3.2222 \(\int \frac{f+g x}{(d+e x)^3 (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=284 \[ \frac{16 c^2 (b+2 c x) (-7 b e g+6 c d g+8 c e f)}{35 e (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{4 c (-7 b e g+6 c d g+8 c e f)}{35 e^2 (d+e x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (-7 b e g+6 c d g+8 c e f)}{35 e^2 (d+e x)^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (e f-d g)}{7 e^2 (d+e x)^3 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

(16*c^2*(8*c*e*f + 6*c*d*g - 7*b*e*g)*(b + 2*c*x))/(35*e*(2*c*d - b*e)^5*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2]) - (2*(e*f - d*g))/(7*e^2*(2*c*d - b*e)*(d + e*x)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(8*c*
e*f + 6*c*d*g - 7*b*e*g))/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (4*
c*(8*c*e*f + 6*c*d*g - 7*b*e*g))/(35*e^2*(2*c*d - b*e)^3*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])

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Rubi [A]  time = 0.389914, antiderivative size = 284, 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, 613} \[ \frac{16 c^2 (b+2 c x) (-7 b e g+6 c d g+8 c e f)}{35 e (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{4 c (-7 b e g+6 c d g+8 c e f)}{35 e^2 (d+e x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (-7 b e g+6 c d g+8 c e f)}{35 e^2 (d+e x)^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (e f-d g)}{7 e^2 (d+e x)^3 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.231404, size = 331, normalized size = 1.17 \[ \frac{2 \left (4 b^2 c^2 e^2 \left (2 d^2 e (23 f+53 g x)+31 d^3 g+d e^2 x (20 f+59 g x)+2 e^3 x^2 (2 f+7 g x)\right )-2 b^3 c e^3 \left (11 d^2 g+d e (24 f+38 g x)+e^2 x (4 f+7 g x)\right )+b^4 e^4 (2 d g+5 e f+7 e g x)-8 b c^3 e \left (d^2 e^2 x (52 f-11 g x)+d^3 e (48 f+46 g x)+15 d^4 g+4 d e^3 x^2 (8 f-9 g x)+2 e^4 x^3 (4 f-7 g x)\right )+16 c^4 \left (-2 d^2 e^3 x^2 (10 f+9 g x)+d^3 e^2 x (4 f-15 g x)+d^4 e (13 f+3 g x)+d^5 g-6 d e^4 x^3 (4 f+g x)-8 e^5 f x^4\right )\right )}{35 e^2 (d+e x)^3 (b e-2 c d)^5 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b^4*e^4*(5*e*f + 2*d*g + 7*e*g*x) + 16*c^4*(d^5*g - 8*e^5*f*x^4 + d^3*e^2*x*(4*f - 15*g*x) - 6*d*e^4*x^3*(
4*f + g*x) + d^4*e*(13*f + 3*g*x) - 2*d^2*e^3*x^2*(10*f + 9*g*x)) - 2*b^3*c*e^3*(11*d^2*g + e^2*x*(4*f + 7*g*x
) + d*e*(24*f + 38*g*x)) - 8*b*c^3*e*(15*d^4*g + d^2*e^2*x*(52*f - 11*g*x) + 4*d*e^3*x^2*(8*f - 9*g*x) + 2*e^4
*x^3*(4*f - 7*g*x) + d^3*e*(48*f + 46*g*x)) + 4*b^2*c^2*e^2*(31*d^3*g + 2*e^3*x^2*(2*f + 7*g*x) + 2*d^2*e*(23*
f + 53*g*x) + d*e^2*x*(20*f + 59*g*x))))/(35*e^2*(-2*c*d + b*e)^5*(d + e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d -
e*x))])

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Maple [B]  time = 0.011, size = 564, normalized size = 2. \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 112\,b{c}^{3}{e}^{5}g{x}^{4}-96\,{c}^{4}d{e}^{4}g{x}^{4}-128\,{c}^{4}{e}^{5}f{x}^{4}+56\,{b}^{2}{c}^{2}{e}^{5}g{x}^{3}+288\,b{c}^{3}d{e}^{4}g{x}^{3}-64\,b{c}^{3}{e}^{5}f{x}^{3}-288\,{c}^{4}{d}^{2}{e}^{3}g{x}^{3}-384\,{c}^{4}d{e}^{4}f{x}^{3}-14\,{b}^{3}c{e}^{5}g{x}^{2}+236\,{b}^{2}{c}^{2}d{e}^{4}g{x}^{2}+16\,{b}^{2}{c}^{2}{e}^{5}f{x}^{2}+88\,b{c}^{3}{d}^{2}{e}^{3}g{x}^{2}-256\,b{c}^{3}d{e}^{4}f{x}^{2}-240\,{c}^{4}{d}^{3}{e}^{2}g{x}^{2}-320\,{c}^{4}{d}^{2}{e}^{3}f{x}^{2}+7\,{b}^{4}{e}^{5}gx-76\,{b}^{3}cd{e}^{4}gx-8\,{b}^{3}c{e}^{5}fx+424\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}gx+80\,{b}^{2}{c}^{2}d{e}^{4}fx-368\,b{c}^{3}{d}^{3}{e}^{2}gx-416\,b{c}^{3}{d}^{2}{e}^{3}fx+48\,{c}^{4}{d}^{4}egx+64\,{c}^{4}{d}^{3}{e}^{2}fx+2\,{b}^{4}d{e}^{4}g+5\,{b}^{4}{e}^{5}f-22\,{b}^{3}c{d}^{2}{e}^{3}g-48\,{b}^{3}cd{e}^{4}f+124\,{b}^{2}{c}^{2}{d}^{3}{e}^{2}g+184\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}f-120\,b{c}^{3}{d}^{4}eg-384\,b{c}^{3}{d}^{3}{e}^{2}f+16\,{c}^{4}{d}^{5}g+208\,{c}^{4}{d}^{4}ef \right ) }{35\,{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 ( ex+d \right ) ^{2}} \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)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

-2/35*(c*e*x+b*e-c*d)*(112*b*c^3*e^5*g*x^4-96*c^4*d*e^4*g*x^4-128*c^4*e^5*f*x^4+56*b^2*c^2*e^5*g*x^3+288*b*c^3
*d*e^4*g*x^3-64*b*c^3*e^5*f*x^3-288*c^4*d^2*e^3*g*x^3-384*c^4*d*e^4*f*x^3-14*b^3*c*e^5*g*x^2+236*b^2*c^2*d*e^4
*g*x^2+16*b^2*c^2*e^5*f*x^2+88*b*c^3*d^2*e^3*g*x^2-256*b*c^3*d*e^4*f*x^2-240*c^4*d^3*e^2*g*x^2-320*c^4*d^2*e^3
*f*x^2+7*b^4*e^5*g*x-76*b^3*c*d*e^4*g*x-8*b^3*c*e^5*f*x+424*b^2*c^2*d^2*e^3*g*x+80*b^2*c^2*d*e^4*f*x-368*b*c^3
*d^3*e^2*g*x-416*b*c^3*d^2*e^3*f*x+48*c^4*d^4*e*g*x+64*c^4*d^3*e^2*f*x+2*b^4*d*e^4*g+5*b^4*e^5*f-22*b^3*c*d^2*
e^3*g-48*b^3*c*d*e^4*f+124*b^2*c^2*d^3*e^2*g+184*b^2*c^2*d^2*e^3*f-120*b*c^3*d^4*e*g-384*b*c^3*d^3*e^2*f+16*c^
4*d^5*g+208*c^4*d^4*e*f)/(e*x+d)^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)/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)

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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)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),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)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),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)/(e*x+d)**3/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")

[Out]

[undef, undef, undef, 1]