3.22.99 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^2} \, dx\) [2199]

Optimal. Leaf size=354 \[ \frac {(2 c d-b e)^2 (10 c e f-4 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^2 e}+\frac {(10 c e f-4 c d g-3 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 c e}+\frac {(10 c e f-4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^2}+\frac {(2 c d-b e)^4 (10 c e f-4 c d g-3 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{256 c^{5/2} e^2} \]

[Out]

1/48*(-3*b*e*g-4*c*d*g+10*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c/e+1/15*(-3*b*e*g-4*c*d*g+1
0*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)+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)^2+1/256*(-b*e+2*c*d)^4*(-3*b*e*g-4*c*d*g+10*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^(5/2)/e^2+1/128*(-b*e+2*c*d)^2*(-3*b*e*g-4*c*d*g+10*c*e*f)*(2*
c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c^2/e

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Rubi [A]
time = 0.38, antiderivative size = 354, normalized size of antiderivative = 1.00, 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 = {806, 678, 626, 635, 210} \begin {gather*} \frac {(2 c d-b e)^4 \text {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-4 c d g+10 c e f)}{256 c^{5/2} e^2}+\frac {(b+2 c x) (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g-4 c d g+10 c e f)}{128 c^2 e}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g-4 c d g+10 c e f)}{48 c e}+\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)^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} (-3 b e g-4 c d g+10 c e f)}{15 e^2 (2 c d-b e)} \end {gather*}

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)^2,x]

[Out]

((2*c*d - b*e)^2*(10*c*e*f - 4*c*d*g - 3*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(128*c^
2*e) + ((10*c*e*f - 4*c*d*g - 3*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(48*c*e) + ((1
0*c*e*f - 4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15*e^2*(2*c*d - b*e)) + (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)^2) + ((2*c*d - b*e)^4*(10*c*e*
f - 4*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])])/(256*c^(
5/2)*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 626

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 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 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*} \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)^2} \, dx &=\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)^2}+\frac {(10 c e f-4 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} \, dx}{3 e (2 c d-b e)}\\ &=\frac {(10 c e f-4 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}}{15 e^2 (2 c d-b e)}+\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)^2}-\frac {((-2 c d+b e) (10 c e f-4 c d g-3 b e g)) \int \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{6 e (2 c d-b e)}\\ &=\frac {(10 c e f-4 c d g-3 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}}{48 c e}+\frac {(10 c e f-4 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}}{15 e^2 (2 c d-b e)}+\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)^2}+\frac {\left ((2 c d-b e)^2 (10 c e f-4 c d g-3 b e g)\right ) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{32 c e}\\ &=\frac {(2 c d-b e)^2 (10 c e f-4 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^2 e}+\frac {(10 c e f-4 c d g-3 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}}{48 c e}+\frac {(10 c e f-4 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}}{15 e^2 (2 c d-b e)}+\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)^2}+\frac {\left ((2 c d-b e)^4 (10 c e f-4 c d g-3 b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{256 c^2 e}\\ &=\frac {(2 c d-b e)^2 (10 c e f-4 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^2 e}+\frac {(10 c e f-4 c d g-3 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}}{48 c e}+\frac {(10 c e f-4 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}}{15 e^2 (2 c d-b e)}+\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)^2}+\frac {\left ((2 c d-b e)^4 (10 c e f-4 c d g-3 b e g)\right ) \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 )}{128 c^2 e}\\ &=\frac {(2 c d-b e)^2 (10 c e f-4 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{128 c^2 e}+\frac {(10 c e f-4 c d g-3 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}}{48 c e}+\frac {(10 c e f-4 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}}{15 e^2 (2 c d-b e)}+\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)^2}+\frac {(2 c d-b e)^4 (10 c e f-4 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 )}{256 c^{5/2} e^2}\\ \end {align*}

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Mathematica [A]
time = 0.98, size = 376, normalized size = 1.06 \begin {gather*} \frac {(-2 c d+b e)^4 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {\sqrt {c} \left (-45 b^4 e^4 g+30 b^3 c e^3 (5 e f+8 d g+e g x)-16 c^4 \left (56 d^4 g+20 d e^3 x^2 (4 f+3 g x)-10 d^3 e (8 f+3 g x)-6 e^4 x^3 (5 f+4 g x)-d^2 e^2 x (45 f+32 g x)\right )+8 b c^3 e \left (174 d^3 g+2 e^3 x^2 (85 f+63 g x)-d^2 e (195 f+71 g x)-2 d e^2 x (125 f+82 g x)\right )+4 b^2 c^2 e^2 \left (-199 d^2 g+d e (70 f+32 g x)+e^2 x (295 f+186 g x)\right )\right )}{(-2 c d+b e)^4 (-c d+b e+c e x)^2}-\frac {15 (10 c e f-4 c d g-3 b e g) \tan ^{-1}\left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{\sqrt {d+e x} (-b e+c (d-e x))^{5/2}}\right )}{1920 c^{5/2} e^2} \end {gather*}

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)^2,x]

[Out]

((-2*c*d + b*e)^4*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((Sqrt[c]*(-45*b^4*e^4*g + 3
0*b^3*c*e^3*(5*e*f + 8*d*g + e*g*x) - 16*c^4*(56*d^4*g + 20*d*e^3*x^2*(4*f + 3*g*x) - 10*d^3*e*(8*f + 3*g*x) -
6*e^4*x^3*(5*f + 4*g*x) - d^2*e^2*x*(45*f + 32*g*x)) + 8*b*c^3*e*(174*d^3*g + 2*e^3*x^2*(85*f + 63*g*x) - d^2
*e*(195*f + 71*g*x) - 2*d*e^2*x*(125*f + 82*g*x)) + 4*b^2*c^2*e^2*(-199*d^2*g + d*e*(70*f + 32*g*x) + e^2*x*(2
95*f + 186*g*x))))/((-2*c*d + b*e)^4*(-(c*d) + b*e + c*e*x)^2) - (15*(10*c*e*f - 4*c*d*g - 3*b*e*g)*ArcTan[Sqr
t[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])])/(Sqrt[d + e*x]*(-(b*e) + c*(d - e*x))^(5/2))))/(1920*c^(5/2)*e^
2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(750\) vs. \(2(328)=656\).
time = 0.04, size = 751, normalized size = 2.12

method	result	size
default	\(\frac {g \left (\frac {\left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (-b \,e^{2}+2 c d e \right ) \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right ) \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c \,e^{2}}+\frac {3 \left (-b \,e^{2}+2 c d e \right )^{2} \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{4 c \,e^{2}}+\frac {\left (-b \,e^{2}+2 c d e \right )^{2} \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 c d e}{2 c \,e^{2}}\right )}{\sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{16 c \,e^{2}}\right )}{2}\right )}{e^{2}}+\frac {\left (-d g +e f \right ) \left (\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {10 c \,e^{2} \left (\frac {\left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (-b \,e^{2}+2 c d e \right ) \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right ) \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c \,e^{2}}+\frac {3 \left (-b \,e^{2}+2 c d e \right )^{2} \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{4 c \,e^{2}}+\frac {\left (-b \,e^{2}+2 c d e \right )^{2} \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 c d e}{2 c \,e^{2}}\right )}{\sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{16 c \,e^{2}}\right )}{2}\right )}{3 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{3}}\)	\(751\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

g/e^2*(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^3*(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))))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1775 vs. \(2 (333) = 666\).
time = 0.65, size = 1775, normalized size = 5.01 \begin {gather*} \frac {c^{4} d^{5} g \arcsin \left (\frac {2 \, c x e}{2 \, c d - b e} + \frac {4 \, c d}{2 \, c d - b e} - \frac {b e}{2 \, c d - b e}\right ) e^{\left (-2\right )}}{4 \, \left (-c\right )^{\frac {3}{2}}} - \frac {5 \, c^{4} d^{4} f \arcsin \left (\frac {2 \, c x e}{2 \, c d - b e} + \frac {4 \, c d}{2 \, c d - b e} - \frac {b e}{2 \, c d - b e}\right ) e^{\left (-1\right )}}{8 \, \left (-c\right )^{\frac {3}{2}}} - \frac {5 \, b c^{3} d^{4} g \arcsin \left (\frac {2 \, c x e}{2 \, c d - b e} + \frac {4 \, c d}{2 \, c d - b e} - \frac {b e}{2 \, c d - b e}\right ) e^{\left (-1\right )}}{16 \, \left (-c\right )^{\frac {3}{2}}} - \frac {1}{4} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} c^{2} d^{3} g x e^{\left (-1\right )} - \frac {1}{2} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} c^{2} d^{4} g e^{\left (-2\right )} + \frac {5 \, b c^{3} d^{3} f \arcsin \left (\frac {2 \, c x e}{2 \, c d - b e} + \frac {4 \, c d}{2 \, c d - b e} - \frac {b e}{2 \, c d - b e}\right )}{4 \, \left (-c\right )^{\frac {3}{2}}} - \frac {15 \, b^{2} c^{2} d^{2} f \arcsin \left (\frac {2 \, c x e}{2 \, c d - b e} + \frac {4 \, c d}{2 \, c d - b e} - \frac {b e}{2 \, c d - b e}\right ) e}{16 \, \left (-c\right )^{\frac {3}{2}}} + \frac {5 \, b^{3} c d^{2} g \arcsin \left (\frac {2 \, c x e}{2 \, c d - b e} + \frac {4 \, c d}{2 \, c d - b e} - \frac {b e}{2 \, c d - b e}\right ) e}{32 \, \left (-c\right )^{\frac {3}{2}}} + \frac {5}{4} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} c^{2} d^{3} f e^{\left (-1\right )} + \frac {1}{4} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b c d^{3} g e^{\left (-1\right )} + \frac {5}{8} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} c^{2} d^{2} f x + \frac {1}{16} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b c d^{2} g x + \frac {5 \, b^{3} c d f \arcsin \left (\frac {2 \, c x e}{2 \, c d - b e} + \frac {4 \, c d}{2 \, c d - b e} - \frac {b e}{2 \, c d - b e}\right ) e^{2}}{16 \, \left (-c\right )^{\frac {3}{2}}} - \frac {5 \, b^{4} d g \arcsin \left (\frac {2 \, c x e}{2 \, c d - b e} + \frac {4 \, c d}{2 \, c d - b e} - \frac {b e}{2 \, c d - b e}\right ) e^{2}}{64 \, \left (-c\right )^{\frac {3}{2}}} - \frac {5}{8} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b c d f x e + \frac {1}{8} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b^{2} d g x e + \frac {1}{4} \, {\left (-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e\right )}^{\frac {3}{2}} c d g x e^{\left (-1\right )} - \frac {5}{12} \, {\left (-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e\right )}^{\frac {3}{2}} c d^{2} g e^{\left (-2\right )} - \frac {25}{16} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b c d^{2} f + \frac {7}{32} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b^{2} d^{2} g - \frac {5 \, b^{4} f \arcsin \left (\frac {2 \, c x e}{2 \, c d - b e} + \frac {4 \, c d}{2 \, c d - b e} - \frac {b e}{2 \, c d - b e}\right ) e^{3}}{128 \, \left (-c\right )^{\frac {3}{2}}} + \frac {3 \, b^{5} g \arcsin \left (\frac {2 \, c x e}{2 \, c d - b e} + \frac {4 \, c d}{2 \, c d - b e} - \frac {b e}{2 \, c d - b e}\right ) e^{3}}{256 \, \left (-c\right )^{\frac {3}{2}} c} + \frac {5}{32} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b^{2} f x e^{2} - \frac {3 \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b^{3} g x e^{2}}{64 \, c} + \frac {5}{8} \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b^{2} d f e - \frac {5 \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b^{3} d g e}{32 \, c} + \frac {5}{12} \, {\left (-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e\right )}^{\frac {3}{2}} c d f e^{\left (-1\right )} + \frac {1}{3} \, {\left (-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e\right )}^{\frac {3}{2}} b d g e^{\left (-1\right )} - \frac {1}{8} \, {\left (-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e\right )}^{\frac {3}{2}} b g x - \frac {5 \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b^{3} f e^{2}}{64 \, c} + \frac {3 \, \sqrt {c x^{2} e^{2} + 4 \, c d x e + 3 \, c d^{2} - b x e^{2} - b d e} b^{4} g e^{2}}{128 \, c^{2}} + \frac {1}{5} \, {\left (-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e\right )}^{\frac {5}{2}} g e^{\left (-2\right )} - \frac {5}{24} \, {\left (-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e\right )}^{\frac {3}{2}} b f - \frac {{\left (-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e\right )}^{\frac {3}{2}} b^{2} g}{16 \, c} - \frac {{\left (-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e\right )}^{\frac {5}{2}} d g}{4 \, {\left (x e^{3} + d e^{2}\right )}} + \frac {{\left (-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e\right )}^{\frac {5}{2}} f}{4 \, {\left (x e^{2} + d e\right )}} \end {gather*}

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)^2,x, algorithm="maxima")

[Out]

1/4*c^4*d^5*g*arcsin(2*c*x*e/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))*e^(-2)/(-c)^(3/2) - 5/8*
c^4*d^4*f*arcsin(2*c*x*e/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))*e^(-1)/(-c)^(3/2) - 5/16*b*c
^3*d^4*g*arcsin(2*c*x*e/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))*e^(-1)/(-c)^(3/2) - 1/4*sqrt(
c*x^2*e^2 + 4*c*d*x*e + 3*c*d^2 - b*x*e^2 - b*d*e)*c^2*d^3*g*x*e^(-1) - 1/2*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*c*d
^2 - b*x*e^2 - b*d*e)*c^2*d^4*g*e^(-2) + 5/4*b*c^3*d^3*f*arcsin(2*c*x*e/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) -
b*e/(2*c*d - b*e))/(-c)^(3/2) - 15/16*b^2*c^2*d^2*f*arcsin(2*c*x*e/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(
2*c*d - b*e))*e/(-c)^(3/2) + 5/32*b^3*c*d^2*g*arcsin(2*c*x*e/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d
- b*e))*e/(-c)^(3/2) + 5/4*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*c*d^2 - b*x*e^2 - b*d*e)*c^2*d^3*f*e^(-1) + 1/4*sqrt
(c*x^2*e^2 + 4*c*d*x*e + 3*c*d^2 - b*x*e^2 - b*d*e)*b*c*d^3*g*e^(-1) + 5/8*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*c*d^
2 - b*x*e^2 - b*d*e)*c^2*d^2*f*x + 1/16*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*c*d^2 - b*x*e^2 - b*d*e)*b*c*d^2*g*x +
5/16*b^3*c*d*f*arcsin(2*c*x*e/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))*e^2/(-c)^(3/2) - 5/64*b
^4*d*g*arcsin(2*c*x*e/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))*e^2/(-c)^(3/2) - 5/8*sqrt(c*x^2
*e^2 + 4*c*d*x*e + 3*c*d^2 - b*x*e^2 - b*d*e)*b*c*d*f*x*e + 1/8*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*c*d^2 - b*x*e^2
- b*d*e)*b^2*d*g*x*e + 1/4*(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)^(3/2)*c*d*g*x*e^(-1) - 5/12*(-c*x^2*e^2 + c
*d^2 - b*x*e^2 - b*d*e)^(3/2)*c*d^2*g*e^(-2) - 25/16*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*c*d^2 - b*x*e^2 - b*d*e)*b
*c*d^2*f + 7/32*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*c*d^2 - b*x*e^2 - b*d*e)*b^2*d^2*g - 5/128*b^4*f*arcsin(2*c*x*e
/(2*c*d - b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))*e^3/(-c)^(3/2) + 3/256*b^5*g*arcsin(2*c*x*e/(2*c*d -
b*e) + 4*c*d/(2*c*d - b*e) - b*e/(2*c*d - b*e))*e^3/((-c)^(3/2)*c) + 5/32*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*c*d^
2 - b*x*e^2 - b*d*e)*b^2*f*x*e^2 - 3/64*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*c*d^2 - b*x*e^2 - b*d*e)*b^3*g*x*e^2/c
+ 5/8*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*c*d^2 - b*x*e^2 - b*d*e)*b^2*d*f*e - 5/32*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*
c*d^2 - b*x*e^2 - b*d*e)*b^3*d*g*e/c + 5/12*(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)^(3/2)*c*d*f*e^(-1) + 1/3*(-
c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)^(3/2)*b*d*g*e^(-1) - 1/8*(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)^(3/2)*b*g
*x - 5/64*sqrt(c*x^2*e^2 + 4*c*d*x*e + 3*c*d^2 - b*x*e^2 - b*d*e)*b^3*f*e^2/c + 3/128*sqrt(c*x^2*e^2 + 4*c*d*x
*e + 3*c*d^2 - b*x*e^2 - b*d*e)*b^4*g*e^2/c^2 + 1/5*(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)^(5/2)*g*e^(-2) - 5/
24*(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)^(3/2)*b*f - 1/16*(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)^(3/2)*b^2*g/
c - 1/4*(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)^(5/2)*d*g/(x*e^3 + d*e^2) + 1/4*(-c*x^2*e^2 + c*d^2 - b*x*e^2 -
b*d*e)^(5/2)*f/(x*e^2 + d*e)

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Fricas [A]
time = 2.38, size = 1024, normalized size = 2.89 \begin {gather*} \left [-\frac {{\left (15 \, {\left (64 \, c^{5} d^{5} g + 320 \, b c^{4} d^{3} f e^{2} - {\left (10 \, b^{4} c f - 3 \, b^{5} g\right )} e^{5} + 20 \, {\left (4 \, b^{3} c^{2} d f - b^{4} c d g\right )} e^{4} - 40 \, {\left (6 \, b^{2} c^{3} d^{2} f - b^{3} c^{2} d^{2} g\right )} e^{3} - 80 \, {\left (2 \, c^{5} d^{4} f + b c^{4} d^{4} g\right )} e\right )} \sqrt {-c} \log \left (-4 \, c^{2} d^{2} + 4 \, b c d e + 4 \, \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} {\left (2 \, c x + b\right )} \sqrt {-c} e + {\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2}\right )} e^{2}\right ) + 4 \, {\left (896 \, c^{5} d^{4} g - {\left (384 \, c^{5} g x^{4} + 150 \, b^{3} c^{2} f - 45 \, b^{4} c g + 48 \, {\left (10 \, c^{5} f + 21 \, b c^{4} g\right )} x^{3} + 8 \, {\left (170 \, b c^{4} f + 93 \, b^{2} c^{3} g\right )} x^{2} + 10 \, {\left (118 \, b^{2} c^{3} f + 3 \, b^{3} c^{2} g\right )} x\right )} e^{4} + 8 \, {\left (120 \, c^{5} d g x^{3} - 35 \, b^{2} c^{3} d f - 30 \, b^{3} c^{2} d g + 4 \, {\left (40 \, c^{5} d f + 41 \, b c^{4} d g\right )} x^{2} + 2 \, {\left (125 \, b c^{4} d f - 8 \, b^{2} c^{3} d g\right )} x\right )} e^{3} - 4 \, {\left (128 \, c^{5} d^{2} g x^{2} - 390 \, b c^{4} d^{2} f - 199 \, b^{2} c^{3} d^{2} g + 2 \, {\left (90 \, c^{5} d^{2} f - 71 \, b c^{4} d^{2} g\right )} x\right )} e^{2} - 16 \, {\left (30 \, c^{5} d^{3} g x + 80 \, c^{5} d^{3} f + 87 \, b c^{4} d^{3} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}\right )} e^{\left (-2\right )}}{7680 \, c^{3}}, \frac {{\left (15 \, {\left (64 \, c^{5} d^{5} g + 320 \, b c^{4} d^{3} f e^{2} - {\left (10 \, b^{4} c f - 3 \, b^{5} g\right )} e^{5} + 20 \, {\left (4 \, b^{3} c^{2} d f - b^{4} c d g\right )} e^{4} - 40 \, {\left (6 \, b^{2} c^{3} d^{2} f - b^{3} c^{2} d^{2} g\right )} e^{3} - 80 \, {\left (2 \, c^{5} d^{4} f + b c^{4} d^{4} g\right )} e\right )} \sqrt {c} \arctan \left (-\frac {\sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} {\left (2 \, c x + b\right )} \sqrt {c} e}{2 \, {\left (c^{2} d^{2} - b c d e - {\left (c^{2} x^{2} + b c x\right )} e^{2}\right )}}\right ) - 2 \, {\left (896 \, c^{5} d^{4} g - {\left (384 \, c^{5} g x^{4} + 150 \, b^{3} c^{2} f - 45 \, b^{4} c g + 48 \, {\left (10 \, c^{5} f + 21 \, b c^{4} g\right )} x^{3} + 8 \, {\left (170 \, b c^{4} f + 93 \, b^{2} c^{3} g\right )} x^{2} + 10 \, {\left (118 \, b^{2} c^{3} f + 3 \, b^{3} c^{2} g\right )} x\right )} e^{4} + 8 \, {\left (120 \, c^{5} d g x^{3} - 35 \, b^{2} c^{3} d f - 30 \, b^{3} c^{2} d g + 4 \, {\left (40 \, c^{5} d f + 41 \, b c^{4} d g\right )} x^{2} + 2 \, {\left (125 \, b c^{4} d f - 8 \, b^{2} c^{3} d g\right )} x\right )} e^{3} - 4 \, {\left (128 \, c^{5} d^{2} g x^{2} - 390 \, b c^{4} d^{2} f - 199 \, b^{2} c^{3} d^{2} g + 2 \, {\left (90 \, c^{5} d^{2} f - 71 \, b c^{4} d^{2} g\right )} x\right )} e^{2} - 16 \, {\left (30 \, c^{5} d^{3} g x + 80 \, c^{5} d^{3} f + 87 \, b c^{4} d^{3} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}\right )} e^{\left (-2\right )}}{3840 \, c^{3}}\right ] \end {gather*}

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)^2,x, algorithm="fricas")

[Out]

[-1/7680*(15*(64*c^5*d^5*g + 320*b*c^4*d^3*f*e^2 - (10*b^4*c*f - 3*b^5*g)*e^5 + 20*(4*b^3*c^2*d*f - b^4*c*d*g)
*e^4 - 40*(6*b^2*c^3*d^2*f - b^3*c^2*d^2*g)*e^3 - 80*(2*c^5*d^4*f + b*c^4*d^4*g)*e)*sqrt(-c)*log(-4*c^2*d^2 +
4*b*c*d*e + 4*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*(2*c*x + b)*sqrt(-c)*e + (8*c^2*x^2 + 8*b*c*x + b^2)*e^2
) + 4*(896*c^5*d^4*g - (384*c^5*g*x^4 + 150*b^3*c^2*f - 45*b^4*c*g + 48*(10*c^5*f + 21*b*c^4*g)*x^3 + 8*(170*b
*c^4*f + 93*b^2*c^3*g)*x^2 + 10*(118*b^2*c^3*f + 3*b^3*c^2*g)*x)*e^4 + 8*(120*c^5*d*g*x^3 - 35*b^2*c^3*d*f - 3
0*b^3*c^2*d*g + 4*(40*c^5*d*f + 41*b*c^4*d*g)*x^2 + 2*(125*b*c^4*d*f - 8*b^2*c^3*d*g)*x)*e^3 - 4*(128*c^5*d^2*
g*x^2 - 390*b*c^4*d^2*f - 199*b^2*c^3*d^2*g + 2*(90*c^5*d^2*f - 71*b*c^4*d^2*g)*x)*e^2 - 16*(30*c^5*d^3*g*x +
80*c^5*d^3*f + 87*b*c^4*d^3*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2))*e^(-2)/c^3, 1/3840*(15*(64*c^5*d^5*
g + 320*b*c^4*d^3*f*e^2 - (10*b^4*c*f - 3*b^5*g)*e^5 + 20*(4*b^3*c^2*d*f - b^4*c*d*g)*e^4 - 40*(6*b^2*c^3*d^2*
f - b^3*c^2*d^2*g)*e^3 - 80*(2*c^5*d^4*f + b*c^4*d^4*g)*e)*sqrt(c)*arctan(-1/2*sqrt(c*d^2 - b*d*e - (c*x^2 + b
*x)*e^2)*(2*c*x + b)*sqrt(c)*e/(c^2*d^2 - b*c*d*e - (c^2*x^2 + b*c*x)*e^2)) - 2*(896*c^5*d^4*g - (384*c^5*g*x^
4 + 150*b^3*c^2*f - 45*b^4*c*g + 48*(10*c^5*f + 21*b*c^4*g)*x^3 + 8*(170*b*c^4*f + 93*b^2*c^3*g)*x^2 + 10*(118
*b^2*c^3*f + 3*b^3*c^2*g)*x)*e^4 + 8*(120*c^5*d*g*x^3 - 35*b^2*c^3*d*f - 30*b^3*c^2*d*g + 4*(40*c^5*d*f + 41*b
*c^4*d*g)*x^2 + 2*(125*b*c^4*d*f - 8*b^2*c^3*d*g)*x)*e^3 - 4*(128*c^5*d^2*g*x^2 - 390*b*c^4*d^2*f - 199*b^2*c^
3*d^2*g + 2*(90*c^5*d^2*f - 71*b*c^4*d^2*g)*x)*e^2 - 16*(30*c^5*d^3*g*x + 80*c^5*d^3*f + 87*b*c^4*d^3*g)*e)*sq
rt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2))*e^(-2)/c^3]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )^{2}}\, dx \end {gather*}

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)**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3788 vs. \(2 (333) = 666\).
time = 2.51, size = 3788, normalized size = 10.70 \begin {gather*} \text {Too large to display} \end {gather*}

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)^2,x, algorithm="giac")

[Out]

1/1920*(15*(64*c^5*d^5*g*sgn(1/(x*e + d)) - 160*c^5*d^4*f*e*sgn(1/(x*e + d)) - 80*b*c^4*d^4*g*e*sgn(1/(x*e + d
)) + 320*b*c^4*d^3*f*e^2*sgn(1/(x*e + d)) - 240*b^2*c^3*d^2*f*e^3*sgn(1/(x*e + d)) + 40*b^3*c^2*d^2*g*e^3*sgn(
1/(x*e + d)) + 80*b^3*c^2*d*f*e^4*sgn(1/(x*e + d)) - 20*b^4*c*d*g*e^4*sgn(1/(x*e + d)) - 10*b^4*c*f*e^5*sgn(1/
(x*e + d)) + 3*b^5*g*e^5*sgn(1/(x*e + d)))*arctan(sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))/sqrt(c))*e^(-3)/c
^(5/2) + (960*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^4*c^5*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^5*g*sgn
(1/(x*e + d)) + 16000*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^6*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d
^5*g*sgn(1/(x*e + d)) - 8192*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^7*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e
+ d))*d^5*g*sgn(1/(x*e + d)) - 960*c^9*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^5*g*sgn(1/(x*e + d)) - 448
0*c^8*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^5*g*sgn(1/(x*e + d)) - 2400*(c - 2*c*d/(x*e + d) + b*e/(x
*e + d))^4*c^5*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^4*f*e*sgn(1/(x*e + d)) - 9280*(c - 2*c*d/(x*e + d)
+ b*e/(x*e + d))^3*c^6*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^4*f*e*sgn(1/(x*e + d)) + 20480*(c - 2*c*d
/(x*e + d) + b*e/(x*e + d))^2*c^7*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^4*f*e*sgn(1/(x*e + d)) + 2400*c
^9*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^4*f*e*sgn(1/(x*e + d)) + 11200*c^8*(-c + 2*c*d/(x*e + d) - b*e
/(x*e + d))^(3/2)*d^4*f*e*sgn(1/(x*e + d)) - 1200*b*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^4*c^4*sqrt(-c + 2*c*
d/(x*e + d) - b*e/(x*e + d))*d^4*g*e*sgn(1/(x*e + d)) - 35360*b*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^5*sq
rt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^4*g*e*sgn(1/(x*e + d)) + 10240*b*(c - 2*c*d/(x*e + d) + b*e/(x*e +
d))^2*c^6*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^4*g*e*sgn(1/(x*e + d)) + 1200*b*c^8*sqrt(-c + 2*c*d/(x*
e + d) - b*e/(x*e + d))*d^4*g*e*sgn(1/(x*e + d)) + 5600*b*c^7*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^4
*g*e*sgn(1/(x*e + d)) + 4800*b*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^4*c^4*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*
e + d))*d^3*f*e^2*sgn(1/(x*e + d)) + 18560*b*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^5*sqrt(-c + 2*c*d/(x*e
+ d) - b*e/(x*e + d))*d^3*f*e^2*sgn(1/(x*e + d)) - 40960*b*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^6*sqrt(-c
+ 2*c*d/(x*e + d) - b*e/(x*e + d))*d^3*f*e^2*sgn(1/(x*e + d)) - 4800*b*c^8*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x
*e + d))*d^3*f*e^2*sgn(1/(x*e + d)) - 22400*b*c^7*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^3*f*e^2*sgn(1
/(x*e + d)) + 30720*b^2*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^4*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))
*d^3*g*e^2*sgn(1/(x*e + d)) - 3600*b^2*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^4*c^3*sqrt(-c + 2*c*d/(x*e + d) -
b*e/(x*e + d))*d^2*f*e^3*sgn(1/(x*e + d)) - 13920*b^2*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^4*sqrt(-c + 2
*c*d/(x*e + d) - b*e/(x*e + d))*d^2*f*e^3*sgn(1/(x*e + d)) + 30720*b^2*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2
*c^5*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^2*f*e^3*sgn(1/(x*e + d)) + 3600*b^2*c^7*sqrt(-c + 2*c*d/(x*e
+ d) - b*e/(x*e + d))*d^2*f*e^3*sgn(1/(x*e + d)) + 16800*b^2*c^6*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)
*d^2*f*e^3*sgn(1/(x*e + d)) + 600*b^3*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^4*c^2*sqrt(-c + 2*c*d/(x*e + d) -
b*e/(x*e + d))*d^2*g*e^3*sgn(1/(x*e + d)) - 13040*b^3*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^3*sqrt(-c + 2*
c*d/(x*e + d) - b*e/(x*e + d))*d^2*g*e^3*sgn(1/(x*e + d)) - 5120*b^3*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c
^4*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^2*g*e^3*sgn(1/(x*e + d)) - 600*b^3*c^6*sqrt(-c + 2*c*d/(x*e +
d) - b*e/(x*e + d))*d^2*g*e^3*sgn(1/(x*e + d)) - 2800*b^3*c^5*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^2
*g*e^3*sgn(1/(x*e + d)) + 1200*b^3*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^4*c^2*sqrt(-c + 2*c*d/(x*e + d) - b*e
/(x*e + d))*d*f*e^4*sgn(1/(x*e + d)) + 4640*b^3*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^3*sqrt(-c + 2*c*d/(x
*e + d) - b*e/(x*e + d))*d*f*e^4*sgn(1/(x*e + d)) - 10240*b^3*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^4*sqrt
(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d*f*e^4*sgn(1/(x*e + d)) - 1200*b^3*c^6*sqrt(-c + 2*c*d/(x*e + d) - b*e
/(x*e + d))*d*f*e^4*sgn(1/(x*e + d)) - 5600*b^3*c^5*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d*f*e^4*sgn(1
/(x*e + d)) - 300*b^4*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^4*c*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d*g
*e^4*sgn(1/(x*e + d)) + 2680*b^4*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^3*c^2*sqrt(-c + 2*c*d/(x*e + d) - b*e/(
x*e + d))*d*g*e^4*sgn(1/(x*e + d)) + 2560*b^4*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^3*sqrt(-c + 2*c*d/(x*e
+ d) - b*e/(x*e + d))*d*g*e^4*sgn(1/(x*e + d)) + 300*b^4*c^5*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d*g*e
^4*sgn(1/(x*e + d)) + 1400*b^4*c^4*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d*g*e^4*sgn(1/(x*e + d)) - 150
*b^4*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^4*c*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*f*e^5*sgn(1/(x*e + d
)) - 580*b^4*(c - 2*c*d/(x*e + d) + b*e/(x*e + ...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)^2,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)^2, x)

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