3.20.50 \(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=285 \[ -\frac {16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{315 e^2 (d+e x)^3 (2 c d-b e)^4}-\frac {8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{105 e^2 (d+e x)^4 (2 c d-b e)^3}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{21 e^2 (d+e x)^5 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 e^2 (d+e x)^6 (2 c d-b e)} \]

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Rubi [A]  time = 0.46, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {16 c^2 \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+2 c e f)}{315 e^2 (d+e x)^3 (2 c d-b e)^4}-\frac {8 c \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+2 c e f)}{105 e^2 (d+e x)^4 (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} (-3 b e g+4 c d g+2 c e f)}{21 e^2 (d+e x)^5 (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}}{9 e^2 (d+e x)^6 (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)^6,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(9*e^2*(2*c*d - b*e)*(d + e*x)^6) - (2*(2*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)^(3/2))/(21*e^2*(2*c*d - b*e)^2*(d + e*x)^5) - (8*c*(
2*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)^(3/2))/(105*e^2*(2*c*d - b*e)^3*(d + e*x)^4
) - (16*c^2*(2*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)^(3/2))/(315*e^2*(2*c*d - b*e)^
4*(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)^6} \, 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}}{9 e^2 (2 c d-b e) (d+e x)^6}+\frac {(2 c e f+4 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)^5} \, dx}{3 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}}{9 e^2 (2 c d-b e) (d+e x)^6}-\frac {2 (2 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 )^{3/2}}{21 e^2 (2 c d-b e)^2 (d+e x)^5}+\frac {(4 c (2 c e f+4 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)^4} \, dx}{21 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}}{9 e^2 (2 c d-b e) (d+e x)^6}-\frac {2 (2 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 )^{3/2}}{21 e^2 (2 c d-b e)^2 (d+e x)^5}-\frac {8 c (2 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 )^{3/2}}{105 e^2 (2 c d-b e)^3 (d+e x)^4}+\frac {\left (8 c^2 (2 c e f+4 c d g-3 b e g)\right ) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx}{105 e (2 c d-b e)^3}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 e^2 (2 c d-b e) (d+e x)^6}-\frac {2 (2 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 )^{3/2}}{21 e^2 (2 c d-b e)^2 (d+e x)^5}-\frac {8 c (2 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 )^{3/2}}{105 e^2 (2 c d-b e)^3 (d+e x)^4}-\frac {16 c^2 (2 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 )^{3/2}}{315 e^2 (2 c d-b e)^4 (d+e x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 232, normalized size = 0.81 \begin {gather*} -\frac {2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (-5 b^3 e^3 (2 d g+7 e f+9 e g x)+6 b^2 c e^2 \left (11 d^2 g+d e (40 f+52 g x)+e^2 x (5 f+6 g x)\right )-12 b c^2 e \left (12 d^3 g+d^2 e (47 f+61 g x)+2 d e^2 x (7 f+8 g x)+2 e^3 x^2 (f+g x)\right )+8 c^3 \left (11 d^4 g+d^3 e (58 f+66 g x)+3 d^2 e^2 x (11 f+8 g x)+4 d e^3 x^2 (3 f+g x)+2 e^4 f x^3\right )\right )}{315 e^2 (d+e x)^6 (b e-2 c d)^4} \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)^6,x]

[Out]

(-2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-5*b^3*e^3*(7*e*f + 2*d*g + 9*e*g*x) + 6*b^2*c*e^2*(11*d^2*g + e
^2*x*(5*f + 6*g*x) + d*e*(40*f + 52*g*x)) - 12*b*c^2*e*(12*d^3*g + 2*e^3*x^2*(f + g*x) + 2*d*e^2*x*(7*f + 8*g*
x) + d^2*e*(47*f + 61*g*x)) + 8*c^3*(11*d^4*g + 2*e^4*f*x^3 + 4*d*e^3*x^2*(3*f + g*x) + 3*d^2*e^2*x*(11*f + 8*
g*x) + d^3*e*(58*f + 66*g*x))))/(315*e^2*(-2*c*d + b*e)^4*(d + e*x)^6)

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IntegrateAlgebraic [B]  time = 108.88, size = 15269, normalized size = 53.58 \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)^6,x]

[Out]

Result too large to show

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)^6,x, algorithm="fricas")

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)^6,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.06, size = 382, normalized size = 1.34 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (24 b \,c^{2} e^{4} g \,x^{3}-32 c^{3} d \,e^{3} g \,x^{3}-16 c^{3} e^{4} f \,x^{3}-36 b^{2} c \,e^{4} g \,x^{2}+192 b \,c^{2} d \,e^{3} g \,x^{2}+24 b \,c^{2} e^{4} f \,x^{2}-192 c^{3} d^{2} e^{2} g \,x^{2}-96 c^{3} d \,e^{3} f \,x^{2}+45 b^{3} e^{4} g x -312 b^{2} c d \,e^{3} g x -30 b^{2} c \,e^{4} f x +732 b \,c^{2} d^{2} e^{2} g x +168 b \,c^{2} d \,e^{3} f x -528 c^{3} d^{3} e g x -264 c^{3} d^{2} e^{2} f x +10 b^{3} d \,e^{3} g +35 b^{3} e^{4} f -66 b^{2} c \,d^{2} e^{2} g -240 b^{2} c d \,e^{3} f +144 b \,c^{2} d^{3} e g +564 b \,c^{2} d^{2} e^{2} f -88 c^{3} d^{4} g -464 c^{3} d^{3} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{315 \left (e x +d \right )^{5} \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 ) 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)^6,x)

[Out]

-2/315*(c*e*x+b*e-c*d)*(24*b*c^2*e^4*g*x^3-32*c^3*d*e^3*g*x^3-16*c^3*e^4*f*x^3-36*b^2*c*e^4*g*x^2+192*b*c^2*d*
e^3*g*x^2+24*b*c^2*e^4*f*x^2-192*c^3*d^2*e^2*g*x^2-96*c^3*d*e^3*f*x^2+45*b^3*e^4*g*x-312*b^2*c*d*e^3*g*x-30*b^
2*c*e^4*f*x+732*b*c^2*d^2*e^2*g*x+168*b*c^2*d*e^3*f*x-528*c^3*d^3*e*g*x-264*c^3*d^2*e^2*f*x+10*b^3*d*e^3*g+35*
b^3*e^4*f-66*b^2*c*d^2*e^2*g-240*b^2*c*d*e^3*f+144*b*c^2*d^3*e*g+564*b*c^2*d^2*e^2*f-88*c^3*d^4*g-464*c^3*d^3*
e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^5/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)

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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)^6,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 = 13.95, size = 4962, normalized size = 17.41

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)^6,x)

[Out]

(((d*((32*c^5*e*f - 160*c^5*d*g + 96*b*c^4*e*g)/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)
))/e - (8*b*c^3*(5*b*e*g - 10*c*d*g + 2*c*e*f))/(945*e*(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) - (((d*((32*c^5*e*f - 320*c^5*d*g + 176*b*c^4*e*g)/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(9
45*e*(b*e - 2*c*d)^5)))/e - (16*b*c^3*(5*b*e*g - 10*c*d*g + c*e*f))/(945*e*(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) - (((d*((32*c^5*e*f - 384*c^5*d*g + 208*b*c^4*e*g)/(945*e*(b*e - 2*c*d)^
5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e - (16*b*c^3*(6*b*e*g - 12*c*d*g + c*e*f))/(945*e*(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) - (((d*((32*c^5*e*f - 448*c^5*d*g + 240*b*c^4*e*g)/(
945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e - (16*b*c^3*(7*b*e*g - 14*c*d*g + c*e*f))/(9
45*e*(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) + (((d*((32*c^5*e*f - 544*c^5*d*
g + 288*b*c^4*e*g)/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e - (8*b*c^3*(17*b*e*g - 3
4*c*d*g + 2*c*e*f))/(945*e*(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) + (((d*((3
2*c^5*e*f - 608*c^5*d*g + 320*b*c^4*e*g)/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e -
(8*b*c^3*(19*b*e*g - 38*c*d*g + 2*c*e*f))/(945*e*(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) + (((d*((32*c^5*e*f - 672*c^5*d*g + 352*b*c^4*e*g)/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(
b*e - 2*c*d)^5)))/e - (8*b*c^3*(21*b*e*g - 42*c*d*g + 2*c*e*f))/(945*e*(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) - (((d*((32*c^5*e*f - 832*c^5*d*g + 432*b*c^4*e*g)/(945*e*(b*e - 2*c*d)^5) -
(32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e - (16*b*c^3*(13*b*e*g - 26*c*d*g + c*e*f))/(945*e*(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) - (((d*((4*c^2*e*f - 8*c^2*d*g + 6*b*c*e*g)/(9*(7*b*e^2
- 14*c*d*e)*(b*e - 2*c*d)) - (4*c^2*d*g)/(9*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d))))/e - (2*b*(b*e*g - 2*c*d*g +
c*e*f))/(9*(7*b*e^2 - 14*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)^4 - ((
(2*f*(b*e - c*d))/(9*b*e^2 - 18*c*d*e) - (d*((2*b*e*g - 2*c*d*g + 2*c*e*f)/(9*b*e^2 - 18*c*d*e) - (2*c*d*g)/(9
*b*e^2 - 18*c*d*e)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^5 - (((d*((4*c^2*(11*b*e*g - 20
*c*d*g + 2*c*e*f))/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c
*d)^2)))/e - (96*c^3*d^2*g - 32*c^3*d*e*f + 20*b*c^2*e^2*f + 36*b^2*c*e^2*g - 120*b*c^2*d*e*g)/(63*e*(5*b*e^2
- 10*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)^3 + (((d*((8*c^2*(10*b*e*
g - 19*c*d*g + c*e*f))/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e -
2*c*d)^2)))/e - (8*c*(b*e - c*d)*(9*b*e*g - 18*c*d*g + c*e*f))/(63*e*(5*b*e^2 - 10*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)^3 - (((d*((16*c^4*e*f - 64*c^4*d*g + 40*b*c^3*e*g)/(315*
(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (8*b*c^2*(
2*b*e*g - 4*c*d*g + c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(
1/2))/(d + e*x)^2 + (((d*((16*c^4*e*f - 176*c^4*d*g + 96*b*c^3*e*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)
- (16*c^4*d*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (4*b*c^2*(11*b*e*g - 22*c*d*g + 2*c*e*f))/(315*
(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((16*c^4
*e*f - 208*c^4*d*g + 112*b*c^3*e*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(315*(3*b*e^2 - 6
*c*d*e)*(b*e - 2*c*d)^3)))/e - (4*b*c^2*(13*b*e*g - 26*c*d*g + 2*c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d
)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 - (((d*((16*c^4*e*f - 320*c^4*d*g + 168*b*c^3*e
*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (
8*b*c^2*(10*b*e*g - 20*c*d*g + c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e -
b*e^2*x)^(1/2))/(d + e*x)^2 + (((d*((4*c*(6*b*e*g - 11*c*d*g + c*e*f))/(9*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d))
- (4*c^2*d*g)/(9*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d))))/e - (4*(b*e - c*d)*(5*b*e*g - 10*c*d*g + c*e*f))/(9*e*
(7*b*e^2 - 14*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)^4 + (((d*((16*c^3*
(5*b*e*g - 9*c*d*g + c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(315*(3*b*e^2 - 6*c*d*e)
*(b*e - 2*c*d)^3)))/e - (192*c^4*d^2*g + 72*b^2*c^2*e^2*g - 48*c^4*d*e*f + 32*b*c^3*e^2*f - 240*b*c^3*d*e*g)/(
315*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3))*(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*(17*b*e*g - 32*c*d*g + 2*c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(315*(3*b*e^2
- 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (144*c^4*d^2*g + 120*b^2*c^2*e^2*g + 80*c^4*d*e*f - 32*b*c^3*e^2*f - 312*b*c
^3*d*e*g)/(315*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3))*(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*(19*b*e*g - 36*c*d*g + 2*c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(31
5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (624*c^4*d^2*g + 216*b^2*c^2*e^2*g - 48*c^4*d*e*f + 32*b*c^3*e^2*
f - 744*b*c^3*d*e*g)/(315*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))
/(d + e*x)^2 + (((d*((16*c^3*(13*b*e*g - 25*c*d*g + c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^
4*d*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (16*c^2*(b*e - c*d)*(12*b*e*g - 24*c*d*g + c*e*f))/(315
*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((320*c^5
*d^2*g + 128*b^2*c^3*e^2*g - 64*c^5*d*e*f + 48*b*c^4*e^2*f - 416*b*c^4*d*e*g)/(945*e^2*(b*e - 2*c*d)^5) - (d*(
(16*c^4*(9*b*e*g - 16*c*d*g + 2*c*e*f))/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e)*(c
*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((768*c^5*d^2*g + 296*b^2*c^3*e^2*g - 64*c^5*d*e*f + 4
8*b*c^4*e^2*f - 976*b*c^4*d*e*g)/(945*e^2*(b*e - 2*c*d)^5) - (d*((32*c^4*(8*b*e*g - 15*c*d*g + c*e*f))/(945*e*
(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d +
e*x) + (((800*c^5*d^2*g + 304*b^2*c^3*e^2*g - 224*c^5*d*e*f + 128*b*c^4*e^2*f - 1008*b*c^4*d*e*g)/(945*e^2*(b
*e - 2*c*d)^5) - (d*((16*c^4*(21*b*e*g - 40*c*d*g + 2*c*e*f))/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b
*e - 2*c*d)^5)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((896*c^5*d^2*g + 344*b^2*c^3*e^
2*g - 64*c^5*d*e*f + 48*b*c^4*e^2*f - 1136*b*c^4*d*e*g)/(945*e^2*(b*e - 2*c*d)^5) - (d*((32*c^4*(9*b*e*g - 17*
c*d*g + c*e*f))/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e
- b*e^2*x)^(1/2))/(d + e*x) + (((1344*c^5*d^2*g + 512*b^2*c^3*e^2*g - 64*c^5*d*e*f + 48*b*c^4*e^2*f - 1696*b*
c^4*d*e*g)/(945*e^2*(b*e - 2*c*d)^5) - (d*((16*c^4*(25*b*e*g - 48*c*d*g + 2*c*e*f))/(945*e*(b*e - 2*c*d)^5) -
(32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((32*c
^4*(15*b*e*g - 29*c*d*g + c*e*f))/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e - (32*c^3
*(b*e - c*d)*(14*b*e*g - 28*c*d*g + c*e*f))/(945*e^2*(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) + (((d*((8*c^3*e*f - 24*c^3*d*g + 16*b*c^2*e*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (8
*c^3*d*g)/(63*(5*b*e^2 - 10*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))/(63*(5*b*e^2 -
10*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)^3 - (((d*((8*c^3*e*f - 96*
c^3*d*g + 52*b*c^2*e*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (8*c^3*d*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e
- 2*c*d)^2)))/e - (4*b*c*(6*b*e*g - 12*c*d*g + c*e*f))/(63*(5*b*e^2 - 10*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)^3 - (((d*((16*c^4*(23*b*e*g - 44*c*d*g + 2*c*e*f))/(945*e*(b*e - 2*
c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e + (32*c^5*d^2*g - 176*b^2*c^3*e^2*g - 32*c^5*d*e*f + 336*b*
c^4*d*e*g)/(945*e^2*(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) + (((d*((32*c^4*(
7*b*e*g - 13*c*d*g + c*e*f))/(945*e*(b*e - 2*c*d)^5) - (32*c^5*d*g)/(945*e*(b*e - 2*c*d)^5)))/e - (256*c^5*d^2
*g + 176*b^2*c^3*e^2*g + 32*c^5*d*e*f - 480*b*c^4*d*e*g)/(945*e^2*(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)

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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 )^{6}}\, 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)**6,x)

[Out]

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**6, x)

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