3.2275 \(\int \frac{f+g x}{(d+e x)^{5/2} (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=387 \[ \frac{5 c^2 \sqrt{d+e x} (-6 b e g+5 c d g+7 c e f)}{8 e^2 (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 c^2 (-6 b e g+5 c d g+7 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{9/2}}-\frac{5 c (-6 b e g+5 c d g+7 c e f)}{24 e^2 \sqrt{d+e x} (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{-6 b e g+5 c d g+7 c e f}{12 e^2 (d+e x)^{3/2} (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{e f-d g}{3 e^2 (d+e x)^{5/2} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

-(e*f - d*g)/(3*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (7*c*e*f + 5*c*
d*g - 6*b*e*g)/(12*e^2*(2*c*d - b*e)^2*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (5*c*(7*c*
e*f + 5*c*d*g - 6*b*e*g))/(24*e^2*(2*c*d - b*e)^3*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (
5*c^2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*Sqrt[d + e*x])/(8*e^2*(2*c*d - b*e)^4*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2]) - (5*c^2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d -
b*e]*Sqrt[d + e*x])])/(8*e^2*(2*c*d - b*e)^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.628522, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109, Rules used = {792, 672, 666, 660, 208} \[ \frac{5 c^2 \sqrt{d+e x} (-6 b e g+5 c d g+7 c e f)}{8 e^2 (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 c^2 (-6 b e g+5 c d g+7 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{9/2}}-\frac{5 c (-6 b e g+5 c d g+7 c e f)}{24 e^2 \sqrt{d+e x} (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{-6 b e g+5 c d g+7 c e f}{12 e^2 (d+e x)^{3/2} (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{e f-d g}{3 e^2 (d+e x)^{5/2} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)/((d + e*x)^(5/2)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]

[Out]

-(e*f - d*g)/(3*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (7*c*e*f + 5*c*
d*g - 6*b*e*g)/(12*e^2*(2*c*d - b*e)^2*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (5*c*(7*c*
e*f + 5*c*d*g - 6*b*e*g))/(24*e^2*(2*c*d - b*e)^3*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (
5*c^2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*Sqrt[d + e*x])/(8*e^2*(2*c*d - b*e)^4*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2]) - (5*c^2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d -
b*e]*Sqrt[d + e*x])])/(8*e^2*(2*c*d - b*e)^(9/2))

Rule 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rule 672

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*(m + 2*p + 2))/((m + p + 1)*(2*c*d - b*e)), I
nt[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ
[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rule 666

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((2*c*d - b*e)*(d +
e*x)^m*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*c*d - b*e)*(m + 2*p + 2))/((p + 1)*
(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ[0, m, 1] && IntegerQ[2*p]

Rule 660

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

Rule 208

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

Rubi steps

\begin{align*} \int \frac{f+g x}{(d+e x)^{5/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{(7 c e f+5 c d g-6 b e g) \int \frac{1}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{6 e (2 c d-b e)}\\ &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{7 c e f+5 c d g-6 b e g}{12 e^2 (2 c d-b e)^2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{(5 c (7 c e f+5 c d g-6 b e g)) \int \frac{1}{\sqrt{d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{24 e (2 c d-b e)^2}\\ &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{7 c e f+5 c d g-6 b e g}{12 e^2 (2 c d-b e)^2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 c (7 c e f+5 c d g-6 b e g)}{24 e^2 (2 c d-b e)^3 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{\left (5 c^2 (7 c e f+5 c d g-6 b e g)\right ) \int \frac{\sqrt{d+e x}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{16 e (2 c d-b e)^3}\\ &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{7 c e f+5 c d g-6 b e g}{12 e^2 (2 c d-b e)^2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 c (7 c e f+5 c d g-6 b e g)}{24 e^2 (2 c d-b e)^3 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{5 c^2 (7 c e f+5 c d g-6 b e g) \sqrt{d+e x}}{8 e^2 (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{\left (5 c^2 (7 c e f+5 c d g-6 b e g)\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{16 e (2 c d-b e)^4}\\ &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{7 c e f+5 c d g-6 b e g}{12 e^2 (2 c d-b e)^2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 c (7 c e f+5 c d g-6 b e g)}{24 e^2 (2 c d-b e)^3 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{5 c^2 (7 c e f+5 c d g-6 b e g) \sqrt{d+e x}}{8 e^2 (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{\left (5 c^2 (7 c e f+5 c d g-6 b e g)\right ) \operatorname{Subst}\left (\int \frac{1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt{d+e x}}\right )}{8 (2 c d-b e)^4}\\ &=-\frac{e f-d g}{3 e^2 (2 c d-b e) (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{7 c e f+5 c d g-6 b e g}{12 e^2 (2 c d-b e)^2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 c (7 c e f+5 c d g-6 b e g)}{24 e^2 (2 c d-b e)^3 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{5 c^2 (7 c e f+5 c d g-6 b e g) \sqrt{d+e x}}{8 e^2 (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 c^2 (7 c e f+5 c d g-6 b e g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{2 c d-b e} \sqrt{d+e x}}\right )}{8 e^2 (2 c d-b e)^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.129055, size = 127, normalized size = 0.33 \[ \frac{\frac{c^2 (d+e x)^3 (-6 b e g+5 c d g+7 c e f) \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )}{(2 c d-b e)^3}+d g-e f}{3 e^2 (d+e x)^{5/2} (2 c d-b e) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)/((d + e*x)^(5/2)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]

[Out]

(-(e*f) + d*g + (c^2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*(d + e*x)^3*Hypergeometric2F1[-1/2, 3, 1/2, (-(c*d) + b*e +
c*e*x)/(-2*c*d + b*e)])/(2*c*d - b*e)^3)/(3*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)*Sqrt[(d + e*x)*(-(b*e) + c*(d -
e*x))])

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Maple [B]  time = 0.036, size = 1224, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24/(e*x+d)^(7/2)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(-105*(b*e-2*c*d)^(1/2)*x^3*c^3*e^4*f+14*(b*e-2*c*d)
^(1/2)*x*b^2*c*e^4*f-85*(b*e-2*c*d)^(1/2)*x*c^3*d^3*e*g-119*(b*e-2*c*d)^(1/2)*x*c^3*d^2*e^2*f-105*arctan((-c*e
*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^3*d^3*e*f-105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e
-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x^3*c^3*e^4*f+90*(b*e-2*c*d)^(1/2)*x^3*b*c^2*e^4*g-75*(b*e-2*c*d)^(1/2)*
x^3*c^3*d*e^3*g+30*(b*e-2*c*d)^(1/2)*x^2*b^2*c*e^4*g-35*(b*e-2*c*d)^(1/2)*x^2*b*c^2*e^4*f-175*(b*e-2*c*d)^(1/2
)*x^2*c^3*d^2*e^2*g-245*(b*e-2*c*d)^(1/2)*x^2*c^3*d*e^3*f+40*(b*e-2*c*d)^(1/2)*b^2*c*d^2*e^2*g+62*(b*e-2*c*d)^
(1/2)*b^2*c*d*e^3*f+13*(b*e-2*c*d)^(1/2)*b*c^2*d^3*e*g-187*(b*e-2*c*d)^(1/2)*b*c^2*d^2*e^2*f-8*(b*e-2*c*d)^(1/
2)*b^3*e^4*f-49*(b*e-2*c*d)^(1/2)*c^3*d^4*g-12*(b*e-2*c*d)^(1/2)*x*b^3*e^4*g-75*arctan((-c*e*x-b*e+c*d)^(1/2)/
(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^3*d^4*g-4*(b*e-2*c*d)^(1/2)*b^3*d*e^3*g+85*(b*e-2*c*d)^(1/2)*c^3*d
^3*e*f-75*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x^3*c^3*d*e^3*g-225*arctan((
-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x^2*c^3*d^2*e^2*g-315*arctan((-c*e*x-b*e+c*d)^
(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x^2*c^3*d*e^3*f+185*(b*e-2*c*d)^(1/2)*x^2*b*c^2*d*e^3*g-225*ar
ctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x*c^3*d^3*e*g-315*arctan((-c*e*x-b*e+c*d
)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x*c^3*d^2*e^2*f+118*(b*e-2*c*d)^(1/2)*x*b^2*c*d*e^3*g+12*(b*
e-2*c*d)^(1/2)*x*b*c^2*d^2*e^2*g-126*(b*e-2*c*d)^(1/2)*x*b*c^2*d*e^3*f+90*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2
*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*b*c^2*d^3*e*g+270*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x
-b*e+c*d)^(1/2)*x^2*b*c^2*d*e^3*g+270*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*
x*b*c^2*d^2*e^2*g+90*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x^3*b*c^2*e^4*g)/
(c*e*x+b*e-c*d)/e^2/(b*e-2*c*d)^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(e*x + d)^(5/2)), x)

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Fricas [B]  time = 2.024, size = 5883, normalized size = 15.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

[1/48*(15*((7*c^4*e^6*f + (5*c^4*d*e^5 - 6*b*c^3*e^6)*g)*x^5 + (7*(3*c^4*d*e^5 + b*c^3*e^6)*f + (15*c^4*d^2*e^
4 - 13*b*c^3*d*e^5 - 6*b^2*c^2*e^6)*g)*x^4 + 2*(7*(c^4*d^2*e^4 + 2*b*c^3*d*e^5)*f + (5*c^4*d^3*e^3 + 4*b*c^3*d
^2*e^4 - 12*b^2*c^2*d*e^5)*g)*x^3 - 2*(7*(c^4*d^3*e^3 - 3*b*c^3*d^2*e^4)*f + (5*c^4*d^4*e^2 - 21*b*c^3*d^3*e^3
+ 18*b^2*c^2*d^2*e^4)*g)*x^2 - 7*(c^4*d^5*e - b*c^3*d^4*e^2)*f - (5*c^4*d^6 - 11*b*c^3*d^5*e + 6*b^2*c^2*d^4*
e^2)*g - (7*(3*c^4*d^4*e^2 - 4*b*c^3*d^3*e^3)*f + (15*c^4*d^5*e - 38*b*c^3*d^4*e^2 + 24*b^2*c^2*d^3*e^3)*g)*x)
*sqrt(2*c*d - b*e)*log(-(c*e^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c
*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^
2 - b*d*e)*(15*(7*(2*c^4*d*e^4 - b*c^3*e^5)*f + (10*c^4*d^2*e^3 - 17*b*c^3*d*e^4 + 6*b^2*c^2*e^5)*g)*x^3 + 5*(
7*(14*c^4*d^2*e^3 - 5*b*c^3*d*e^4 - b^2*c^2*e^5)*f + (70*c^4*d^3*e^2 - 109*b*c^3*d^2*e^3 + 25*b^2*c^2*d*e^4 +
6*b^3*c*e^5)*g)*x^2 - (170*c^4*d^4*e - 459*b*c^3*d^3*e^2 + 311*b^2*c^2*d^2*e^3 - 78*b^3*c*d*e^4 + 8*b^4*e^5)*f
+ (98*c^4*d^5 - 75*b*c^3*d^4*e - 67*b^2*c^2*d^3*e^2 + 48*b^3*c*d^2*e^3 - 4*b^4*d*e^4)*g + (7*(34*c^4*d^3*e^2
+ 19*b*c^3*d^2*e^3 - 22*b^2*c^2*d*e^4 + 2*b^3*c*e^5)*f + (170*c^4*d^4*e - 109*b*c^3*d^3*e^2 - 224*b^2*c^2*d^2*
e^3 + 142*b^3*c*d*e^4 - 12*b^4*e^5)*g)*x)*sqrt(e*x + d))/(32*c^6*d^10*e^2 - 112*b*c^5*d^9*e^3 + 160*b^2*c^4*d^
8*e^4 - 120*b^3*c^3*d^7*e^5 + 50*b^4*c^2*d^6*e^6 - 11*b^5*c*d^5*e^7 + b^6*d^4*e^8 - (32*c^6*d^5*e^7 - 80*b*c^5
*d^4*e^8 + 80*b^2*c^4*d^3*e^9 - 40*b^3*c^3*d^2*e^10 + 10*b^4*c^2*d*e^11 - b^5*c*e^12)*x^5 - (96*c^6*d^6*e^6 -
208*b*c^5*d^5*e^7 + 160*b^2*c^4*d^4*e^8 - 40*b^3*c^3*d^3*e^9 - 10*b^4*c^2*d^2*e^10 + 7*b^5*c*d*e^11 - b^6*e^12
)*x^4 - 2*(32*c^6*d^7*e^5 - 16*b*c^5*d^6*e^6 - 80*b^2*c^4*d^5*e^7 + 120*b^3*c^3*d^4*e^8 - 70*b^4*c^2*d^3*e^9 +
19*b^5*c*d^2*e^10 - 2*b^6*d*e^11)*x^3 + 2*(32*c^6*d^8*e^4 - 176*b*c^5*d^7*e^5 + 320*b^2*c^4*d^6*e^6 - 280*b^3
*c^3*d^5*e^7 + 130*b^4*c^2*d^4*e^8 - 31*b^5*c*d^3*e^9 + 3*b^6*d^2*e^10)*x^2 + (96*c^6*d^9*e^3 - 368*b*c^5*d^8*
e^4 + 560*b^2*c^4*d^7*e^5 - 440*b^3*c^3*d^6*e^6 + 190*b^4*c^2*d^5*e^7 - 43*b^5*c*d^4*e^8 + 4*b^6*d^3*e^9)*x),
1/24*(15*((7*c^4*e^6*f + (5*c^4*d*e^5 - 6*b*c^3*e^6)*g)*x^5 + (7*(3*c^4*d*e^5 + b*c^3*e^6)*f + (15*c^4*d^2*e^4
- 13*b*c^3*d*e^5 - 6*b^2*c^2*e^6)*g)*x^4 + 2*(7*(c^4*d^2*e^4 + 2*b*c^3*d*e^5)*f + (5*c^4*d^3*e^3 + 4*b*c^3*d^
2*e^4 - 12*b^2*c^2*d*e^5)*g)*x^3 - 2*(7*(c^4*d^3*e^3 - 3*b*c^3*d^2*e^4)*f + (5*c^4*d^4*e^2 - 21*b*c^3*d^3*e^3
+ 18*b^2*c^2*d^2*e^4)*g)*x^2 - 7*(c^4*d^5*e - b*c^3*d^4*e^2)*f - (5*c^4*d^6 - 11*b*c^3*d^5*e + 6*b^2*c^2*d^4*e
^2)*g - (7*(3*c^4*d^4*e^2 - 4*b*c^3*d^3*e^3)*f + (15*c^4*d^5*e - 38*b*c^3*d^4*e^2 + 24*b^2*c^2*d^3*e^3)*g)*x)*
sqrt(-2*c*d + b*e)*arctan(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*sqrt(e*x + d)/(c*e^2*x
^2 + b*e^2*x - c*d^2 + b*d*e)) + sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(15*(7*(2*c^4*d*e^4 - b*c^3*e^5)*f
+ (10*c^4*d^2*e^3 - 17*b*c^3*d*e^4 + 6*b^2*c^2*e^5)*g)*x^3 + 5*(7*(14*c^4*d^2*e^3 - 5*b*c^3*d*e^4 - b^2*c^2*e
^5)*f + (70*c^4*d^3*e^2 - 109*b*c^3*d^2*e^3 + 25*b^2*c^2*d*e^4 + 6*b^3*c*e^5)*g)*x^2 - (170*c^4*d^4*e - 459*b*
c^3*d^3*e^2 + 311*b^2*c^2*d^2*e^3 - 78*b^3*c*d*e^4 + 8*b^4*e^5)*f + (98*c^4*d^5 - 75*b*c^3*d^4*e - 67*b^2*c^2*
d^3*e^2 + 48*b^3*c*d^2*e^3 - 4*b^4*d*e^4)*g + (7*(34*c^4*d^3*e^2 + 19*b*c^3*d^2*e^3 - 22*b^2*c^2*d*e^4 + 2*b^3
*c*e^5)*f + (170*c^4*d^4*e - 109*b*c^3*d^3*e^2 - 224*b^2*c^2*d^2*e^3 + 142*b^3*c*d*e^4 - 12*b^4*e^5)*g)*x)*sqr
t(e*x + d))/(32*c^6*d^10*e^2 - 112*b*c^5*d^9*e^3 + 160*b^2*c^4*d^8*e^4 - 120*b^3*c^3*d^7*e^5 + 50*b^4*c^2*d^6*
e^6 - 11*b^5*c*d^5*e^7 + b^6*d^4*e^8 - (32*c^6*d^5*e^7 - 80*b*c^5*d^4*e^8 + 80*b^2*c^4*d^3*e^9 - 40*b^3*c^3*d^
2*e^10 + 10*b^4*c^2*d*e^11 - b^5*c*e^12)*x^5 - (96*c^6*d^6*e^6 - 208*b*c^5*d^5*e^7 + 160*b^2*c^4*d^4*e^8 - 40*
b^3*c^3*d^3*e^9 - 10*b^4*c^2*d^2*e^10 + 7*b^5*c*d*e^11 - b^6*e^12)*x^4 - 2*(32*c^6*d^7*e^5 - 16*b*c^5*d^6*e^6
- 80*b^2*c^4*d^5*e^7 + 120*b^3*c^3*d^4*e^8 - 70*b^4*c^2*d^3*e^9 + 19*b^5*c*d^2*e^10 - 2*b^6*d*e^11)*x^3 + 2*(3
2*c^6*d^8*e^4 - 176*b*c^5*d^7*e^5 + 320*b^2*c^4*d^6*e^6 - 280*b^3*c^3*d^5*e^7 + 130*b^4*c^2*d^4*e^8 - 31*b^5*c
*d^3*e^9 + 3*b^6*d^2*e^10)*x^2 + (96*c^6*d^9*e^3 - 368*b*c^5*d^8*e^4 + 560*b^2*c^4*d^7*e^5 - 440*b^3*c^3*d^6*e
^6 + 190*b^4*c^2*d^5*e^7 - 43*b^5*c*d^4*e^8 + 4*b^6*d^3*e^9)*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((g*x+f)/(e*x+d)**(5/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, 1]