3.2258 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=372 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (4 b e g-11 c d g+3 c e f)}{4 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-11 c d g+3 c e f)}{12 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac{5 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-11 c d g+3 c e f)}{4 e^2 \sqrt{d+e x}}-\frac{5 c \sqrt{2 c d-b e} (4 b e g-11 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2} \]

[Out]

(5*c*(3*c*e*f - 11*c*d*g + 4*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2*Sqrt[d + e*x]) + (5*c*(3
*c*e*f - 11*c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(12*e^2*(2*c*d - b*e)*(d + e*x)^(3/2
)) + ((3*c*e*f - 11*c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(4*e^2*(2*c*d - b*e)*(d + e*
x)^(7/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)^(11/2)) -
(5*c*Sqrt[2*c*d - b*e]*(3*c*e*f - 11*c*d*g + 4*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])])/(4*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.60362, antiderivative size = 372, 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, 662, 664, 660, 208} \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (4 b e g-11 c d g+3 c e f)}{4 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-11 c d g+3 c e f)}{12 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac{5 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-11 c d g+3 c e f)}{4 e^2 \sqrt{d+e x}}-\frac{5 c \sqrt{2 c d-b e} (4 b e g-11 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2} \]

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)^(11/2),x]

[Out]

(5*c*(3*c*e*f - 11*c*d*g + 4*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2*Sqrt[d + e*x]) + (5*c*(3
*c*e*f - 11*c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(12*e^2*(2*c*d - b*e)*(d + e*x)^(3/2
)) + ((3*c*e*f - 11*c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(4*e^2*(2*c*d - b*e)*(d + e*
x)^(7/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)^(11/2)) -
(5*c*Sqrt[2*c*d - b*e]*(3*c*e*f - 11*c*d*g + 4*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])])/(4*e^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 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 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)^{11/2}} \, dx &=-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac{(3 c e f-11 c d g+4 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)^{9/2}} \, dx}{4 e (2 c d-b e)}\\ &=\frac{(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac{(5 c (3 c e f-11 c d g+4 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)^{5/2}} \, dx}{8 e (2 c d-b e)}\\ &=\frac{5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac{(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac{(5 c (3 c e f-11 c d g+4 b e g)) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{8 e}\\ &=\frac{5 c (3 c e f-11 c d g+4 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 \sqrt{d+e x}}+\frac{5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac{(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac{(5 c (2 c d-b e) (3 c e f-11 c d g+4 b e g)) \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e}\\ &=\frac{5 c (3 c e f-11 c d g+4 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 \sqrt{d+e x}}+\frac{5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac{(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac{1}{4} (5 c (2 c d-b e) (3 c e f-11 c d g+4 b e g)) \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 )\\ &=\frac{5 c (3 c e f-11 c d g+4 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 \sqrt{d+e x}}+\frac{5 c (3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac{(3 c e f-11 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac{5 c \sqrt{2 c d-b e} (3 c e f-11 c d g+4 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 )}{4 e^2}\\ \end{align*}

Mathematica [C]  time = 0.288284, size = 130, normalized size = 0.35 \[ \frac{((d+e x) (c (d-e x)-b e))^{7/2} \left (\frac{c (d+e x)^2 (4 b e g-11 c d g+3 c e f) \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )}{e (b e-2 c d)^2}+\frac{7 d g}{e}-7 f\right )}{14 e (d+e x)^{11/2} (2 c d-b e)} \]

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)^(11/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(7/2)*(-7*f + (7*d*g)/e + (c*(3*c*e*f - 11*c*d*g + 4*b*e*g)*(d + e*x)^2*Hy
pergeometric2F1[2, 7/2, 9/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])/(e*(-2*c*d + b*e)^2)))/(14*e*(2*c*d - b*e
)*(d + e*x)^(11/2))

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Maple [B]  time = 0.031, size = 1189, 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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(11/2),x)

[Out]

-1/12*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(-180*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^3*d^2*
e^2*f-285*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^3*e*g-24*x^2*c^2*e^3*f*(-c*e*x-b*e+c*d)^(1/
2)*(b*e-2*c*d)^(1/2)+12*x*b^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+330*arctan((-c*e*x-b*e+c*d)^(1/2)
/(b*e-2*c*d)^(1/2))*x^2*c^3*d^2*e^2*g-90*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^3*d*e^3*f+660*
arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^3*d^3*e*g+6*b^2*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^
(1/2)+45*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^2*e^2*f+60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*
e-2*c*d)^(1/2))*b^2*c*d^2*e^2*g+60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b^2*c*e^4*g+45*arctan(
(-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^2*e^4*f-8*x^3*c^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(
1/2)+6*b^2*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-54*c^2*d^2*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^
(1/2)-187*x*b*c*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+330*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)
^(1/2))*c^3*d^4*g+206*c^2*d^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-90*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e
-2*c*d)^(1/2))*c^3*d^3*e*f-56*x^2*b*c*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+112*x^2*c^2*d*e^2*g*(-c*e
*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+27*x*b*c*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+350*x*c^2*d^2*e*g*
(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-102*x*c^2*d*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-107*b*c*d^
2*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+3*b*c*d*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-285*arct
an((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^2*d*e^3*g-570*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(
1/2))*x*b*c^2*d^2*e^2*g+120*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b^2*c*d*e^3*g+90*arctan((-c*e*x
-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^2*d*e^3*f)/(e*x+d)^(5/2)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*c*d)^(1/2)

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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{11}{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)^(11/2),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)^(11/2), x)

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Fricas [A]  time = 1.79935, size = 2021, normalized size = 5.43 \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)^(11/2),x, algorithm="fricas")

[Out]

[1/24*(15*(3*c^2*d^3*e*f + (3*c^2*e^4*f - (11*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(3*c^2*d*e^3*f - (11*c^2*d^2*e
^2 - 4*b*c*d*e^3)*g)*x^2 - (11*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(3*c^2*d^2*e^2*f - (11*c^2*d^3*e - 4*b*c*d^2*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*(8*c^2*e^3*g*x^3 + 8*(3*c^2
*e^3*f - 7*(2*c^2*d*e^2 - b*c*e^3)*g)*x^2 + 3*(18*c^2*d^2*e - b*c*d*e^2 - 2*b^2*e^3)*f - (206*c^2*d^3 - 107*b*
c*d^2*e + 6*b^2*d*e^2)*g + (3*(34*c^2*d*e^2 - 9*b*c*e^3)*f - (350*c^2*d^2*e - 187*b*c*d*e^2 + 12*b^2*e^3)*g)*x
)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2), -
1/12*(15*(3*c^2*d^3*e*f + (3*c^2*e^4*f - (11*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(3*c^2*d*e^3*f - (11*c^2*d^2*e^
2 - 4*b*c*d*e^3)*g)*x^2 - (11*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(3*c^2*d^2*e^2*f - (11*c^2*d^3*e - 4*b*c*d^2*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)) - (8*c^2*e^3*g*x^3 + 8*(3*c^2*e^3*f - 7*(2*c^2*d*e^2 - b*c*e^3)*g)*x^2 + 3
*(18*c^2*d^2*e - b*c*d*e^2 - 2*b^2*e^3)*f - (206*c^2*d^3 - 107*b*c*d^2*e + 6*b^2*d*e^2)*g + (3*(34*c^2*d*e^2 -
9*b*c*e^3)*f - (350*c^2*d^2*e - 187*b*c*d*e^2 + 12*b^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*
sqrt(e*x + d))/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*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)**(11/2),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)^(11/2),x, algorithm="giac")

[Out]

Timed out