3.20.49 \(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=210 \[ -\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+10 c d g+4 c e f)}{105 e^2 (d+e x)^3 (2 c d-b e)^3}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+10 c d g+4 c e f)}{35 e^2 (d+e x)^4 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 e^2 (d+e x)^5 (2 c d-b e)} \]

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Rubi [A]  time = 0.34, antiderivative size = 210, 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, 650} \begin {gather*} -\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+10 c d g+4 c e f)}{105 e^2 (d+e x)^3 (2 c d-b e)^3}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-7 b e g+10 c d g+4 c e f)}{35 e^2 (d+e x)^4 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 e^2 (d+e x)^5 (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^5,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(7*e^2*(2*c*d - b*e)*(d + e*x)^5) - (2*(4*c*e*f +
10*c*d*g - 7*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^4) - (4*c*
(4*c*e*f + 10*c*d*g - 7*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(105*e^2*(2*c*d - b*e)^3*(d + e*x)
^3)

Rule 650

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))/((p + 1)*(2*c*d - b*e)), 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] && EqQ[m + 2*p + 2, 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) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 e^2 (2 c d-b e) (d+e x)^5}+\frac {(4 c e f+10 c d g-7 b e g) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx}{7 e (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 )^{3/2}}{7 e^2 (2 c d-b e) (d+e x)^5}-\frac {2 (4 c e f+10 c d g-7 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{35 e^2 (2 c d-b e)^2 (d+e x)^4}+\frac {(2 c (4 c e f+10 c d g-7 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx}{35 e (2 c d-b e)^2}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 e^2 (2 c d-b e) (d+e x)^5}-\frac {2 (4 c e f+10 c d g-7 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{35 e^2 (2 c d-b e)^2 (d+e x)^4}-\frac {4 c (4 c e f+10 c d g-7 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{105 e^2 (2 c d-b e)^3 (d+e x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 154, normalized size = 0.73 \begin {gather*} \frac {2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (3 b^2 e^2 (2 d g+5 e f+7 e g x)-2 b c e \left (13 d^2 g+d e (36 f+50 g x)+e^2 x (6 f+7 g x)\right )+4 c^2 \left (5 d^3 g+d^2 e (23 f+25 g x)+5 d e^2 x (2 f+g x)+2 e^3 f x^2\right )\right )}{105 e^2 (d+e x)^5 (b e-2 c d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^5,x]

[Out]

(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(3*b^2*e^2*(5*e*f + 2*d*g + 7*e*g*x) + 4*c^2*(5*d^3*g + 2*e^3*f*x^
2 + 5*d*e^2*x*(2*f + g*x) + d^2*e*(23*f + 25*g*x)) - 2*b*c*e*(13*d^2*g + e^2*x*(6*f + 7*g*x) + d*e*(36*f + 50*
g*x))))/(105*e^2*(-2*c*d + b*e)^3*(d + e*x)^5)

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IntegrateAlgebraic [B]  time = 53.11, size = 9683, normalized size = 46.11 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^5,x]

[Out]

Result too large to show

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fricas [B]  time = 47.06, size = 540, normalized size = 2.57 \begin {gather*} \frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, c^{3} e^{4} f + {\left (10 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}\right )} g\right )} x^{3} + {\left (4 \, {\left (8 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} f + {\left (80 \, c^{3} d^{2} e^{2} - 66 \, b c^{2} d e^{3} + 7 \, b^{2} c e^{4}\right )} g\right )} x^{2} - {\left (92 \, c^{3} d^{3} e - 164 \, b c^{2} d^{2} e^{2} + 87 \, b^{2} c d e^{3} - 15 \, b^{3} e^{4}\right )} f - 2 \, {\left (10 \, c^{3} d^{4} - 23 \, b c^{2} d^{3} e + 16 \, b^{2} c d^{2} e^{2} - 3 \, b^{3} d e^{3}\right )} g + {\left ({\left (52 \, c^{3} d^{2} e^{2} - 20 \, b c^{2} d e^{3} + 3 \, b^{2} c e^{4}\right )} f - {\left (80 \, c^{3} d^{3} e - 174 \, b c^{2} d^{2} e^{2} + 115 \, b^{2} c d e^{3} - 21 \, b^{3} e^{4}\right )} g\right )} x\right )}}{105 \, {\left (8 \, c^{3} d^{7} e^{2} - 12 \, b c^{2} d^{6} e^{3} + 6 \, b^{2} c d^{5} e^{4} - b^{3} d^{4} e^{5} + {\left (8 \, c^{3} d^{3} e^{6} - 12 \, b c^{2} d^{2} e^{7} + 6 \, b^{2} c d e^{8} - b^{3} e^{9}\right )} x^{4} + 4 \, {\left (8 \, c^{3} d^{4} e^{5} - 12 \, b c^{2} d^{3} e^{6} + 6 \, b^{2} c d^{2} e^{7} - b^{3} d e^{8}\right )} x^{3} + 6 \, {\left (8 \, c^{3} d^{5} e^{4} - 12 \, b c^{2} d^{4} e^{5} + 6 \, b^{2} c d^{3} e^{6} - b^{3} d^{2} e^{7}\right )} x^{2} + 4 \, {\left (8 \, c^{3} d^{6} e^{3} - 12 \, b c^{2} d^{5} e^{4} + 6 \, b^{2} c d^{4} e^{5} - b^{3} d^{3} e^{6}\right )} x\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)^(1/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

2/105*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^3*e^4*f + (10*c^3*d*e^3 - 7*b*c^2*e^4)*g)*x^3 + (4*(8
*c^3*d*e^3 - b*c^2*e^4)*f + (80*c^3*d^2*e^2 - 66*b*c^2*d*e^3 + 7*b^2*c*e^4)*g)*x^2 - (92*c^3*d^3*e - 164*b*c^2
*d^2*e^2 + 87*b^2*c*d*e^3 - 15*b^3*e^4)*f - 2*(10*c^3*d^4 - 23*b*c^2*d^3*e + 16*b^2*c*d^2*e^2 - 3*b^3*d*e^3)*g
+ ((52*c^3*d^2*e^2 - 20*b*c^2*d*e^3 + 3*b^2*c*e^4)*f - (80*c^3*d^3*e - 174*b*c^2*d^2*e^2 + 115*b^2*c*d*e^3 -
21*b^3*e^4)*g)*x)/(8*c^3*d^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5 + (8*c^3*d^3*e^6 - 12*b*c^
2*d^2*e^7 + 6*b^2*c*d*e^8 - b^3*e^9)*x^4 + 4*(8*c^3*d^4*e^5 - 12*b*c^2*d^3*e^6 + 6*b^2*c*d^2*e^7 - b^3*d*e^8)*
x^3 + 6*(8*c^3*d^5*e^4 - 12*b*c^2*d^4*e^5 + 6*b^2*c*d^3*e^6 - b^3*d^2*e^7)*x^2 + 4*(8*c^3*d^6*e^3 - 12*b*c^2*d
^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \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)^(1/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{2,[1,
0,1,0,3,0]%%%}+%%%{-2,[1,0,1,0,1,1]%%%}+%%%{2,[1,0,1,0,1,0]%%%}+%%%{2,[1,0,0,1,0,1]%%%}+%%%{-2,[0,1,2,0,2,0]%%
%}+%%%{2,[0,1,2,0,0,1]%%%}+%%%{-2,[0,1,2,0,0,0]%%%}+%%%{2,[0,1,0,2,0,1]%%%},0,%%%{1,[2,0,2,0,6,0]%%%}+%%%{-2,[
2,0,2,0,4,1]%%%}+%%%{-2,[2,0,2,0,4,0]%%%}+%%%{1,[2,0,2,0,2,2]%%%}+%%%{2,[2,0,2,0,2,1]%%%}+%%%{1,[2,0,2,0,2,0]%
%%}+%%%{-2,[2,0,1,1,3,1]%%%}+%%%{2,[2,0,1,1,1,2]%%%}+%%%{2,[2,0,1,1,1,1]%%%}+%%%{1,[2,0,0,2,0,2]%%%}+%%%{-2,[1
,1,3,0,5,0]%%%}+%%%{4,[1,1,3,0,3,1]%%%}+%%%{4,[1,1,3,0,3,0]%%%}+%%%{-2,[1,1,3,0,1,2]%%%}+%%%{-4,[1,1,3,0,1,1]%
%%}+%%%{-2,[1,1,3,0,1,0]%%%}+%%%{2,[1,1,2,1,2,1]%%%}+%%%{-2,[1,1,2,1,0,2]%%%}+%%%{-2,[1,1,2,1,0,1]%%%}+%%%{-2,
[1,1,1,2,3,1]%%%}+%%%{2,[1,1,1,2,1,2]%%%}+%%%{2,[1,1,1,2,1,1]%%%}+%%%{2,[1,1,0,3,0,2]%%%}+%%%{1,[0,2,4,0,4,0]%
%%}+%%%{-2,[0,2,4,0,2,1]%%%}+%%%{-2,[0,2,4,0,2,0]%%%}+%%%{1,[0,2,4,0,0,2]%%%}+%%%{2,[0,2,4,0,0,1]%%%}+%%%{1,[0
,2,4,0,0,0]%%%}+%%%{2,[0,2,2,2,2,1]%%%}+%%%{-2,[0,2,2,2,0,2]%%%}+%%%{-2,[0,2,2,2,0,1]%%%}+%%%{1,[0,2,0,4,0,2]%
%%}] at parameters values [-62,52,82,66,20,-30]exp(1)^2*(2*(-(exp(1)*x+d)^-1/exp(1)*(-(exp(1)*x+d)^-1/exp(1)*(
-(21*b^3*g*sign((exp(1)*x+d)^-1)*exp(1)^12+3*b^2*c*f*sign((exp(1)*x+d)^-1)*exp(1)^12+12*c^3*d^2*f*sign((exp(1)
*x+d)^-1)*exp(1)^10-180*c^3*d^3*g*sign((exp(1)*x+d)^-1)*exp(1)^9-12*b*c^2*d*f*sign((exp(1)*x+d)^-1)*exp(1)^11+
264*b*c^2*d^2*g*sign((exp(1)*x+d)^-1)*exp(1)^10-129*b^2*c*d*g*sign((exp(1)*x+d)^-1)*exp(1)^11)/(105*b^3*exp(1)
^15-840*c^3*d^3*exp(1)^12+1260*b*c^2*d^2*exp(1)^13-630*b^2*c*d*exp(1)^14)+(exp(1)*x+d)^-1/exp(1)*(-15*b^3*f*si
gn((exp(1)*x+d)^-1)*exp(1)^14+15*b^3*d*g*sign((exp(1)*x+d)^-1)*exp(1)^13+120*c^3*d^3*f*sign((exp(1)*x+d)^-1)*e
xp(1)^11-120*c^3*d^4*g*sign((exp(1)*x+d)^-1)*exp(1)^10-180*b*c^2*d^2*f*sign((exp(1)*x+d)^-1)*exp(1)^12+180*b*c
^2*d^3*g*sign((exp(1)*x+d)^-1)*exp(1)^11+90*b^2*c*d*f*sign((exp(1)*x+d)^-1)*exp(1)^13-90*b^2*c*d^2*g*sign((exp
(1)*x+d)^-1)*exp(1)^12)/(105*b^3*exp(1)^15-840*c^3*d^3*exp(1)^12+1260*b*c^2*d^2*exp(1)^13-630*b^2*c*d*exp(1)^1
4))-(4*b*c^2*f*sign((exp(1)*x+d)^-1)*exp(1)^10-7*b^2*c*g*sign((exp(1)*x+d)^-1)*exp(1)^10-8*c^3*d*f*sign((exp(1
)*x+d)^-1)*exp(1)^9-20*c^3*d^2*g*sign((exp(1)*x+d)^-1)*exp(1)^8+24*b*c^2*d*g*sign((exp(1)*x+d)^-1)*exp(1)^9)/(
105*b^3*exp(1)^15-840*c^3*d^3*exp(1)^12+1260*b*c^2*d^2*exp(1)^13-630*b^2*c*d*exp(1)^14))-(8*c^3*f*sign((exp(1)
*x+d)^-1)*exp(1)^8-14*b*c^2*g*sign((exp(1)*x+d)^-1)*exp(1)^8+20*c^3*d*g*sign((exp(1)*x+d)^-1)*exp(1)^7)/(105*b
^3*exp(1)^15-840*c^3*d^3*exp(1)^12+1260*b*c^2*d^2*exp(1)^13-630*b^2*c*d*exp(1)^14)-C_0*(210*b^2*exp(1)^11+840*
c^2*d^2*exp(1)^9-840*b*c*d*exp(1)^10)/(105*b^3*exp(1)^15-840*c^3*d^3*exp(1)^12+1260*b*c^2*d^2*exp(1)^13-630*b^
2*c*d*exp(1)^14))*sqrt(-c*exp(2)-b*d*(-(exp(1)*x+d)^-1/exp(1))^2*exp(1)^5-b*(exp(1)*x+d)^-1/exp(1)*exp(1)^2*ex
p(2)+c*d^2*(-(exp(1)*x+d)^-1/exp(1))^2*exp(1)^4+b*d*(-(exp(1)*x+d)^-1/exp(1))^2*exp(1)^3*exp(2)+2*c*d*(exp(1)*
x+d)^-1/exp(1)*exp(1)*exp(2)-c*d^2*(-(exp(1)*x+d)^-1/exp(1))^2*exp(1)^2*exp(2))-2*C_0*ln(abs(-2*((exp(1)*x+d)^
-1/exp(1)*sqrt(-b*d*exp(1)^5+c*d^2*exp(1)^4+b*d*exp(1)^3*exp(2)-c*d^2*exp(1)^2*exp(2))+sqrt(-c*exp(2)-b*d*(-(e
xp(1)*x+d)^-1/exp(1))^2*exp(1)^5-b*(exp(1)*x+d)^-1/exp(1)*exp(1)^2*exp(2)+c*d^2*(-(exp(1)*x+d)^-1/exp(1))^2*ex
p(1)^4+b*d*(-(exp(1)*x+d)^-1/exp(1))^2*exp(1)^3*exp(2)+2*c*d*(exp(1)*x+d)^-1/exp(1)*exp(1)*exp(2)-c*d^2*(-(exp
(1)*x+d)^-1/exp(1))^2*exp(1)^2*exp(2)))*sqrt(-b*d*exp(1)^3+c*d^2*exp(1)^2-c*d^2*exp(2)+b*d*exp(1)*exp(2))+b*ex
p(1)*exp(2)-2*c*d*exp(2)))/sqrt(-b*d*exp(1)^3+c*d^2*exp(1)^2-c*d^2*exp(2)+b*d*exp(1)*exp(2))/exp(1)-(28*b*c^2*
g*sqrt(-c*exp(2))*exp(1)-40*c^3*d*g*sqrt(-c*exp(2))-16*c^3*f*sqrt(-c*exp(2))*exp(1))/(105*b^3*exp(1)^8-630*b^2
*c*d*exp(1)^7+1260*b*c^2*d^2*exp(1)^6-840*c^3*d^3*exp(1)^5)*sign((exp(1)*x+d)^-1))

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maple [A]  time = 0.06, size = 236, normalized size = 1.12 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-14 b c \,e^{3} g \,x^{2}+20 c^{2} d \,e^{2} g \,x^{2}+8 c^{2} e^{3} f \,x^{2}+21 b^{2} e^{3} g x -100 b c d \,e^{2} g x -12 b c \,e^{3} f x +100 c^{2} d^{2} e g x +40 c^{2} d \,e^{2} f x +6 b^{2} d \,e^{2} g +15 b^{2} e^{3} f -26 b c \,d^{2} e g -72 b c d \,e^{2} f +20 c^{2} d^{3} g +92 c^{2} d^{2} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{105 \left (e x +d \right )^{4} \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}} \end {gather*}

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)^(1/2)/(e*x+d)^5,x)

[Out]

-2/105*(c*e*x+b*e-c*d)*(-14*b*c*e^3*g*x^2+20*c^2*d*e^2*g*x^2+8*c^2*e^3*f*x^2+21*b^2*e^3*g*x-100*b*c*d*e^2*g*x-
12*b*c*e^3*f*x+100*c^2*d^2*e*g*x+40*c^2*d*e^2*f*x+6*b^2*d*e^2*g+15*b^2*e^3*f-26*b*c*d^2*e*g-72*b*c*d*e^2*f+20*
c^2*d^3*g+92*c^2*d^2*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4/(b^3*e^3-6*b^2*c*d*e^2+12*b*c^2*d^2
*e-8*c^3*d^3)/e^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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(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 details)Is b*e-2*c*d zero or nonzero?

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mupad [B]  time = 6.89, size = 2325, normalized size = 11.07

result too large to display

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)^(1/2))/(d + e*x)^5,x)

[Out]

(((d*((16*c^4*e*f - 144*c^4*d*g + 80*b*c^3*e*g)/(105*e*(b*e - 2*c*d)^4) - (16*c^4*d*g)/(105*e*(b*e - 2*c*d)^4)
))/e - (4*b*c^2*(9*b*e*g - 18*c*d*g + 2*c*e*f))/(105*e*(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) - (((d*((16*c^4*e*f - 64*c^4*d*g + 40*b*c^3*e*g)/(105*e*(b*e - 2*c*d)^4) - (16*c^4*d*g)/(105
*e*(b*e - 2*c*d)^4)))/e - (8*b*c^2*(2*b*e*g - 4*c*d*g + c*e*f))/(105*e*(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) + (((d*((16*c^4*e*f - 176*c^4*d*g + 96*b*c^3*e*g)/(105*e*(b*e - 2*c*d)^4) -
(16*c^4*d*g)/(105*e*(b*e - 2*c*d)^4)))/e - (4*b*c^2*(11*b*e*g - 22*c*d*g + 2*c*e*f))/(105*e*(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) - (((d*((16*c^4*e*f - 256*c^4*d*g + 136*b*c^3*e*g)/(105
*e*(b*e - 2*c*d)^4) - (16*c^4*d*g)/(105*e*(b*e - 2*c*d)^4)))/e - (8*b*c^2*(8*b*e*g - 16*c*d*g + c*e*f))/(105*e
*(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) - (((d*((4*c^2*e*f - 8*c^2*d*g + 6*b
*c*e*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)) - (4*c^2*d*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d))))/e - (2*b
*(b*e*g - 2*c*d*g + c*e*f))/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2
))/(d + e*x)^3 - (((2*f*(b*e - c*d))/(7*b*e^2 - 14*c*d*e) - (d*((2*b*e*g - 2*c*d*g + 2*c*e*f)/(7*b*e^2 - 14*c*
d*e) - (2*c*d*g)/(7*b*e^2 - 14*c*d*e)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^4 - (((d*((4
*c^2*(9*b*e*g - 16*c*d*g + 2*c*e*f))/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(35*(3*b*e^2 - 6*c
*d*e)*(b*e - 2*c*d)^2)))/e - (72*c^3*d^2*g - 24*c^3*d*e*f + 16*b*c^2*e^2*f + 28*b^2*c*e^2*g - 92*b*c^2*d*e*g)/
(35*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 + (((d*((
8*c^2*(8*b*e*g - 15*c*d*g + c*e*f))/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(35*(3*b*e^2 - 6*c*
d*e)*(b*e - 2*c*d)^2)))/e - (8*c*(b*e - c*d)*(7*b*e*g - 14*c*d*g + c*e*f))/(35*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 + (((d*((4*c*(5*b*e*g - 9*c*d*g + c*e*f))/(7
*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)) - (4*c^2*d*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d))))/e - (4*(b*e - c*d)
*(4*b*e*g - 8*c*d*g + c*e*f))/(7*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^
(1/2))/(d + e*x)^3 - (((128*c^4*d^2*g + 52*b^2*c^2*e^2*g - 32*c^4*d*e*f + 24*b*c^3*e^2*f - 168*b*c^3*d*e*g)/(1
05*e^2*(b*e - 2*c*d)^4) - (d*((16*c^3*(4*b*e*g - 7*c*d*g + c*e*f))/(105*e*(b*e - 2*c*d)^4) - (16*c^4*d*g)/(105
*e*(b*e - 2*c*d)^4)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((352*c^4*d^2*g + 136*b^2*c
^2*e^2*g - 32*c^4*d*e*f + 24*b*c^3*e^2*f - 448*b*c^3*d*e*g)/(105*e^2*(b*e - 2*c*d)^4) - (d*((8*c^3*(15*b*e*g -
28*c*d*g + 2*c*e*f))/(105*e*(b*e - 2*c*d)^4) - (16*c^4*d*g)/(105*e*(b*e - 2*c*d)^4)))/e)*(c*d^2 - c*e^2*x^2 -
b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((16*c^3*(10*b*e*g - 19*c*d*g + c*e*f))/(105*e*(b*e - 2*c*d)^4) - (1
6*c^4*d*g)/(105*e*(b*e - 2*c*d)^4)))/e - (16*c^2*(b*e - c*d)*(9*b*e*g - 18*c*d*g + c*e*f))/(105*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) + (((d*((8*c^3*e*f - 24*c^3*d*g + 16*b*c^2*e*g)
/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2)))/e - (2*b*c*
(3*b*e*g - 6*c*d*g + 2*c*e*f))/(35*(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 - (((d*((8*c^3*e*f - 80*c^3*d*g + 44*b*c^2*e*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2) -
(8*c^3*d*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2)))/e - (4*b*c*(5*b*e*g - 10*c*d*g + c*e*f))/(35*(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 - (((d*((8*c^3*(13*b*e*g
- 24*c*d*g + 2*c*e*f))/(105*e*(b*e - 2*c*d)^4) - (16*c^4*d*g)/(105*e*(b*e - 2*c*d)^4)))/e - (144*c^4*d^2*g +
88*b^2*c^2*e^2*g + 16*c^4*d*e*f - 248*b*c^3*d*e*g)/(105*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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{5}}\, 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)**(1/2)/(e*x+d)**5,x)

[Out]

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

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