3.2193 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=285 \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{3465 e^2 (d+e x)^5 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{693 e^2 (d+e x)^6 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{99 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (d+e x)^8 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(11*e^2*(2*c*d - b*e)*(d + e*x)^8) - (2*(6*c*e*f
+ 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^7) - (8*
c*(6*c*e*f + 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(693*e^2*(2*c*d - b*e)^3*(d + e
*x)^6) - (16*c^2*(6*c*e*f + 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3465*e^2*(2*c*d
- b*e)^4*(d + e*x)^5)

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Rubi [A]  time = 0.447909, 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} \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{3465 e^2 (d+e x)^5 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{693 e^2 (d+e x)^6 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+16 c d g+6 c e f)}{99 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 e^2 (d+e x)^8 (2 c d-b e)} \]

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)^8,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(11*e^2*(2*c*d - b*e)*(d + e*x)^8) - (2*(6*c*e*f
+ 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(99*e^2*(2*c*d - b*e)^2*(d + e*x)^7) - (8*
c*(6*c*e*f + 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(693*e^2*(2*c*d - b*e)^3*(d + e
*x)^6) - (16*c^2*(6*c*e*f + 16*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3465*e^2*(2*c*d
- b*e)^4*(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)^8} \, 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}}{11 e^2 (2 c d-b e) (d+e x)^8}+\frac{(6 c e f+16 c d g-11 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}{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 )^{5/2}}{11 e^2 (2 c d-b e) (d+e x)^8}-\frac{2 (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^7}+\frac{(4 c (6 c e f+16 c d g-11 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}{99 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}}{11 e^2 (2 c d-b e) (d+e x)^8}-\frac{2 (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac{8 c (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{693 e^2 (2 c d-b e)^3 (d+e x)^6}+\frac{\left (8 c^2 (6 c e f+16 c d g-11 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}{693 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}}{11 e^2 (2 c d-b e) (d+e x)^8}-\frac{2 (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^7}-\frac{8 c (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{693 e^2 (2 c d-b e)^3 (d+e x)^6}-\frac{16 c^2 (6 c e f+16 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3465 e^2 (2 c d-b e)^4 (d+e x)^5}\\ \end{align*}

Mathematica [A]  time = 0.207446, size = 249, normalized size = 0.87 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (10 b^2 c e^2 \left (43 d^2 g+d e (210 f+254 g x)+e^2 x (21 f+22 g x)\right )-35 b^3 e^3 (2 d g+9 e f+11 e g x)-4 b c^2 e \left (d^2 e (1185 f+1391 g x)+212 d^3 g+2 d e^2 x (135 f+128 g x)+2 e^3 x^2 (15 f+11 g x)\right )+8 c^3 \left (d^2 e^2 x (183 f+128 g x)+8 d^3 e (57 f+61 g x)+61 d^4 g+16 d e^3 x^2 (3 f+g x)+6 e^4 f x^3\right )\right )}{3465 e^2 (d+e x)^6 (b e-2 c d)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-35*b^3*e^3*(9*e*f + 2*d*g + 11*e*g*x) +
8*c^3*(61*d^4*g + 6*e^4*f*x^3 + 16*d*e^3*x^2*(3*f + g*x) + 8*d^3*e*(57*f + 61*g*x) + d^2*e^2*x*(183*f + 128*g*
x)) + 10*b^2*c*e^2*(43*d^2*g + e^2*x*(21*f + 22*g*x) + d*e*(210*f + 254*g*x)) - 4*b*c^2*e*(212*d^3*g + 2*e^3*x
^2*(15*f + 11*g*x) + 2*d*e^2*x*(135*f + 128*g*x) + d^2*e*(1185*f + 1391*g*x))))/(3465*e^2*(-2*c*d + b*e)^4*(d
+ e*x)^6)

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Maple [A]  time = 0.011, size = 382, normalized size = 1.3 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 88\,b{c}^{2}{e}^{4}g{x}^{3}-128\,{c}^{3}d{e}^{3}g{x}^{3}-48\,{c}^{3}{e}^{4}f{x}^{3}-220\,{b}^{2}c{e}^{4}g{x}^{2}+1024\,b{c}^{2}d{e}^{3}g{x}^{2}+120\,b{c}^{2}{e}^{4}f{x}^{2}-1024\,{c}^{3}{d}^{2}{e}^{2}g{x}^{2}-384\,{c}^{3}d{e}^{3}f{x}^{2}+385\,{b}^{3}{e}^{4}gx-2540\,{b}^{2}cd{e}^{3}gx-210\,{b}^{2}c{e}^{4}fx+5564\,b{c}^{2}{d}^{2}{e}^{2}gx+1080\,b{c}^{2}d{e}^{3}fx-3904\,{c}^{3}{d}^{3}egx-1464\,{c}^{3}{d}^{2}{e}^{2}fx+70\,{b}^{3}d{e}^{3}g+315\,{b}^{3}{e}^{4}f-430\,{b}^{2}c{d}^{2}{e}^{2}g-2100\,{b}^{2}cd{e}^{3}f+848\,b{c}^{2}{d}^{3}eg+4740\,b{c}^{2}{d}^{2}{e}^{2}f-488\,{c}^{3}{d}^{4}g-3648\,{c}^{3}{d}^{3}ef \right ) }{3465\, \left ( ex+d \right ) ^{7}{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{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)^8,x)

[Out]

-2/3465*(c*e*x+b*e-c*d)*(88*b*c^2*e^4*g*x^3-128*c^3*d*e^3*g*x^3-48*c^3*e^4*f*x^3-220*b^2*c*e^4*g*x^2+1024*b*c^
2*d*e^3*g*x^2+120*b*c^2*e^4*f*x^2-1024*c^3*d^2*e^2*g*x^2-384*c^3*d*e^3*f*x^2+385*b^3*e^4*g*x-2540*b^2*c*d*e^3*
g*x-210*b^2*c*e^4*f*x+5564*b*c^2*d^2*e^2*g*x+1080*b*c^2*d*e^3*f*x-3904*c^3*d^3*e*g*x-1464*c^3*d^2*e^2*f*x+70*b
^3*d*e^3*g+315*b^3*e^4*f-430*b^2*c*d^2*e^2*g-2100*b^2*c*d*e^3*f+848*b*c^2*d^3*e*g+4740*b*c^2*d^2*e^2*f-488*c^3
*d^4*g-3648*c^3*d^3*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^7/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)^(3/2)/(e*x+d)^8,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)^8,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)**8,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)^8,x, algorithm="giac")

[Out]

Timed out