3.20.61 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=138 \[ \frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-2 c (6 d g+e f))}{35 e^2 (d+e x)^5 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e^2 (d+e x)^6 (2 c d-b e)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {792, 650} \begin {gather*} \frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-2 c (6 d g+e f))}{35 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 )^{5/2}}{7 e^2 (d+e x)^6 (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)^(3/2))/(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)^(5/2))/(7*e^2*(2*c*d - b*e)*(d + e*x)^6) + (2*(7*b*e*g -
2*c*(e*f + 6*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^5)

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 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) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(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 )^{5/2}}{7 e^2 (2 c d-b e) (d+e x)^6}-\frac {\left (2 \left (\frac {5}{2} e \left (-2 c e^2 f+b e^2 g\right )-6 \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx}{7 e \left (-2 c d e^2+b e^3\right )}\\ &=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e^2 (2 c d-b e) (d+e x)^6}+\frac {2 (7 b e g-2 c (e f+6 d g)) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 e^2 (2 c d-b e)^2 (d+e x)^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 104, normalized size = 0.75 \begin {gather*} -\frac {2 (b e-c d+c e x)^2 \sqrt {(d+e x) (c (d-e x)-b e)} \left (2 c \left (d^2 g+6 d e (f+g x)+e^2 f x\right )-b e (2 d g+5 e f+7 e g x)\right )}{35 e^2 (d+e x)^4 (b e-2 c d)^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)^(3/2))/(d + e*x)^6,x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-(b*e*(5*e*f + 2*d*g + 7*e*g*x)) + 2*c*(d
^2*g + e^2*f*x + 6*d*e*(f + g*x))))/(35*e^2*(-2*c*d + b*e)^2*(d + e*x)^4)

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 80.88, size = 10737, normalized size = 77.80 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^6,x]

[Out]

Result too large to show

________________________________________________________________________________________

fricas [B]  time = 47.46, size = 464, normalized size = 3.36 \begin {gather*} -\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (2 \, c^{3} e^{4} f + {\left (12 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}\right )} g\right )} x^{3} + {\left ({\left (8 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} f - 2 \, {\left (11 \, c^{3} d^{2} e^{2} - 18 \, b c^{2} d e^{3} + 7 \, b^{2} c e^{4}\right )} g\right )} x^{2} + {\left (12 \, c^{3} d^{3} e - 29 \, b c^{2} d^{2} e^{2} + 22 \, b^{2} c d e^{3} - 5 \, b^{3} e^{4}\right )} f + 2 \, {\left (c^{3} d^{4} - 3 \, b c^{2} d^{3} e + 3 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} g - {\left (2 \, {\left (11 \, c^{3} d^{2} e^{2} - 15 \, b c^{2} d e^{3} + 4 \, b^{2} c e^{4}\right )} f - {\left (8 \, c^{3} d^{3} e - 23 \, b c^{2} d^{2} e^{2} + 22 \, b^{2} c d e^{3} - 7 \, b^{3} e^{4}\right )} g\right )} x\right )}}{35 \, {\left (4 \, c^{2} d^{6} e^{2} - 4 \, b c d^{5} e^{3} + b^{2} d^{4} e^{4} + {\left (4 \, c^{2} d^{2} e^{6} - 4 \, b c d e^{7} + b^{2} e^{8}\right )} x^{4} + 4 \, {\left (4 \, c^{2} d^{3} e^{5} - 4 \, b c d^{2} e^{6} + b^{2} d e^{7}\right )} x^{3} + 6 \, {\left (4 \, c^{2} d^{4} e^{4} - 4 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}\right )} x^{2} + 4 \, {\left (4 \, c^{2} d^{5} e^{3} - 4 \, b c d^{4} e^{4} + b^{2} d^{3} e^{5}\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)^(3/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

-2/35*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((2*c^3*e^4*f + (12*c^3*d*e^3 - 7*b*c^2*e^4)*g)*x^3 + ((8*c^3
*d*e^3 - b*c^2*e^4)*f - 2*(11*c^3*d^2*e^2 - 18*b*c^2*d*e^3 + 7*b^2*c*e^4)*g)*x^2 + (12*c^3*d^3*e - 29*b*c^2*d^
2*e^2 + 22*b^2*c*d*e^3 - 5*b^3*e^4)*f + 2*(c^3*d^4 - 3*b*c^2*d^3*e + 3*b^2*c*d^2*e^2 - b^3*d*e^3)*g - (2*(11*c
^3*d^2*e^2 - 15*b*c^2*d*e^3 + 4*b^2*c*e^4)*f - (8*c^3*d^3*e - 23*b*c^2*d^2*e^2 + 22*b^2*c*d*e^3 - 7*b^3*e^4)*g
)*x)/(4*c^2*d^6*e^2 - 4*b*c*d^5*e^3 + b^2*d^4*e^4 + (4*c^2*d^2*e^6 - 4*b*c*d*e^7 + b^2*e^8)*x^4 + 4*(4*c^2*d^3
*e^5 - 4*b*c*d^2*e^6 + b^2*d*e^7)*x^3 + 6*(4*c^2*d^4*e^4 - 4*b*c*d^3*e^5 + b^2*d^2*e^6)*x^2 + 4*(4*c^2*d^5*e^3
- 4*b*c*d^4*e^4 + b^2*d^3*e^5)*x)

________________________________________________________________________________________

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)^(3/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.06, size = 128, normalized size = 0.93 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (7 b \,e^{2} g x -12 c d e g x -2 c \,e^{2} f x +2 b d e g +5 b \,e^{2} f -2 c \,d^{2} g -12 c d e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{5} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\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)^(3/2)/(e*x+d)^6,x)

[Out]

-2/35*(c*e*x+b*e-c*d)*(7*b*e^2*g*x-12*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g+5*b*e^2*f-2*c*d^2*g-12*c*d*e*f)*(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^5/e^2/(b^2*e^2-4*b*c*d*e+4*c^2*d^2)

________________________________________________________________________________________

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)^(3/2)/(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?

________________________________________________________________________________________

mupad [B]  time = 9.50, size = 3763, normalized size = 27.27

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)^(3/2))/(d + e*x)^6,x)

[Out]

(((d*((d*((16*c^4*(6*b*e*g - 10*c*d*g + c*e*f))/(105*(b*e - 2*c*d)^4) - (16*c^5*d*g)/(105*(b*e - 2*c*d)^4)))/e
- (608*c^5*d^2*g + 196*b^2*c^3*e^2*g - 160*c^5*d*e*f + 96*b*c^4*e^2*f - 688*b*c^4*d*e*g)/(105*e*(b*e - 2*c*d)
^4)))/e + (4*b*c^2*(19*b^2*e^2*g + 76*c^2*d^2*g + 11*b*c*e^2*f - 20*c^2*d*e*f - 76*b*c*d*e*g))/(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*((d*((8*c^4*(7*b*e*g - 10*c*d*g + 2*c*
e*f))/(105*(b*e - 2*c*d)^4) - (16*c^5*d*g)/(105*(b*e - 2*c*d)^4)))/e - (208*c^5*d^2*g + 76*b^2*c^3*e^2*g - 80*
c^5*d*e*f + 56*b*c^4*e^2*f - 248*b*c^4*d*e*g)/(105*e*(b*e - 2*c*d)^4)))/e + (2*b*c^2*(13*b^2*e^2*g + 52*c^2*d^
2*g + 12*b*c*e^2*f - 20*c^2*d*e*f - 52*b*c*d*e*g))/(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*((d*((16*c^4*(7*b*e*g - 12*c*d*g + c*e*f))/(105*(b*e - 2*c*d)^4) - (16*c^5*d*g)/(1
05*(b*e - 2*c*d)^4)))/e - (768*c^5*d^2*g + 244*b^2*c^3*e^2*g - 192*c^5*d*e*f + 112*b*c^4*e^2*f - 864*b*c^4*d*e
*g)/(105*e*(b*e - 2*c*d)^4)))/e + (4*b*c^2*(24*b^2*e^2*g + 96*c^2*d^2*g + 13*b*c*e^2*f - 24*c^2*d*e*f - 96*b*c
*d*e*g))/(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*((d*((8*c^4*(1
9*b*e*g - 34*c*d*g + 2*c*e*f))/(105*(b*e - 2*c*d)^4) - (16*c^5*d*g)/(105*(b*e - 2*c*d)^4)))/e - (1728*c^5*d^2*
g + 504*b^2*c^3*e^2*g - 272*c^5*d*e*f + 152*b*c^4*e^2*f - 1864*b*c^4*d*e*g)/(105*e*(b*e - 2*c*d)^4)))/e + (8*b
*c^2*(27*b^2*e^2*g + 108*c^2*d^2*g + 9*b*c*e^2*f - 17*c^2*d*e*f - 108*b*c*d*e*g))/(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*((d*((8*c^3*e*(9*b*e*g - 16*c*d*g + c*e*f))/(35*(3*
b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2) - (8*c^4*d*e*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2)))/e - (216*b^2*c^2*
e^3*g + 72*b*c^3*e^3*f - 128*c^4*d*e^2*f + 728*c^4*d^2*e*g - 792*b*c^3*d*e^2*g)/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e
- 2*c*d)^2)))/e + (8*c*(b*e - c*d)*(19*b^2*e^2*g + 76*c^2*d^2*g + 8*b*c*e^2*f - 15*c^2*d*e*f - 76*b*c*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*(
(d*((8*c^3*e*(3*b*e*g - 4*c*d*g + c*e*f))/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2) - (8*c^4*d*e*g)/(35*(3*b*e^
2 - 6*c*d*e)*(b*e - 2*c*d)^2)))/e - (64*c^4*d^2*g + 26*b^2*c^2*e^2*g - 32*c^4*d*e*f + 24*b*c^3*e^2*f - 80*b*c^
3*d*e*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2)))/e + (2*b*c*(4*b^2*e^2*g + 16*c^2*d^2*g + 5*b*c*e^2*f - 8*c
^2*d*e*f - 16*b*c*d*e*g))/(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*((d*((4*c^3*e*(13*b*e*g - 22*c*d*g + 2*c*e*f))/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2)
- (8*c^4*d*e*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2)))/e - (288*c^4*d^2*g + 96*b^2*c^2*e^2*g - 88*c^4*d*e*
f + 52*b*c^3*e^2*f - 332*b*c^3*d*e*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2)))/e + (4*b*c*(9*b^2*e^2*g + 36*
c^2*d^2*g + 6*b*c*e^2*f - 11*c^2*d*e*f - 36*b*c*d*e*g))/(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 + (((76*b^2*c^3*e^3*f - 416*c^5*d^3*g + 88*b^3*c^2*e^3*g + 160*c^
5*d^2*e*f - 224*b*c^4*d*e^2*f + 768*b*c^4*d^2*e*g - 456*b^2*c^3*d*e^2*g)/(105*e^2*(b*e - 2*c*d)^4) + (d*((d*((
16*c^4*(5*b*e*g - 8*c*d*g + c*e*f))/(105*(b*e - 2*c*d)^4) - (16*c^5*d*g)/(105*(b*e - 2*c*d)^4)))/e - (148*b^2*
c^3*e^3*g + 80*b*c^4*e^3*f - 128*c^5*d*e^2*f + 448*c^5*d^2*e*g - 512*b*c^4*d*e^2*g)/(105*e^2*(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) - (((88*b^2*c^3*e^3*f - 832*c^5*d^3*g + 224*b^3*c^
2*e^3*g + 128*c^5*d^2*e*f - 232*b*c^4*d*e^2*f + 1728*b*c^4*d^2*e*g - 1104*b^2*c^3*d*e^2*g)/(105*e^2*(b*e - 2*c
*d)^4) + (d*((d*((8*c^4*(15*b*e*g - 26*c*d*g + 2*c*e*f))/(105*(b*e - 2*c*d)^4) - (16*c^5*d*g)/(105*(b*e - 2*c*
d)^4)))/e - (328*b^2*c^3*e^3*g + 120*b*c^4*e^3*f - 208*c^5*d*e^2*f + 1088*c^5*d^2*e*g - 1192*b*c^4*d*e^2*g)/(1
05*e^2*(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) - (((160*b^2*c^3*e^3*f - 1
536*c^5*d^3*g + 312*b^3*c^2*e^3*g + 384*c^5*d^2*e*f - 504*b*c^4*d*e^2*f + 2784*b*c^4*d^2*e*g - 1632*b^2*c^3*d*
e^2*g)/(105*e^2*(b*e - 2*c*d)^4) + (d*((d*((8*c^4*(17*b*e*g - 30*c*d*g + 2*c*e*f))/(105*(b*e - 2*c*d)^4) - (16
*c^5*d*g)/(105*(b*e - 2*c*d)^4)))/e - (400*b^2*c^3*e^3*g + 136*b*c^4*e^3*f - 240*c^5*d*e^2*f + 1344*c^5*d^2*e*
g - 1464*b*c^4*d*e^2*g)/(105*e^2*(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*((d*((4*c^2*e*(6*b*e*g - 10*c*d*g + c*e*f))/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)) - (4*c^3*d*e*g)/(7*(
5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d))))/e - (4*c*(3*b*e - 5*c*d)*(3*b*e*g - 5*c*d*g + 2*c*e*f))/(7*(5*b*e^2 - 10*
c*d*e)*(b*e - 2*c*d))))/e + (4*(b*e - c*d)*(4*b^2*e^2*g + 16*c^2*d^2*g + 5*b*c*e^2*f - 9*c^2*d*e*f - 16*b*c*d*
e*g))/(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 - (((
2*b^3*e^2*g + 8*b*c^2*d^2*g + 4*b^2*c*e^2*f - 6*b*c^2*d*e*f - 8*b^2*c*d*e*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*
c*d)) - (d*((16*c^3*d^2*g - 12*c^3*d*e*f + 10*b*c^2*e^2*f + 8*b^2*c*e^2*g - 22*b*c^2*d*e*g)/(7*(5*b*e^2 - 10*c
*d*e)*(b*e - 2*c*d)) - (d*((2*c^2*e*(5*b*e*g - 6*c*d*g + 2*c*e*f))/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)) - (4
*c^3*d*e*g)/(7*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d))))/e))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d +
e*x)^3 + (((d*((d*((16*c^4*(11*b*e*g - 20*c*d*g + c*e*f))/(105*(b*e - 2*c*d)^4) - (16*c^5*d*g)/(105*(b*e - 2*c
*d)^4)))/e - (16*c^3*(45*b^2*e^2*g + 159*c^2*d^2*g + 11*b*c*e^2*f - 20*c^2*d*e*f - 169*b*c*d*e*g))/(105*e*(b*e
- 2*c*d)^4)))/e + (16*c^2*(b*e - c*d)*(35*b^2*e^2*g + 140*c^2*d^2*g + 10*b*c*e^2*f - 19*c^2*d*e*f - 140*b*c*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) + (((2*f*(b*e - c*d)^
2)/(7*b*e^2 - 14*c*d*e) + (d*((d*((2*c*e*(2*b*e*g - 2*c*d*g + c*e*f))/(7*b*e^2 - 14*c*d*e) - (2*c^2*d*e*g)/(7*
b*e^2 - 14*c*d*e)))/e - (2*(b*e - c*d)*(b*e*g - c*d*g + 2*c*e*f))/(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*((d*((4*c^3*e*(11*b*e*g - 18*c*d*g + 2*c*e*f))/(35*(3*b*e^2 - 6*
c*d*e)*(b*e - 2*c*d)^2) - (8*c^4*d*e*g)/(35*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2)))/e - (76*b^2*c^2*e^3*g + 44*
b*c^3*e^3*f - 72*c^4*d*e^2*f + 224*c^4*d^2*e*g - 260*b*c^3*d*e^2*g)/(35*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2)
))/e + (44*b^2*c^2*e^3*f - 192*c^4*d^3*g + 40*b^3*c*e^3*g + 96*c^4*d^2*e*f - 132*b*c^3*d*e^2*f + 352*b*c^3*d^2
*e*g - 208*b^2*c^2*d*e^2*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

________________________________________________________________________________________

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 {3}{2}} \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)**(3/2)/(e*x+d)**6,x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**6, x)

________________________________________________________________________________________