3.2260 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=383 \[ -\frac{5 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+15 c d g+c e f)}{64 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac{5 c^3 (-8 b e g+15 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{64 e^2 (2 c d-b e)^{3/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (d+e x)^{15/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+15 c d g+c e f)}{24 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+15 c d g+c e f)}{96 e^2 (d+e x)^{7/2} (2 c d-b e)} \]

[Out]

(-5*c^2*(c*e*f + 15*c*d*g - 8*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(64*e^2*(2*c*d - b*e)*(d + e*x
)^(3/2)) + (5*c*(c*e*f + 15*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(96*e^2*(2*c*d - b*e
)*(d + e*x)^(7/2)) - ((c*e*f + 15*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(24*e^2*(2*c*d
- b*e)*(d + e*x)^(11/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(4*e^2*(2*c*d - b*e)*(d
+ e*x)^(15/2)) + (5*c^3*(c*e*f + 15*c*d*g - 8*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])])/(64*e^2*(2*c*d - b*e)^(3/2))

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Rubi [A]  time = 0.594049, antiderivative size = 383, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {792, 662, 660, 208} \[ -\frac{5 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+15 c d g+c e f)}{64 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac{5 c^3 (-8 b e g+15 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{64 e^2 (2 c d-b e)^{3/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (d+e x)^{15/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+15 c d g+c e f)}{24 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+15 c d g+c e f)}{96 e^2 (d+e x)^{7/2} (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(15/2),x]

[Out]

(-5*c^2*(c*e*f + 15*c*d*g - 8*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(64*e^2*(2*c*d - b*e)*(d + e*x
)^(3/2)) + (5*c*(c*e*f + 15*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(96*e^2*(2*c*d - b*e
)*(d + e*x)^(7/2)) - ((c*e*f + 15*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(24*e^2*(2*c*d
- b*e)*(d + e*x)^(11/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(4*e^2*(2*c*d - b*e)*(d
+ e*x)^(15/2)) + (5*c^3*(c*e*f + 15*c*d*g - 8*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])])/(64*e^2*(2*c*d - b*e)^(3/2))

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 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(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] && GtQ[
p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] && 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) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx &=-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (2 c d-b e) (d+e x)^{15/2}}+\frac{(c e f+15 c d g-8 b e g) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx}{8 e (2 c d-b e)}\\ &=-\frac{(c e f+15 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{24 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (2 c d-b e) (d+e x)^{15/2}}-\frac{(5 c (c e f+15 c d g-8 b e g)) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx}{48 e (2 c d-b e)}\\ &=\frac{5 c (c e f+15 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{96 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(c e f+15 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{24 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (2 c d-b e) (d+e x)^{15/2}}+\frac{\left (5 c^2 (c e f+15 c d g-8 b e g)\right ) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx}{64 e (2 c d-b e)}\\ &=-\frac{5 c^2 (c e f+15 c d g-8 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac{5 c (c e f+15 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{96 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(c e f+15 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{24 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (2 c d-b e) (d+e x)^{15/2}}-\frac{\left (5 c^3 (c e f+15 c d g-8 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}{128 e (2 c d-b e)}\\ &=-\frac{5 c^2 (c e f+15 c d g-8 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac{5 c (c e f+15 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{96 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(c e f+15 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{24 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (2 c d-b e) (d+e x)^{15/2}}-\frac{\left (5 c^3 (c e f+15 c d g-8 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 )}{64 (2 c d-b e)}\\ &=-\frac{5 c^2 (c e f+15 c d g-8 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac{5 c (c e f+15 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{96 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(c e f+15 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{24 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (2 c d-b e) (d+e x)^{15/2}}+\frac{5 c^3 (c e f+15 c d g-8 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 )}{64 e^2 (2 c d-b e)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 4.06304, size = 303, normalized size = 0.79 \[ -\frac{((d+e x) (c (d-e x)-b e))^{5/2} \left (-\frac{e (d+e x) (-8 b e g+15 c d g+c e f) \left (15 c^3 \sqrt{e} (d+e x)^3 \sqrt{c (d-e x)-b e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{c (d-e x)-b e}}{\sqrt{e (b e-2 c d)}}\right )-\sqrt{e (b e-2 c d)} \left (2 b^2 c e^2 (17 e x-7 d)+8 b^3 e^3+b c^2 e \left (19 d^2-18 d e x+59 e^2 x^2\right )+c^3 \left (-\left (d^2 e x+13 d^3+19 d e^2 x^2-33 e^3 x^3\right )\right )\right )\right )}{3 \sqrt{e (b e-2 c d)}}-16 e (e f-d g) (b e-c d+c e x)^4\right )}{64 e^3 (d+e x)^{13/2} (2 c d-b e) (b e-c d+c e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(15/2),x]

[Out]

-(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(-16*e*(e*f - d*g)*(-(c*d) + b*e + c*e*x)^4 - (e*(c*e*f + 15*c*d*g
- 8*b*e*g)*(d + e*x)*(-(Sqrt[e*(-2*c*d + b*e)]*(8*b^3*e^3 + 2*b^2*c*e^2*(-7*d + 17*e*x) + b*c^2*e*(19*d^2 - 18
*d*e*x + 59*e^2*x^2) - c^3*(13*d^3 + d^2*e*x + 19*d*e^2*x^2 - 33*e^3*x^3))) + 15*c^3*Sqrt[e]*(d + e*x)^3*Sqrt[
-(b*e) + c*(d - e*x)]*ArcTan[(Sqrt[e]*Sqrt[-(b*e) + c*(d - e*x)])/Sqrt[e*(-2*c*d + b*e)]]))/(3*Sqrt[e*(-2*c*d
+ b*e)])))/(64*e^3*(2*c*d - b*e)*(d + e*x)^(13/2)*(-(c*d) + b*e + c*e*x)^3)

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Maple [B]  time = 0.035, size = 1517, normalized size = 4. \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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(15/2),x)

[Out]

-1/192*(-61*c^3*d^3*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^
(1/2))*x^4*c^4*e^5*f+64*x*b^3*e^4*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+16*b^3*d*e^3*g*(-c*e*x-b*e+c*d)^(
1/2)*(b*e-2*c*d)^(1/2)+120*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^4*b*c^3*e^5*g-225*arctan((-c*e*x
-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^4*c^4*d*e^4*g-900*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^
4*d^2*e^3*g-60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^4*d*e^4*f-1350*arctan((-c*e*x-b*e+c*d)^(
1/2)/(b*e-2*c*d)^(1/2))*x^2*c^4*d^3*e^2*g-90*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^4*d^2*e^3*
f-900*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^4*d^4*e*g-60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c
*d)^(1/2))*x*c^4*d^3*e^2*f+120*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d^4*e*g+15*x^3*c^3*e^4*f
*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-225*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^5*g-15*ar
ctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^4*e*f+48*b^3*e^4*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/
2)-147*c^3*d^4*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-308*x*b*c^2*d*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*
d)^(1/2)+117*x*c^3*d^2*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+50*b*c^2*d^3*e*g*(-c*e*x-b*e+c*d)^(1/2)*
(b*e-2*c*d)^(1/2)+118*x^2*b*c^2*e^4*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-24*b^2*c*d^2*e^2*g*(-c*e*x-b*e+
c*d)^(1/2)*(b*e-2*c*d)^(1/2)-152*b^2*c*d*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+720*arctan((-c*e*x-b*e
+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^3*d^2*e^3*g+480*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^
3*d^3*e^2*g+208*x^2*b^2*c*e^4*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-561*x^2*c^3*d^2*e^2*g*(-c*e*x-b*e+c*d
)^(1/2)*(b*e-2*c*d)^(1/2)-191*x^2*c^3*d*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+136*x*b^2*c*e^4*f*(-c*e
*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-549*x*c^3*d^3*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-158*x^2*b*c^2*d
*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-104*x*b^2*c*d*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+2
04*x*b*c^2*d^2*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+150*b*c^2*d^2*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-
2*c*d)^(1/2)+264*x^3*b*c^2*e^4*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-543*x^3*c^3*d*e^3*g*(-c*e*x-b*e+c*d)
^(1/2)*(b*e-2*c*d)^(1/2)+480*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*b*c^3*d*e^4*g)*(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(1/2)/(b*e-2*c*d)^(3/2)/e^2/(-c*e*x-b*e+c*d)^(1/2)/(e*x+d)^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{5}{2}}{\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac{15}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(15/2),x, algorithm="maxima")

[Out]

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(15/2), x)

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Fricas [B]  time = 1.80167, size = 4012, normalized size = 10.48 \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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(15/2),x, algorithm="fricas")

[Out]

[1/384*(15*(c^4*d^5*e*f + (c^4*e^6*f + (15*c^4*d*e^5 - 8*b*c^3*e^6)*g)*x^5 + 5*(c^4*d*e^5*f + (15*c^4*d^2*e^4
- 8*b*c^3*d*e^5)*g)*x^4 + 10*(c^4*d^2*e^4*f + (15*c^4*d^3*e^3 - 8*b*c^3*d^2*e^4)*g)*x^3 + 10*(c^4*d^3*e^3*f +
(15*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3)*g)*x^2 + (15*c^4*d^6 - 8*b*c^3*d^5*e)*g + 5*(c^4*d^4*e^2*f + (15*c^4*d^5*e
- 8*b*c^3*d^4*e^2)*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)*(3*(5*(2*c^4*d*e^4 - b*c^3*e^5)*f - (362*c^4*d^2*e^3 - 357*b*c^3*d*e^4 + 88
*b^2*c^2*e^5)*g)*x^3 - ((382*c^4*d^2*e^3 - 427*b*c^3*d*e^4 + 118*b^2*c^2*e^5)*f + (1122*c^4*d^3*e^2 - 245*b*c^
3*d^2*e^3 - 574*b^2*c^2*d*e^4 + 208*b^3*c*e^5)*g)*x^2 - (122*c^4*d^4*e - 361*b*c^3*d^3*e^2 + 454*b^2*c^2*d^2*e
^3 - 248*b^3*c*d*e^4 + 48*b^4*e^5)*f - (294*c^4*d^5 - 247*b*c^3*d^4*e + 98*b^2*c^2*d^3*e^2 - 56*b^3*c*d^2*e^3
+ 16*b^4*d*e^4)*g + ((234*c^4*d^3*e^2 - 733*b*c^3*d^2*e^3 + 580*b^2*c^2*d*e^4 - 136*b^3*c*e^5)*f - (1098*c^4*d
^4*e - 957*b*c^3*d^3*e^2 + 412*b^2*c^2*d^2*e^3 - 232*b^3*c*d*e^4 + 64*b^4*e^5)*g)*x)*sqrt(e*x + d))/(4*c^2*d^7
*e^2 - 4*b*c*d^6*e^3 + b^2*d^5*e^4 + (4*c^2*d^2*e^7 - 4*b*c*d*e^8 + b^2*e^9)*x^5 + 5*(4*c^2*d^3*e^6 - 4*b*c*d^
2*e^7 + b^2*d*e^8)*x^4 + 10*(4*c^2*d^4*e^5 - 4*b*c*d^3*e^6 + b^2*d^2*e^7)*x^3 + 10*(4*c^2*d^5*e^4 - 4*b*c*d^4*
e^5 + b^2*d^3*e^6)*x^2 + 5*(4*c^2*d^6*e^3 - 4*b*c*d^5*e^4 + b^2*d^4*e^5)*x), 1/192*(15*(c^4*d^5*e*f + (c^4*e^6
*f + (15*c^4*d*e^5 - 8*b*c^3*e^6)*g)*x^5 + 5*(c^4*d*e^5*f + (15*c^4*d^2*e^4 - 8*b*c^3*d*e^5)*g)*x^4 + 10*(c^4*
d^2*e^4*f + (15*c^4*d^3*e^3 - 8*b*c^3*d^2*e^4)*g)*x^3 + 10*(c^4*d^3*e^3*f + (15*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3)
*g)*x^2 + (15*c^4*d^6 - 8*b*c^3*d^5*e)*g + 5*(c^4*d^4*e^2*f + (15*c^4*d^5*e - 8*b*c^3*d^4*e^2)*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)*(3*(5*(2*c^4*d*e^4 - b*c^3*e^5)*f - (362*c^
4*d^2*e^3 - 357*b*c^3*d*e^4 + 88*b^2*c^2*e^5)*g)*x^3 - ((382*c^4*d^2*e^3 - 427*b*c^3*d*e^4 + 118*b^2*c^2*e^5)*
f + (1122*c^4*d^3*e^2 - 245*b*c^3*d^2*e^3 - 574*b^2*c^2*d*e^4 + 208*b^3*c*e^5)*g)*x^2 - (122*c^4*d^4*e - 361*b
*c^3*d^3*e^2 + 454*b^2*c^2*d^2*e^3 - 248*b^3*c*d*e^4 + 48*b^4*e^5)*f - (294*c^4*d^5 - 247*b*c^3*d^4*e + 98*b^2
*c^2*d^3*e^2 - 56*b^3*c*d^2*e^3 + 16*b^4*d*e^4)*g + ((234*c^4*d^3*e^2 - 733*b*c^3*d^2*e^3 + 580*b^2*c^2*d*e^4
- 136*b^3*c*e^5)*f - (1098*c^4*d^4*e - 957*b*c^3*d^3*e^2 + 412*b^2*c^2*d^2*e^3 - 232*b^3*c*d*e^4 + 64*b^4*e^5)
*g)*x)*sqrt(e*x + d))/(4*c^2*d^7*e^2 - 4*b*c*d^6*e^3 + b^2*d^5*e^4 + (4*c^2*d^2*e^7 - 4*b*c*d*e^8 + b^2*e^9)*x
^5 + 5*(4*c^2*d^3*e^6 - 4*b*c*d^2*e^7 + b^2*d*e^8)*x^4 + 10*(4*c^2*d^4*e^5 - 4*b*c*d^3*e^6 + b^2*d^2*e^7)*x^3
+ 10*(4*c^2*d^5*e^4 - 4*b*c*d^4*e^5 + b^2*d^3*e^6)*x^2 + 5*(4*c^2*d^6*e^3 - 4*b*c*d^5*e^4 + b^2*d^4*e^5)*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)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(15/2),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((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(15/2),x, algorithm="giac")

[Out]

Timed out