3.2216 \(\int \frac{f+g x}{(d+e x)^5 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=360 \[ -\frac{32 c^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x) (2 c d-b e)^5}-\frac{16 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x)^2 (2 c d-b e)^4}-\frac{4 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{105 e^2 (d+e x)^3 (2 c d-b e)^3}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{63 e^2 (d+e x)^4 (2 c d-b e)^2}-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (d+e x)^5 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(9*e^2*(2*c*d - b*e)*(d + e*x)^5) - (2*(8*c*e*f + 1
0*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(63*e^2*(2*c*d - b*e)^2*(d + e*x)^4) - (4*c*(8*c
*e*f + 10*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^3*(d + e*x)^3) -
(16*c^2*(8*c*e*f + 10*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(315*e^2*(2*c*d - b*e)^4*(d
+ e*x)^2) - (32*c^3*(8*c*e*f + 10*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(315*e^2*(2*c*d
- b*e)^5*(d + e*x))

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Rubi [A]  time = 0.558267, 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} \[ -\frac{32 c^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x) (2 c d-b e)^5}-\frac{16 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{315 e^2 (d+e x)^2 (2 c d-b e)^4}-\frac{4 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{105 e^2 (d+e x)^3 (2 c d-b e)^3}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+10 c d g+8 c e f)}{63 e^2 (d+e x)^4 (2 c d-b e)^2}-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (d+e x)^5 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(9*e^2*(2*c*d - b*e)*(d + e*x)^5) - (2*(8*c*e*f + 1
0*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(63*e^2*(2*c*d - b*e)^2*(d + e*x)^4) - (4*c*(8*c
*e*f + 10*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^3*(d + e*x)^3) -
(16*c^2*(8*c*e*f + 10*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(315*e^2*(2*c*d - b*e)^4*(d
+ e*x)^2) - (32*c^3*(8*c*e*f + 10*c*d*g - 9*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(315*e^2*(2*c*d
- b*e)^5*(d + e*x))

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}{(d+e x)^5 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}+\frac{(8 c e f+10 c d g-9 b e g) \int \frac{1}{(d+e x)^4 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{9 e (2 c d-b e)}\\ &=-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac{2 (8 c e f+10 c d g-9 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}+\frac{(2 c (8 c e f+10 c d g-9 b e g)) \int \frac{1}{(d+e x)^3 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{21 e (2 c d-b e)^2}\\ &=-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac{2 (8 c e f+10 c d g-9 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}-\frac{4 c (8 c e f+10 c d g-9 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^3}+\frac{\left (8 c^2 (8 c e f+10 c d g-9 b e g)\right ) \int \frac{1}{(d+e x)^2 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{105 e (2 c d-b e)^3}\\ &=-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac{2 (8 c e f+10 c d g-9 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}-\frac{4 c (8 c e f+10 c d g-9 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^3}-\frac{16 c^2 (8 c e f+10 c d g-9 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{315 e^2 (2 c d-b e)^4 (d+e x)^2}+\frac{\left (16 c^3 (8 c e f+10 c d g-9 b e g)\right ) \int \frac{1}{(d+e x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{315 e (2 c d-b e)^4}\\ &=-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{9 e^2 (2 c d-b e) (d+e x)^5}-\frac{2 (8 c e f+10 c d g-9 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{63 e^2 (2 c d-b e)^2 (d+e x)^4}-\frac{4 c (8 c e f+10 c d g-9 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{105 e^2 (2 c d-b e)^3 (d+e x)^3}-\frac{16 c^2 (8 c e f+10 c d g-9 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{315 e^2 (2 c d-b e)^4 (d+e x)^2}-\frac{32 c^3 (8 c e f+10 c d g-9 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{315 e^2 (2 c d-b e)^5 (d+e x)}\\ \end{align*}

Mathematica [A]  time = 0.247476, size = 348, normalized size = 0.97 \[ -\frac{2 (b e-c d+c e x) \left (12 b^2 c^2 e^2 \left (2 d^2 e (47 f+67 g x)+29 d^3 g+d e^2 x (28 f+41 g x)+2 e^3 x^2 (2 f+3 g x)\right )-2 b^3 c e^3 \left (47 d^2 g+2 d e (80 f+107 g x)+e^2 x (20 f+27 g x)\right )+5 b^4 e^4 (2 d g+7 e f+9 e g x)-8 b c^3 e \left (3 d^2 e^2 x (44 f+83 g x)+d^3 e (232 f+390 g x)+83 d^4 g+4 d e^3 x^2 (12 f+25 g x)+2 e^4 x^3 (4 f+9 g x)\right )+16 c^4 \left (2 d^2 e^3 x^2 (42 f+25 g x)+5 d^3 e^2 x (20 f+21 g x)+d^4 e (83 f+125 g x)+25 d^5 g+10 d e^4 x^3 (4 f+g x)+8 e^5 f x^4\right )\right )}{315 e^2 (d+e x)^4 (b e-2 c d)^5 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-(c*d) + b*e + c*e*x)*(5*b^4*e^4*(7*e*f + 2*d*g + 9*e*g*x) + 12*b^2*c^2*e^2*(29*d^3*g + 2*e^3*x^2*(2*f +
3*g*x) + d*e^2*x*(28*f + 41*g*x) + 2*d^2*e*(47*f + 67*g*x)) - 2*b^3*c*e^3*(47*d^2*g + e^2*x*(20*f + 27*g*x) +
2*d*e*(80*f + 107*g*x)) + 16*c^4*(25*d^5*g + 8*e^5*f*x^4 + 10*d*e^4*x^3*(4*f + g*x) + 5*d^3*e^2*x*(20*f + 21*g
*x) + 2*d^2*e^3*x^2*(42*f + 25*g*x) + d^4*e*(83*f + 125*g*x)) - 8*b*c^3*e*(83*d^4*g + 2*e^4*x^3*(4*f + 9*g*x)
+ 4*d*e^3*x^2*(12*f + 25*g*x) + 3*d^2*e^2*x*(44*f + 83*g*x) + d^3*e*(232*f + 390*g*x))))/(315*e^2*(-2*c*d + b*
e)^5*(d + e*x)^4*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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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 ( -144\,b{c}^{3}{e}^{5}g{x}^{4}+160\,{c}^{4}d{e}^{4}g{x}^{4}+128\,{c}^{4}{e}^{5}f{x}^{4}+72\,{b}^{2}{c}^{2}{e}^{5}g{x}^{3}-800\,b{c}^{3}d{e}^{4}g{x}^{3}-64\,b{c}^{3}{e}^{5}f{x}^{3}+800\,{c}^{4}{d}^{2}{e}^{3}g{x}^{3}+640\,{c}^{4}d{e}^{4}f{x}^{3}-54\,{b}^{3}c{e}^{5}g{x}^{2}+492\,{b}^{2}{c}^{2}d{e}^{4}g{x}^{2}+48\,{b}^{2}{c}^{2}{e}^{5}f{x}^{2}-1992\,b{c}^{3}{d}^{2}{e}^{3}g{x}^{2}-384\,b{c}^{3}d{e}^{4}f{x}^{2}+1680\,{c}^{4}{d}^{3}{e}^{2}g{x}^{2}+1344\,{c}^{4}{d}^{2}{e}^{3}f{x}^{2}+45\,{b}^{4}{e}^{5}gx-428\,{b}^{3}cd{e}^{4}gx-40\,{b}^{3}c{e}^{5}fx+1608\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}gx+336\,{b}^{2}{c}^{2}d{e}^{4}fx-3120\,b{c}^{3}{d}^{3}{e}^{2}gx-1056\,b{c}^{3}{d}^{2}{e}^{3}fx+2000\,{c}^{4}{d}^{4}egx+1600\,{c}^{4}{d}^{3}{e}^{2}fx+10\,{b}^{4}d{e}^{4}g+35\,{b}^{4}{e}^{5}f-94\,{b}^{3}c{d}^{2}{e}^{3}g-320\,{b}^{3}cd{e}^{4}f+348\,{b}^{2}{c}^{2}{d}^{3}{e}^{2}g+1128\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}f-664\,b{c}^{3}{d}^{4}eg-1856\,b{c}^{3}{d}^{3}{e}^{2}f+400\,{c}^{4}{d}^{5}g+1328\,{c}^{4}{d}^{4}ef \right ) }{315\, \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 ){e}^{2} \left ( ex+d \right ) ^{4}}{\frac{1}{\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)/(e*x+d)^5/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

-2/315*(c*e*x+b*e-c*d)*(-144*b*c^3*e^5*g*x^4+160*c^4*d*e^4*g*x^4+128*c^4*e^5*f*x^4+72*b^2*c^2*e^5*g*x^3-800*b*
c^3*d*e^4*g*x^3-64*b*c^3*e^5*f*x^3+800*c^4*d^2*e^3*g*x^3+640*c^4*d*e^4*f*x^3-54*b^3*c*e^5*g*x^2+492*b^2*c^2*d*
e^4*g*x^2+48*b^2*c^2*e^5*f*x^2-1992*b*c^3*d^2*e^3*g*x^2-384*b*c^3*d*e^4*f*x^2+1680*c^4*d^3*e^2*g*x^2+1344*c^4*
d^2*e^3*f*x^2+45*b^4*e^5*g*x-428*b^3*c*d*e^4*g*x-40*b^3*c*e^5*f*x+1608*b^2*c^2*d^2*e^3*g*x+336*b^2*c^2*d*e^4*f
*x-3120*b*c^3*d^3*e^2*g*x-1056*b*c^3*d^2*e^3*f*x+2000*c^4*d^4*e*g*x+1600*c^4*d^3*e^2*f*x+10*b^4*d*e^4*g+35*b^4
*e^5*f-94*b^3*c*d^2*e^3*g-320*b^3*c*d*e^4*f+348*b^2*c^2*d^3*e^2*g+1128*b^2*c^2*d^2*e^3*f-664*b*c^3*d^4*e*g-185
6*b*c^3*d^3*e^2*f+400*c^4*d^5*g+1328*c^4*d^4*e*f)/(e*x+d)^4/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)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)

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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)/(e*x+d)^5/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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)/(e*x+d)^5/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f + g x}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)**5/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

Integral((f + g*x)/(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(d + e*x)**5), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^5/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")

[Out]

sage0*x