3.2181 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=360 \[ -\frac{32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{3465 e^2 (d+e x)^3 (2 c d-b e)^5}-\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{1155 e^2 (d+e x)^4 (2 c d-b e)^4}-\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{231 e^2 (d+e x)^5 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{99 e^2 (d+e x)^6 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 e^2 (d+e x)^7 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(11*e^2*(2*c*d - b*e)*(d + e*x)^7) - (2*(8*c*e*f
+ 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^6) - (4*
c*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(231*e^2*(2*c*d - b*e)^3*(d + e
*x)^5) - (16*c^2*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(1155*e^2*(2*c*d
- b*e)^4*(d + e*x)^4) - (32*c^3*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/
(3465*e^2*(2*c*d - b*e)^5*(d + e*x)^3)

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Rubi [A]  time = 0.575811, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068, Rules used = {792, 658, 650} \[ -\frac{32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{3465 e^2 (d+e x)^3 (2 c d-b e)^5}-\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{1155 e^2 (d+e x)^4 (2 c d-b e)^4}-\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{231 e^2 (d+e x)^5 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+14 c d g+8 c e f)}{99 e^2 (d+e x)^6 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 e^2 (d+e x)^7 (2 c d-b e)} \]

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)^7,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(11*e^2*(2*c*d - b*e)*(d + e*x)^7) - (2*(8*c*e*f
+ 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^6) - (4*
c*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(231*e^2*(2*c*d - b*e)^3*(d + e
*x)^5) - (16*c^2*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(1155*e^2*(2*c*d
- b*e)^4*(d + e*x)^4) - (32*c^3*(8*c*e*f + 14*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/
(3465*e^2*(2*c*d - b*e)^5*(d + e*x)^3)

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]

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 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]

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)^7} \, 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}}{11 e^2 (2 c d-b e) (d+e x)^7}+\frac{(8 c e f+14 c d g-11 b e g) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx}{11 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}}{11 e^2 (2 c d-b e) (d+e x)^7}-\frac{2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^6}+\frac{(2 c (8 c e f+14 c d g-11 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}{33 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}}{11 e^2 (2 c d-b e) (d+e x)^7}-\frac{2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^6}-\frac{4 c (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{231 e^2 (2 c d-b e)^3 (d+e x)^5}+\frac{\left (8 c^2 (8 c e f+14 c d g-11 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)^4} \, dx}{231 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}}{11 e^2 (2 c d-b e) (d+e x)^7}-\frac{2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^6}-\frac{4 c (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{231 e^2 (2 c d-b e)^3 (d+e x)^5}-\frac{16 c^2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1155 e^2 (2 c d-b e)^4 (d+e x)^4}+\frac{\left (16 c^3 (8 c e f+14 c d g-11 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}{1155 e (2 c d-b e)^4}\\ &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 e^2 (2 c d-b e) (d+e x)^7}-\frac{2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^6}-\frac{4 c (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{231 e^2 (2 c d-b e)^3 (d+e x)^5}-\frac{16 c^2 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1155 e^2 (2 c d-b e)^4 (d+e x)^4}-\frac{32 c^3 (8 c e f+14 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3465 e^2 (2 c d-b e)^5 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.304279, size = 335, normalized size = 0.93 \[ \frac{2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (12 b^2 c^2 e^2 \left (d^2 e (790 f+986 g x)+167 d^3 g+d e^2 x (180 f+211 g x)+2 e^3 x^2 (10 f+11 g x)\right )-10 b^3 c e^3 \left (61 d^2 g+d e (280 f+346 g x)+e^2 x (28 f+33 g x)\right )+35 b^4 e^4 (2 d g+9 e f+11 e g x)-8 b c^3 e \left (3 d^2 e^2 x (244 f+277 g x)+2 d^3 e (912 f+1141 g x)+365 d^4 g+4 d e^3 x^2 (48 f+49 g x)+2 e^4 x^3 (12 f+11 g x)\right )+16 c^4 \left (2 d^2 e^3 x^2 (90 f+49 g x)+7 d^3 e^2 x (52 f+45 g x)+d^4 e (547 f+637 g x)+91 d^5 g+14 d e^4 x^3 (4 f+g x)+8 e^5 f x^4\right )\right )}{3465 e^2 (d+e x)^7 (b e-2 c d)^5} \]

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)^7,x]

[Out]

(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(35*b^4*e^4*(9*e*f + 2*d*g + 11*e*g*x) - 10*b^3*c*e^3*(61*d^2*g +
e^2*x*(28*f + 33*g*x) + d*e*(280*f + 346*g*x)) + 16*c^4*(91*d^5*g + 8*e^5*f*x^4 + 14*d*e^4*x^3*(4*f + g*x) + 7
*d^3*e^2*x*(52*f + 45*g*x) + 2*d^2*e^3*x^2*(90*f + 49*g*x) + d^4*e*(547*f + 637*g*x)) + 12*b^2*c^2*e^2*(167*d^
3*g + 2*e^3*x^2*(10*f + 11*g*x) + d*e^2*x*(180*f + 211*g*x) + d^2*e*(790*f + 986*g*x)) - 8*b*c^3*e*(365*d^4*g
+ 2*e^4*x^3*(12*f + 11*g*x) + 4*d*e^3*x^2*(48*f + 49*g*x) + 3*d^2*e^2*x*(244*f + 277*g*x) + 2*d^3*e*(912*f + 1
141*g*x))))/(3465*e^2*(-2*c*d + b*e)^5*(d + e*x)^7)

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Maple [A]  time = 0.011, size = 564, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -176\,b{c}^{3}{e}^{5}g{x}^{4}+224\,{c}^{4}d{e}^{4}g{x}^{4}+128\,{c}^{4}{e}^{5}f{x}^{4}+264\,{b}^{2}{c}^{2}{e}^{5}g{x}^{3}-1568\,b{c}^{3}d{e}^{4}g{x}^{3}-192\,b{c}^{3}{e}^{5}f{x}^{3}+1568\,{c}^{4}{d}^{2}{e}^{3}g{x}^{3}+896\,{c}^{4}d{e}^{4}f{x}^{3}-330\,{b}^{3}c{e}^{5}g{x}^{2}+2532\,{b}^{2}{c}^{2}d{e}^{4}g{x}^{2}+240\,{b}^{2}{c}^{2}{e}^{5}f{x}^{2}-6648\,b{c}^{3}{d}^{2}{e}^{3}g{x}^{2}-1536\,b{c}^{3}d{e}^{4}f{x}^{2}+5040\,{c}^{4}{d}^{3}{e}^{2}g{x}^{2}+2880\,{c}^{4}{d}^{2}{e}^{3}f{x}^{2}+385\,{b}^{4}{e}^{5}gx-3460\,{b}^{3}cd{e}^{4}gx-280\,{b}^{3}c{e}^{5}fx+11832\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}gx+2160\,{b}^{2}{c}^{2}d{e}^{4}fx-18256\,b{c}^{3}{d}^{3}{e}^{2}gx-5856\,b{c}^{3}{d}^{2}{e}^{3}fx+10192\,{c}^{4}{d}^{4}egx+5824\,{c}^{4}{d}^{3}{e}^{2}fx+70\,{b}^{4}d{e}^{4}g+315\,{b}^{4}{e}^{5}f-610\,{b}^{3}c{d}^{2}{e}^{3}g-2800\,{b}^{3}cd{e}^{4}f+2004\,{b}^{2}{c}^{2}{d}^{3}{e}^{2}g+9480\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}f-2920\,b{c}^{3}{d}^{4}eg-14592\,b{c}^{3}{d}^{3}{e}^{2}f+1456\,{c}^{4}{d}^{5}g+8752\,{c}^{4}{d}^{4}ef \right ) }{3465\, \left ( ex+d \right ) ^{6}{e}^{2} \left ({b}^{5}{e}^{5}-10\,{b}^{4}cd{e}^{4}+40\,{b}^{3}{c}^{2}{d}^{2}{e}^{3}-80\,{b}^{2}{c}^{3}{d}^{3}{e}^{2}+80\,b{c}^{4}{d}^{4}e-32\,{c}^{5}{d}^{5} \right ) }\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}} \end{align*}

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)^7,x)

[Out]

-2/3465*(c*e*x+b*e-c*d)*(-176*b*c^3*e^5*g*x^4+224*c^4*d*e^4*g*x^4+128*c^4*e^5*f*x^4+264*b^2*c^2*e^5*g*x^3-1568
*b*c^3*d*e^4*g*x^3-192*b*c^3*e^5*f*x^3+1568*c^4*d^2*e^3*g*x^3+896*c^4*d*e^4*f*x^3-330*b^3*c*e^5*g*x^2+2532*b^2
*c^2*d*e^4*g*x^2+240*b^2*c^2*e^5*f*x^2-6648*b*c^3*d^2*e^3*g*x^2-1536*b*c^3*d*e^4*f*x^2+5040*c^4*d^3*e^2*g*x^2+
2880*c^4*d^2*e^3*f*x^2+385*b^4*e^5*g*x-3460*b^3*c*d*e^4*g*x-280*b^3*c*e^5*f*x+11832*b^2*c^2*d^2*e^3*g*x+2160*b
^2*c^2*d*e^4*f*x-18256*b*c^3*d^3*e^2*g*x-5856*b*c^3*d^2*e^3*f*x+10192*c^4*d^4*e*g*x+5824*c^4*d^3*e^2*f*x+70*b^
4*d*e^4*g+315*b^4*e^5*f-610*b^3*c*d^2*e^3*g-2800*b^3*c*d*e^4*f+2004*b^2*c^2*d^3*e^2*g+9480*b^2*c^2*d^2*e^3*f-2
920*b*c^3*d^4*e*g-14592*b*c^3*d^3*e^2*f+1456*c^4*d^5*g+8752*c^4*d^4*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2
)/(e*x+d)^6/e^2/(b^5*e^5-10*b^4*c*d*e^4+40*b^3*c^2*d^2*e^3-80*b^2*c^3*d^3*e^2+80*b*c^4*d^4*e-32*c^5*d^5)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)^7,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**7,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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)^7,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError