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

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

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Rubi [A]  time = 0.28, 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 = {792, 658, 613} \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 613

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 658

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 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

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 = 0.13, size = 233, normalized size = 1.11 \begin {gather*} \frac {2 \left (b^3 e^3 (2 d g+3 e f+5 e g x)-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 )+4 b c^2 e \left (4 d^3 g+7 d^2 e (3 f+g x)+2 d e^2 x (9 f-8 g x)+2 e^3 x^2 (3 f-5 g x)\right )+8 c^3 \left (d^4 g+d^3 e (2 g x-6 f)+d^2 e^2 x (3 f+8 g x)+4 d e^3 x^2 (3 f+g x)+6 e^4 f x^3\right )\right )}{15 e^2 (d+e x)^2 (b e-2 c d)^4 \sqrt {(d+e x) (c (d-e x)-b e)}} \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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IntegrateAlgebraic [F]  time = 180.16, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(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]

$Aborted

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fricas [B]  time = 28.90, size = 649, normalized size = 3.11 \begin {gather*} \frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (8 \, {\left (6 \, c^{3} e^{4} f + {\left (4 \, c^{3} d e^{3} - 5 \, b c^{2} e^{4}\right )} g\right )} x^{3} + 4 \, {\left (6 \, {\left (4 \, c^{3} d e^{3} + b c^{2} e^{4}\right )} f + {\left (16 \, c^{3} d^{2} e^{2} - 16 \, b c^{2} d e^{3} - 5 \, b^{2} c e^{4}\right )} g\right )} x^{2} - 3 \, {\left (16 \, c^{3} d^{3} e - 28 \, b c^{2} d^{2} e^{2} + 8 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} f + 2 \, {\left (4 \, c^{3} d^{4} + 8 \, b c^{2} d^{3} e - 13 \, b^{2} c d^{2} e^{2} + b^{3} d e^{3}\right )} g + {\left (6 \, {\left (4 \, c^{3} d^{2} e^{2} + 12 \, b c^{2} d e^{3} - b^{2} c e^{4}\right )} f + {\left (16 \, c^{3} d^{3} e + 28 \, b c^{2} d^{2} e^{2} - 64 \, b^{2} c d e^{3} + 5 \, b^{3} e^{4}\right )} g\right )} x\right )}}{15 \, {\left (16 \, c^{5} d^{8} e^{2} - 48 \, b c^{4} d^{7} e^{3} + 56 \, b^{2} c^{3} d^{6} e^{4} - 32 \, b^{3} c^{2} d^{5} e^{5} + 9 \, b^{4} c d^{4} e^{6} - b^{5} d^{3} e^{7} - {\left (16 \, c^{5} d^{4} e^{6} - 32 \, b c^{4} d^{3} e^{7} + 24 \, b^{2} c^{3} d^{2} e^{8} - 8 \, b^{3} c^{2} d e^{9} + b^{4} c e^{10}\right )} x^{4} - {\left (32 \, c^{5} d^{5} e^{5} - 48 \, b c^{4} d^{4} e^{6} + 16 \, b^{2} c^{3} d^{3} e^{7} + 8 \, b^{3} c^{2} d^{2} e^{8} - 6 \, b^{4} c d e^{9} + b^{5} e^{10}\right )} x^{3} - 3 \, {\left (16 \, b c^{4} d^{5} e^{5} - 32 \, b^{2} c^{3} d^{4} e^{6} + 24 \, b^{3} c^{2} d^{3} e^{7} - 8 \, b^{4} c d^{2} e^{8} + b^{5} d e^{9}\right )} x^{2} + {\left (32 \, c^{5} d^{7} e^{3} - 112 \, b c^{4} d^{6} e^{4} + 144 \, b^{2} c^{3} d^{5} e^{5} - 88 \, b^{3} c^{2} d^{4} e^{6} + 26 \, b^{4} c d^{3} e^{7} - 3 \, b^{5} d^{2} e^{8}\right )} x\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*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(8*(6*c^3*e^4*f + (4*c^3*d*e^3 - 5*b*c^2*e^4)*g)*x^3 + 4*(6*(4
*c^3*d*e^3 + b*c^2*e^4)*f + (16*c^3*d^2*e^2 - 16*b*c^2*d*e^3 - 5*b^2*c*e^4)*g)*x^2 - 3*(16*c^3*d^3*e - 28*b*c^
2*d^2*e^2 + 8*b^2*c*d*e^3 - b^3*e^4)*f + 2*(4*c^3*d^4 + 8*b*c^2*d^3*e - 13*b^2*c*d^2*e^2 + b^3*d*e^3)*g + (6*(
4*c^3*d^2*e^2 + 12*b*c^2*d*e^3 - b^2*c*e^4)*f + (16*c^3*d^3*e + 28*b*c^2*d^2*e^2 - 64*b^2*c*d*e^3 + 5*b^3*e^4)
*g)*x)/(16*c^5*d^8*e^2 - 48*b*c^4*d^7*e^3 + 56*b^2*c^3*d^6*e^4 - 32*b^3*c^2*d^5*e^5 + 9*b^4*c*d^4*e^6 - b^5*d^
3*e^7 - (16*c^5*d^4*e^6 - 32*b*c^4*d^3*e^7 + 24*b^2*c^3*d^2*e^8 - 8*b^3*c^2*d*e^9 + b^4*c*e^10)*x^4 - (32*c^5*
d^5*e^5 - 48*b*c^4*d^4*e^6 + 16*b^2*c^3*d^3*e^7 + 8*b^3*c^2*d^2*e^8 - 6*b^4*c*d*e^9 + b^5*e^10)*x^3 - 3*(16*b*
c^4*d^5*e^5 - 32*b^2*c^3*d^4*e^6 + 24*b^3*c^2*d^3*e^7 - 8*b^4*c*d^2*e^8 + b^5*d*e^9)*x^2 + (32*c^5*d^7*e^3 - 1
12*b*c^4*d^6*e^4 + 144*b^2*c^3*d^5*e^5 - 88*b^3*c^2*d^4*e^6 + 26*b^4*c*d^3*e^7 - 3*b^5*d^2*e^8)*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \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]

sage0*x

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maple [A]  time = 0.06, size = 382, normalized size = 1.83 \begin {gather*} -\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 ) \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}} e^{2}} \end {gather*}

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)

[Out]

-2/15*(c*e*x+b*e-c*d)*(-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-2
6*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)/(e*x+d)/e^2
/(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)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)

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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(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?

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mupad [B]  time = 4.59, size = 2126, normalized size = 10.17

result too large to display

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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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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