3.2198 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=346 \[ -\frac{(b+2 c x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (5 b e g-2 c (6 e f-d g))}{512 c^3 e}-\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-2 c (6 e f-d g))}{192 c^2 e}-\frac{(2 c d-b e)^5 (5 b e g-2 c (6 e f-d g)) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{7/2} e^2}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-5 b e g-2 c d g+12 c e f)}{60 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6 c e^2 (d+e x)} \]

[Out]

-((2*c*d - b*e)^3*(5*b*e*g - 2*c*(6*e*f - d*g))*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(512*c^
3*e) - ((2*c*d - b*e)*(5*b*e*g - 2*c*(6*e*f - d*g))*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(
192*c^2*e) + ((12*c*e*f - 2*c*d*g - 5*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(60*c*e^2) - (g*(d*(
c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(6*c*e^2*(d + e*x)) - ((2*c*d - b*e)^5*(5*b*e*g - 2*c*(6*e*f - d*g))*
ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(1024*c^(7/2)*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.527541, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {794, 664, 612, 621, 204} \[ -\frac{(b+2 c x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (5 b e g-2 c (6 e f-d g))}{512 c^3 e}-\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-2 c (6 e f-d g))}{192 c^2 e}-\frac{(2 c d-b e)^5 (5 b e g-2 c (6 e f-d g)) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{7/2} e^2}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-5 b e g-2 c d g+12 c e f)}{60 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6 c e^2 (d+e x)} \]

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),x]

[Out]

-((2*c*d - b*e)^3*(5*b*e*g - 2*c*(6*e*f - d*g))*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(512*c^
3*e) - ((2*c*d - b*e)*(5*b*e*g - 2*c*(6*e*f - d*g))*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(
192*c^2*e) + ((12*c*e*f - 2*c*d*g - 5*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(60*c*e^2) - (g*(d*(
c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(6*c*e^2*(d + e*x)) - ((2*c*d - b*e)^5*(5*b*e*g - 2*c*(6*e*f - d*g))*
ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(1024*c^(7/2)*e^2)

Rule 794

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

Rule 664

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 + 2*p + 1)), x] - Dist[(p*(2*c*d - b*e))/(e^2*(m + 2*p + 1)), 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] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 612

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

Rule 621

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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} \, dx &=-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6 c e^2 (d+e x)}-\frac{\left (c e^3 f-\left (-c d e^2+b e^3\right ) g+\frac{7}{2} e \left (-2 c e^2 f+b e^2 g\right )\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{d+e x} \, dx}{6 c e^3}\\ &=\frac{(12 c e f-2 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6 c e^2 (d+e x)}+\frac{((2 c d-b e) (12 c e f-2 c d g-5 b e g)) \int \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{24 c e}\\ &=\frac{(2 c d-b e) (12 c e f-2 c d g-5 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^2 e}+\frac{(12 c e f-2 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6 c e^2 (d+e x)}+\frac{\left ((2 c d-b e)^3 (12 c e f-2 c d g-5 b e g)\right ) \int \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{128 c^2 e}\\ &=\frac{(2 c d-b e)^3 (12 c e f-2 c d g-5 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^3 e}+\frac{(2 c d-b e) (12 c e f-2 c d g-5 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^2 e}+\frac{(12 c e f-2 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6 c e^2 (d+e x)}+\frac{\left ((2 c d-b e)^5 (12 c e f-2 c d g-5 b e g)\right ) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{1024 c^3 e}\\ &=\frac{(2 c d-b e)^3 (12 c e f-2 c d g-5 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^3 e}+\frac{(2 c d-b e) (12 c e f-2 c d g-5 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^2 e}+\frac{(12 c e f-2 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6 c e^2 (d+e x)}+\frac{\left ((2 c d-b e)^5 (12 c e f-2 c d g-5 b e g)\right ) \operatorname{Subst}\left (\int \frac{1}{-4 c e^2-x^2} \, dx,x,\frac{-b e^2-2 c e^2 x}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{512 c^3 e}\\ &=\frac{(2 c d-b e)^3 (12 c e f-2 c d g-5 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^3 e}+\frac{(2 c d-b e) (12 c e f-2 c d g-5 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{192 c^2 e}+\frac{(12 c e f-2 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{60 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6 c e^2 (d+e x)}+\frac{(2 c d-b e)^5 (12 c e f-2 c d g-5 b e g) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{7/2} e^2}\\ \end{align*}

Mathematica [B]  time = 6.22465, size = 1230, normalized size = 3.55 \[ -\frac{(c d e+(c d-b e) e)^2 \left (-6 c f e^2-\left (\frac{5}{2} e (c d-b e)-\frac{7 c d e}{2}\right ) g\right ) ((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{15 (c d e+(c d-b e) e)^3 \left (-\frac{4 c^2 (d+e x)^2 e^4}{3 (c d e+(c d-b e) e)^2 \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )^2}-\frac{2 c (d+e x) e^2}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}+\frac{2 \sqrt{c} \sqrt{d+e x} \sin ^{-1}\left (\frac{\sqrt{c} e \sqrt{d+e x}}{\sqrt{c d e+(c d-b e) e} \sqrt{\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}}}\right ) e}{\sqrt{c d e+(c d-b e) e} \sqrt{\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}} \sqrt{1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}}}\right ) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )^3}{512 c^3 e^6 (d+e x)^3 \left (1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^3}+\frac{1}{2} \left (\frac{1}{1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}}+\frac{5}{8 \left (1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^2}+\frac{5}{16 \left (1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^3}\right )\right ) \left (1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^{7/2}}{15 c e^5 \left (\frac{e}{\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}}\right )^{5/2} (c d-b e-c e x)^2 \sqrt{\frac{e (c d-b e-c e x)}{c d e+(c d-b e) e}}}-\frac{g (c d-b e-c e x) ((d+e x) (c (d-e x)-b e))^{5/2}}{6 c e^2} \]

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),x]

[Out]

-(g*(c*d - b*e - c*e*x)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))/(6*c*e^2) - ((c*d*e + e*(c*d - b*e))^2*(-6*c
*e^2*f - ((-7*c*d*e)/2 + (5*e*(c*d - b*e))/2)*g)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(1 - (c*e^2*(d + e*x
))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))))^
(7/2)*((5/(16*(1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d -
b*e))/(c*d*e + e*(c*d - b*e)))))^3) + 5/(8*(1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e
+ e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))))^2) + (1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d -
b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))))^(-1))/2 + (15*(c*d*e
+ e*(c*d - b*e))^3*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))^3*((-2*c*e^
2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d -
b*e)))) - (4*c^2*e^4*(d + e*x)^2)/(3*(c*d*e + e*(c*d - b*e))^2*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d
- b*e))/(c*d*e + e*(c*d - b*e)))^2) + (2*Sqrt[c]*e*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*e*Sqrt[d + e*x])/(Sqrt[c*d*e
+ e*(c*d - b*e)]*Sqrt[(c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e))])])/(Sqrt
[c*d*e + e*(c*d - b*e)]*Sqrt[(c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e))]*Sq
rt[1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*
e + e*(c*d - b*e))))])))/(512*c^3*e^6*(d + e*x)^3*(1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(
c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))))^3)))/(15*c*e^5*(e/((c*d*e^2)/(c*d*e + e*
(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e))))^(5/2)*(c*d - b*e - c*e*x)^2*Sqrt[(e*(c*d - b*e - c*
e*x))/(c*d*e + e*(c*d - b*e))])

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 3533, normalized size = 10.2 \begin{align*} \text{output 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),x)

[Out]

-5/48*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^2*d-5/24*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b*d-5/6
4*g*e^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^4*d+15/64*g*e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^
3*d^2+5/96*g*e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b^2+5/24*g/e*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2
)*x*d^2+5/256*g*e^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b^4+15/32*g*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(1/2)*x*b^2*d^2-25/32*g*e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2))*b^3*d^3+5/1024*g*e^5/c^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(1/2))*b^6-15/16*g*c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)
)*b*d^5-5/8*g*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b*d^3+5/32*g/e*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/
2)*b*d^4+9/64*e^2*b^3/c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d*f-3/256*e^5*b^5/c^2/(c*e^2)^(1/2)*
arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*f+1
5/32*e^2*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2
*c*d*e)*(x+d/e))^(1/2))*d^3*g+5/16*g/e*c^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(1/2))*d^6+5/16*g/e*c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4-9/32*e*b^2*(-(x+d/e)^2*c*e^2
+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d^2*g+9/32*e^2*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d*f-
9/64*e*b^3/c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d^2*g+9/16*b*c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*
e)*(x+d/e))^(1/2)*x*d^3*g+15/16*b*c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-
(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^5*g-1/4/e*c*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))
^(3/2)*x*g+3/128*e^2*b^4/c^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d*g-15/32*e^3*b^3/(c*e^2)^(1/2)
*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^
2*f-3/8/e*c^2*d^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*g-3/16/e*c*d^4*(-(x+d/e)^2*c*e^2+(-b*e^2
+2*c*d*e)*(x+d/e))^(1/2)*b*g-3/8/e*c^3*d^6/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^
2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*g-3/64*e^3*b^3/c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d
/e))^(1/2)*x*f+1/8*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*b*f+9/32*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+
2*c*d*e)*(x+d/e))^(1/2)*d^3*g-5/16*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d^3+1/6*g/e*(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(5/2)*x-1/5/e^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*d*g+1/5/e*(-(x+d/e)^2*c*e^2+(
-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*f-9/32*e*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d^2*f-1/8*e*b*(-
(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*x*f+1/4*c*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*
x*f+1/16*b^2/c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*d*g+1/8*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*
(x+d/e))^(3/2)*x*d*g+3/8*c^2*d^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*f+3/16*c*d^3*(-(x+d/e)^2*
c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*f+5/512*g*e^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^5+5/48*g/e*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^2+3/8*c^3*d^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2
+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*f-1/8/e*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*
d*e)*(x+d/e))^(3/2)*b*g-1/16*e*b^2/c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*f-3/128*e^3*b^4/c^2*(-(
x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*f+1/12*g/e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b+5/192*g*e
/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^3-5/32*g*e^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b^3*d+75
/64*g*e*c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^2*d^4-15/25
6*g*e^4/c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^5*d+75/25
6*g*e^3/c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^4*d^2+15/12
8*e^4*b^4/c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*
c*d*e)*(x+d/e))^(1/2))*d*f+15/16*e^2*b^2*c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^
2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^3*f-15/16*e*b*c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(
x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^4*f+3/64*e^2*b^3/c*(-(x
+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d*g+3/256*e^4*b^5/c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/
e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d*g-9/16*e*b*c*(-(x+d/e)^2*c*
e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d^2*f-15/128*e^3*b^4/c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b
*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^2*g-15/16*e*b^2*c/(c*e^2)^(1/2)*arct
an((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^4*g

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 8.34609, size = 3163, normalized size = 9.14 \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),x, algorithm="fricas")

[Out]

[-1/30720*(15*(12*(32*c^6*d^5*e - 80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c^2*d*e^
5 - b^5*c*e^6)*f - (64*c^6*d^6 - 240*b^2*c^4*d^4*e^2 + 320*b^3*c^3*d^3*e^3 - 180*b^4*c^2*d^2*e^4 + 48*b^5*c*d*
e^5 - 5*b^6*e^6)*g)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 - 4*sqrt(-c*e^2
*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*(1280*c^6*e^5*g*x^5 + 128*(12*c^6*e^5*f - (12*c^
6*d*e^4 - 25*b*c^5*e^5)*g)*x^4 - 16*(12*(10*c^6*d*e^4 - 21*b*c^5*e^5)*f + (140*c^6*d^2*e^3 - 8*b*c^5*d*e^4 - 1
35*b^2*c^4*e^5)*g)*x^3 - 8*(12*(32*c^6*d^2*e^3 - 2*b*c^5*d*e^4 - 31*b^2*c^4*e^5)*f - (384*c^6*d^3*e^2 - 804*b*
c^5*d^2*e^3 + 408*b^2*c^4*d*e^4 + 5*b^3*c^3*e^5)*g)*x^2 + 12*(128*c^6*d^4*e - 56*b*c^5*d^3*e^2 - 172*b^2*c^4*d
^2*e^3 + 130*b^3*c^3*d*e^4 - 15*b^4*c^2*e^5)*f - (1536*c^6*d^5 - 3312*b*c^5*d^4*e + 3216*b^2*c^4*d^3*e^2 - 188
0*b^3*c^3*d^2*e^3 + 620*b^4*c^2*d*e^4 - 75*b^5*c*e^5)*g + 2*(12*(200*c^6*d^3*e^2 - 428*b*c^5*d^2*e^3 + 218*b^2
*c^4*d*e^4 + 5*b^3*c^3*e^5)*f + (240*c^6*d^4*e - 144*b*c^5*d^3*e^2 - 216*b^2*c^4*d^2*e^3 + 180*b^3*c^3*d*e^4 -
25*b^4*c^2*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^4*e^2), -1/15360*(15*(12*(32*c^6*d^5*e -
80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c^2*d*e^5 - b^5*c*e^6)*f - (64*c^6*d^6 -
240*b^2*c^4*d^4*e^2 + 320*b^3*c^3*d^3*e^3 - 180*b^4*c^2*d^2*e^4 + 48*b^5*c*d*e^5 - 5*b^6*e^6)*g)*sqrt(c)*arcta
n(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 +
b*c*d*e)) - 2*(1280*c^6*e^5*g*x^5 + 128*(12*c^6*e^5*f - (12*c^6*d*e^4 - 25*b*c^5*e^5)*g)*x^4 - 16*(12*(10*c^6*
d*e^4 - 21*b*c^5*e^5)*f + (140*c^6*d^2*e^3 - 8*b*c^5*d*e^4 - 135*b^2*c^4*e^5)*g)*x^3 - 8*(12*(32*c^6*d^2*e^3 -
2*b*c^5*d*e^4 - 31*b^2*c^4*e^5)*f - (384*c^6*d^3*e^2 - 804*b*c^5*d^2*e^3 + 408*b^2*c^4*d*e^4 + 5*b^3*c^3*e^5)
*g)*x^2 + 12*(128*c^6*d^4*e - 56*b*c^5*d^3*e^2 - 172*b^2*c^4*d^2*e^3 + 130*b^3*c^3*d*e^4 - 15*b^4*c^2*e^5)*f -
(1536*c^6*d^5 - 3312*b*c^5*d^4*e + 3216*b^2*c^4*d^3*e^2 - 1880*b^3*c^3*d^2*e^3 + 620*b^4*c^2*d*e^4 - 75*b^5*c
*e^5)*g + 2*(12*(200*c^6*d^3*e^2 - 428*b*c^5*d^2*e^3 + 218*b^2*c^4*d*e^4 + 5*b^3*c^3*e^5)*f + (240*c^6*d^4*e -
144*b*c^5*d^3*e^2 - 216*b^2*c^4*d^2*e^3 + 180*b^3*c^3*d*e^4 - 25*b^4*c^2*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x
+ c*d^2 - b*d*e))/(c^4*e^2)]

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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),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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),x, algorithm="giac")

[Out]

Exception raised: TypeError