrt(c*d
*g)*a^3*c^2*d^3*e^5*f^2*g^4*abs(g) + 12*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^2*c^3*d^6*e^2*f*g^5*abs(g) - 8*sqrt(
e^2*f - d*e*g)*sqrt(c*d*g)*a^3*c^2*d^4*e^4*f*g^5*abs(g) - 5*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*c*d^2*e^6*f*g^
5*abs(g) + sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^5*e^8*f*g^5*abs(g) - 4*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^3*c^2*d^
5*e^3*g^6*abs(g) + 5*sqrt(e^2*f - d*e*g)*sqrt(c*d*g)*a^4*c*d^3*e^5*g^6*abs(g) - sqrt(e^2*f - d*e*g)*sqrt(c*d*g
)*a^5*d*e^7*g^6*abs(g) + sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^5*d^5*e^3*f^6*abs(g) - 6*sqrt(-c*d^2*e*g^2 + a*e^3*g
^2)*c^5*d^6*e^2*f^5*g*abs(g) + 9*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^5*d^7*e*f^4*g^2*abs(g) + 12*sqrt(-c*d^2*e*g^
2 + a*e^3*g^2)*a*c^4*d^5*e^3*f^4*g^2*abs(g) - 6*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^3*d^3*e^5*f^4*g^2*abs(g)
- 4*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*c^5*d^8*f^3*g^3*abs(g) - 24*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a*c^4*d^6*e^2*f^
3*g^3*abs(g) + 8*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^3*c^2*d^2*e^6*f^3*g^3*abs(g) + 12*sqrt(-c*d^2*e*g^2 + a*e^3*
g^2)*a*c^4*d^7*e*f^2*g^4*abs(g) + 18*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^3*d^5*e^3*f^2*g^4*abs(g) - 12*sqrt(-
c*d^2*e*g^2 + a*e^3*g^2)*a^3*c^2*d^3*e^5*f^2*g^4*abs(g) - 3*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^4*c*d*e^7*f^2*g^4
*abs(g) - 12*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^2*c^3*d^6*e^2*f*g^5*abs(g) + 6*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^
4*c*d^2*e^6*f*g^5*abs(g) + 4*sqrt(-c*d^2*e*g^2 + a*e^3*g^2)*a^3*c^2*d^5*e^3*g^6*abs(g) - 3*sqrt(-c*d^2*e*g^2 +
a*e^3*g^2)*a^4*c*d^3*e^5*g^6*abs(g))

Mupad [B] (verification not implemented)

Time = 13.98 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.40 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\left (\frac {8\,x\,\left (a\,e\,g+3\,c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {\sqrt {d+e\,x}\,\left (-2\,a^2\,e^2\,g^2+12\,a\,c\,d\,e\,f\,g+6\,c^2\,d^2\,f^2\right )}{3\,c\,d\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c\,d\,g\,x^2\,\sqrt {d+e\,x}}{3\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^3\,\sqrt {f+g\,x}+\frac {a\,f\,\sqrt {f+g\,x}}{c\,g}+\frac {x\,\sqrt {f+g\,x}\,\left (c\,f\,d^2+a\,g\,d\,e+a\,f\,e^2\right )}{c\,d\,e\,g}+\frac {x^2\,\sqrt {f+g\,x}\,\left (c\,g\,d^2+c\,f\,d\,e+a\,g\,e^2\right )}{c\,d\,e\,g}} \]

[In]

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

[Out]

(((8*x*(a*e*g + 3*c*d*f)*(d + e*x)^(1/2))/(3*e*(a*e*g - c*d*f)^3) + ((d + e*x)^(1/2)*(6*c^2*d^2*f^2 - 2*a^2*e^
2*g^2 + 12*a*c*d*e*f*g))/(3*c*d*e*g*(a*e*g - c*d*f)^3) + (16*c*d*g*x^2*(d + e*x)^(1/2))/(3*e*(a*e*g - c*d*f)^3
))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(x^3*(f + g*x)^(1/2) + (a*f*(f + g*x)^(1/2))/(c*g) + (x*(f +
g*x)^(1/2)*(a*e^2*f + c*d^2*f + a*d*e*g))/(c*d*e*g) + (x^2*(f + g*x)^(1/2)*(a*e^2*g + c*d^2*g + c*d*e*f))/(c*
d*e*g))