3.2220 \(\int \frac{f+g x}{(d+e x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=136 \[ \frac{2 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{3 e (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (e f-d g)}{3 e^2 (d+e x) (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

(2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x))/(3*e*(2*c*d - b*e)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])
- (2*(e*f - d*g))/(3*e^2*(2*c*d - b*e)*(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 = 0.144947, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {792, 613} \[ \frac{2 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{3 e (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (e f-d g)}{3 e^2 (d+e x) (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x))/(3*e*(2*c*d - b*e)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])
- (2*(e*f - d*g))/(3*e^2*(2*c*d - b*e)*(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 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) \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)}{3 e^2 (2 c d-b e) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{(4 c e f+2 c d g-3 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}{3 e (2 c d-b e)}\\ &=\frac{2 (4 c e f+2 c d g-3 b e g) (b+2 c x)}{3 e (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (e f-d g)}{3 e^2 (2 c d-b e) (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.121711, size = 149, normalized size = 1.1 \[ \frac{2 b^2 e^2 (2 d g+e (f+3 g x))+4 b c e \left (d^2 g+d e (2 g x-4 f)+e^2 x (3 g x-2 f)\right )-8 c^2 \left (d^2 e (g x-f)+d^3 g+d e^2 x (2 f+g x)+2 e^3 f x^2\right )}{3 e^2 (d+e x) (b e-2 c d)^3 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 228, normalized size = 1.7 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 6\,bc{e}^{3}g{x}^{2}-4\,{c}^{2}d{e}^{2}g{x}^{2}-8\,{c}^{2}{e}^{3}f{x}^{2}+3\,{b}^{2}{e}^{3}gx+4\,bcd{e}^{2}gx-4\,bc{e}^{3}fx-4\,{c}^{2}{d}^{2}egx-8\,{c}^{2}d{e}^{2}fx+2\,{b}^{2}d{e}^{2}g+{b}^{2}{e}^{3}f+2\,bc{d}^{2}eg-8\,bcd{e}^{2}f-4\,{c}^{2}{d}^{3}g+4\,{c}^{2}{d}^{2}ef \right ) }{ \left ( 3\,{b}^{3}{e}^{3}-18\,{b}^{2}cd{e}^{2}+36\,b{c}^{2}{d}^{2}e-24\,{c}^{3}{d}^{3} \right ){e}^{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)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

-2/3*(c*e*x+b*e-c*d)*(6*b*c*e^3*g*x^2-4*c^2*d*e^2*g*x^2-8*c^2*e^3*f*x^2+3*b^2*e^3*g*x+4*b*c*d*e^2*g*x-4*b*c*e^
3*f*x-4*c^2*d^2*e*g*x-8*c^2*d*e^2*f*x+2*b^2*d*e^2*g+b^2*e^3*f+2*b*c*d^2*e*g-8*b*c*d*e^2*f-4*c^2*d^3*g+4*c^2*d^
2*e*f)/(b^3*e^3-6*b^2*c*d*e^2+12*b*c^2*d^2*e-8*c^3*d^3)/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)/(-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 [B]  time = 44.0539, size = 810, normalized size = 5.96 \begin{align*} \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \,{\left (4 \, c^{2} e^{3} f +{\left (2 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} g\right )} x^{2} -{\left (4 \, c^{2} d^{2} e - 8 \, b c d e^{2} + b^{2} e^{3}\right )} f + 2 \,{\left (2 \, c^{2} d^{3} - b c d^{2} e - b^{2} d e^{2}\right )} g +{\left (4 \,{\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} f +{\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} - 3 \, b^{2} e^{3}\right )} g\right )} x\right )}}{3 \,{\left (8 \, c^{4} d^{6} e^{2} - 20 \, b c^{3} d^{5} e^{3} + 18 \, b^{2} c^{2} d^{4} e^{4} - 7 \, b^{3} c d^{3} e^{5} + b^{4} d^{2} e^{6} -{\left (8 \, c^{4} d^{3} e^{5} - 12 \, b c^{3} d^{2} e^{6} + 6 \, b^{2} c^{2} d e^{7} - b^{3} c e^{8}\right )} x^{3} -{\left (8 \, c^{4} d^{4} e^{4} - 4 \, b c^{3} d^{3} e^{5} - 6 \, b^{2} c^{2} d^{2} e^{6} + 5 \, b^{3} c d e^{7} - b^{4} e^{8}\right )} x^{2} +{\left (8 \, c^{4} d^{5} e^{3} - 28 \, b c^{3} d^{4} e^{4} + 30 \, b^{2} c^{2} d^{3} e^{5} - 13 \, b^{3} c d^{2} e^{6} + 2 \, b^{4} d e^{7}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^2*e^3*f + (2*c^2*d*e^2 - 3*b*c*e^3)*g)*x^2 - (4*c^2*d^2
*e - 8*b*c*d*e^2 + b^2*e^3)*f + 2*(2*c^2*d^3 - b*c*d^2*e - b^2*d*e^2)*g + (4*(2*c^2*d*e^2 + b*c*e^3)*f + (4*c^
2*d^2*e - 4*b*c*d*e^2 - 3*b^2*e^3)*g)*x)/(8*c^4*d^6*e^2 - 20*b*c^3*d^5*e^3 + 18*b^2*c^2*d^4*e^4 - 7*b^3*c*d^3*
e^5 + b^4*d^2*e^6 - (8*c^4*d^3*e^5 - 12*b*c^3*d^2*e^6 + 6*b^2*c^2*d*e^7 - b^3*c*e^8)*x^3 - (8*c^4*d^4*e^4 - 4*
b*c^3*d^3*e^5 - 6*b^2*c^2*d^2*e^6 + 5*b^3*c*d*e^7 - b^4*e^8)*x^2 + (8*c^4*d^5*e^3 - 28*b*c^3*d^4*e^4 + 30*b^2*
c^2*d^3*e^5 - 13*b^3*c*d^2*e^6 + 2*b^4*d*e^7)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Integral((f + g*x)/((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(d + e*x)), x)

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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)/(-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]