3.791 \(\int \frac{(d+e x)^{3/2}}{(f+g x)^4 \sqrt{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=351 \[ -\frac{c^2 d^2 \left (6 a e^2 g-c d (5 d g+e f)\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{8 g^{3/2} (c d f-a e g)^{7/2}}-\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{8 g \sqrt{d+e x} (f+g x) (c d f-a e g)^3}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{12 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac{(e f-d g) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt{d+e x} (f+g x)^3 (c d f-a e g)} \]

[Out]

-((e*f - d*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*g*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) - (
(6*a*e^2*g - c*d*(e*f + 5*d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*g*(c*d*f - a*e*g)^2*Sqrt[d +
e*x]*(f + g*x)^2) - (c*d*(6*a*e^2*g - c*d*(e*f + 5*d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*g*(c*
d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) - (c^2*d^2*(6*a*e^2*g - c*d*(e*f + 5*d*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])])/(8*g^(3/2)*(c*d*f - a*e*g)^(7/2))

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Rubi [A]  time = 0.557881, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {878, 872, 874, 205} \[ -\frac{c^2 d^2 \left (6 a e^2 g-c d (5 d g+e f)\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{8 g^{3/2} (c d f-a e g)^{7/2}}-\frac{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{8 g \sqrt{d+e x} (f+g x) (c d f-a e g)^3}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{12 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac{(e f-d g) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt{d+e x} (f+g x)^3 (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*f - d*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*g*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) - (
(6*a*e^2*g - c*d*(e*f + 5*d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(12*g*(c*d*f - a*e*g)^2*Sqrt[d +
e*x]*(f + g*x)^2) - (c*d*(6*a*e^2*g - c*d*(e*f + 5*d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(8*g*(c*
d*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)) - (c^2*d^2*(6*a*e^2*g - c*d*(e*f + 5*d*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])])/(8*g^(3/2)*(c*d*f - a*e*g)^(7/2))

Rule 878

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(e^2*(e*f - d*g)*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/(g*(n + 1)*(c*e*f + c*d*g
- b*e*g)), x] - Dist[(e*(b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3)))/(g*(n + 1)*(c*e*f + c*d*g - b*
e*g)), Int[(d + e*x)^(m - 1)*(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 - 1, 0] && LtQ[n, -1] && IntegerQ[2*p]

Rule 872

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 874

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]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{(d+e x)^{3/2}}{(f+g x)^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt{d+e x} (f+g x)^3}+\frac{\left (e \left (\frac{1}{2} c d e^2 f+\frac{11}{2} c d^2 e g-3 e \left (c d^2+a e^2\right ) g\right )\right ) \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}{3 g \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )}\\ &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt{d+e x} (f+g x)^3}-\frac{\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^2}-\frac{\left (c d \left (6 a e^2 g-c d (e f+5 d g)\right )\right ) \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}{8 g (c d f-a e g)^2}\\ &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt{d+e x} (f+g x)^3}-\frac{\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^2}-\frac{c d \left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^3 \sqrt{d+e x} (f+g x)}-\frac{\left (c^2 d^2 \left (6 a e^2 g-c d (e f+5 d g)\right )\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}{16 g (c d f-a e g)^3}\\ &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt{d+e x} (f+g x)^3}-\frac{\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^2}-\frac{c d \left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^3 \sqrt{d+e x} (f+g x)}-\frac{\left (c^2 d^2 e^2 \left (6 a e^2 g-c d (e f+5 d g)\right )\right ) \operatorname{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 )}{8 g (c d f-a e g)^3}\\ &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt{d+e x} (f+g x)^3}-\frac{\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt{d+e x} (f+g x)^2}-\frac{c d \left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^3 \sqrt{d+e x} (f+g x)}-\frac{c^2 d^2 \left (6 a e^2 g-c d (e f+5 d g)\right ) \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 )}{8 g^{3/2} (c d f-a e g)^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.117978, size = 132, normalized size = 0.38 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{e f-d g}{(f+g x)^3}-\frac{c^2 d^2 \left (c d (5 d g+e f)-6 a e^2 g\right ) \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )}{(c d f-a e g)^3}\right )}{3 g \sqrt{d+e x} (a e g-c d f)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((e*f - d*g)/(f + g*x)^3 - (c^2*d^2*(-6*a*e^2*g + c*d*(e*f + 5*d*g))*Hypergeome
tric2F1[1/2, 3, 3/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)])/(c*d*f - a*e*g)^3))/(3*g*(-(c*d*f) + a*e*g)*Sqrt[d
+ e*x])

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Maple [B]  time = 0.35, size = 1142, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

-1/24*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(-15*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x^3*c^
3*d^4*g^4+4*a^2*e^3*f*g^2*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)-50*x*a*c*d*e^2*f*g^2*((a*e*g-c*d*f)*g)^(1/
2)*(c*d*x+a*e)^(1/2)-15*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^4*f^3*g-3*arctanh((c*d*x+a*
e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*c^3*d^3*e*f^4-16*a*c*d*e^2*f^2*g*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)
+54*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x*a*c^2*d^2*e^2*f^2*g^2-18*x^2*a*c*d*e^2*g^3*((a*e*g-
c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)+3*x^2*c^2*d^2*e*f*g^2*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)-10*x*a*c*d^2
*e*g^3*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)+8*x*c^2*d^2*e*f^2*g*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)
-26*a*c*d^2*e*f*g^2*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)-9*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^
(1/2))*x^2*c^3*d^3*e*f^2*g^2-9*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x*c^3*d^3*e*f^3*g+18*arcta
nh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*a*c^2*d^2*e^2*f^3*g+40*x*c^2*d^3*f*g^2*((a*e*g-c*d*f)*g)^(1/2)
*(c*d*x+a*e)^(1/2)+18*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x^3*a*c^2*d^2*e^2*g^4-3*arctanh((c*
d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x^3*c^3*d^3*e*f*g^3+54*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)
^(1/2))*x^2*a*c^2*d^2*e^2*f*g^3+12*x*a^2*e^3*g^3*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)+8*a^2*d*e^2*g^3*((a
*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)+33*c^2*d^3*f^2*g*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)-3*c^2*d^2*e*
f^3*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)-45*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x^2*c^3*
d^4*f*g^3-45*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x*c^3*d^4*f^2*g^2+15*x^2*c^2*d^3*g^3*((a*e*g
-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2))/(e*x+d)^(1/2)/((a*e*g-c*d*f)*g)^(1/2)/(g*x+f)^3/g/(a*e*g-c*d*f)/(a^2*e^2*g
^2-2*a*c*d*e*f*g+c^2*d^2*f^2)/(c*d*x+a*e)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^4), x)

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Fricas [B]  time = 1.78885, size = 5411, normalized size = 15.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/48*(3*(c^3*d^4*e*f^4 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)*f^3*g + (c^3*d^3*e^2*f*g^3 + (5*c^3*d^4*e - 6*a*c^2*d
^2*e^3)*g^4)*x^4 + (3*c^3*d^3*e^2*f^2*g^2 + 2*(8*c^3*d^4*e - 9*a*c^2*d^2*e^3)*f*g^3 + (5*c^3*d^5 - 6*a*c^2*d^3
*e^2)*g^4)*x^3 + 3*(c^3*d^3*e^2*f^3*g + 6*(c^3*d^4*e - a*c^2*d^2*e^3)*f^2*g^2 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)*
f*g^3)*x^2 + (c^3*d^3*e^2*f^4 + 2*(4*c^3*d^4*e - 3*a*c^2*d^2*e^3)*f^3*g + 3*(5*c^3*d^5 - 6*a*c^2*d^3*e^2)*f^2*
g^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x -
2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f +
d*g)*x)) + 2*(3*c^3*d^3*e*f^4*g + 8*a^3*d*e^3*g^5 - (33*c^3*d^4 - 13*a*c^2*d^2*e^2)*f^3*g^2 + (59*a*c^2*d^3*e
- 20*a^2*c*d*e^3)*f^2*g^3 - 2*(17*a^2*c*d^2*e^2 - 2*a^3*e^4)*f*g^4 - 3*(c^3*d^3*e*f^2*g^3 + (5*c^3*d^4 - 7*a*
c^2*d^2*e^2)*f*g^4 - (5*a*c^2*d^3*e - 6*a^2*c*d*e^3)*g^5)*x^2 - 2*(4*c^3*d^3*e*f^3*g^2 + (20*c^3*d^4 - 29*a*c^
2*d^2*e^2)*f^2*g^3 - (25*a*c^2*d^3*e - 31*a^2*c*d*e^3)*f*g^4 + (5*a^2*c*d^2*e^2 - 6*a^3*e^4)*g^5)*x)*sqrt(c*d*
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^5*f^7*g^2 - 4*a*c^3*d^4*e*f^6*g^3 + 6*a^2*c^2*d^3*e^2
*f^5*g^4 - 4*a^3*c*d^2*e^3*f^4*g^5 + a^4*d*e^4*f^3*g^6 + (c^4*d^4*e*f^4*g^5 - 4*a*c^3*d^3*e^2*f^3*g^6 + 6*a^2*
c^2*d^2*e^3*f^2*g^7 - 4*a^3*c*d*e^4*f*g^8 + a^4*e^5*g^9)*x^4 + (3*c^4*d^4*e*f^5*g^4 + a^4*d*e^4*g^9 + (c^4*d^5
- 12*a*c^3*d^3*e^2)*f^4*g^5 - 2*(2*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^3*g^6 + 6*(a^2*c^2*d^3*e^2 - 2*a^3*c*d*
e^4)*f^2*g^7 - (4*a^3*c*d^2*e^3 - 3*a^4*e^5)*f*g^8)*x^3 + 3*(c^4*d^4*e*f^6*g^3 + a^4*d*e^4*f*g^8 + (c^4*d^5 -
4*a*c^3*d^3*e^2)*f^5*g^4 - 2*(2*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^4*g^5 + 2*(3*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^
4)*f^3*g^6 - (4*a^3*c*d^2*e^3 - a^4*e^5)*f^2*g^7)*x^2 + (c^4*d^4*e*f^7*g^2 + 3*a^4*d*e^4*f^2*g^7 + (3*c^4*d^5
- 4*a*c^3*d^3*e^2)*f^6*g^3 - 6*(2*a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^5*g^4 + 2*(9*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^
4)*f^4*g^5 - (12*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g^6)*x), -1/24*(3*(c^3*d^4*e*f^4 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)
*f^3*g + (c^3*d^3*e^2*f*g^3 + (5*c^3*d^4*e - 6*a*c^2*d^2*e^3)*g^4)*x^4 + (3*c^3*d^3*e^2*f^2*g^2 + 2*(8*c^3*d^4
*e - 9*a*c^2*d^2*e^3)*f*g^3 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)*g^4)*x^3 + 3*(c^3*d^3*e^2*f^3*g + 6*(c^3*d^4*e - a
*c^2*d^2*e^3)*f^2*g^2 + (5*c^3*d^5 - 6*a*c^2*d^3*e^2)*f*g^3)*x^2 + (c^3*d^3*e^2*f^4 + 2*(4*c^3*d^4*e - 3*a*c^2
*d^2*e^3)*f^3*g + 3*(5*c^3*d^5 - 6*a*c^2*d^3*e^2)*f^2*g^2)*x)*sqrt(c*d*f*g - a*e*g^2)*arctan(sqrt(c*d*e*x^2 +
a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*f*g - a*e*g^2)*sqrt(e*x + d)/(c*d*e*g*x^2 + a*d*e*g + (c*d^2 + a*e^2)*g*x)
) + (3*c^3*d^3*e*f^4*g + 8*a^3*d*e^3*g^5 - (33*c^3*d^4 - 13*a*c^2*d^2*e^2)*f^3*g^2 + (59*a*c^2*d^3*e - 20*a^2*
c*d*e^3)*f^2*g^3 - 2*(17*a^2*c*d^2*e^2 - 2*a^3*e^4)*f*g^4 - 3*(c^3*d^3*e*f^2*g^3 + (5*c^3*d^4 - 7*a*c^2*d^2*e^
2)*f*g^4 - (5*a*c^2*d^3*e - 6*a^2*c*d*e^3)*g^5)*x^2 - 2*(4*c^3*d^3*e*f^3*g^2 + (20*c^3*d^4 - 29*a*c^2*d^2*e^2)
*f^2*g^3 - (25*a*c^2*d^3*e - 31*a^2*c*d*e^3)*f*g^4 + (5*a^2*c*d^2*e^2 - 6*a^3*e^4)*g^5)*x)*sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^4*d^5*f^7*g^2 - 4*a*c^3*d^4*e*f^6*g^3 + 6*a^2*c^2*d^3*e^2*f^5*g^4 -
4*a^3*c*d^2*e^3*f^4*g^5 + a^4*d*e^4*f^3*g^6 + (c^4*d^4*e*f^4*g^5 - 4*a*c^3*d^3*e^2*f^3*g^6 + 6*a^2*c^2*d^2*e^
3*f^2*g^7 - 4*a^3*c*d*e^4*f*g^8 + a^4*e^5*g^9)*x^4 + (3*c^4*d^4*e*f^5*g^4 + a^4*d*e^4*g^9 + (c^4*d^5 - 12*a*c^
3*d^3*e^2)*f^4*g^5 - 2*(2*a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^3*g^6 + 6*(a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^2*g
^7 - (4*a^3*c*d^2*e^3 - 3*a^4*e^5)*f*g^8)*x^3 + 3*(c^4*d^4*e*f^6*g^3 + a^4*d*e^4*f*g^8 + (c^4*d^5 - 4*a*c^3*d^
3*e^2)*f^5*g^4 - 2*(2*a*c^3*d^4*e - 3*a^2*c^2*d^2*e^3)*f^4*g^5 + 2*(3*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^3*g^6
- (4*a^3*c*d^2*e^3 - a^4*e^5)*f^2*g^7)*x^2 + (c^4*d^4*e*f^7*g^2 + 3*a^4*d*e^4*f^2*g^7 + (3*c^4*d^5 - 4*a*c^3*
d^3*e^2)*f^6*g^3 - 6*(2*a*c^3*d^4*e - a^2*c^2*d^2*e^3)*f^5*g^4 + 2*(9*a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4)*f^4*g^5
- (12*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g^6)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)/(g*x+f)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

Timed out