3.2229 \(\int \frac{f+g x}{(d+e x)^2 (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=283 \[ \frac{128 c^2 (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^6 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 c (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (-7 b e g+4 c d g+10 c e f)}{35 e^2 (d+e x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{7 e^2 (d+e x)^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(16*c*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(b + 2*c*x))/(105*e*(2*c*d - b*e)^4*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2)) - (2*(e*f - d*g))/(7*e^2*(2*c*d - b*e)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(
10*c*e*f + 4*c*d*g - 7*b*e*g))/(35*e^2*(2*c*d - b*e)^2*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))
+ (128*c^2*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(b + 2*c*x))/(105*e*(2*c*d - b*e)^6*Sqrt[d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.395133, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {792, 658, 614, 613} \[ \frac{128 c^2 (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^6 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 c (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (-7 b e g+4 c d g+10 c e f)}{35 e^2 (d+e x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{7 e^2 (d+e x)^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*c*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(b + 2*c*x))/(105*e*(2*c*d - b*e)^4*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2)) - (2*(e*f - d*g))/(7*e^2*(2*c*d - b*e)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(
10*c*e*f + 4*c*d*g - 7*b*e*g))/(35*e^2*(2*c*d - b*e)^2*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))
+ (128*c^2*(10*c*e*f + 4*c*d*g - 7*b*e*g)*(b + 2*c*x))/(105*e*(2*c*d - b*e)^6*Sqrt[d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2])

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 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=-\frac{2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(10 c e f+4 c d g-7 b e g) \int \frac{1}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{7 e (2 c d-b e)}\\ &=-\frac{2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (10 c e f+4 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(8 c (10 c e f+4 c d g-7 b e g)) \int \frac{1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{35 e (2 c d-b e)^2}\\ &=\frac{16 c (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (10 c e f+4 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{\left (64 c^2 (10 c e f+4 c d g-7 b e g)\right ) \int \frac{1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{105 e (2 c d-b e)^4}\\ &=\frac{16 c (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (10 c e f+4 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{128 c^2 (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^6 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.348365, size = 468, normalized size = 1.65 \[ \frac{96 b^2 c^3 e^2 \left (d^2 e^2 x (65 f-54 g x)+20 d^3 e (3 f+2 g x)+17 d^4 g+40 d e^3 x^2 (f-2 g x)+2 e^4 x^3 (5 f-14 g x)\right )-16 b^3 c^2 e^3 \left (d^2 e (115 f+293 g x)+88 d^3 g+2 d e^2 x (25 f+86 g x)+2 e^3 x^2 (5 f+21 g x)\right )+4 b^4 c e^4 \left (43 d^2 g+2 d e (45 f+73 g x)+e^2 x (15 f+28 g x)\right )-6 b^5 e^5 (2 d g+5 e f+7 e g x)+32 b c^4 e \left (4 d^2 e^3 x^2 (75 f+43 g x)+12 d^3 e^2 x (24 g x-5 f)-39 d^4 e (5 f-g x)+6 d^5 g+8 d e^4 x^3 (45 f-8 g x)+8 e^5 x^4 (15 f-7 g x)\right )-64 c^5 \left (8 d^3 e^3 x^2 (15 f+g x)+4 d^2 e^4 x^3 (5 f-8 g x)+3 d^4 e^2 x (15 f+16 g x)-6 d^5 e (5 f-3 g x)+9 d^6 g-16 d e^5 x^4 (5 f+g x)-40 e^6 f x^5\right )}{105 e^2 (d+e x)^3 (b e-2 c d)^6 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*b^5*e^5*(5*e*f + 2*d*g + 7*e*g*x) + 96*b^2*c^3*e^2*(17*d^4*g + d^2*e^2*x*(65*f - 54*g*x) + 2*e^4*x^3*(5*f
- 14*g*x) + 40*d*e^3*x^2*(f - 2*g*x) + 20*d^3*e*(3*f + 2*g*x)) - 64*c^5*(9*d^6*g - 40*e^6*f*x^5 + 4*d^2*e^4*x^
3*(5*f - 8*g*x) - 6*d^5*e*(5*f - 3*g*x) - 16*d*e^5*x^4*(5*f + g*x) + 8*d^3*e^3*x^2*(15*f + g*x) + 3*d^4*e^2*x*
(15*f + 16*g*x)) + 32*b*c^4*e*(6*d^5*g + 8*d*e^4*x^3*(45*f - 8*g*x) + 8*e^5*x^4*(15*f - 7*g*x) - 39*d^4*e*(5*f
- g*x) + 12*d^3*e^2*x*(-5*f + 24*g*x) + 4*d^2*e^3*x^2*(75*f + 43*g*x)) + 4*b^4*c*e^4*(43*d^2*g + e^2*x*(15*f
+ 28*g*x) + 2*d*e*(45*f + 73*g*x)) - 16*b^3*c^2*e^3*(88*d^3*g + 2*e^3*x^2*(5*f + 21*g*x) + 2*d*e^2*x*(25*f + 8
6*g*x) + d^2*e*(115*f + 293*g*x)))/(105*e^2*(-2*c*d + b*e)^6*(d + e*x)^3*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)
*(-(b*e) + c*(d - e*x))])

________________________________________________________________________________________

Maple [B]  time = 0.012, size = 782, normalized size = 2.8 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 896\,b{c}^{4}{e}^{6}g{x}^{5}-512\,{c}^{5}d{e}^{5}g{x}^{5}-1280\,{c}^{5}{e}^{6}f{x}^{5}+1344\,{b}^{2}{c}^{3}{e}^{6}g{x}^{4}+1024\,b{c}^{4}d{e}^{5}g{x}^{4}-1920\,b{c}^{4}{e}^{6}f{x}^{4}-1024\,{c}^{5}{d}^{2}{e}^{4}g{x}^{4}-2560\,{c}^{5}d{e}^{5}f{x}^{4}+336\,{b}^{3}{c}^{2}{e}^{6}g{x}^{3}+3840\,{b}^{2}{c}^{3}d{e}^{5}g{x}^{3}-480\,{b}^{2}{c}^{3}{e}^{6}f{x}^{3}-2752\,b{c}^{4}{d}^{2}{e}^{4}g{x}^{3}-5760\,b{c}^{4}d{e}^{5}f{x}^{3}+256\,{c}^{5}{d}^{3}{e}^{3}g{x}^{3}+640\,{c}^{5}{d}^{2}{e}^{4}f{x}^{3}-56\,{b}^{4}c{e}^{6}g{x}^{2}+1376\,{b}^{3}{c}^{2}d{e}^{5}g{x}^{2}+80\,{b}^{3}{c}^{2}{e}^{6}f{x}^{2}+2592\,{b}^{2}{c}^{3}{d}^{2}{e}^{4}g{x}^{2}-1920\,{b}^{2}{c}^{3}d{e}^{5}f{x}^{2}-4608\,b{c}^{4}{d}^{3}{e}^{3}g{x}^{2}-4800\,b{c}^{4}{d}^{2}{e}^{4}f{x}^{2}+1536\,{c}^{5}{d}^{4}{e}^{2}g{x}^{2}+3840\,{c}^{5}{d}^{3}{e}^{3}f{x}^{2}+21\,{b}^{5}{e}^{6}gx-292\,{b}^{4}cd{e}^{5}gx-30\,{b}^{4}c{e}^{6}fx+2344\,{b}^{3}{c}^{2}{d}^{2}{e}^{4}gx+400\,{b}^{3}{c}^{2}d{e}^{5}fx-1920\,{b}^{2}{c}^{3}{d}^{3}{e}^{3}gx-3120\,{b}^{2}{c}^{3}{d}^{2}{e}^{4}fx-624\,b{c}^{4}{d}^{4}{e}^{2}gx+960\,b{c}^{4}{d}^{3}{e}^{3}fx+576\,{c}^{5}{d}^{5}egx+1440\,{c}^{5}{d}^{4}{e}^{2}fx+6\,{b}^{5}d{e}^{5}g+15\,{b}^{5}{e}^{6}f-86\,{b}^{4}c{d}^{2}{e}^{4}g-180\,{b}^{4}cd{e}^{5}f+704\,{b}^{3}{c}^{2}{d}^{3}{e}^{3}g+920\,{b}^{3}{c}^{2}{d}^{2}{e}^{4}f-816\,{b}^{2}{c}^{3}{d}^{4}{e}^{2}g-2880\,{b}^{2}{c}^{3}{d}^{3}{e}^{3}f-96\,b{c}^{4}{d}^{5}eg+3120\,b{c}^{4}{d}^{4}{e}^{2}f+288\,{c}^{5}{d}^{6}g-960\,{c}^{5}{d}^{5}ef \right ) }{ \left ( 105\,ex+105\,d \right ) \left ({b}^{6}{e}^{6}-12\,{b}^{5}cd{e}^{5}+60\,{b}^{4}{c}^{2}{d}^{2}{e}^{4}-160\,{b}^{3}{c}^{3}{d}^{3}{e}^{3}+240\,{b}^{2}{c}^{4}{d}^{4}{e}^{2}-192\,b{c}^{5}{d}^{5}e+64\,{c}^{6}{d}^{6} \right ){e}^{2}} \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)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

-2/105*(c*e*x+b*e-c*d)*(896*b*c^4*e^6*g*x^5-512*c^5*d*e^5*g*x^5-1280*c^5*e^6*f*x^5+1344*b^2*c^3*e^6*g*x^4+1024
*b*c^4*d*e^5*g*x^4-1920*b*c^4*e^6*f*x^4-1024*c^5*d^2*e^4*g*x^4-2560*c^5*d*e^5*f*x^4+336*b^3*c^2*e^6*g*x^3+3840
*b^2*c^3*d*e^5*g*x^3-480*b^2*c^3*e^6*f*x^3-2752*b*c^4*d^2*e^4*g*x^3-5760*b*c^4*d*e^5*f*x^3+256*c^5*d^3*e^3*g*x
^3+640*c^5*d^2*e^4*f*x^3-56*b^4*c*e^6*g*x^2+1376*b^3*c^2*d*e^5*g*x^2+80*b^3*c^2*e^6*f*x^2+2592*b^2*c^3*d^2*e^4
*g*x^2-1920*b^2*c^3*d*e^5*f*x^2-4608*b*c^4*d^3*e^3*g*x^2-4800*b*c^4*d^2*e^4*f*x^2+1536*c^5*d^4*e^2*g*x^2+3840*
c^5*d^3*e^3*f*x^2+21*b^5*e^6*g*x-292*b^4*c*d*e^5*g*x-30*b^4*c*e^6*f*x+2344*b^3*c^2*d^2*e^4*g*x+400*b^3*c^2*d*e
^5*f*x-1920*b^2*c^3*d^3*e^3*g*x-3120*b^2*c^3*d^2*e^4*f*x-624*b*c^4*d^4*e^2*g*x+960*b*c^4*d^3*e^3*f*x+576*c^5*d
^5*e*g*x+1440*c^5*d^4*e^2*f*x+6*b^5*d*e^5*g+15*b^5*e^6*f-86*b^4*c*d^2*e^4*g-180*b^4*c*d*e^5*f+704*b^3*c^2*d^3*
e^3*g+920*b^3*c^2*d^2*e^4*f-816*b^2*c^3*d^4*e^2*g-2880*b^2*c^3*d^3*e^3*f-96*b*c^4*d^5*e*g+3120*b*c^4*d^4*e^2*f
+288*c^5*d^6*g-960*c^5*d^5*e*f)/(e*x+d)/(b^6*e^6-12*b^5*c*d*e^5+60*b^4*c^2*d^2*e^4-160*b^3*c^3*d^3*e^3+240*b^2
*c^4*d^4*e^2-192*b*c^5*d^5*e+64*c^6*d^6)/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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)/(e*x+d)**2/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

sage0*x