\(\int \frac {f+g x}{\sqrt {d+e x} (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\) [2283]
Optimal result
Integrand size = 46, antiderivative size = 378 \[
\int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {e f-d g}{2 e^2 (2 c d-b e) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(7 c e f+c d g-4 b e g) \sqrt {d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {5 (7 c e f+c d g-4 b e g)}{12 e^2 (2 c d-b e)^3 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {5 c (7 c e f+c d g-4 b e g) \sqrt {d+e x}}{4 e^2 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 c (7 c e f+c d g-4 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 (2 c d-b e)^{9/2}}
\]
[Out]
-5/4*c*(-4*b*e*g+c*d*g+7*c*e*f)*arctanh((d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/(-b*e+2*c*d)^(1/2)/(e*x+d)^(1/2
))/e^2/(-b*e+2*c*d)^(9/2)+1/2*(d*g-e*f)/e^2/(-b*e+2*c*d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/(e*x+d)^(1/2)+
1/6*(-4*b*e*g+c*d*g+7*c*e*f)*(e*x+d)^(1/2)/e^2/(-b*e+2*c*d)^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)-5/12*(-4*
b*e*g+c*d*g+7*c*e*f)/e^2/(-b*e+2*c*d)^3/(e*x+d)^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)+5/4*c*(-4*b*e*g+c
*d*g+7*c*e*f)*(e*x+d)^(1/2)/e^2/(-b*e+2*c*d)^4/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)
Rubi [A] (verified)
Time = 0.39 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {806, 680, 686, 674, 214}
\[
\int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {5 c \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right ) (-4 b e g+c d g+7 c e f)}{4 e^2 (2 c d-b e)^{9/2}}-\frac {e f-d g}{2 e^2 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {5 c \sqrt {d+e x} (-4 b e g+c d g+7 c e f)}{4 e^2 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 (-4 b e g+c d g+7 c e f)}{12 e^2 \sqrt {d+e x} (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {\sqrt {d+e x} (-4 b e g+c d g+7 c e f)}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}
\]
[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]
-1/2*(e*f - d*g)/(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 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 674
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 680
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 686
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))),
Int[(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] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]
Rule 806
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]
Rubi steps \begin{align*}
\text {integral}& = -\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)) \text {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 [A] (verified)
Time = 1.07 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.87
\[
\int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {c (d+e x)^{5/2} \left (\frac {(-b e+c (d-e x)) \left (6 b^3 e^3 (d g+e (f+2 g x))+c^3 \left (61 d^4 g-105 e^4 f x^3+d^2 e^2 x (161 f-5 g x)-5 d e^3 x^2 (7 f+3 g x)+d^3 e (43 f+23 g x)\right )-4 b c^2 e \left (33 d^3 g+49 d e^2 f x+5 e^3 x^2 (7 f-3 g x)+d^2 e (-4 f+30 g x)\right )+b^2 c e^2 \left (65 d^2 g+e^2 x (-21 f+80 g x)+d e (-57 f+109 g x)\right )\right )}{c (-2 c d+b e)^4 (d+e x)^2}+\frac {15 (7 c e f+c d g-4 b e g) (-b e+c (d-e x))^{5/2} \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{9/2}}\right )}{12 e^2 ((d+e x) (-b e+c (d-e x)))^{5/2}}
\]
[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]
(c*(d + e*x)^(5/2)*(((-(b*e) + c*(d - e*x))*(6*b^3*e^3*(d*g + e*(f + 2*g*x)) + c^3*(61*d^4*g - 105*e^4*f*x^3 +
d^2*e^2*x*(161*f - 5*g*x) - 5*d*e^3*x^2*(7*f + 3*g*x) + d^3*e*(43*f + 23*g*x)) - 4*b*c^2*e*(33*d^3*g + 49*d*e
^2*f*x + 5*e^3*x^2*(7*f - 3*g*x) + d^2*e*(-4*f + 30*g*x)) + b^2*c*e^2*(65*d^2*g + e^2*x*(-21*f + 80*g*x) + d*e
*(-57*f + 109*g*x))))/(c*(-2*c*d + b*e)^4*(d + e*x)^2) + (15*(7*c*e*f + c*d*g - 4*b*e*g)*(-(b*e) + c*(d - e*x)
)^(5/2)*ArcTan[Sqrt[-(b*e) + c*(d - e*x)]/Sqrt[-2*c*d + b*e]])/(-2*c*d + b*e)^(9/2)))/(12*e^2*((d + e*x)*(-(b*
e) + c*(d - e*x)))^(5/2))
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1519\) vs. \(2(344)=688\).
Time = 0.37 (sec) , antiderivative size = 1520, normalized size of antiderivative =
4.02
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method | result | size |
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default |
\(\text {Expression too large to display}\) |
\(1520\) |
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[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,method=_RETURNVERBOSE)
[Out]
1/12*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-105*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(
1/2))*c^3*e^4*f*x^3+60*(b*e-2*c*d)^(1/2)*b*c^2*e^4*g*x^3-15*(b*e-2*c*d)^(1/2)*c^3*d*e^3*g*x^3+80*(b*e-2*c*d)^(
1/2)*b^2*c*e^4*g*x^2-140*(b*e-2*c*d)^(1/2)*b*c^2*e^4*f*x^2-5*(b*e-2*c*d)^(1/2)*c^3*d^2*e^2*g*x^2-35*(b*e-2*c*d
)^(1/2)*c^3*d*e^3*f*x^2+105*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*
f-21*(b*e-2*c*d)^(1/2)*b^2*c*e^4*f*x+15*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)
)*c^3*d^4*g+12*(b*e-2*c*d)^(1/2)*b^3*e^4*g*x+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))*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)+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+60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*e^4*g*x^2*(-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*e^4*f*x^2*(-c*e*x-b*e+c*d)^(1/2)+45*(-c*e*x-b*
e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d*e^3*g*x^2-90*(-c*e*x-b*e+c*d)^(1/2)*arct
an((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^2*e^2*g*x+60*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*
d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*e^4*g*x^3-15*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d
)^(1/2))*c^3*d*e^3*g*x^3-15*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^2*e^
2*g*x^2-105*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d*e^3*f*x^2+15*(-c*e*x
-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*g*x+105*(-c*e*x-b*e+c*d)^(1/2)*arct
an((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^2*e^2*f*x-75*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)
^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^3*e*g+109*(b*e-2*c*d)^(1/2)*b^2*c*d*e^3*g*x-120*(b*e-2*c*d)^(1/2)*b*c^2*d^2*
e^2*g*x-196*(b*e-2*c*d)^(1/2)*b*c^2*d*e^3*f*x+120*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d*e^3
*g*x*(-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))*b*c^2*d*e^3*f*x*(-c*e*x-b*e+c
*d)^(1/2)+23*(b*e-2*c*d)^(1/2)*c^3*d^3*e*g*x+161*(b*e-2*c*d)^(1/2)*c^3*d^2*e^2*f*x+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-105*(b*e-2*c*d)^(1/2)*c^3*e^4*f*x^3)/(e*x+d)^(5/2)/(c*e*x+b*e-c*d)^2/e^2/(b*e-2*c*d)^(9/2)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1532 vs. \(2 (344) = 688\).
Time = 2.58 (sec) , antiderivative size = 3096, normalized size of antiderivative = 8.19
\[
\int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display}
\]
[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)]
Sympy [F]
\[
\int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \sqrt {d + e x}}\, dx
\]
[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]
Integral((f + g*x)/((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*sqrt(d + e*x)), x)
Maxima [F]
\[
\int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int { \frac {g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x }
\]
[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)
Giac [A] (verification not implemented)
none
Time = 0.61 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.63
\[
\int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {5 \, {\left (7 \, c^{2} e f + c^{2} d g - 4 \, b c e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{4 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} \sqrt {-2 \, c d + b e}} - \frac {2 \, {\left (2 \, c^{3} d e f - b c^{2} e^{2} f + 2 \, c^{3} d^{2} g - 3 \, b c^{2} d e g + b^{2} c e^{2} g - 9 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c^{2} e f - 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c^{2} d g + 6 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b c e g\right )}}{3 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}} - \frac {26 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d e f - 13 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e^{2} f - 10 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g - 3 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} d e g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c e^{2} g - 11 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} e f + 3 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 4 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c e g}{4 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} {\left (e x + d\right )}^{2} c^{2}}
\]
[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]
5/4*(7*c^2*e*f + c^2*d*g - 4*b*c*e*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/((16*c^4*d^4
*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d^2*e^4 - 8*b^3*c*d*e^5 + b^4*e^6)*sqrt(-2*c*d + b*e)) - 2/3*(2*c^3*d*e*f
- b*c^2*e^2*f + 2*c^3*d^2*g - 3*b*c^2*d*e*g + b^2*c*e^2*g - 9*((e*x + d)*c - 2*c*d + b*e)*c^2*e*f - 3*((e*x +
d)*c - 2*c*d + b*e)*c^2*d*g + 6*((e*x + d)*c - 2*c*d + b*e)*b*c*e*g)/((16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24
*b^2*c^2*d^2*e^4 - 8*b^3*c*d*e^5 + b^4*e^6)*((e*x + d)*c - 2*c*d + b*e)*sqrt(-(e*x + d)*c + 2*c*d - b*e)) - 1/
4*(26*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d*e*f - 13*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*e^2*f - 10*sqrt(-
(e*x + d)*c + 2*c*d - b*e)*c^3*d^2*g - 3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*d*e*g + 4*sqrt(-(e*x + d)*c +
2*c*d - b*e)*b^2*c*e^2*g - 11*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*e*f + 3*(-(e*x + d)*c + 2*c*d - b*e)^(3/2
)*c^2*d*g + 4*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*e*g)/((16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d^2
*e^4 - 8*b^3*c*d*e^5 + b^4*e^6)*(e*x + d)^2*c^2)
Mupad [F(-1)]
Timed out. \[
\int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {f+g\,x}{\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}} \,d x
\]
[In]
int((f + g*x)/((d + e*x)^(1/2)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2)),x)
[Out]
int((f + g*x)/((d + e*x)^(1/2)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2)), x)