3.2221 \(\int \frac{f+g x}{(d+e x)^2 (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=209 \[ -\frac{2 (e f-d g)}{5 e^2 (d+e x)^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{8 c (b+2 c x) (-5 b e g+4 c d g+6 c e f)}{15 e (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (-5 b e g+4 c d g+6 c e f)}{15 e^2 (d+e x) (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

(8*c*(6*c*e*f + 4*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^4*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2
]) - (2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(6*c*e*f
+ 4*c*d*g - 5*b*e*g))/(15*e^2*(2*c*d - b*e)^2*(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 = 0.284928, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068, Rules used = {792, 658, 613} \[ -\frac{2 (e f-d g)}{5 e^2 (d+e x)^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{8 c (b+2 c x) (-5 b e g+4 c d g+6 c e f)}{15 e (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (-5 b e g+4 c d g+6 c e f)}{15 e^2 (d+e x) (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*c*(6*c*e*f + 4*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^4*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2
]) - (2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(6*c*e*f
+ 4*c*d*g - 5*b*e*g))/(15*e^2*(2*c*d - b*e)^2*(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)^2 \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)}{5 e^2 (2 c d-b e) (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{(6 c e f+4 c d g-5 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}{5 e (2 c d-b e)}\\ &=-\frac{2 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (6 c e f+4 c d g-5 b e g)}{15 e^2 (2 c d-b e)^2 (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{(4 c (6 c e f+4 c d g-5 b e g)) \int \frac{1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{15 e (2 c d-b e)^2}\\ &=\frac{8 c (6 c e f+4 c d g-5 b e g) (b+2 c x)}{15 e (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (6 c e f+4 c d g-5 b e g)}{15 e^2 (2 c d-b e)^2 (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.156048, size = 233, normalized size = 1.11 \[ \frac{2 \left (-2 b^2 c e^2 \left (13 d^2 g+4 d e (3 f+8 g x)+e^2 x (3 f+10 g x)\right )+b^3 e^3 (2 d g+3 e f+5 e g x)+4 b c^2 e \left (7 d^2 e (3 f+g x)+4 d^3 g+2 d e^2 x (9 f-8 g x)+2 e^3 x^2 (3 f-5 g x)\right )+8 c^3 \left (d^2 e^2 x (3 f+8 g x)+d^3 e (2 g x-6 f)+d^4 g+4 d e^3 x^2 (3 f+g x)+6 e^4 f x^3\right )\right )}{15 e^2 (d+e x)^2 (b e-2 c d)^4 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

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)^(3/2)),x]

[Out]

(2*(b^3*e^3*(3*e*f + 2*d*g + 5*e*g*x) + 4*b*c^2*e*(4*d^3*g + 2*d*e^2*x*(9*f - 8*g*x) + 2*e^3*x^2*(3*f - 5*g*x)
+ 7*d^2*e*(3*f + g*x)) + 8*c^3*(d^4*g + 6*e^4*f*x^3 + 4*d*e^3*x^2*(3*f + g*x) + d^3*e*(-6*f + 2*g*x) + d^2*e^
2*x*(3*f + 8*g*x)) - 2*b^2*c*e^2*(13*d^2*g + 4*d*e*(3*f + 8*g*x) + e^2*x*(3*f + 10*g*x))))/(15*e^2*(-2*c*d + b
*e)^4*(d + e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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Maple [A]  time = 0.01, size = 382, normalized size = 1.8 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -40\,b{c}^{2}{e}^{4}g{x}^{3}+32\,{c}^{3}d{e}^{3}g{x}^{3}+48\,{c}^{3}{e}^{4}f{x}^{3}-20\,{b}^{2}c{e}^{4}g{x}^{2}-64\,b{c}^{2}d{e}^{3}g{x}^{2}+24\,b{c}^{2}{e}^{4}f{x}^{2}+64\,{c}^{3}{d}^{2}{e}^{2}g{x}^{2}+96\,{c}^{3}d{e}^{3}f{x}^{2}+5\,{b}^{3}{e}^{4}gx-64\,{b}^{2}cd{e}^{3}gx-6\,{b}^{2}c{e}^{4}fx+28\,b{c}^{2}{d}^{2}{e}^{2}gx+72\,b{c}^{2}d{e}^{3}fx+16\,{c}^{3}{d}^{3}egx+24\,{c}^{3}{d}^{2}{e}^{2}fx+2\,{b}^{3}d{e}^{3}g+3\,{b}^{3}{e}^{4}f-26\,{b}^{2}c{d}^{2}{e}^{2}g-24\,{b}^{2}cd{e}^{3}f+16\,b{c}^{2}{d}^{3}eg+84\,b{c}^{2}{d}^{2}{e}^{2}f+8\,{c}^{3}{d}^{4}g-48\,{c}^{3}{d}^{3}ef \right ) }{ \left ( 15\,ex+15\,d \right ){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)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

-2/15*(c*e*x+b*e-c*d)*(-40*b*c^2*e^4*g*x^3+32*c^3*d*e^3*g*x^3+48*c^3*e^4*f*x^3-20*b^2*c*e^4*g*x^2-64*b*c^2*d*e
^3*g*x^2+24*b*c^2*e^4*f*x^2+64*c^3*d^2*e^2*g*x^2+96*c^3*d*e^3*f*x^2+5*b^3*e^4*g*x-64*b^2*c*d*e^3*g*x-6*b^2*c*e
^4*f*x+28*b*c^2*d^2*e^2*g*x+72*b*c^2*d*e^3*f*x+16*c^3*d^3*e*g*x+24*c^3*d^2*e^2*f*x+2*b^3*d*e^3*g+3*b^3*e^4*f-2
6*b^2*c*d^2*e^2*g-24*b^2*c*d*e^3*f+16*b*c^2*d^3*e*g+84*b*c^2*d^2*e^2*f+8*c^3*d^4*g-48*c^3*d^3*e*f)/(e*x+d)/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)/(-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)^2/(-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)^2/(-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)**2/(-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*} \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)^(3/2),x, algorithm="giac")

[Out]

sage0*x