3.2207 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=285 \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{9009 e^2 (d+e x)^7 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{1287 e^2 (d+e x)^8 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{143 e^2 (d+e x)^9 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e^2 (d+e x)^{10} (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(13*e^2*(2*c*d - b*e)*(d + e*x)^10) - (2*(6*c*e*f
+ 20*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(143*e^2*(2*c*d - b*e)^2*(d + e*x)^9) - (
8*c*(6*c*e*f + 20*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(1287*e^2*(2*c*d - b*e)^3*(d
+ e*x)^8) - (16*c^2*(6*c*e*f + 20*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9009*e^2*(2*
c*d - b*e)^4*(d + e*x)^7)

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Rubi [A]  time = 0.436946, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068, Rules used = {792, 658, 650} \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{9009 e^2 (d+e x)^7 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{1287 e^2 (d+e x)^8 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+20 c d g+6 c e f)}{143 e^2 (d+e x)^9 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e^2 (d+e x)^{10} (2 c d-b e)} \]

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)^10,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(13*e^2*(2*c*d - b*e)*(d + e*x)^10) - (2*(6*c*e*f
+ 20*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(143*e^2*(2*c*d - b*e)^2*(d + e*x)^9) - (
8*c*(6*c*e*f + 20*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(1287*e^2*(2*c*d - b*e)^3*(d
+ e*x)^8) - (16*c^2*(6*c*e*f + 20*c*d*g - 13*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9009*e^2*(2*
c*d - b*e)^4*(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)^{10}} \, 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}}{13 e^2 (2 c d-b e) (d+e x)^{10}}+\frac{(6 c e f+20 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 )^{5/2}}{(d+e x)^9} \, 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 )^{7/2}}{13 e^2 (2 c d-b e) (d+e x)^{10}}-\frac{2 (6 c e f+20 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^9}+\frac{(4 c (6 c e f+20 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 )^{5/2}}{(d+e x)^8} \, 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 )^{7/2}}{13 e^2 (2 c d-b e) (d+e x)^{10}}-\frac{2 (6 c e f+20 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^9}-\frac{8 c (6 c e f+20 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^8}+\frac{\left (8 c^2 (6 c e f+20 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 )^{5/2}}{(d+e x)^7} \, dx}{1287 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}}{13 e^2 (2 c d-b e) (d+e x)^{10}}-\frac{2 (6 c e f+20 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^9}-\frac{8 c (6 c e f+20 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^8}-\frac{16 c^2 (6 c e f+20 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9009 e^2 (2 c d-b e)^4 (d+e x)^7}\\ \end{align*}

Mathematica [A]  time = 0.250729, size = 250, normalized size = 0.88 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (14 b^2 c e^2 \left (53 d^2 g+4 d e (81 f+94 g x)+e^2 x (27 f+26 g x)\right )-63 b^3 e^3 (2 d g+11 e f+13 e g x)-4 b c^2 e \left (d^2 e (2499 f+2801 g x)+348 d^3 g+2 d e^2 x (231 f+200 g x)+2 e^3 x^2 (21 f+13 g x)\right )+8 c^3 \left (d^2 e^2 x (291 f+200 g x)+10 d^3 e (93 f+97 g x)+97 d^4 g+20 d e^3 x^2 (3 f+g x)+6 e^4 f x^3\right )\right )}{9009 e^2 (d+e x)^7 (b e-2 c d)^4} \]

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)^10,x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-63*b^3*e^3*(11*e*f + 2*d*g + 13*e*g*x) +
14*b^2*c*e^2*(53*d^2*g + e^2*x*(27*f + 26*g*x) + 4*d*e*(81*f + 94*g*x)) + 8*c^3*(97*d^4*g + 6*e^4*f*x^3 + 20*d
*e^3*x^2*(3*f + g*x) + 10*d^3*e*(93*f + 97*g*x) + d^2*e^2*x*(291*f + 200*g*x)) - 4*b*c^2*e*(348*d^3*g + 2*e^3*
x^2*(21*f + 13*g*x) + 2*d*e^2*x*(231*f + 200*g*x) + d^2*e*(2499*f + 2801*g*x))))/(9009*e^2*(-2*c*d + b*e)^4*(d
+ e*x)^7)

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Maple [A]  time = 0.012, size = 382, normalized size = 1.3 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 104\,b{c}^{2}{e}^{4}g{x}^{3}-160\,{c}^{3}d{e}^{3}g{x}^{3}-48\,{c}^{3}{e}^{4}f{x}^{3}-364\,{b}^{2}c{e}^{4}g{x}^{2}+1600\,b{c}^{2}d{e}^{3}g{x}^{2}+168\,b{c}^{2}{e}^{4}f{x}^{2}-1600\,{c}^{3}{d}^{2}{e}^{2}g{x}^{2}-480\,{c}^{3}d{e}^{3}f{x}^{2}+819\,{b}^{3}{e}^{4}gx-5264\,{b}^{2}cd{e}^{3}gx-378\,{b}^{2}c{e}^{4}fx+11204\,b{c}^{2}{d}^{2}{e}^{2}gx+1848\,b{c}^{2}d{e}^{3}fx-7760\,{c}^{3}{d}^{3}egx-2328\,{c}^{3}{d}^{2}{e}^{2}fx+126\,{b}^{3}d{e}^{3}g+693\,{b}^{3}{e}^{4}f-742\,{b}^{2}c{d}^{2}{e}^{2}g-4536\,{b}^{2}cd{e}^{3}f+1392\,b{c}^{2}{d}^{3}eg+9996\,b{c}^{2}{d}^{2}{e}^{2}f-776\,{c}^{3}{d}^{4}g-7440\,{c}^{3}{d}^{3}ef \right ) }{9009\, \left ( ex+d \right ) ^{9}{e}^{2} \left ({b}^{4}{e}^{4}-8\,{b}^{3}cd{e}^{3}+24\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}-32\,b{c}^{3}{d}^{3}e+16\,{c}^{4}{d}^{4} \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)^10,x)

[Out]

-2/9009*(c*e*x+b*e-c*d)*(104*b*c^2*e^4*g*x^3-160*c^3*d*e^3*g*x^3-48*c^3*e^4*f*x^3-364*b^2*c*e^4*g*x^2+1600*b*c
^2*d*e^3*g*x^2+168*b*c^2*e^4*f*x^2-1600*c^3*d^2*e^2*g*x^2-480*c^3*d*e^3*f*x^2+819*b^3*e^4*g*x-5264*b^2*c*d*e^3
*g*x-378*b^2*c*e^4*f*x+11204*b*c^2*d^2*e^2*g*x+1848*b*c^2*d*e^3*f*x-7760*c^3*d^3*e*g*x-2328*c^3*d^2*e^2*f*x+12
6*b^3*d*e^3*g+693*b^3*e^4*f-742*b^2*c*d^2*e^2*g-4536*b^2*c*d*e^3*f+1392*b*c^2*d^3*e*g+9996*b*c^2*d^2*e^2*f-776
*c^3*d^4*g-7440*c^3*d^3*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^9/e^2/(b^4*e^4-8*b^3*c*d*e^3+24*b^
2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*c^4*d^4)

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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)^10,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)^10,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)**10,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)^10,x, algorithm="giac")

[Out]

Timed out