3.2208 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=360 \[ -\frac{32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{45045 e^2 (d+e x)^7 (2 c d-b e)^5}-\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{6435 e^2 (d+e x)^8 (2 c d-b e)^4}-\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{715 e^2 (d+e x)^9 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{195 e^2 (d+e x)^{10} (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (d+e x)^{11} (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(15*e^2*(2*c*d - b*e)*(d + e*x)^11) - (2*(8*c*e*f
+ 22*c*d*g - 15*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(195*e^2*(2*c*d - b*e)^2*(d + e*x)^10) -
(4*c*(8*c*e*f + 22*c*d*g - 15*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(715*e^2*(2*c*d - b*e)^3*(d
+ e*x)^9) - (16*c^2*(8*c*e*f + 22*c*d*g - 15*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(6435*e^2*(2*
c*d - b*e)^4*(d + e*x)^8) - (32*c^3*(8*c*e*f + 22*c*d*g - 15*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2
))/(45045*e^2*(2*c*d - b*e)^5*(d + e*x)^7)

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Rubi [A]  time = 0.580618, 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} \[ -\frac{32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{45045 e^2 (d+e x)^7 (2 c d-b e)^5}-\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{6435 e^2 (d+e x)^8 (2 c d-b e)^4}-\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{715 e^2 (d+e x)^9 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-15 b e g+22 c d g+8 c e f)}{195 e^2 (d+e x)^{10} (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (d+e x)^{11} (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^11,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(15*e^2*(2*c*d - b*e)*(d + e*x)^11) - (2*(8*c*e*f
+ 22*c*d*g - 15*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(195*e^2*(2*c*d - b*e)^2*(d + e*x)^10) -
(4*c*(8*c*e*f + 22*c*d*g - 15*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(715*e^2*(2*c*d - b*e)^3*(d
+ e*x)^9) - (16*c^2*(8*c*e*f + 22*c*d*g - 15*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(6435*e^2*(2*
c*d - b*e)^4*(d + e*x)^8) - (32*c^3*(8*c*e*f + 22*c*d*g - 15*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2
))/(45045*e^2*(2*c*d - b*e)^5*(d + e*x)^7)

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) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{11}} \, dx &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 e^2 (2 c d-b e) (d+e x)^{11}}+\frac{(8 c e f+22 c d g-15 b e g) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx}{15 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 )^{7/2}}{15 e^2 (2 c d-b e) (d+e x)^{11}}-\frac{2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 e^2 (2 c d-b e)^2 (d+e x)^{10}}+\frac{(2 c (8 c e f+22 c d g-15 b e g)) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^9} \, dx}{65 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 )^{7/2}}{15 e^2 (2 c d-b e) (d+e x)^{11}}-\frac{2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 e^2 (2 c d-b e)^2 (d+e x)^{10}}-\frac{4 c (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 e^2 (2 c d-b e)^3 (d+e x)^9}+\frac{\left (8 c^2 (8 c e f+22 c d g-15 b e g)\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx}{715 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 )^{7/2}}{15 e^2 (2 c d-b e) (d+e x)^{11}}-\frac{2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 e^2 (2 c d-b e)^2 (d+e x)^{10}}-\frac{4 c (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 e^2 (2 c d-b e)^3 (d+e x)^9}-\frac{16 c^2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6435 e^2 (2 c d-b e)^4 (d+e x)^8}+\frac{\left (16 c^3 (8 c e f+22 c d g-15 b e g)\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx}{6435 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 )^{7/2}}{15 e^2 (2 c d-b e) (d+e x)^{11}}-\frac{2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{195 e^2 (2 c d-b e)^2 (d+e x)^{10}}-\frac{4 c (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{715 e^2 (2 c d-b e)^3 (d+e x)^9}-\frac{16 c^2 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6435 e^2 (2 c d-b e)^4 (d+e x)^8}-\frac{32 c^3 (8 c e f+22 c d g-15 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{45045 e^2 (2 c d-b e)^5 (d+e x)^7}\\ \end{align*}

Mathematica [A]  time = 0.353064, size = 351, normalized size = 0.98 \[ -\frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (84 b^2 c^2 e^2 \left (6 d^2 e (167 f+189 g x)+133 d^3 g+3 d e^2 x (52 f+51 g x)+2 e^3 x^2 (6 f+5 g x)\right )-42 b^3 c e^3 \left (89 d^2 g+d e (616 f+706 g x)+e^2 x (44 f+45 g x)\right )+231 b^4 e^4 (2 d g+13 e f+15 e g x)-8 b c^3 e \left (3 d^2 e^2 x (1316 f+1201 g x)+2 d^3 e (7672 f+8481 g x)+1801 d^4 g+4 d e^3 x^2 (168 f+121 g x)+2 e^4 x^3 (28 f+15 g x)\right )+16 c^4 \left (2 d^2 e^3 x^2 (234 f+121 g x)+11 d^3 e^2 x (148 f+117 g x)+d^4 e (4243 f+4477 g x)+407 d^5 g+22 d e^4 x^3 (4 f+g x)+8 e^5 f x^4\right )\right )}{45045 e^2 (d+e x)^8 (b e-2 c d)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^11,x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(231*b^4*e^4*(13*e*f + 2*d*g + 15*e*g*x) +
84*b^2*c^2*e^2*(133*d^3*g + 2*e^3*x^2*(6*f + 5*g*x) + 3*d*e^2*x*(52*f + 51*g*x) + 6*d^2*e*(167*f + 189*g*x))
- 42*b^3*c*e^3*(89*d^2*g + e^2*x*(44*f + 45*g*x) + d*e*(616*f + 706*g*x)) + 16*c^4*(407*d^5*g + 8*e^5*f*x^4 +
22*d*e^4*x^3*(4*f + g*x) + 11*d^3*e^2*x*(148*f + 117*g*x) + 2*d^2*e^3*x^2*(234*f + 121*g*x) + d^4*e*(4243*f +
4477*g*x)) - 8*b*c^3*e*(1801*d^4*g + 2*e^4*x^3*(28*f + 15*g*x) + 4*d*e^3*x^2*(168*f + 121*g*x) + 3*d^2*e^2*x*(
1316*f + 1201*g*x) + 2*d^3*e*(7672*f + 8481*g*x))))/(45045*e^2*(-2*c*d + b*e)^5*(d + e*x)^8)

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Maple [A]  time = 0.012, size = 564, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -240\,b{c}^{3}{e}^{5}g{x}^{4}+352\,{c}^{4}d{e}^{4}g{x}^{4}+128\,{c}^{4}{e}^{5}f{x}^{4}+840\,{b}^{2}{c}^{2}{e}^{5}g{x}^{3}-3872\,b{c}^{3}d{e}^{4}g{x}^{3}-448\,b{c}^{3}{e}^{5}f{x}^{3}+3872\,{c}^{4}{d}^{2}{e}^{3}g{x}^{3}+1408\,{c}^{4}d{e}^{4}f{x}^{3}-1890\,{b}^{3}c{e}^{5}g{x}^{2}+12852\,{b}^{2}{c}^{2}d{e}^{4}g{x}^{2}+1008\,{b}^{2}{c}^{2}{e}^{5}f{x}^{2}-28824\,b{c}^{3}{d}^{2}{e}^{3}g{x}^{2}-5376\,b{c}^{3}d{e}^{4}f{x}^{2}+20592\,{c}^{4}{d}^{3}{e}^{2}g{x}^{2}+7488\,{c}^{4}{d}^{2}{e}^{3}f{x}^{2}+3465\,{b}^{4}{e}^{5}gx-29652\,{b}^{3}cd{e}^{4}gx-1848\,{b}^{3}c{e}^{5}fx+95256\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}gx+13104\,{b}^{2}{c}^{2}d{e}^{4}fx-135696\,b{c}^{3}{d}^{3}{e}^{2}gx-31584\,b{c}^{3}{d}^{2}{e}^{3}fx+71632\,{c}^{4}{d}^{4}egx+26048\,{c}^{4}{d}^{3}{e}^{2}fx+462\,{b}^{4}d{e}^{4}g+3003\,{b}^{4}{e}^{5}f-3738\,{b}^{3}c{d}^{2}{e}^{3}g-25872\,{b}^{3}cd{e}^{4}f+11172\,{b}^{2}{c}^{2}{d}^{3}{e}^{2}g+84168\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}f-14408\,b{c}^{3}{d}^{4}eg-122752\,b{c}^{3}{d}^{3}{e}^{2}f+6512\,{c}^{4}{d}^{5}g+67888\,{c}^{4}{d}^{4}ef \right ) }{45045\, \left ( ex+d \right ) ^{10}{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 ) } \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{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)^(5/2)/(e*x+d)^11,x)

[Out]

-2/45045*(c*e*x+b*e-c*d)*(-240*b*c^3*e^5*g*x^4+352*c^4*d*e^4*g*x^4+128*c^4*e^5*f*x^4+840*b^2*c^2*e^5*g*x^3-387
2*b*c^3*d*e^4*g*x^3-448*b*c^3*e^5*f*x^3+3872*c^4*d^2*e^3*g*x^3+1408*c^4*d*e^4*f*x^3-1890*b^3*c*e^5*g*x^2+12852
*b^2*c^2*d*e^4*g*x^2+1008*b^2*c^2*e^5*f*x^2-28824*b*c^3*d^2*e^3*g*x^2-5376*b*c^3*d*e^4*f*x^2+20592*c^4*d^3*e^2
*g*x^2+7488*c^4*d^2*e^3*f*x^2+3465*b^4*e^5*g*x-29652*b^3*c*d*e^4*g*x-1848*b^3*c*e^5*f*x+95256*b^2*c^2*d^2*e^3*
g*x+13104*b^2*c^2*d*e^4*f*x-135696*b*c^3*d^3*e^2*g*x-31584*b*c^3*d^2*e^3*f*x+71632*c^4*d^4*e*g*x+26048*c^4*d^3
*e^2*f*x+462*b^4*d*e^4*g+3003*b^4*e^5*f-3738*b^3*c*d^2*e^3*g-25872*b^3*c*d*e^4*f+11172*b^2*c^2*d^3*e^2*g+84168
*b^2*c^2*d^2*e^3*f-14408*b*c^3*d^4*e*g-122752*b*c^3*d^3*e^2*f+6512*c^4*d^5*g+67888*c^4*d^4*e*f)*(-c*e^2*x^2-b*
e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^10/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)^(5/2)/(e*x+d)^11,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)^(5/2)/(e*x+d)^11,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)**(5/2)/(e*x+d)**11,x)

[Out]

Timed out

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Giac [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)^(5/2)/(e*x+d)^11,x, algorithm="giac")

[Out]

Timed out