3.20.62 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=210 \[ -\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+14 c d g+4 c e f)}{315 e^2 (d+e x)^5 (2 c d-b e)^3}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+14 c d g+4 c e f)}{63 e^2 (d+e x)^6 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 e^2 (d+e x)^7 (2 c d-b e)} \]

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Rubi [A]  time = 0.35, 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 )^{5/2} (-9 b e g+14 c d g+4 c e f)}{315 e^2 (d+e x)^5 (2 c d-b e)^3}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+14 c d g+4 c e f)}{63 e^2 (d+e x)^6 (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}}{9 e^2 (d+e x)^7 (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)^7,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9*e^2*(2*c*d - b*e)*(d + e*x)^7) - (2*(4*c*e*f +
14*c*d*g - 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(63*e^2*(2*c*d - b*e)^2*(d + e*x)^6) - (4*c*
(4*c*e*f + 14*c*d*g - 9*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(315*e^2*(2*c*d - b*e)^3*(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 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) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^7} \, 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}}{9 e^2 (2 c d-b e) (d+e x)^7}+\frac {(4 c e f+14 c d g-9 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx}{9 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 )^{5/2}}{9 e^2 (2 c d-b e) (d+e x)^7}-\frac {2 (4 c e f+14 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{63 e^2 (2 c d-b e)^2 (d+e x)^6}+\frac {(2 c (4 c e f+14 c d g-9 b e g)) \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}{63 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 )^{5/2}}{9 e^2 (2 c d-b e) (d+e x)^7}-\frac {2 (4 c e f+14 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{63 e^2 (2 c d-b e)^2 (d+e x)^6}-\frac {4 c (4 c e f+14 c d g-9 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{315 e^2 (2 c d-b e)^3 (d+e x)^5}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 168, normalized size = 0.80 \begin {gather*} \frac {2 (b e-c d+c e x)^2 \sqrt {(d+e x) (c (d-e x)-b e)} \left (5 b^2 e^2 (2 d g+7 e f+9 e g x)-2 b c e \left (19 d^2 g+d e (80 f+98 g x)+e^2 x (10 f+9 g x)\right )+4 c^2 \left (7 d^3 g+d^2 e (47 f+49 g x)+7 d e^2 x (2 f+g x)+2 e^3 f x^2\right )\right )}{315 e^2 (d+e x)^5 (b e-2 c d)^3} \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)^7,x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(5*b^2*e^2*(7*e*f + 2*d*g + 9*e*g*x) + 4*c^
2*(7*d^3*g + 2*e^3*f*x^2 + 7*d*e^2*x*(2*f + g*x) + d^2*e*(47*f + 49*g*x)) - 2*b*c*e*(19*d^2*g + e^2*x*(10*f +
9*g*x) + d*e*(80*f + 98*g*x))))/(315*e^2*(-2*c*d + b*e)^3*(d + e*x)^5)

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IntegrateAlgebraic [F]  time = 180.20, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[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)^7,x]

[Out]

$Aborted

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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)^(3/2)/(e*x+d)^7,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)^(3/2)/(e*x+d)^7,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.05, size = 236, normalized size = 1.12 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-18 b c \,e^{3} g \,x^{2}+28 c^{2} d \,e^{2} g \,x^{2}+8 c^{2} e^{3} f \,x^{2}+45 b^{2} e^{3} g x -196 b c d \,e^{2} g x -20 b c \,e^{3} f x +196 c^{2} d^{2} e g x +56 c^{2} d \,e^{2} f x +10 b^{2} d \,e^{2} g +35 b^{2} e^{3} f -38 b c \,d^{2} e g -160 b c d \,e^{2} f +28 c^{2} d^{3} g +188 c^{2} d^{2} e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{315 \left (e x +d \right )^{6} \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)^(3/2)/(e*x+d)^7,x)

[Out]

-2/315*(c*e*x+b*e-c*d)*(-18*b*c*e^3*g*x^2+28*c^2*d*e^2*g*x^2+8*c^2*e^3*f*x^2+45*b^2*e^3*g*x-196*b*c*d*e^2*g*x-
20*b*c*e^3*f*x+196*c^2*d^2*e*g*x+56*c^2*d*e^2*f*x+10*b^2*d*e^2*g+35*b^2*e^3*f-38*b*c*d^2*e*g-160*b*c*d*e^2*f+2
8*c^2*d^3*g+188*c^2*d^2*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^6/(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)^(3/2)/(e*x+d)^7,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 = 19.83, size = 8039, normalized size = 38.28

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)^7,x)

[Out]

(((d*((d*((32*c^5*(4*b*e*g - 6*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e
- (608*c^6*d^2*g + 208*b^2*c^4*e^2*g - 192*c^6*d*e*f + 128*b*c^5*e^2*f - 704*b*c^5*d*e*g)/(945*e*(b*e - 2*c*d)
^5)))/e + (4*b*c^3*(19*b^2*e^2*g + 76*c^2*d^2*g + 14*b*c*e^2*f - 24*c^2*d*e*f - 76*b*c*d*e*g))/(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*((d*((16*c^5*(13*b*e*g - 22*c*d*g + 2*
c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (1568*c^6*d^2*g + 488*b^2*c^4*e^2*g -
352*c^6*d*e*f + 208*b*c^5*e^2*f - 1744*b*c^5*d*e*g)/(945*e*(b*e - 2*c*d)^5)))/e + (4*b*c^3*(49*b^2*e^2*g + 19
6*c^2*d^2*g + 24*b*c*e^2*f - 44*c^2*d*e*f - 196*b*c*d*e*g))/(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*((d*((16*c^5*(15*b*e*g - 26*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32
*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (1952*c^6*d^2*g + 600*b^2*c^4*e^2*g - 416*c^6*d*e*f + 240*b*c^5*e^2*f -
2160*b*c^5*d*e*g)/(945*e*(b*e - 2*c*d)^5)))/e + (4*b*c^3*(61*b^2*e^2*g + 244*c^2*d^2*g + 28*b*c*e^2*f - 52*c^2
*d*e*f - 244*b*c*d*e*g))/(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*((d*((16*c^5*(17*b*e*g - 30*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e
- (2336*c^6*d^2*g + 712*b^2*c^4*e^2*g - 480*c^6*d*e*f + 272*b*c^5*e^2*f - 2576*b*c^5*d*e*g)/(945*e*(b*e - 2*c
*d)^5)))/e + (4*b*c^3*(73*b^2*e^2*g + 292*c^2*d^2*g + 32*b*c*e^2*f - 60*c^2*d*e*f - 292*b*c*d*e*g))/(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*((d*((32*c^5*(10*b*e*g - 18*c*d*g
+ c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (4032*c^6*d^2*g + 1160*b^2*c^4*e^2
*g - 576*c^6*d*e*f + 320*b*c^5*e^2*f - 4320*b*c^5*d*e*g)/(945*e*(b*e - 2*c*d)^5)))/e + (8*b*c^3*(63*b^2*e^2*g
+ 252*c^2*d^2*g + 19*b*c*e^2*f - 36*c^2*d*e*f - 252*b*c*d*e*g))/(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*((d*((32*c^5*(11*b*e*g - 20*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (
32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (4736*c^6*d^2*g + 1352*b^2*c^4*e^2*g - 640*c^6*d*e*f + 352*b*c^5*e^2*f
- 5056*b*c^5*d*e*g)/(945*e*(b*e - 2*c*d)^5)))/e + (8*b*c^3*(74*b^2*e^2*g + 296*c^2*d^2*g + 21*b*c*e^2*f - 40*
c^2*d*e*f - 296*b*c*d*e*g))/(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*((d*((32*c^5*(12*b*e*g - 22*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/
e - (5696*c^6*d^2*g + 1608*b^2*c^4*e^2*g - 704*c^6*d*e*f + 384*b*c^5*e^2*f - 6048*b*c^5*d*e*g)/(945*e*(b*e - 2
*c*d)^5)))/e + (8*b*c^3*(89*b^2*e^2*g + 356*c^2*d^2*g + 23*b*c*e^2*f - 44*c^2*d*e*f - 356*b*c*d*e*g))/(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*((d*((16*c^5*(29*b*e*g - 54*c*d
*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (9216*c^6*d^2*g + 2528*b^2*c^4
*e^2*g - 864*c^6*d*e*f + 464*b*c^5*e^2*f - 9648*b*c^5*d*e*g)/(945*e*(b*e - 2*c*d)^5)))/e + (16*b*c^3*(72*b^2*e
^2*g + 288*c^2*d^2*g + 14*b*c*e^2*f - 27*c^2*d*e*f - 288*b*c*d*e*g))/(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*((d*((8*c^3*e*(11*b*e*g - 20*c*d*g + c*e*f))/(63*(5*b*e^2 - 10*c
*d*e)*(b*e - 2*c*d)^2) - (8*c^4*d*e*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e - (312*b^2*c^2*e^3*g + 88
*b*c^3*e^3*f - 160*c^4*d*e^2*f + 1080*c^4*d^2*e*g - 1160*b*c^3*d*e^2*g)/(63*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*
d)^2)))/e + (8*c*(b*e - c*d)*(29*b^2*e^2*g + 116*c^2*d^2*g + 10*b*c*e^2*f - 19*c^2*d*e*f - 116*b*c*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*((d*
((8*c^3*e*(3*b*e*g - 4*c*d*g + c*e*f))/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (8*c^4*d*e*g)/(63*(5*b*e^2
- 10*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)/(63*(5*b*e^2 - 10*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))/(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*((d*((4*c^3*e*(15*b*e*g - 26*c*d*g + 2*c*e*f))/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2
) - (8*c^4*d*e*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e - (352*c^4*d^2*g + 116*b^2*c^2*e^2*g - 104*c^4
*d*e*f + 60*b*c^3*e^2*f - 404*b*c^3*d*e*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e + (4*b*c*(11*b^2*e^2*
g + 44*c^2*d^2*g + 7*b*c*e^2*f - 13*c^2*d*e*f - 44*b*c*d*e*g))/(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 - (((176*b^2*c^4*e^3*f - 1216*c^6*d^3*g + 252*b^3*c^3*e^3
*g + 384*c^6*d^2*e*f - 528*b*c^5*d*e^2*f + 2224*b*c^5*d^2*e*g - 1312*b^2*c^4*d*e^2*g)/(945*e^2*(b*e - 2*c*d)^5
) + (d*((d*((16*c^5*(11*b*e*g - 18*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5
)))/e - (376*b^2*c^4*e^3*g + 176*b*c^5*e^3*f - 288*c^6*d*e^2*f + 1184*c^6*d^2*e*g - 1328*b*c^5*d*e^2*g)/(945*e
^2*(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) + (((176*b^2*c^4*e^3*f - 1888*
c^6*d^3*g + 536*b^3*c^3*e^3*g + 224*c^6*d^2*e*f - 448*b*c^5*d*e^2*f + 4032*b*c^5*d^2*e*g - 2616*b^2*c^4*d*e^2*
g)/(945*e^2*(b*e - 2*c*d)^5) + (d*((d*((32*c^5*(8*b*e*g - 14*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d
*g)/(945*(b*e - 2*c*d)^5)))/e - (776*b^2*c^4*e^3*g + 256*b*c^5*e^3*f - 448*c^6*d*e^2*f + 2624*c^6*d^2*e*g - 28
48*b*c^5*d*e^2*g)/(945*e^2*(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) + (((3
44*b^2*c^4*e^3*f - 3904*c^6*d^3*g + 784*b^3*c^3*e^3*g + 832*c^6*d^2*e*f - 1088*b*c^5*d*e^2*f + 7040*b*c^5*d^2*
e*g - 4112*b^2*c^4*d*e^2*g)/(945*e^2*(b*e - 2*c*d)^5) + (d*((d*((32*c^5*(9*b*e*g - 16*c*d*g + c*e*f))/(945*(b*
e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (936*b^2*c^4*e^3*g + 288*b*c^5*e^3*f - 512*c^6*d*e^2*
f + 3200*c^6*d^2*e*g - 3456*b*c^5*d*e^2*g)/(945*e^2*(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) + (((392*b^2*c^4*e^3*f - 4672*c^6*d^3*g + 936*b^3*c^3*e^3*g + 960*c^6*d^2*e*f - 1248*b*c^5*
d*e^2*f + 8416*b*c^5*d^2*e*g - 4912*b^2*c^4*d*e^2*g)/(945*e^2*(b*e - 2*c*d)^5) + (d*((d*((32*c^5*(10*b*e*g - 1
8*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (1096*b^2*c^4*e^3*g + 320*b
*c^5*e^3*f - 576*c^6*d*e^2*f + 3776*c^6*d^2*e*g - 4064*b*c^5*d*e^2*g)/(945*e^2*(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*((d*((16*c^4*e*(6*b*e*g - 10*c*d*g + c*e*f))/(315*(3*b*e^
2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (196*b^2*c^3*e^
3*g + 96*b*c^4*e^3*f - 160*c^5*d*e^2*f + 608*c^5*d^2*e*g - 688*b*c^4*d*e^2*g)/(315*e*(3*b*e^2 - 6*c*d*e)*(b*e
- 2*c*d)^3)))/e + (104*b^2*c^3*e^3*f - 624*c^5*d^3*g + 124*b^3*c^2*e^3*g + 240*c^5*d^2*e*f - 320*b*c^4*d*e^2*f
+ 1120*b*c^4*d^2*e*g - 652*b^2*c^3*d*e^2*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*((d*((8*c^4*e*(19*b*e*g - 34*c*d*g + 2*c*e*f))/(315*(3*b*e^2 - 6*c
*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (504*b^2*c^3*e^3*g + 1
52*b*c^4*e^3*f - 272*c^5*d*e^2*f + 1728*c^5*d^2*e*g - 1864*b*c^4*d*e^2*g)/(315*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*
c*d)^3)))/e + (88*b^2*c^3*e^3*f - 1280*c^5*d^3*g + 368*b^3*c^2*e^3*g + 64*c^5*d^2*e*f - 200*b*c^4*d*e^2*f + 27
52*b*c^4*d^2*e*g - 1792*b^2*c^3*d*e^2*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*((d*((8*c^4*e*(21*b*e*g - 38*c*d*g + 2*c*e*f))/(315*(3*b*e^2 - 6*c*d*e
)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (592*b^2*c^3*e^3*g + 168*b
*c^4*e^3*f - 304*c^5*d*e^2*f + 2048*c^5*d^2*e*g - 2200*b*c^4*d*e^2*g)/(315*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)
^3)))/e + (248*b^2*c^3*e^3*f - 2784*c^5*d^3*g + 520*b^3*c^2*e^3*g + 672*c^5*d^2*e*f - 824*b*c^4*d*e^2*f + 4864
*b*c^4*d^2*e*g - 2776*b^2*c^3*d*e^2*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 + (((3904*c^6*d^3*g - 176*b^2*c^4*e^3*f - 1328*b^3*c^3*e^3*g + 64*c^6*d^2*e*f +
304*b*c^5*d*e^2*f - 9216*b*c^5*d^2*e*g + 6288*b^2*c^4*d*e^2*g)/(945*e^2*(b*e - 2*c*d)^5) - (d*((d*((16*c^5*(2
5*b*e*g - 46*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (16*c^4*(116*b
^2*e^2*g + 416*c^2*d^2*g + 25*b*c*e^2*f - 46*c^2*d*e*f - 439*b*c*d*e*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) - (((432*b^2*c^4*e^3*f - 5504*c^6*d^3*g + 1280*b^3*c^3*e^3*g +
1024*c^6*d^2*e*f - 1360*b*c^5*d*e^2*f + 10624*b*c^5*d^2*e*g - 6496*b^2*c^4*d*e^2*g)/(945*e^2*(b*e - 2*c*d)^5)
+ (d*((d*((16*c^5*(23*b*e*g - 42*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5))
)/e - (16*c^4*(101*b^2*e^2*g + 360*c^2*d^2*g + 23*b*c*e^2*f - 42*c^2*d*e*f - 381*b*c*d*e*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) - (((560*b^2*c^4*e^3*f - 11392*c^6*d^3*g +
2224*b^3*c^3*e^3*g + 1408*c^6*d^2*e*f - 1808*b*c^5*d*e^2*f + 20288*b*c^5*d^2*e*g - 11744*b^2*c^4*d*e^2*g)/(945
*e^2*(b*e - 2*c*d)^5) + (d*((d*((16*c^5*(27*b*e*g - 50*c*d*g + 2*c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/
(945*(b*e - 2*c*d)^5)))/e - (16*c^4*(135*b^2*e^2*g + 488*c^2*d^2*g + 27*b*c*e^2*f - 50*c^2*d*e*f - 513*b*c*d*e
*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*((d*((4*c^2*e*
(7*b*e*g - 12*c*d*g + c*e*f))/(9*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d)) - (4*c^3*d*e*g)/(9*(7*b*e^2 - 14*c*d*e)*(
b*e - 2*c*d))))/e - (28*b*c^2*e^3*f + 44*b^2*c*e^3*g - 48*c^3*d*e^2*f + 124*c^3*d^2*e*g - 148*b*c^2*d*e^2*g)/(
9*e*(7*b*e^2 - 14*c*d*e)*(b*e - 2*c*d))))/e + (4*(b*e - c*d)*(5*b^2*e^2*g + 20*c^2*d^2*g + 6*b*c*e^2*f - 11*c^
2*d*e*f - 20*b*c*d*e*g))/(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 - (((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)/(9*(7*b*e^2 -
14*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)
/(9*(7*b*e^2 - 14*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))/(9*(7*b*e^2 - 14*c*d*e)*
(b*e - 2*c*d)) - (4*c^3*d*e*g)/(9*(7*b*e^2 - 14*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)^4 - (((d*((d*((8*c^4*e*(7*b*e*g - 10*c*d*g + 2*c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e -
2*c*d)^3) - (16*c^5*d*e*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/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)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e + (2*b*c^2*(1
3*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))/(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*((d*((16*c^4*e*(7*b*e*g - 12*c*d*g + c*
e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/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)/(315*(3*b*e^2 - 6*c
*d*e)*(b*e - 2*c*d)^3)))/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))/(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
*((d*((16*c^4*e*(8*b*e*g - 14*c*d*g + c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(315*
(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (928*c^5*d^2*g + 292*b^2*c^3*e^2*g - 224*c^5*d*e*f + 128*b*c^4*e^2*
f - 1040*b*c^4*d*e*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e + (4*b*c^2*(29*b^2*e^2*g + 116*c^2*d^2*g +
15*b*c*e^2*f - 28*c^2*d*e*f - 116*b*c*d*e*g))/(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*((d*((8*c^4*e*(23*b*e*g - 42*c*d*g + 2*c*e*f))/(315*(3*b*e^2 - 6*c
*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (2496*c^5*d^2*g + 712*
b^2*c^3*e^2*g - 336*c^5*d*e*f + 184*b*c^4*e^2*f - 2664*b*c^4*d*e*g)/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3))
)/e + (8*b*c^2*(39*b^2*e^2*g + 156*c^2*d^2*g + 11*b*c*e^2*f - 21*c^2*d*e*f - 156*b*c*d*e*g))/(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*((d*((32*c^5*(16*b*e
*g - 30*c*d*g + c*e*f))/(945*(b*e - 2*c*d)^5) - (32*c^6*d*g)/(945*(b*e - 2*c*d)^5)))/e - (32*c^4*(100*b^2*e^2*
g + 369*c^2*d^2*g + 16*b*c*e^2*f - 30*c^2*d*e*f - 384*b*c*d*e*g))/(945*e*(b*e - 2*c*d)^5)))/e + (32*c^3*(b*e -
c*d)*(85*b^2*e^2*g + 340*c^2*d^2*g + 15*b*c*e^2*f - 29*c^2*d*e*f - 340*b*c*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) + (((2*f*(b*e - c*d)^2)/(9*b*e^2 - 18*c*d*e) + (d*((d*
((2*c*e*(2*b*e*g - 2*c*d*g + c*e*f))/(9*b*e^2 - 18*c*d*e) - (2*c^2*d*e*g)/(9*b*e^2 - 18*c*d*e)))/e - (2*(b*e -
c*d)*(b*e*g - c*d*g + 2*c*e*f))/(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*((d*((4*c^3*e*(13*b*e*g - 22*c*d*g + 2*c*e*f))/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2) - (8*c^4
*d*e*g)/(63*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e - (96*b^2*c^2*e^3*g + 52*b*c^3*e^3*f - 88*c^4*d*e^2*f +
288*c^4*d^2*e*g - 332*b*c^3*d*e^2*g)/(63*e*(5*b*e^2 - 10*c*d*e)*(b*e - 2*c*d)^2)))/e + (56*b^2*c^2*e^3*f - 256
*c^4*d^3*g + 52*b^3*c*e^3*g + 128*c^4*d^2*e*f - 172*b*c^3*d*e^2*f + 464*b*c^3*d^2*e*g - 272*b^2*c^2*d*e^2*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*((
d*((16*c^4*e*(14*b*e*g - 26*c*d*g + c*e*f))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3) - (16*c^5*d*e*g)/(315*(3
*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e - (16*c^3*(72*b^2*e^2*g + 261*c^2*d^2*g + 14*b*c*e^2*f - 26*c^2*d*e*f -
274*b*c*d*e*g))/(315*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^3)))/e + (16*c^2*(b*e - c*d)*(59*b^2*e^2*g + 236*c^2*d
^2*g + 13*b*c*e^2*f - 25*c^2*d*e*f - 236*b*c*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

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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 )^{7}}\, 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)**7,x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**7, x)

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