\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^5} \, dx\) [2202]

Optimal result

Integrand size = 44, antiderivative size = 350 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {5 \sqrt {c} (2 c d-b e) (4 c e f-10 c d g+3 b e g) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2} \]

[Out]

5/6*c*(3*b*e*g-10*c*d*g+4*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(-b*e+2*c*d)/(e*x+d)+2/3*(3*b*e*g-
10*c*d*g+4*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2/(-b*e+2*c*d)/(e*x+d)^3-2/3*(-d*g+e*f)*(d*(-b*e+c*
d)-b*e^2*x-c*e^2*x^2)^(7/2)/e^2/(-b*e+2*c*d)/(e*x+d)^5+5/8*(-b*e+2*c*d)*(3*b*e*g-10*c*d*g+4*c*e*f)*arctan(1/2*
e*(2*c*x+b)/c^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))*c^(1/2)/e^2+5/4*c*(3*b*e*g-10*c*d*g+4*c*e*f)*(d*(-
b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {806, 676, 678, 635, 210} \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {5 \sqrt {c} (2 c d-b e) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (3 b e g-10 c d g+4 c e f)}{8 e^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (3 b e g-10 c d g+4 c e f)}{3 e^2 (d+e x)^3 (2 c d-b e)}+\frac {5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (3 b e g-10 c d g+4 c e f)}{6 e^2 (d+e x) (2 c d-b e)}+\frac {5 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-10 c d g+4 c e f)}{4 e^2} \]

[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)^5,x]

[Out]

(5*c*(4*c*e*f - 10*c*d*g + 3*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2) + (5*c*(4*c*e*f - 10*c*
d*g + 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(6*e^2*(2*c*d - b*e)*(d + e*x)) + (2*(4*c*e*f - 10
*c*d*g + 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^3) - (2*(e*f - d
*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^5) + (5*Sqrt[c]*(2*c*d - b*e)*
(4*c*e*f - 10*c*d*g + 3*b*e*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])])/
(8*e^2)

Rule 210

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

Rule 635

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 676

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 678

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 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 {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}-\frac {(4 c e f-10 c d g+3 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)^4} \, dx}{3 e (2 c d-b e)} \\ & = \frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (4 c e f-10 c d g+3 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)^2} \, dx}{3 e (2 c d-b e)} \\ & = \frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (4 c e f-10 c d g+3 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx}{4 e} \\ & = \frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (2 c d-b e) (4 c e f-10 c d g+3 b e g)) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e} \\ & = \frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {(5 c (2 c d-b e) (4 c e f-10 c d g+3 b e g)) \text {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 )}{4 e} \\ & = \frac {5 c (4 c e f-10 c d g+3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac {5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac {2 (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac {5 \sqrt {c} (2 c d-b e) (4 c e f-10 c d g+3 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 )}{8 e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.24 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.76 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\frac {((d+e x) (-b e+c (d-e x)))^{5/2} \left (-\sqrt {-b e+c (d-e x)} \left (b c e \left (-147 d^2 g+d e (24 f-206 g x)+e^2 x (56 f-27 g x)\right )+8 b^2 e^2 (2 d g+e (f+3 g x))+2 c^2 \left (118 d^3 g-23 d^2 e (2 f-7 g x)-3 e^3 x^2 (2 f+g x)+4 d e^2 x (-17 f+6 g x)\right )\right )+15 \sqrt {-c} (2 c d-b e) (4 c e f-10 c d g+3 b e g) (d+e x)^{3/2} \log \left (-\sqrt {-c} \sqrt {d+e x}+\sqrt {c d-b e-c e x}\right )\right )}{12 e^2 (d+e x)^4 (-b e+c (d-e x))^{5/2}} \]

[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)^5,x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(-(Sqrt[-(b*e) + c*(d - e*x)]*(b*c*e*(-147*d^2*g + d*e*(24*f - 206*g
*x) + e^2*x*(56*f - 27*g*x)) + 8*b^2*e^2*(2*d*g + e*(f + 3*g*x)) + 2*c^2*(118*d^3*g - 23*d^2*e*(2*f - 7*g*x) -
3*e^3*x^2*(2*f + g*x) + 4*d*e^2*x*(-17*f + 6*g*x)))) + 15*Sqrt[-c]*(2*c*d - b*e)*(4*c*e*f - 10*c*d*g + 3*b*e*
g)*(d + e*x)^(3/2)*Log[-(Sqrt[-c]*Sqrt[d + e*x]) + Sqrt[c*d - b*e - c*e*x]]))/(12*e^2*(d + e*x)^4*(-(b*e) + c*
(d - e*x))^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1254\) vs. \(2(324)=648\).

Time = 2.27 (sec) , antiderivative size = 1255, normalized size of antiderivative = 3.59

method	result	size
default	\(\text {Expression too large to display}\)	\(1255\)

[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)^5,x,method=_RETURNVERBOSE)

[Out]

g/e^5*(-2/(-b*e^2+2*c*d*e)/(x+d/e)^4*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)-6*c*e^2/(-b*e^2+2*c*d*e
)*(2/(-b*e^2+2*c*d*e)/(x+d/e)^3*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)+8*c*e^2/(-b*e^2+2*c*d*e)*(2/
3/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)+10/3*c*e^2/(-b*e^2+2*c*d*e)*(1/
5*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)+1/2*(-b*e^2+2*c*d*e)*(-1/8*(-2*c*e^2*(x+d/e)-b*e^2+2*c*d*e
)/c/e^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)+3/16*(-b*e^2+2*c*d*e)^2/c/e^2*(-1/4*(-2*c*e^2*(x+d/e
)-b*e^2+2*c*d*e)/c/e^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+1/8*(-b*e^2+2*c*d*e)^2/c/e^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)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2
))))))))+(-d*g+e*f)/e^6*(-2/3/(-b*e^2+2*c*d*e)/(x+d/e)^5*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)-4/3
*c*e^2/(-b*e^2+2*c*d*e)*(-2/(-b*e^2+2*c*d*e)/(x+d/e)^4*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)-6*c*e
^2/(-b*e^2+2*c*d*e)*(2/(-b*e^2+2*c*d*e)/(x+d/e)^3*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)+8*c*e^2/(-
b*e^2+2*c*d*e)*(2/3/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)+10/3*c*e^2/(-
b*e^2+2*c*d*e)*(1/5*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)+1/2*(-b*e^2+2*c*d*e)*(-1/8*(-2*c*e^2*(x+
d/e)-b*e^2+2*c*d*e)/c/e^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)+3/16*(-b*e^2+2*c*d*e)^2/c/e^2*(-1/
4*(-2*c*e^2*(x+d/e)-b*e^2+2*c*d*e)/c/e^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+1/8*(-b*e^2+2*c*d*e
)^2/c/e^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)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*
d*e)*(x+d/e))^(1/2)))))))))

Fricas [A] (verification not implemented)

none

Time = 2.40 (sec) , antiderivative size = 951, normalized size of antiderivative = 2.72 \[ \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)^5} \, dx=\left [-\frac {15 \, {\left ({\left (4 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (20 \, c^{2} d^{2} e^{2} - 16 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} g\right )} x^{2} + 4 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (20 \, c^{2} d^{4} - 16 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (4 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (20 \, c^{2} d^{3} e - 16 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} g\right )} x\right )} \sqrt {-c} \log \left (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} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (6 \, c^{2} e^{3} g x^{3} + 3 \, {\left (4 \, c^{2} e^{3} f - {\left (16 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} g\right )} x^{2} + 4 \, {\left (23 \, c^{2} d^{2} e - 6 \, b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (236 \, c^{2} d^{3} - 147 \, b c d^{2} e + 16 \, b^{2} d e^{2}\right )} g + 2 \, {\left (4 \, {\left (17 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (161 \, c^{2} d^{2} e - 103 \, b c d e^{2} + 12 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{48 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, -\frac {15 \, {\left ({\left (4 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (20 \, c^{2} d^{2} e^{2} - 16 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} g\right )} x^{2} + 4 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (20 \, c^{2} d^{4} - 16 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (4 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (20 \, c^{2} d^{3} e - 16 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} g\right )} x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, {\left (6 \, c^{2} e^{3} g x^{3} + 3 \, {\left (4 \, c^{2} e^{3} f - {\left (16 \, c^{2} d e^{2} - 9 \, b c e^{3}\right )} g\right )} x^{2} + 4 \, {\left (23 \, c^{2} d^{2} e - 6 \, b c d e^{2} - 2 \, b^{2} e^{3}\right )} f - {\left (236 \, c^{2} d^{3} - 147 \, b c d^{2} e + 16 \, b^{2} d e^{2}\right )} g + 2 \, {\left (4 \, {\left (17 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (161 \, c^{2} d^{2} e - 103 \, b c d e^{2} + 12 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{24 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \]

[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)^5,x, algorithm="fricas")

[Out]

[-1/48*(15*((4*(2*c^2*d*e^3 - b*c*e^4)*f - (20*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*g)*x^2 + 4*(2*c^2*d^3*e
- b*c*d^2*e^2)*f - (20*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2)*g + 2*(4*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (20*c
^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*g)*x)*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*(6*c^2*e^3*g*x^3 + 3*
(4*c^2*e^3*f - (16*c^2*d*e^2 - 9*b*c*e^3)*g)*x^2 + 4*(23*c^2*d^2*e - 6*b*c*d*e^2 - 2*b^2*e^3)*f - (236*c^2*d^3
- 147*b*c*d^2*e + 16*b^2*d*e^2)*g + 2*(4*(17*c^2*d*e^2 - 7*b*c*e^3)*f - (161*c^2*d^2*e - 103*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))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2), -1/24*(15*((4*(2*c^
2*d*e^3 - b*c*e^4)*f - (20*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*g)*x^2 + 4*(2*c^2*d^3*e - b*c*d^2*e^2)*f -
(20*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2)*g + 2*(4*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (20*c^2*d^3*e - 16*b*c*d^
2*e^2 + 3*b^2*d*e^3)*g)*x)*sqrt(c)*arctan(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*(6*c^2*e^3*g*x^3 + 3*(4*c^2*e^3*f - (16*c^2*d*e^2 - 9*b*
c*e^3)*g)*x^2 + 4*(23*c^2*d^2*e - 6*b*c*d*e^2 - 2*b^2*e^3)*f - (236*c^2*d^3 - 147*b*c*d^2*e + 16*b^2*d*e^2)*g
+ 2*(4*(17*c^2*d*e^2 - 7*b*c*e^3)*f - (161*c^2*d^2*e - 103*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))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)]

Sympy [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{5}}\, dx \]

[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)**5,x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \]

[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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` f
or more deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1076 vs. \(2 (324) = 648\).

Time = 0.69 (sec) , antiderivative size = 1076, normalized size of antiderivative = 3.07 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\text {Too large to display} \]

[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)^5,x, algorithm="giac")

[Out]

-1/12*(15*(8*c^3*d*e*f*sgn(1/(e*x + d))*sgn(e) - 4*b*c^2*e^2*f*sgn(1/(e*x + d))*sgn(e) - 20*c^3*d^2*g*sgn(1/(e
*x + d))*sgn(e) + 16*b*c^2*d*e*g*sgn(1/(e*x + d))*sgn(e) - 3*b^2*c*e^2*g*sgn(1/(e*x + d))*sgn(e))*arctan(sqrt(
-c + 2*c*d/(e*x + d) - b*e/(e*x + d))/sqrt(c))/(sqrt(c)*e^3) - 3*(8*c^4*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x +
d))*d*e*f*sgn(1/(e*x + d))*sgn(e) + 8*c^3*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d*e*f*sgn(1/(e*x + d))
*sgn(e) - 4*b*c^3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*e^2*f*sgn(1/(e*x + d))*sgn(e) - 4*b*c^2*(-c + 2*c
*d/(e*x + d) - b*e/(e*x + d))^(3/2)*e^2*f*sgn(1/(e*x + d))*sgn(e) - 36*c^4*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*
x + d))*d^2*g*sgn(1/(e*x + d))*sgn(e) - 44*c^3*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d^2*g*sgn(1/(e*x +
d))*sgn(e) + 32*b*c^3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d*e*g*sgn(1/(e*x + d))*sgn(e) + 40*b*c^2*(-c
+ 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d*e*g*sgn(1/(e*x + d))*sgn(e) - 7*b^2*c^2*sqrt(-c + 2*c*d/(e*x + d)
- b*e/(e*x + d))*e^2*g*sgn(1/(e*x + d))*sgn(e) - 9*b^2*c*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*e^2*g*sg
n(1/(e*x + d))*sgn(e))/((2*c*d/(e*x + d) - b*e/(e*x + d))^2*e^3) - 8*(12*c^2*sqrt(-c + 2*c*d/(e*x + d) - b*e/(
e*x + d))*d*e^7*f*sgn(1/(e*x + d))*sgn(e) - 2*c*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d*e^7*f*sgn(1/(e*
x + d))*sgn(e) - 6*b*c*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*e^8*f*sgn(1/(e*x + d))*sgn(e) + b*(-c + 2*c*
d/(e*x + d) - b*e/(e*x + d))^(3/2)*e^8*f*sgn(1/(e*x + d))*sgn(e) - 24*c^2*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x
+ d))*d^2*e^6*g*sgn(1/(e*x + d))*sgn(e) + 2*c*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d^2*e^6*g*sgn(1/(e
*x + d))*sgn(e) + 18*b*c*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d*e^7*g*sgn(1/(e*x + d))*sgn(e) - b*(-c +
2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d*e^7*g*sgn(1/(e*x + d))*sgn(e) - 3*b^2*sqrt(-c + 2*c*d/(e*x + d) - b*e
/(e*x + d))*e^8*g*sgn(1/(e*x + d))*sgn(e))/e^9)*abs(e)

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^5} \,d x \]

[In]

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

[Out]

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