\(\int \frac {(d+e x)^{3/2}}{(f+g x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [671]

Optimal result

Integrand size = 46, antiderivative size = 274 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {5 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {15 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {15 c^2 d^2 \sqrt {g} \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 (c d f-a e g)^{7/2}} \]

[Out]

-15/4*c^2*d^2*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)^(7/2)-2*(e*x+d)^(1/2)/(-a*e*g+c*d*f)/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-5/2*g
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^2/(g*x+f)^2/(e*x+d)^(1/2)-15/4*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)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {882, 886, 888, 211} \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {15 c^2 d^2 \sqrt {g} \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{4 (c d f-a e g)^{7/2}}-\frac {15 c d g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 \sqrt {d+e x} (f+g x) (c d f-a e g)^3}-\frac {5 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]

[In]

Int[(d + e*x)^(3/2)/((f + g*x)^3*(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)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (5*g*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*(c*d*f - a*e*g)^2*Sqrt[d + e*x]*(f + g*x)^2) - (15*c*d*g*Sqrt[a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2])/(4*(c*d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) - (15*c^2*d^2*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])])/(4*(c*d*f - a*e*g)^(7/2
))

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)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(5 g) \int \frac {\sqrt {d+e x}}{(f+g x)^3 \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)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {5 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {(15 c d 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}{4 (c d f-a e g)^2} \\ & = -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {5 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {15 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (15 c^2 d^2 g\right ) \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}{8 (c d f-a e g)^3} \\ & = -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {5 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {15 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (15 c^2 d^2 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 )}{4 (c d f-a e g)^3} \\ & = -\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {5 g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {15 c d g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {15 c^2 d^2 \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 )}{4 (c d f-a e g)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.68 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {\sqrt {d+e x} \left (\sqrt {c d f-a e g} \left (-2 a^2 e^2 g^2+a c d e g (9 f+5 g x)+c^2 d^2 \left (8 f^2+25 f g x+15 g^2 x^2\right )\right )+15 c^2 d^2 \sqrt {g} \sqrt {a e+c d x} (f+g x)^2 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{4 (c d f-a e g)^{7/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)^2} \]

[In]

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

[Out]

-1/4*(Sqrt[d + e*x]*(Sqrt[c*d*f - a*e*g]*(-2*a^2*e^2*g^2 + a*c*d*e*g*(9*f + 5*g*x) + c^2*d^2*(8*f^2 + 25*f*g*x
+ 15*g^2*x^2)) + 15*c^2*d^2*Sqrt[g]*Sqrt[a*e + c*d*x]*(f + g*x)^2*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c*d
*f - a*e*g]]))/((c*d*f - a*e*g)^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^2)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.35

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) \sqrt {c d x +a e}\, c^{2} d^{2} g^{3} x^{2}+30 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) \sqrt {c d x +a e}\, c^{2} d^{2} f \,g^{2} x -15 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} g^{2} x^{2}+15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -c d f \right ) g}}\right ) \sqrt {c d x +a e}\, c^{2} d^{2} f^{2} g -5 \sqrt {\left (a e g -c d f \right ) g}\, a c d e \,g^{2} x -25 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f g x +2 \sqrt {\left (a e g -c d f \right ) g}\, a^{2} e^{2} g^{2}-9 \sqrt {\left (a e g -c d f \right ) g}\, a c d e f g -8 \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} f^{2}\right )}{4 \sqrt {e x +d}\, \left (c d x +a e \right ) \left (a e g -c d f \right )^{3} \left (g x +f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}}\)	\(369\)

[In]

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

[Out]

-1/4/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*(c*d*x
+a*e)^(1/2)*c^2*d^2*g^3*x^2+30*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*(c*d*x+a*e)^(1/2)*c^2*d^2*
f*g^2*x-15*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*g^2*x^2+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*(c*
d*x+a*e)^(1/2)*c^2*d^2*f^2*g-5*((a*e*g-c*d*f)*g)^(1/2)*a*c*d*e*g^2*x-25*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*f*g*x+
2*((a*e*g-c*d*f)*g)^(1/2)*a^2*e^2*g^2-9*((a*e*g-c*d*f)*g)^(1/2)*a*c*d*e*f*g-8*((a*e*g-c*d*f)*g)^(1/2)*c^2*d^2*
f^2)/(c*d*x+a*e)/(a*e*g-c*d*f)^3/(g*x+f)^2/((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. 910 vs. \(2 (244) = 488\).

Time = 0.49 (sec) , antiderivative size = 1863, normalized size of antiderivative = 6.80 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/8*(15*(c^3*d^3*e*g^2*x^4 + a*c^2*d^3*e*f^2 + (2*c^3*d^3*e*f*g + (c^3*d^4 + a*c^2*d^2*e^2)*g^2)*x^3 + (c^3*
d^3*e*f^2 + a*c^2*d^3*e*g^2 + 2*(c^3*d^4 + a*c^2*d^2*e^2)*f*g)*x^2 + (2*a*c^2*d^3*e*f*g + (c^3*d^4 + a*c^2*d^2
*e^2)*f^2)*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*(15*c^2*d^2*g^2*x^2 + 8*c^2*d^2*f^2 + 9*a*c*d*e*f*g - 2*a^2*e^2*g^2 + 5*(5
*c^2*d^2*f*g + a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(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), -1/4*(15*(c^3*d^3*e*g^2*x^4 + a*c^2*d^3
*e*f^2 + (2*c^3*d^3*e*f*g + (c^3*d^4 + a*c^2*d^2*e^2)*g^2)*x^3 + (c^3*d^3*e*f^2 + a*c^2*d^3*e*g^2 + 2*(c^3*d^4
+ a*c^2*d^2*e^2)*f*g)*x^2 + (2*a*c^2*d^3*e*f*g + (c^3*d^4 + a*c^2*d^2*e^2)*f^2)*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)) + (15*c^2*d^2*g^2*x^2 + 8*c^2*d^2*f^2 + 9*a*c*d*e*f*g - 2*a^2*e^2*g^2
+ 5*(5*c^2*d^2*f*g + a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(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. \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1390 vs. \(2 (244) = 488\).

Time = 0.70 (sec) , antiderivative size = 1390, normalized size of antiderivative = 5.07 \[ \int \frac {(d+e x)^{3/2}}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*(15*c^2*d^2*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^3*d^3*e^2
*f^3*abs(e) - 3*a*c^2*d^2*e^3*f^2*g*abs(e) + 3*a^2*c*d*e^4*f*g^2*abs(e) - a^3*e^5*g^3*abs(e))*sqrt(c*d*f*g - a
*e*g^2)*e) + 8*c^2*d^2/((c^3*d^3*e^2*f^3*abs(e) - 3*a*c^2*d^2*e^3*f^2*g*abs(e) + 3*a^2*c*d*e^4*f*g^2*abs(e) -
a^3*e^5*g^3*abs(e))*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)) + (9*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^3*
d^3*e^2*f*g - 9*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^2*d^2*e^3*g^2 + 7*((e*x + d)*c*d*e - c*d^2*e + a*e
^3)^(3/2)*c^2*d^2*g^2)/((c^3*d^3*e^2*f^3*abs(e) - 3*a*c^2*d^2*e^3*f^2*g*abs(e) + 3*a^2*c*d*e^4*f*g^2*abs(e) -
a^3*e^5*g^3*abs(e))*(c*d*e^2*f - a*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g)^2))*e^4 + 1/4*(15*sqrt(-c*d^
2*e + a*e^3)*c^2*d^2*e^3*f^2*g*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) - 30*sqrt(-c*d^2*e
+ a*e^3)*c^2*d^3*e^2*f*g^2*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 15*sqrt(-c*d^2*e +
a*e^3)*c^2*d^4*e*g^3*arctan(sqrt(-c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g^2)*e)) + 8*sqrt(c*d*f*g - a*e*g^2)*
c^2*d^2*e^4*f^2 - 25*sqrt(c*d*f*g - a*e*g^2)*c^2*d^3*e^3*f*g + 9*sqrt(c*d*f*g - a*e*g^2)*a*c*d*e^5*f*g + 15*sq
rt(c*d*f*g - a*e*g^2)*c^2*d^4*e^2*g^2 - 5*sqrt(c*d*f*g - a*e*g^2)*a*c*d^2*e^4*g^2 - 2*sqrt(c*d*f*g - a*e*g^2)*
a^2*e^6*g^2)/(sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^3*d^3*e^2*f^5*abs(e) - 2*sqrt(-c*d^2*e + a*e^3)
*sqrt(c*d*f*g - a*e*g^2)*c^3*d^4*e*f^4*g*abs(e) - 3*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^2*e
^3*f^4*g*abs(e) + sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*c^3*d^5*f^3*g^2*abs(e) + 6*sqrt(-c*d^2*e + a*
e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^3*e^2*f^3*g^2*abs(e) + 3*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a
^2*c*d*e^4*f^3*g^2*abs(e) - 3*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a*c^2*d^4*e*f^2*g^3*abs(e) - 6*sq
rt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^2*e^3*f^2*g^3*abs(e) - sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*
g - a*e*g^2)*a^3*e^5*f^2*g^3*abs(e) + 3*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^2*c*d^3*e^2*f*g^4*abs
(e) + 2*sqrt(-c*d^2*e + a*e^3)*sqrt(c*d*f*g - a*e*g^2)*a^3*d*e^4*f*g^4*abs(e) - sqrt(-c*d^2*e + a*e^3)*sqrt(c*
d*f*g - a*e*g^2)*a^3*d^2*e^3*g^5*abs(e))

Mupad [F(-1)]

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

[In]

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