3.2283 \(\int \frac{f+g x}{\sqrt{d+e x} (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=378 \[ -\frac{e f-d g}{2 e^2 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{5 c \sqrt{d+e x} (-4 b e g+c d g+7 c e f)}{4 e^2 (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 (-4 b e g+c d g+7 c e f)}{12 e^2 \sqrt{d+e x} (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{\sqrt{d+e x} (-4 b e g+c d g+7 c e f)}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{5 c (-4 b e g+c d g+7 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{9/2}} \]

[Out]

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

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Rubi [A]  time = 0.589985, antiderivative size = 378, 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, 666, 672, 660, 208} \[ -\frac{e f-d g}{2 e^2 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{5 c \sqrt{d+e x} (-4 b e g+c d g+7 c e f)}{4 e^2 (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 (-4 b e g+c d g+7 c e f)}{12 e^2 \sqrt{d+e x} (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{\sqrt{d+e x} (-4 b e g+c d g+7 c e f)}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{5 c (-4 b e g+c d g+7 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*f - d*g)/(2*e^2*(2*c*d - b*e)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + ((7*c*e*f + c*d
*g - 4*b*e*g)*Sqrt[d + e*x])/(6*e^2*(2*c*d - b*e)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (5*(7*c*e*f
+ c*d*g - 4*b*e*g))/(12*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*(
7*c*e*f + c*d*g - 4*b*e*g)*Sqrt[d + e*x])/(4*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*(7*c*e*f + 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*(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 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 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 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}{\sqrt{d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=-\frac{e f-d g}{2 e^2 (2 c d-b e) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(7 c e f+c d g-4 b e g) \int \frac{\sqrt{d+e x}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{4 e (2 c d-b e)}\\ &=-\frac{e f-d g}{2 e^2 (2 c d-b e) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(7 c e f+c d g-4 b e g) \sqrt{d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(5 (7 c e f+c d g-4 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}{12 e (2 c d-b e)^2}\\ &=-\frac{e f-d g}{2 e^2 (2 c d-b e) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(7 c e f+c d g-4 b e g) \sqrt{d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{5 (7 c e f+c d g-4 b e g)}{12 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 (7 c e f+c d g-4 b e g)) \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}{8 e (2 c d-b e)^3}\\ &=-\frac{e f-d g}{2 e^2 (2 c d-b e) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(7 c e f+c d g-4 b e g) \sqrt{d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{5 (7 c e f+c d g-4 b e g)}{12 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 (7 c e f+c d g-4 b e g) \sqrt{d+e x}}{4 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 (7 c e f+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 (2 c d-b e)^4}\\ &=-\frac{e f-d g}{2 e^2 (2 c d-b e) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(7 c e f+c d g-4 b e g) \sqrt{d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{5 (7 c e f+c d g-4 b e g)}{12 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 (7 c e f+c d g-4 b e g) \sqrt{d+e x}}{4 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 (7 c e f+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 )}{4 (2 c d-b e)^4}\\ &=-\frac{e f-d g}{2 e^2 (2 c d-b e) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(7 c e f+c d g-4 b e g) \sqrt{d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{5 (7 c e f+c d g-4 b e g)}{12 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 (7 c e f+c d g-4 b e g) \sqrt{d+e x}}{4 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 (7 c e f+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 (2 c d-b e)^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.134515, size = 129, normalized size = 0.34 \[ \frac{\frac{c (d+e x)^2 (-4 b e g+c d g+7 c e f) \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )}{e (b e-2 c d)^2}+\frac{3 d g}{e}-3 f}{6 e \sqrt{d+e x} (2 c d-b e) ((d+e x) (c (d-e x)-b e))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*f + (3*d*g)/e + (c*(7*c*e*f + c*d*g - 4*b*e*g)*(d + e*x)^2*Hypergeometric2F1[-3/2, 2, -1/2, (-(c*d) + b*e
+ c*e*x)/(-2*c*d + b*e)])/(e*(-2*c*d + b*e)^2))/(6*e*(2*c*d - b*e)*Sqrt[d + e*x]*((d + e*x)*(-(b*e) + c*(d - e
*x)))^(3/2))

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Maple [B]  time = 0.035, size = 1528, 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)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

1/12*(-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-21*(b*e-2*c*d)^(1/2)*x*b^2*c
*e^4*f+23*(b*e-2*c*d)^(1/2)*x*c^3*d^3*e*g+161*(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+60*(b*e-2*c*d)^(1/2)*x^3*b*c^2*e^4*g-15*(b*e-2*c*d)^(1/2)*x^3*c^3*d*e^3*
g+80*(b*e-2*c*d)^(1/2)*x^2*b^2*c*e^4*g-140*(b*e-2*c*d)^(1/2)*x^2*b*c^2*e^4*f-5*(b*e-2*c*d)^(1/2)*x^2*c^3*d^2*e
^2*g-35*(b*e-2*c*d)^(1/2)*x^2*c^3*d*e^3*f+65*(b*e-2*c*d)^(1/2)*b^2*c*d^2*e^2*g-57*(b*e-2*c*d)^(1/2)*b^2*c*d*e^
3*f-132*(b*e-2*c*d)^(1/2)*b*c^2*d^3*e*g+16*(b*e-2*c*d)^(1/2)*b*c^2*d^2*e^2*f+6*(b*e-2*c*d)^(1/2)*b^3*e^4*f+61*
(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+15*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+6*(b*e-2*c*d)^(1/2)*b^3*d*e^3*g+43*(b*e-2*c*d)^(1/2)*c^3*d^3*e*f+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*(-c*e*x-b*e+c*d)^(1/2)-105*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*(-c*e*x-b*e+c*d)^(1/2)+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*(-c*e*x-b*e+c*d)^(1/2)-105*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*(-c*e*x-b*e+c*d)^(1/2)+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*(-c*e*x-b*e+c*
d)^(1/2)-210*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*(-c*e*x-b*e+c*d)^(1/2)-15*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-15*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-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^2*c^3*d*e^3*f+15*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^3*e*g+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*c^3*d^
2*e^2*f+109*(b*e-2*c*d)^(1/2)*x*b^2*c*d*e^3*g-120*(b*e-2*c*d)^(1/2)*x*b*c^2*d^2*e^2*g-196*(b*e-2*c*d)^(1/2)*x*
b*c^2*d*e^3*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)*b*c^2*d^3*e*g+45*arct
an((-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-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*b*c^2*d^2*e^2*g+60*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)/(e*x+d)^(5/2)/(c*e*x+b*e-c*d)^2/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{5}{2}} \sqrt{e x + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.22733, size = 6388, normalized size = 16.9 \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)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/24*(15*((7*c^4*e^6*f + (c^4*d*e^5 - 4*b*c^3*e^6)*g)*x^5 + (7*(c^4*d*e^5 + 2*b*c^3*e^6)*f + (c^4*d^2*e^4 -
2*b*c^3*d*e^5 - 8*b^2*c^2*e^6)*g)*x^4 - (7*(2*c^4*d^2*e^4 - 4*b*c^3*d*e^5 - b^2*c^2*e^6)*f + (2*c^4*d^3*e^3 -
12*b*c^3*d^2*e^4 + 15*b^2*c^2*d*e^5 + 4*b^3*c*e^6)*g)*x^3 - (7*(2*c^4*d^3*e^3 - 3*b^2*c^2*d*e^5)*f + (2*c^4*d^
4*e^2 - 8*b*c^3*d^3*e^3 - 3*b^2*c^2*d^2*e^4 + 12*b^3*c*d*e^5)*g)*x^2 + 7*(c^4*d^5*e - 2*b*c^3*d^4*e^2 + b^2*c^
2*d^3*e^3)*f + (c^4*d^6 - 6*b*c^3*d^5*e + 9*b^2*c^2*d^4*e^2 - 4*b^3*c*d^3*e^3)*g + (7*(c^4*d^4*e^2 - 4*b*c^3*d
^3*e^3 + 3*b^2*c^2*d^2*e^4)*f + (c^4*d^5*e - 8*b*c^3*d^4*e^2 + 19*b^2*c^2*d^3*e^3 - 12*b^3*c*d^2*e^4)*g)*x)*sq
rt(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 + (2*c^4*d^2*e^3 - 9*b*c^3*d*e^4 + 4*b^2*c^2*e^5)*g)*x^3 + 5*(7*(2*
c^4*d^2*e^3 + 7*b*c^3*d*e^4 - 4*b^2*c^2*e^5)*f + (2*c^4*d^3*e^2 - b*c^3*d^2*e^3 - 32*b^2*c^2*d*e^4 + 16*b^3*c*
e^5)*g)*x^2 - (86*c^4*d^4*e - 11*b*c^3*d^3*e^2 - 130*b^2*c^2*d^2*e^3 + 69*b^3*c*d*e^4 - 6*b^4*e^5)*f - (122*c^
4*d^5 - 325*b*c^3*d^4*e + 262*b^2*c^2*d^3*e^2 - 53*b^3*c*d^2*e^3 - 6*b^4*d*e^4)*g - (7*(46*c^4*d^3*e^2 - 79*b*
c^3*d^2*e^3 + 22*b^2*c^2*d*e^4 + 3*b^3*c*e^5)*f + (46*c^4*d^4*e - 263*b*c^3*d^3*e^2 + 338*b^2*c^2*d^2*e^3 - 85
*b^3*c*d*e^4 - 12*b^4*e^5)*g)*x)*sqrt(e*x + d))/(32*c^7*d^10*e^2 - 144*b*c^6*d^9*e^3 + 272*b^2*c^5*d^8*e^4 - 2
80*b^3*c^4*d^7*e^5 + 170*b^4*c^3*d^6*e^6 - 61*b^5*c^2*d^5*e^7 + 12*b^6*c*d^4*e^8 - b^7*d^3*e^9 + (32*c^7*d^5*e
^7 - 80*b*c^6*d^4*e^8 + 80*b^2*c^5*d^3*e^9 - 40*b^3*c^4*d^2*e^10 + 10*b^4*c^3*d*e^11 - b^5*c^2*e^12)*x^5 + (32
*c^7*d^6*e^6 - 16*b*c^6*d^5*e^7 - 80*b^2*c^5*d^4*e^8 + 120*b^3*c^4*d^3*e^9 - 70*b^4*c^3*d^2*e^10 + 19*b^5*c^2*
d*e^11 - 2*b^6*c*e^12)*x^4 - (64*c^7*d^7*e^5 - 288*b*c^6*d^6*e^6 + 448*b^2*c^5*d^5*e^7 - 320*b^3*c^4*d^4*e^8 +
100*b^4*c^3*d^3*e^9 - 2*b^5*c^2*d^2*e^10 - 6*b^6*c*d*e^11 + b^7*e^12)*x^3 - (64*c^7*d^8*e^4 - 160*b*c^6*d^7*e
^5 + 64*b^2*c^5*d^6*e^6 + 160*b^3*c^4*d^5*e^7 - 220*b^4*c^3*d^4*e^8 + 118*b^5*c^2*d^3*e^9 - 30*b^6*c*d^2*e^10
+ 3*b^7*d*e^11)*x^2 + (32*c^7*d^9*e^3 - 208*b*c^6*d^8*e^4 + 496*b^2*c^5*d^7*e^5 - 600*b^3*c^4*d^6*e^6 + 410*b^
4*c^3*d^5*e^7 - 161*b^5*c^2*d^4*e^8 + 34*b^6*c*d^3*e^9 - 3*b^7*d^2*e^10)*x), -1/12*(15*((7*c^4*e^6*f + (c^4*d*
e^5 - 4*b*c^3*e^6)*g)*x^5 + (7*(c^4*d*e^5 + 2*b*c^3*e^6)*f + (c^4*d^2*e^4 - 2*b*c^3*d*e^5 - 8*b^2*c^2*e^6)*g)*
x^4 - (7*(2*c^4*d^2*e^4 - 4*b*c^3*d*e^5 - b^2*c^2*e^6)*f + (2*c^4*d^3*e^3 - 12*b*c^3*d^2*e^4 + 15*b^2*c^2*d*e^
5 + 4*b^3*c*e^6)*g)*x^3 - (7*(2*c^4*d^3*e^3 - 3*b^2*c^2*d*e^5)*f + (2*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3 - 3*b^2*c^
2*d^2*e^4 + 12*b^3*c*d*e^5)*g)*x^2 + 7*(c^4*d^5*e - 2*b*c^3*d^4*e^2 + b^2*c^2*d^3*e^3)*f + (c^4*d^6 - 6*b*c^3*
d^5*e + 9*b^2*c^2*d^4*e^2 - 4*b^3*c*d^3*e^3)*g + (7*(c^4*d^4*e^2 - 4*b*c^3*d^3*e^3 + 3*b^2*c^2*d^2*e^4)*f + (c
^4*d^5*e - 8*b*c^3*d^4*e^2 + 19*b^2*c^2*d^3*e^3 - 12*b^3*c*d^2*e^4)*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)) + sqr
t(-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 + (2*c^4*d^2*e^3 - 9*b*c^3*d*e^4 +
4*b^2*c^2*e^5)*g)*x^3 + 5*(7*(2*c^4*d^2*e^3 + 7*b*c^3*d*e^4 - 4*b^2*c^2*e^5)*f + (2*c^4*d^3*e^2 - b*c^3*d^2*e^
3 - 32*b^2*c^2*d*e^4 + 16*b^3*c*e^5)*g)*x^2 - (86*c^4*d^4*e - 11*b*c^3*d^3*e^2 - 130*b^2*c^2*d^2*e^3 + 69*b^3*
c*d*e^4 - 6*b^4*e^5)*f - (122*c^4*d^5 - 325*b*c^3*d^4*e + 262*b^2*c^2*d^3*e^2 - 53*b^3*c*d^2*e^3 - 6*b^4*d*e^4
)*g - (7*(46*c^4*d^3*e^2 - 79*b*c^3*d^2*e^3 + 22*b^2*c^2*d*e^4 + 3*b^3*c*e^5)*f + (46*c^4*d^4*e - 263*b*c^3*d^
3*e^2 + 338*b^2*c^2*d^2*e^3 - 85*b^3*c*d*e^4 - 12*b^4*e^5)*g)*x)*sqrt(e*x + d))/(32*c^7*d^10*e^2 - 144*b*c^6*d
^9*e^3 + 272*b^2*c^5*d^8*e^4 - 280*b^3*c^4*d^7*e^5 + 170*b^4*c^3*d^6*e^6 - 61*b^5*c^2*d^5*e^7 + 12*b^6*c*d^4*e
^8 - b^7*d^3*e^9 + (32*c^7*d^5*e^7 - 80*b*c^6*d^4*e^8 + 80*b^2*c^5*d^3*e^9 - 40*b^3*c^4*d^2*e^10 + 10*b^4*c^3*
d*e^11 - b^5*c^2*e^12)*x^5 + (32*c^7*d^6*e^6 - 16*b*c^6*d^5*e^7 - 80*b^2*c^5*d^4*e^8 + 120*b^3*c^4*d^3*e^9 - 7
0*b^4*c^3*d^2*e^10 + 19*b^5*c^2*d*e^11 - 2*b^6*c*e^12)*x^4 - (64*c^7*d^7*e^5 - 288*b*c^6*d^6*e^6 + 448*b^2*c^5
*d^5*e^7 - 320*b^3*c^4*d^4*e^8 + 100*b^4*c^3*d^3*e^9 - 2*b^5*c^2*d^2*e^10 - 6*b^6*c*d*e^11 + b^7*e^12)*x^3 - (
64*c^7*d^8*e^4 - 160*b*c^6*d^7*e^5 + 64*b^2*c^5*d^6*e^6 + 160*b^3*c^4*d^5*e^7 - 220*b^4*c^3*d^4*e^8 + 118*b^5*
c^2*d^3*e^9 - 30*b^6*c*d^2*e^10 + 3*b^7*d*e^11)*x^2 + (32*c^7*d^9*e^3 - 208*b*c^6*d^8*e^4 + 496*b^2*c^5*d^7*e^
5 - 600*b^3*c^4*d^6*e^6 + 410*b^4*c^3*d^5*e^7 - 161*b^5*c^2*d^4*e^8 + 34*b^6*c*d^3*e^9 - 3*b^7*d^2*e^10)*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)**(1/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x