3.2194 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=360 \[ -\frac{32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+18 c d g+8 c e f)}{15015 e^2 (d+e x)^5 (2 c d-b e)^5}-\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+18 c d g+8 c e f)}{3003 e^2 (d+e x)^6 (2 c d-b e)^4}-\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+18 c d g+8 c e f)}{429 e^2 (d+e x)^7 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+18 c d g+8 c e f)}{143 e^2 (d+e x)^8 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 e^2 (d+e x)^9 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(13*e^2*(2*c*d - b*e)*(d + e*x)^9) - (2*(8*c*e*f
+ 18*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(143*e^2*(2*c*d - b*e)^2*(d + e*x)^8) - (4
*c*(8*c*e*f + 18*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(429*e^2*(2*c*d - b*e)^3*(d +
e*x)^7) - (16*c^2*(8*c*e*f + 18*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3003*e^2*(2*c*
d - b*e)^4*(d + e*x)^6) - (32*c^3*(8*c*e*f + 18*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))
/(15015*e^2*(2*c*d - b*e)^5*(d + e*x)^5)

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Rubi [A]  time = 0.572435, 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} \[ -\frac{32 c^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+18 c d g+8 c e f)}{15015 e^2 (d+e x)^5 (2 c d-b e)^5}-\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+18 c d g+8 c e f)}{3003 e^2 (d+e x)^6 (2 c d-b e)^4}-\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+18 c d g+8 c e f)}{429 e^2 (d+e x)^7 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+18 c d g+8 c e f)}{143 e^2 (d+e x)^8 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{13 e^2 (d+e x)^9 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^9,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(13*e^2*(2*c*d - b*e)*(d + e*x)^9) - (2*(8*c*e*f
+ 18*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(143*e^2*(2*c*d - b*e)^2*(d + e*x)^8) - (4
*c*(8*c*e*f + 18*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(429*e^2*(2*c*d - b*e)^3*(d +
e*x)^7) - (16*c^2*(8*c*e*f + 18*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3003*e^2*(2*c*
d - b*e)^4*(d + e*x)^6) - (32*c^3*(8*c*e*f + 18*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))
/(15015*e^2*(2*c*d - b*e)^5*(d + e*x)^5)

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 )^{3/2}}{(d+e x)^9} \, 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}}{13 e^2 (2 c d-b e) (d+e x)^9}+\frac{(8 c e f+18 c d g-13 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)^8} \, dx}{13 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}}{13 e^2 (2 c d-b e) (d+e x)^9}-\frac{2 (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^8}+\frac{(6 c (8 c e f+18 c d g-13 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)^7} \, dx}{143 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}}{13 e^2 (2 c d-b e) (d+e x)^9}-\frac{2 (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^8}-\frac{4 c (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 e^2 (2 c d-b e)^3 (d+e x)^7}+\frac{\left (8 c^2 (8 c e f+18 c d g-13 b e g)\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)^6} \, dx}{429 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 )^{5/2}}{13 e^2 (2 c d-b e) (d+e x)^9}-\frac{2 (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^8}-\frac{4 c (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 e^2 (2 c d-b e)^3 (d+e x)^7}-\frac{16 c^2 (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3003 e^2 (2 c d-b e)^4 (d+e x)^6}+\frac{\left (16 c^3 (8 c e f+18 c d g-13 b e g)\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}{3003 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 )^{5/2}}{13 e^2 (2 c d-b e) (d+e x)^9}-\frac{2 (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^8}-\frac{4 c (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{429 e^2 (2 c d-b e)^3 (d+e x)^7}-\frac{16 c^2 (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3003 e^2 (2 c d-b e)^4 (d+e x)^6}-\frac{32 c^3 (8 c e f+18 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{15015 e^2 (2 c d-b e)^5 (d+e x)^5}\\ \end{align*}

Mathematica [A]  time = 0.297817, size = 349, normalized size = 0.97 \[ \frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (20 b^2 c^2 e^2 \left (2 d^2 e (833 f+977 g x)+271 d^3 g+d e^2 x (308 f+323 g x)+2 e^3 x^2 (14 f+13 g x)\right )-70 b^3 c e^3 \left (25 d^2 g+2 d e (72 f+85 g x)+e^2 x (12 f+13 g x)\right )+105 b^4 e^4 (2 d g+11 e f+13 e g x)-8 b c^3 e \left (d^2 e^2 x (1940 f+1901 g x)+d^3 e (6200 f+7134 g x)+911 d^4 g+4 d e^3 x^2 (100 f+81 g x)+2 e^4 x^3 (20 f+13 g x)\right )+16 c^4 \left (2 d^2 e^3 x^2 (154 f+81 g x)+3 d^3 e^2 x (284 f+231 g x)+d^4 e (1763 f+1917 g x)+213 d^5 g+18 d e^4 x^3 (4 f+g x)+8 e^5 f x^4\right )\right )}{15015 e^2 (d+e x)^7 (b e-2 c d)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^9,x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(105*b^4*e^4*(11*e*f + 2*d*g + 13*e*g*x) -
70*b^3*c*e^3*(25*d^2*g + e^2*x*(12*f + 13*g*x) + 2*d*e*(72*f + 85*g*x)) + 20*b^2*c^2*e^2*(271*d^3*g + 2*e^3*x^
2*(14*f + 13*g*x) + d*e^2*x*(308*f + 323*g*x) + 2*d^2*e*(833*f + 977*g*x)) + 16*c^4*(213*d^5*g + 8*e^5*f*x^4 +
18*d*e^4*x^3*(4*f + g*x) + 2*d^2*e^3*x^2*(154*f + 81*g*x) + 3*d^3*e^2*x*(284*f + 231*g*x) + d^4*e*(1763*f + 1
917*g*x)) - 8*b*c^3*e*(911*d^4*g + 2*e^4*x^3*(20*f + 13*g*x) + 4*d*e^3*x^2*(100*f + 81*g*x) + d^2*e^2*x*(1940*
f + 1901*g*x) + d^3*e*(6200*f + 7134*g*x))))/(15015*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 ( -208\,b{c}^{3}{e}^{5}g{x}^{4}+288\,{c}^{4}d{e}^{4}g{x}^{4}+128\,{c}^{4}{e}^{5}f{x}^{4}+520\,{b}^{2}{c}^{2}{e}^{5}g{x}^{3}-2592\,b{c}^{3}d{e}^{4}g{x}^{3}-320\,b{c}^{3}{e}^{5}f{x}^{3}+2592\,{c}^{4}{d}^{2}{e}^{3}g{x}^{3}+1152\,{c}^{4}d{e}^{4}f{x}^{3}-910\,{b}^{3}c{e}^{5}g{x}^{2}+6460\,{b}^{2}{c}^{2}d{e}^{4}g{x}^{2}+560\,{b}^{2}{c}^{2}{e}^{5}f{x}^{2}-15208\,b{c}^{3}{d}^{2}{e}^{3}g{x}^{2}-3200\,b{c}^{3}d{e}^{4}f{x}^{2}+11088\,{c}^{4}{d}^{3}{e}^{2}g{x}^{2}+4928\,{c}^{4}{d}^{2}{e}^{3}f{x}^{2}+1365\,{b}^{4}{e}^{5}gx-11900\,{b}^{3}cd{e}^{4}gx-840\,{b}^{3}c{e}^{5}fx+39080\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}gx+6160\,{b}^{2}{c}^{2}d{e}^{4}fx-57072\,b{c}^{3}{d}^{3}{e}^{2}gx-15520\,b{c}^{3}{d}^{2}{e}^{3}fx+30672\,{c}^{4}{d}^{4}egx+13632\,{c}^{4}{d}^{3}{e}^{2}fx+210\,{b}^{4}d{e}^{4}g+1155\,{b}^{4}{e}^{5}f-1750\,{b}^{3}c{d}^{2}{e}^{3}g-10080\,{b}^{3}cd{e}^{4}f+5420\,{b}^{2}{c}^{2}{d}^{3}{e}^{2}g+33320\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}f-7288\,b{c}^{3}{d}^{4}eg-49600\,b{c}^{3}{d}^{3}{e}^{2}f+3408\,{c}^{4}{d}^{5}g+28208\,{c}^{4}{d}^{4}ef \right ) }{15015\, \left ( ex+d \right ) ^{8}{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{3}{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)^(3/2)/(e*x+d)^9,x)

[Out]

-2/15015*(c*e*x+b*e-c*d)*(-208*b*c^3*e^5*g*x^4+288*c^4*d*e^4*g*x^4+128*c^4*e^5*f*x^4+520*b^2*c^2*e^5*g*x^3-259
2*b*c^3*d*e^4*g*x^3-320*b*c^3*e^5*f*x^3+2592*c^4*d^2*e^3*g*x^3+1152*c^4*d*e^4*f*x^3-910*b^3*c*e^5*g*x^2+6460*b
^2*c^2*d*e^4*g*x^2+560*b^2*c^2*e^5*f*x^2-15208*b*c^3*d^2*e^3*g*x^2-3200*b*c^3*d*e^4*f*x^2+11088*c^4*d^3*e^2*g*
x^2+4928*c^4*d^2*e^3*f*x^2+1365*b^4*e^5*g*x-11900*b^3*c*d*e^4*g*x-840*b^3*c*e^5*f*x+39080*b^2*c^2*d^2*e^3*g*x+
6160*b^2*c^2*d*e^4*f*x-57072*b*c^3*d^3*e^2*g*x-15520*b*c^3*d^2*e^3*f*x+30672*c^4*d^4*e*g*x+13632*c^4*d^3*e^2*f
*x+210*b^4*d*e^4*g+1155*b^4*e^5*f-1750*b^3*c*d^2*e^3*g-10080*b^3*c*d*e^4*f+5420*b^2*c^2*d^3*e^2*g+33320*b^2*c^
2*d^2*e^3*f-7288*b*c^3*d^4*e*g-49600*b*c^3*d^3*e^2*f+3408*c^4*d^5*g+28208*c^4*d^4*e*f)*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(3/2)/(e*x+d)^8/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)^(3/2)/(e*x+d)^9,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)^(3/2)/(e*x+d)^9,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)**(3/2)/(e*x+d)**9,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)^(3/2)/(e*x+d)^9,x, algorithm="giac")

[Out]

Timed out