\(\int \frac {(d+e x)^{3/2}}{(f+g x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [670]

Optimal result

Integrand size = 46, antiderivative size = 202 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {3 c d \sqrt {g} \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{(c d f-a e g)^{5/2}} \]

[Out]

-3*c*d*arctan(g^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^(1/2)/(e*x+d)^(1/2))*g^(1/2)/(-a*
e*g+c*d*f)^(5/2)-2*(e*x+d)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-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)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {882, 886, 888, 211} \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {3 c d \sqrt {g} \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{5/2}}-\frac {3 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(f+g x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

[In]

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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]

Rule 888

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[
2*e^2, Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; Fre
eQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(3 g) \int \frac {\sqrt {d+e x}}{(f+g x)^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) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {(3 c d g) \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 (c d f-a e g)^2} \\ & = -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {\left (3 c d e^2 g\right ) \text {Subst}\left (\int \frac {1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{(c d f-a e g)^2} \\ & = -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {3 c d \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{(c d f-a e g)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.70 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {\sqrt {d+e x} \left (\sqrt {c d f-a e g} (a e g+c d (2 f+3 g x))+3 c d \sqrt {g} \sqrt {a e+c d x} (f+g x) \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{(c d f-a e g)^{5/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)} \]

[In]

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

[Out]

-((Sqrt[d + e*x]*(Sqrt[c*d*f - a*e*g]*(a*e*g + c*d*(2*f + 3*g*x)) + 3*c*d*Sqrt[g]*Sqrt[a*e + c*d*x]*(f + g*x)*
ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d*f - a*e*g]]))/((c*d*f - a*e*g)^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]
*(f + g*x)))

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 \sqrt {c d x +a e}\, \operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c d \,g^{2} x +3 \sqrt {c d x +a e}\, \operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) c d f g -3 \sqrt {\left (a e g -c d f \right ) g}\, c d g x -\sqrt {\left (a e g -c d f \right ) g}\, a e g -2 \sqrt {\left (a e g -c d f \right ) g}\, c d f \right )}{\sqrt {e x +d}\, \left (c d x +a e \right ) \left (a e g -c d f \right )^{2} \left (g x +f \right ) \sqrt {\left (a e g -c d f \right ) g}}\)	\(215\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (182) = 364\).

Time = 0.33 (sec) , antiderivative size = 1067, normalized size of antiderivative = 5.28 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (c^{2} d^{2} e g x^{3} + a c d^{2} e f + {\left (c^{2} d^{2} e f + {\left (c^{2} d^{3} + a c d e^{2}\right )} g\right )} x^{2} + {\left (a c d^{2} e g + {\left (c^{2} d^{3} + a c d e^{2}\right )} f\right )} x\right )} \sqrt {-\frac {g}{c d f - a e g}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d f - a e g\right )} \sqrt {e x + d} \sqrt {-\frac {g}{c d f - a e g}} - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right ) - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d g x + 2 \, c d f + a e g\right )} \sqrt {e x + d}}{2 \, {\left (a c^{2} d^{3} e f^{3} - 2 \, a^{2} c d^{2} e^{2} f^{2} g + a^{3} d e^{3} f g^{2} + {\left (c^{3} d^{3} e f^{2} g - 2 \, a c^{2} d^{2} e^{2} f g^{2} + a^{2} c d e^{3} g^{3}\right )} x^{3} + {\left (c^{3} d^{3} e f^{3} + {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f^{2} g - {\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f g^{2} + {\left (a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g^{3}\right )} x^{2} + {\left (a^{3} d e^{3} g^{3} + {\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} f^{3} - {\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} f^{2} g - {\left (a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f g^{2}\right )} x\right )}}, -\frac {3 \, {\left (c^{2} d^{2} e g x^{3} + a c d^{2} e f + {\left (c^{2} d^{2} e f + {\left (c^{2} d^{3} + a c d e^{2}\right )} g\right )} x^{2} + {\left (a c d^{2} e g + {\left (c^{2} d^{3} + a c d e^{2}\right )} f\right )} x\right )} \sqrt {\frac {g}{c d f - a e g}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d f - a e g\right )} \sqrt {e x + d} \sqrt {\frac {g}{c d f - a e g}}}{c d e g x^{2} + a d e g + {\left (c d^{2} + a e^{2}\right )} g x}\right ) + \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d g x + 2 \, c d f + a e g\right )} \sqrt {e x + d}}{a c^{2} d^{3} e f^{3} - 2 \, a^{2} c d^{2} e^{2} f^{2} g + a^{3} d e^{3} f g^{2} + {\left (c^{3} d^{3} e f^{2} g - 2 \, a c^{2} d^{2} e^{2} f g^{2} + a^{2} c d e^{3} g^{3}\right )} x^{3} + {\left (c^{3} d^{3} e f^{3} + {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f^{2} g - {\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f g^{2} + {\left (a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g^{3}\right )} x^{2} + {\left (a^{3} d e^{3} g^{3} + {\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} f^{3} - {\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} f^{2} g - {\left (a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f g^{2}\right )} x}\right ] \]

[In]

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

[Out]

[1/2*(3*(c^2*d^2*e*g*x^3 + a*c*d^2*e*f + (c^2*d^2*e*f + (c^2*d^3 + a*c*d*e^2)*g)*x^2 + (a*c*d^2*e*g + (c^2*d^3
+ a*c*d*e^2)*f)*x)*sqrt(-g/(c*d*f - a*e*g))*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - 2*sqrt(c*d*e*x^2 + a*d*
e + (c*d^2 + a*e^2)*x)*(c*d*f - a*e*g)*sqrt(e*x + d)*sqrt(-g/(c*d*f - a*e*g)) - (c*d*e*f - (c*d^2 + 2*a*e^2)*g
)*x)/(e*g*x^2 + d*f + (e*f + d*g)*x)) - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d*g*x + 2*c*d*f + a
*e*g)*sqrt(e*x + d))/(a*c^2*d^3*e*f^3 - 2*a^2*c*d^2*e^2*f^2*g + a^3*d*e^3*f*g^2 + (c^3*d^3*e*f^2*g - 2*a*c^2*d
^2*e^2*f*g^2 + a^2*c*d*e^3*g^3)*x^3 + (c^3*d^3*e*f^3 + (c^3*d^4 - a*c^2*d^2*e^2)*f^2*g - (2*a*c^2*d^3*e + a^2*
c*d*e^3)*f*g^2 + (a^2*c*d^2*e^2 + a^3*e^4)*g^3)*x^2 + (a^3*d*e^3*g^3 + (c^3*d^4 + a*c^2*d^2*e^2)*f^3 - (a*c^2*
d^3*e + 2*a^2*c*d*e^3)*f^2*g - (a^2*c*d^2*e^2 - a^3*e^4)*f*g^2)*x), -(3*(c^2*d^2*e*g*x^3 + a*c*d^2*e*f + (c^2*
d^2*e*f + (c^2*d^3 + a*c*d*e^2)*g)*x^2 + (a*c*d^2*e*g + (c^2*d^3 + a*c*d*e^2)*f)*x)*sqrt(g/(c*d*f - a*e*g))*ar
ctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*f - a*e*g)*sqrt(e*x + d)*sqrt(g/(c*d*f - a*e*g))/(c*d*e
*g*x^2 + a*d*e*g + (c*d^2 + a*e^2)*g*x)) + sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(3*c*d*g*x + 2*c*d*f +
a*e*g)*sqrt(e*x + d))/(a*c^2*d^3*e*f^3 - 2*a^2*c*d^2*e^2*f^2*g + a^3*d*e^3*f*g^2 + (c^3*d^3*e*f^2*g - 2*a*c^2*
d^2*e^2*f*g^2 + a^2*c*d*e^3*g^3)*x^3 + (c^3*d^3*e*f^3 + (c^3*d^4 - a*c^2*d^2*e^2)*f^2*g - (2*a*c^2*d^3*e + a^2
*c*d*e^3)*f*g^2 + (a^2*c*d^2*e^2 + a^3*e^4)*g^3)*x^2 + (a^3*d*e^3*g^3 + (c^3*d^4 + a*c^2*d^2*e^2)*f^3 - (a*c^2
*d^3*e + 2*a^2*c*d*e^3)*f^2*g - (a^2*c*d^2*e^2 - a^3*e^4)*f*g^2)*x)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (182) = 364\).

Time = 0.50 (sec) , antiderivative size = 753, normalized size of antiderivative = 3.73 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-{\left (\frac {3 \, c d g \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{{\left (c^{2} d^{2} e f^{2} {\left | e \right |} - 2 \, a c d e^{2} f g {\left | e \right |} + a^{2} e^{3} g^{2} {\left | e \right |}\right )} \sqrt {c d f g - a e g^{2}} e} + \frac {2 \, c^{2} d^{2} e^{2} f - 2 \, a c d e^{3} g + 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g}{{\left (c^{2} d^{2} e f^{2} {\left | e \right |} - 2 \, a c d e^{2} f g {\left | e \right |} + a^{2} e^{3} g^{2} {\left | e \right |}\right )} {\left (\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c d e^{2} f - \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} g\right )}}\right )} e^{3} + \frac {3 \, \sqrt {-c d^{2} e + a e^{3}} c d e^{2} f g \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) - 3 \, \sqrt {-c d^{2} e + a e^{3}} c d^{2} e g^{2} \arctan \left (\frac {\sqrt {-c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right ) + 2 \, \sqrt {c d f g - a e g^{2}} c d e^{3} f - 3 \, \sqrt {c d f g - a e g^{2}} c d^{2} e^{2} g + \sqrt {c d f g - a e g^{2}} a e^{4} g}{\sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c^{2} d^{2} e f^{3} {\left | e \right |} - \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} c^{2} d^{3} f^{2} g {\left | e \right |} - 2 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a c d e^{2} f^{2} g {\left | e \right |} + 2 \, \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a c d^{2} e f g^{2} {\left | e \right |} + \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a^{2} e^{3} f g^{2} {\left | e \right |} - \sqrt {-c d^{2} e + a e^{3}} \sqrt {c d f g - a e g^{2}} a^{2} d e^{2} g^{3} {\left | e \right |}} \]

[In]

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

[Out]

-(3*c*d*g*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e))/((c^2*d^2*e*f^2*abs(e)
- 2*a*c*d*e^2*f*g*abs(e) + a^2*e^3*g^2*abs(e))*sqrt(c*d*f*g - a*e*g^2)*e) + (2*c^2*d^2*e^2*f - 2*a*c*d*e^3*g
+ 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d*g)/((c^2*d^2*e*f^2*abs(e) - 2*a*c*d*e^2*f*g*abs(e) + a^2*e^3*g^2*a
bs(e))*(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d*e^2*f - sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*e^3*g +
((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*g)))*e^3 + (3*sqrt(-c*d^2*e + a*e^3)*c*d*e^2*f*g*arctan(sqrt(-c*d^2*
e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 3*sqrt(-c*d^2*e + a*e^3)*c*d^2*e*g^2*arctan(sqrt(-c*d^2*e + a*e^3)
*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 2*sqrt(c*d*f*g - a*e*g^2)*c*d*e^3*f - 3*sqrt(c*d*f*g - a*e*g^2)*c*d^2*e^2*g
+ sqrt(c*d*f*g - a*e*g^2)*a*e^4*g)/(sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^2*d^2*e*f^3*abs(e) - sqrt
(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^2*d^3*f^2*g*abs(e) - 2*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*
g^2)*a*c*d*e^2*f^2*g*abs(e) + 2*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c*d^2*e*f*g^2*abs(e) + sqrt(-
c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*e^3*f*g^2*abs(e) - sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)
*a^2*d*e^2*g^3*abs(e))

Mupad [F(-1)]

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

[In]

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

[Out]

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