3.23.21 \(\int \frac {f+g x}{(d+e x)^2 (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\) [2221]

Optimal. Leaf size=209 \[ \frac {8 c (6 c e f+4 c d g-5 b e g) (b+2 c x)}{15 e (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (6 c e f+4 c d g-5 b e g)}{15 e^2 (2 c d-b e)^2 (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

8/15*c*(-5*b*e*g+4*c*d*g+6*c*e*f)*(2*c*x+b)/e/(-b*e+2*c*d)^4/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-2/5*(-d*g+
e*f)/e^2/(-b*e+2*c*d)/(e*x+d)^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-2/15*(-5*b*e*g+4*c*d*g+6*c*e*f)/e^2/(-b
*e+2*c*d)^2/(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)

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Rubi [A]
time = 0.17, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {806, 672, 627} \begin {gather*} -\frac {2 (e f-d g)}{5 e^2 (d+e x)^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {8 c (b+2 c x) (-5 b e g+4 c d g+6 c e f)}{15 e (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (-5 b e g+4 c d g+6 c e f)}{15 e^2 (d+e x) (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(f + g*x)/((d + e*x)^2*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]

[Out]

(8*c*(6*c*e*f + 4*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^4*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2
]) - (2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(6*c*e*f
+ 4*c*d*g - 5*b*e*g))/(15*e^2*(2*c*d - b*e)^2*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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*(Simplify[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, m, p}, x] && NeQ[b^2 - 4*a
*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]

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}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=-\frac {2 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(6 c e f+4 c d g-5 b e g) \int \frac {1}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{5 e (2 c d-b e)}\\ &=-\frac {2 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (6 c e f+4 c d g-5 b e g)}{15 e^2 (2 c d-b e)^2 (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(4 c (6 c e f+4 c d g-5 b e g)) \int \frac {1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{15 e (2 c d-b e)^2}\\ &=\frac {8 c (6 c e f+4 c d g-5 b e g) (b+2 c x)}{15 e (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (6 c e f+4 c d g-5 b e g)}{15 e^2 (2 c d-b e)^2 (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 233, normalized size = 1.11 \begin {gather*} \frac {2 \left (b^3 e^3 (3 e f+2 d g+5 e g x)+4 b c^2 e \left (4 d^3 g+2 d e^2 x (9 f-8 g x)+2 e^3 x^2 (3 f-5 g x)+7 d^2 e (3 f+g x)\right )+8 c^3 \left (d^4 g+6 e^4 f x^3+4 d e^3 x^2 (3 f+g x)+d^3 e (-6 f+2 g x)+d^2 e^2 x (3 f+8 g x)\right )-2 b^2 c e^2 \left (13 d^2 g+4 d e (3 f+8 g x)+e^2 x (3 f+10 g x)\right )\right )}{15 e^2 (-2 c d+b e)^4 (d+e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)/((d + e*x)^2*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]

[Out]

(2*(b^3*e^3*(3*e*f + 2*d*g + 5*e*g*x) + 4*b*c^2*e*(4*d^3*g + 2*d*e^2*x*(9*f - 8*g*x) + 2*e^3*x^2*(3*f - 5*g*x)
+ 7*d^2*e*(3*f + g*x)) + 8*c^3*(d^4*g + 6*e^4*f*x^3 + 4*d*e^3*x^2*(3*f + g*x) + d^3*e*(-6*f + 2*g*x) + d^2*e^
2*x*(3*f + 8*g*x)) - 2*b^2*c*e^2*(13*d^2*g + 4*d*e*(3*f + 8*g*x) + e^2*x*(3*f + 10*g*x))))/(15*e^2*(-2*c*d + b
*e)^4*(d + e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(396\) vs. \(2(197)=394\).
time = 0.06, size = 397, normalized size = 1.90

method	result	size
trager	\(-\frac {2 \left (-40 b \,c^{2} e^{4} g \,x^{3}+32 c^{3} d \,e^{3} g \,x^{3}+48 c^{3} e^{4} f \,x^{3}-20 b^{2} c \,e^{4} g \,x^{2}-64 b \,c^{2} d \,e^{3} g \,x^{2}+24 b \,c^{2} e^{4} f \,x^{2}+64 c^{3} d^{2} e^{2} g \,x^{2}+96 c^{3} d \,e^{3} f \,x^{2}+5 b^{3} e^{4} g x -64 b^{2} c d \,e^{3} g x -6 b^{2} c \,e^{4} f x +28 b \,c^{2} d^{2} e^{2} g x +72 b \,c^{2} d \,e^{3} f x +16 c^{3} d^{3} e g x +24 c^{3} d^{2} e^{2} f x +2 b^{3} d \,e^{3} g +3 b^{3} e^{4} f -26 b^{2} c \,d^{2} e^{2} g -24 b^{2} c d \,e^{3} f +16 b \,c^{2} d^{3} e g +84 b \,c^{2} d^{2} e^{2} f +8 c^{3} d^{4} g -48 c^{3} d^{3} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{15 \left (c e x +b e -c d \right ) \left (b e -2 c d \right ) \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) e^{2} \left (e x +d \right )^{3}}\)	\(380\)
gosper	\(-\frac {2 \left (c e x +b e -c d \right ) \left (-40 b \,c^{2} e^{4} g \,x^{3}+32 c^{3} d \,e^{3} g \,x^{3}+48 c^{3} e^{4} f \,x^{3}-20 b^{2} c \,e^{4} g \,x^{2}-64 b \,c^{2} d \,e^{3} g \,x^{2}+24 b \,c^{2} e^{4} f \,x^{2}+64 c^{3} d^{2} e^{2} g \,x^{2}+96 c^{3} d \,e^{3} f \,x^{2}+5 b^{3} e^{4} g x -64 b^{2} c d \,e^{3} g x -6 b^{2} c \,e^{4} f x +28 b \,c^{2} d^{2} e^{2} g x +72 b \,c^{2} d \,e^{3} f x +16 c^{3} d^{3} e g x +24 c^{3} d^{2} e^{2} f x +2 b^{3} d \,e^{3} g +3 b^{3} e^{4} f -26 b^{2} c \,d^{2} e^{2} g -24 b^{2} c d \,e^{3} f +16 b \,c^{2} d^{3} e g +84 b \,c^{2} d^{2} e^{2} f +8 c^{3} d^{4} g -48 c^{3} d^{3} e f \right )}{15 \left (e x +d \right ) e^{2} \left (b^{4} e^{4}-8 b^{3} c d \,e^{3}+24 b^{2} c^{2} d^{2} e^{2}-32 b \,c^{3} d^{3} e +16 c^{4} d^{4}\right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}\)	\(382\)
default	\(\frac {g \left (-\frac {2}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}-\frac {8 c \,e^{2} \left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right )}{3 \left (-b \,e^{2}+2 c d e \right )^{3} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{e^{2}}+\frac {\left (-d g +e f \right ) \left (-\frac {2}{5 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}+\frac {6 c \,e^{2} \left (-\frac {2}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}-\frac {8 c \,e^{2} \left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right )}{3 \left (-b \,e^{2}+2 c d e \right )^{3} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{3}}\)	\(397\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

g/e^2*(-2/3/(-b*e^2+2*c*d*e)/(x+d/e)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)-8/3*c*e^2/(-b*e^2+2*c*d
*e)^3*(-2*c*e^2*(x+d/e)-b*e^2+2*c*d*e)/(-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*(-2/5
/(-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))^(1/2)+6/5*c*e^2/(-b*e^2+2*c*d*e)*(-2/3
/(-b*e^2+2*c*d*e)/(x+d/e)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)-8/3*c*e^2/(-b*e^2+2*c*d*e)^3*(-2*c
*e^2*(x+d/e)-b*e^2+2*c*d*e)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),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(2*c*d-%e*b>0)', see `assume?`
for more det

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (201) = 402\).
time = 66.97, size = 614, normalized size = 2.94 \begin {gather*} \frac {2 \, {\left (8 \, c^{3} d^{4} g + {\left (3 \, b^{3} f + 8 \, {\left (6 \, c^{3} f - 5 \, b c^{2} g\right )} x^{3} + 4 \, {\left (6 \, b c^{2} f - 5 \, b^{2} c g\right )} x^{2} - {\left (6 \, b^{2} c f - 5 \, b^{3} g\right )} x\right )} e^{4} + 2 \, {\left (16 \, c^{3} d g x^{3} - 12 \, b^{2} c d f + b^{3} d g + 16 \, {\left (3 \, c^{3} d f - 2 \, b c^{2} d g\right )} x^{2} + 4 \, {\left (9 \, b c^{2} d f - 8 \, b^{2} c d g\right )} x\right )} e^{3} + 2 \, {\left (32 \, c^{3} d^{2} g x^{2} + 42 \, b c^{2} d^{2} f - 13 \, b^{2} c d^{2} g + 2 \, {\left (6 \, c^{3} d^{2} f + 7 \, b c^{2} d^{2} g\right )} x\right )} e^{2} + 16 \, {\left (c^{3} d^{3} g x - 3 \, c^{3} d^{3} f + b c^{2} d^{3} g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}}{15 \, {\left (16 \, c^{5} d^{8} e^{2} - {\left (b^{4} c x^{4} + b^{5} x^{3}\right )} e^{10} + {\left (8 \, b^{3} c^{2} d x^{4} + 6 \, b^{4} c d x^{3} - 3 \, b^{5} d x^{2}\right )} e^{9} - {\left (24 \, b^{2} c^{3} d^{2} x^{4} + 8 \, b^{3} c^{2} d^{2} x^{3} - 24 \, b^{4} c d^{2} x^{2} + 3 \, b^{5} d^{2} x\right )} e^{8} + {\left (32 \, b c^{4} d^{3} x^{4} - 16 \, b^{2} c^{3} d^{3} x^{3} - 72 \, b^{3} c^{2} d^{3} x^{2} + 26 \, b^{4} c d^{3} x - b^{5} d^{3}\right )} e^{7} - {\left (16 \, c^{5} d^{4} x^{4} - 48 \, b c^{4} d^{4} x^{3} - 96 \, b^{2} c^{3} d^{4} x^{2} + 88 \, b^{3} c^{2} d^{4} x - 9 \, b^{4} c d^{4}\right )} e^{6} - 16 \, {\left (2 \, c^{5} d^{5} x^{3} + 3 \, b c^{4} d^{5} x^{2} - 9 \, b^{2} c^{3} d^{5} x + 2 \, b^{3} c^{2} d^{5}\right )} e^{5} - 56 \, {\left (2 \, b c^{4} d^{6} x - b^{2} c^{3} d^{6}\right )} e^{4} + 16 \, {\left (2 \, c^{5} d^{7} x - 3 \, b c^{4} d^{7}\right )} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

2/15*(8*c^3*d^4*g + (3*b^3*f + 8*(6*c^3*f - 5*b*c^2*g)*x^3 + 4*(6*b*c^2*f - 5*b^2*c*g)*x^2 - (6*b^2*c*f - 5*b^
3*g)*x)*e^4 + 2*(16*c^3*d*g*x^3 - 12*b^2*c*d*f + b^3*d*g + 16*(3*c^3*d*f - 2*b*c^2*d*g)*x^2 + 4*(9*b*c^2*d*f -
8*b^2*c*d*g)*x)*e^3 + 2*(32*c^3*d^2*g*x^2 + 42*b*c^2*d^2*f - 13*b^2*c*d^2*g + 2*(6*c^3*d^2*f + 7*b*c^2*d^2*g)
*x)*e^2 + 16*(c^3*d^3*g*x - 3*c^3*d^3*f + b*c^2*d^3*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)/(16*c^5*d^8*
e^2 - (b^4*c*x^4 + b^5*x^3)*e^10 + (8*b^3*c^2*d*x^4 + 6*b^4*c*d*x^3 - 3*b^5*d*x^2)*e^9 - (24*b^2*c^3*d^2*x^4 +
8*b^3*c^2*d^2*x^3 - 24*b^4*c*d^2*x^2 + 3*b^5*d^2*x)*e^8 + (32*b*c^4*d^3*x^4 - 16*b^2*c^3*d^3*x^3 - 72*b^3*c^2
*d^3*x^2 + 26*b^4*c*d^3*x - b^5*d^3)*e^7 - (16*c^5*d^4*x^4 - 48*b*c^4*d^4*x^3 - 96*b^2*c^3*d^4*x^2 + 88*b^3*c^
2*d^4*x - 9*b^4*c*d^4)*e^6 - 16*(2*c^5*d^5*x^3 + 3*b*c^4*d^5*x^2 - 9*b^2*c^3*d^5*x + 2*b^3*c^2*d^5)*e^5 - 56*(
2*b*c^4*d^6*x - b^2*c^3*d^6)*e^4 + 16*(2*c^5*d^7*x - 3*b*c^4*d^7)*e^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)**2/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Integral((f + g*x)/((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(d + e*x)**2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 7465 vs. \(2 (201) = 402\).
time = 1.09, size = 7465, normalized size = 35.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")

[Out]

-2/15*(8*(4*c^3*d*g + 6*c^3*f*e - 5*b*c^2*g*e)*sgn(1/(x*e + d))/(16*sqrt(-c)*c^4*d^4*e - 32*b*sqrt(-c)*c^3*d^3
*e^2 + 24*b^2*sqrt(-c)*c^2*d^2*e^3 - 8*b^3*sqrt(-c)*c*d*e^4 + b^4*sqrt(-c)*e^5) - (196608*(c - 2*c*d/(x*e + d)
+ b*e/(x*e + d))^2*c^16*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^17*g*e^4*sgn(1/(x*e + d))^4 - 983040*c^1
8*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^17*g*e^4*sgn(1/(x*e + d))^4 + 327680*c^17*(-c + 2*c*d/(x*e + d)
- b*e/(x*e + d))^(3/2)*d^17*g*e^4*sgn(1/(x*e + d))^4 - 196608*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^16*sq
rt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^16*f*e^5*sgn(1/(x*e + d))^4 - 2949120*c^18*sqrt(-c + 2*c*d/(x*e + d
) - b*e/(x*e + d))*d^16*f*e^5*sgn(1/(x*e + d))^4 - 983040*c^17*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^
16*f*e^5*sgn(1/(x*e + d))^4 - 1572864*b*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^15*sqrt(-c + 2*c*d/(x*e + d)
- b*e/(x*e + d))*d^16*g*e^5*sgn(1/(x*e + d))^4 + 9830400*b*c^17*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^
16*g*e^5*sgn(1/(x*e + d))^4 - 2293760*b*c^16*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^16*g*e^5*sgn(1/(x*
e + d))^4 + 1572864*b*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^15*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*
d^15*f*e^6*sgn(1/(x*e + d))^4 + 23592960*b*c^17*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^15*f*e^6*sgn(1/(x
*e + d))^4 + 7864320*b*c^16*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^15*f*e^6*sgn(1/(x*e + d))^4 + 58982
40*b^2*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^14*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^15*g*e^6*sgn(
1/(x*e + d))^4 - 45219840*b^2*c^16*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^15*g*e^6*sgn(1/(x*e + d))^4 +
7208960*b^2*c^15*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^15*g*e^6*sgn(1/(x*e + d))^4 - 5898240*b^2*(c -
2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^14*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^14*f*e^7*sgn(1/(x*e + d)
)^4 - 88473600*b^2*c^16*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^14*f*e^7*sgn(1/(x*e + d))^4 - 29491200*b^
2*c^15*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^14*f*e^7*sgn(1/(x*e + d))^4 - 13762560*b^3*(c - 2*c*d/(x
*e + d) + b*e/(x*e + d))^2*c^13*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^14*g*e^7*sgn(1/(x*e + d))^4 + 127
795200*b^3*c^15*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^14*g*e^7*sgn(1/(x*e + d))^4 - 13107200*b^3*c^14*(
-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^14*g*e^7*sgn(1/(x*e + d))^4 + 13762560*b^3*(c - 2*c*d/(x*e + d)
+ b*e/(x*e + d))^2*c^13*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^13*f*e^8*sgn(1/(x*e + d))^4 + 206438400*b
^3*c^15*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^13*f*e^8*sgn(1/(x*e + d))^4 + 68812800*b^3*c^14*(-c + 2*c
*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^13*f*e^8*sgn(1/(x*e + d))^4 + 22364160*b^4*(c - 2*c*d/(x*e + d) + b*e/(x
*e + d))^2*c^12*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^13*g*e^8*sgn(1/(x*e + d))^4 - 249446400*b^4*c^14*
sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^13*g*e^8*sgn(1/(x*e + d))^4 + 14336000*b^4*c^13*(-c + 2*c*d/(x*e
+ d) - b*e/(x*e + d))^(3/2)*d^13*g*e^8*sgn(1/(x*e + d))^4 - 22364160*b^4*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))
^2*c^12*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^12*f*e^9*sgn(1/(x*e + d))^4 - 335462400*b^4*c^14*sqrt(-c
+ 2*c*d/(x*e + d) - b*e/(x*e + d))*d^12*f*e^9*sgn(1/(x*e + d))^4 - 111820800*b^4*c^13*(-c + 2*c*d/(x*e + d) -
b*e/(x*e + d))^(3/2)*d^12*f*e^9*sgn(1/(x*e + d))^4 - 26836992*b^5*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^11
*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))*d^12*g*e^9*sgn(1/(x*e + d))^4 + 357826560*b^5*c^13*sqrt(-c + 2*c*d
/(x*e + d) - b*e/(x*e + d))*d^12*g*e^9*sgn(1/(x*e + d))^4 - 7454720*b^5*c^12*(-c + 2*c*d/(x*e + d) - b*e/(x*e
+ d))^(3/2)*d^12*g*e^9*sgn(1/(x*e + d))^4 + 26836992*b^5*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^11*sqrt(-c
+ 2*c*d/(x*e + d) - b*e/(x*e + d))*d^11*f*e^10*sgn(1/(x*e + d))^4 + 402554880*b^5*c^13*sqrt(-c + 2*c*d/(x*e +
d) - b*e/(x*e + d))*d^11*f*e^10*sgn(1/(x*e + d))^4 + 134184960*b^5*c^12*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))
^(3/2)*d^11*f*e^10*sgn(1/(x*e + d))^4 + 24600576*b^6*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^10*sqrt(-c + 2*
c*d/(x*e + d) - b*e/(x*e + d))*d^11*g*e^10*sgn(1/(x*e + d))^4 - 391372800*b^6*c^12*sqrt(-c + 2*c*d/(x*e + d) -
b*e/(x*e + d))*d^11*g*e^10*sgn(1/(x*e + d))^4 - 3727360*b^6*c^11*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)
*d^11*g*e^10*sgn(1/(x*e + d))^4 - 24600576*b^6*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^10*sqrt(-c + 2*c*d/(x
*e + d) - b*e/(x*e + d))*d^10*f*e^11*sgn(1/(x*e + d))^4 - 369008640*b^6*c^12*sqrt(-c + 2*c*d/(x*e + d) - b*e/(
x*e + d))*d^10*f*e^11*sgn(1/(x*e + d))^4 - 123002880*b^6*c^11*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^1
0*f*e^11*sgn(1/(x*e + d))^4 - 17571840*b^7*(c - 2*c*d/(x*e + d) + b*e/(x*e + d))^2*c^9*sqrt(-c + 2*c*d/(x*e +
d) - b*e/(x*e + d))*d^10*g*e^11*sgn(1/(x*e + d))^4 + 333864960*b^7*c^11*sqrt(-c + 2*c*d/(x*e + d) - b*e/(x*e +
d))*d^10*g*e^11*sgn(1/(x*e + d))^4 + 11714560*b^7*c^10*(-c + 2*c*d/(x*e + d) - b*e/(x*e + d))^(3/2)*d^10*g*e^
11*sgn(1/(x*e + d))^4 + 17571840*b^7*(c - 2*c*d...

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Mupad [B]
time = 4.59, size = 2126, normalized size = 10.17 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)/((d + e*x)^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2)),x)

[Out]

(((d*((16*c^3*f - 16*b*c^2*g)/(15*(b*e - 2*c*d)^5) + (8*c^3*d*g)/(15*e*(b*e - 2*c*d)^5)))/e + (2*b*c*(3*b*g -
4*c*f))/(15*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((4*b*c*g)/(15*e*(b*e
- 2*c*d)^4) - (8*c^2*d*g)/(15*e^2*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (
((2*b*g)/(5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2) - (4*c*d*g)/(5*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2))*(c*d^2
- c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((4*c*g*(3*b*e - 4*c*d))/(15*e^2*(b*e - 2*c*d)^4) - (8*c
^2*d*g)/(15*e^2*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((2*d*g)/(5*b^2*e^
4 + 20*c^2*d^2*e^2 - 20*b*c*d*e^3) - (2*e*f)/(5*b^2*e^4 + 20*c^2*d^2*e^2 - 20*b*c*d*e^3))*(c*d^2 - c*e^2*x^2 -
b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 + (((d*((2*c*e*(3*b*e*g + 2*c*d*g - 4*c*e*f))/(5*(b*e - 2*c*d)^2*(3*b^2*e
^4 + 12*c^2*d^2*e^2 - 12*b*c*d*e^3)) - (4*c^2*d*e*g)/(5*(b*e - 2*c*d)^2*(3*b^2*e^4 + 12*c^2*d^2*e^2 - 12*b*c*d
*e^3))))/e - (12*b^2*e^2*g + 12*c^2*d^2*g - 18*b*c*e^2*f + 28*c^2*d*e*f - 24*b*c*d*e*g)/(5*(b*e - 2*c*d)^2*(3*
b^2*e^4 + 12*c^2*d^2*e^2 - 12*b*c*d*e^3)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - ((x*((d*
(b*e - c*d)*((16*c^5*g*(e*(b*e - c*d) + c*d*e))/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (
16*c^5*e*(c*d*g - 3*b*e*g + 2*c*e*f))/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^5*e^
2*g)/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - ((e*(b*e - c*d) + c*d*e)*(((e*(b*e
- c*d) + c*d*e)*((16*c^5*g*(e*(b*e - c*d) + c*d*e))/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)
) + (16*c^5*e*(c*d*g - 3*b*e*g + 2*c*e*f))/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c
^5*e^2*g)/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (2*c^2*(10*b^2*c^2*e^3*g + 12
*b*c^3*e^3*f - 56*c^4*d*e^2*f + 24*c^4*d^2*e*g - 4*b*c^3*d*e^2*g))/(15*e*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^
2 - 4*b*c^2*d*e)) - (8*b*c^4*e*(c*d*g - 3*b*e*g + 2*c*e*f))/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c
^2*d*e)) - (16*c^5*d*g*(b*e - c*d))/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (2*
c^2*(96*c^4*d^3*g + 58*b^2*c^2*e^3*f - 36*b^3*c*e^3*g + 192*c^4*d^2*e*f - 220*b*c^3*d*e^2*f - 228*b*c^3*d^2*e*
g + 168*b^2*c^2*d*e^2*g))/(15*e*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (b*c*(10*b^2*c^2*e^3*
g + 12*b*c^3*e^3*f - 56*c^4*d*e^2*f + 24*c^4*d^2*e*g - 4*b*c^3*d*e^2*g))/(15*e*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^
2*c*e^2 - 4*b*c^2*d*e))) - (d*(b*e - c*d)*(((e*(b*e - c*d) + c*d*e)*((16*c^5*g*(e*(b*e - c*d) + c*d*e))/(15*(b
*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^5*e*(c*d*g - 3*b*e*g + 2*c*e*f))/(15*(b*e - 2*c*d
)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^5*e^2*g)/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*
c^2*d*e))))/(c*e^2) + (2*c^2*(10*b^2*c^2*e^3*g + 12*b*c^3*e^3*f - 56*c^4*d*e^2*f + 24*c^4*d^2*e*g - 4*b*c^3*d*
e^2*g))/(15*e*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*b*c^4*e*(c*d*g - 3*b*e*g + 2*c*e*f))
/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (16*c^5*d*g*(b*e - c*d))/(15*(b*e - 2*c*d)^4*(4*
c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) + (b*c*(96*c^4*d^3*g + 58*b^2*c^2*e^3*f - 36*b^3*c*e^3*g + 192*c
^4*d^2*e*f - 220*b*c^3*d*e^2*f - 228*b*c^3*d^2*e*g + 168*b^2*c^2*d*e^2*g))/(15*e*(b*e - 2*c*d)^4*(4*c^3*d^2 +
b^2*c*e^2 - 4*b*c^2*d*e)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/((d + e*x)*(b*e - c*d + c*e*x))

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