3.2228 \(\int \frac{f+g x}{(d+e x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=208 \[ -\frac{2 (e f-d g)}{5 e^2 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{16 c (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(8*c*e*f + 2*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2
)) - (2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (16*c*(8*c*
e*f + 2*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^5*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.199452, antiderivative size = 208, 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, 614, 613} \[ -\frac{2 (e f-d g)}{5 e^2 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{16 c (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(8*c*e*f + 2*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2
)) - (2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (16*c*(8*c*
e*f + 2*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^5*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 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) \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)}{5 e^2 (2 c d-b e) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(8 c e f+2 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 )^{5/2}} \, dx}{5 e (2 c d-b e)}\\ &=\frac{2 (8 c e f+2 c d g-5 b e g) (b+2 c x)}{15 e (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(8 c (8 c e f+2 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)^3}\\ &=\frac{2 (8 c e f+2 c d g-5 b e g) (b+2 c x)}{15 e (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{16 c (8 c e f+2 c d g-5 b e g) (b+2 c x)}{15 e (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.214069, size = 345, normalized size = 1.66 \[ -\frac{2 \left (12 b^2 c^2 e^2 \left (2 d^2 e (7 f-g x)+d^3 g+d e^2 x (12 f-19 g x)+2 e^3 x^2 (2 f-5 g x)\right )-2 b^3 c e^3 \left (19 d^2 g+2 d e (8 f+23 g x)+e^2 x (4 f+15 g x)\right )+b^4 e^4 (2 d g+3 e f+5 e g x)+8 b c^3 e \left (3 d^2 e^2 x (4 f+9 g x)-6 d^3 e (4 f-3 g x)+9 d^4 g-4 d e^3 x^2 (g x-12 f)+2 e^4 x^3 (12 f-5 g x)\right )-16 c^4 \left (-2 d^2 e^3 x^2 (g x-6 f)+3 d^3 e^2 x (4 f+g x)-3 d^4 e (f-g x)+3 d^5 g-2 d e^4 x^3 (4 f+g x)-8 e^5 f x^4\right )\right )}{15 e^2 (d+e x)^2 (b e-2 c d)^5 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b^4*e^4*(3*e*f + 2*d*g + 5*e*g*x) + 12*b^2*c^2*e^2*(d^3*g + d*e^2*x*(12*f - 19*g*x) + 2*e^3*x^2*(2*f - 5*
g*x) + 2*d^2*e*(7*f - g*x)) - 16*c^4*(3*d^5*g - 8*e^5*f*x^4 - 3*d^4*e*(f - g*x) - 2*d^2*e^3*x^2*(-6*f + g*x) +
3*d^3*e^2*x*(4*f + g*x) - 2*d*e^4*x^3*(4*f + g*x)) + 8*b*c^3*e*(9*d^4*g + 2*e^4*x^3*(12*f - 5*g*x) - 6*d^3*e*
(4*f - 3*g*x) - 4*d*e^3*x^2*(-12*f + g*x) + 3*d^2*e^2*x*(4*f + 9*g*x)) - 2*b^3*c*e^3*(19*d^2*g + e^2*x*(4*f +
15*g*x) + 2*d*e*(8*f + 23*g*x))))/(15*e^2*(-2*c*d + b*e)^5*(d + e*x)^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(
-(b*e) + c*(d - e*x))])

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 557, normalized size = 2.7 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -80\,b{c}^{3}{e}^{5}g{x}^{4}+32\,{c}^{4}d{e}^{4}g{x}^{4}+128\,{c}^{4}{e}^{5}f{x}^{4}-120\,{b}^{2}{c}^{2}{e}^{5}g{x}^{3}-32\,b{c}^{3}d{e}^{4}g{x}^{3}+192\,b{c}^{3}{e}^{5}f{x}^{3}+32\,{c}^{4}{d}^{2}{e}^{3}g{x}^{3}+128\,{c}^{4}d{e}^{4}f{x}^{3}-30\,{b}^{3}c{e}^{5}g{x}^{2}-228\,{b}^{2}{c}^{2}d{e}^{4}g{x}^{2}+48\,{b}^{2}{c}^{2}{e}^{5}f{x}^{2}+216\,b{c}^{3}{d}^{2}{e}^{3}g{x}^{2}+384\,b{c}^{3}d{e}^{4}f{x}^{2}-48\,{c}^{4}{d}^{3}{e}^{2}g{x}^{2}-192\,{c}^{4}{d}^{2}{e}^{3}f{x}^{2}+5\,{b}^{4}{e}^{5}gx-92\,{b}^{3}cd{e}^{4}gx-8\,{b}^{3}c{e}^{5}fx-24\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}gx+144\,{b}^{2}{c}^{2}d{e}^{4}fx+144\,b{c}^{3}{d}^{3}{e}^{2}gx+96\,b{c}^{3}{d}^{2}{e}^{3}fx-48\,{c}^{4}{d}^{4}egx-192\,{c}^{4}{d}^{3}{e}^{2}fx+2\,{b}^{4}d{e}^{4}g+3\,{b}^{4}{e}^{5}f-38\,{b}^{3}c{d}^{2}{e}^{3}g-32\,{b}^{3}cd{e}^{4}f+12\,{b}^{2}{c}^{2}{d}^{3}{e}^{2}g+168\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}f+72\,b{c}^{3}{d}^{4}eg-192\,b{c}^{3}{d}^{3}{e}^{2}f-48\,{c}^{4}{d}^{5}g+48\,{c}^{4}{d}^{4}ef \right ) }{ \left ( 15\,{b}^{5}{e}^{5}-150\,{b}^{4}cd{e}^{4}+600\,{b}^{3}{c}^{2}{d}^{2}{e}^{3}-1200\,{b}^{2}{c}^{3}{d}^{3}{e}^{2}+1200\,b{c}^{4}{d}^{4}e-480\,{c}^{5}{d}^{5} \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)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

-2/15*(c*e*x+b*e-c*d)*(-80*b*c^3*e^5*g*x^4+32*c^4*d*e^4*g*x^4+128*c^4*e^5*f*x^4-120*b^2*c^2*e^5*g*x^3-32*b*c^3
*d*e^4*g*x^3+192*b*c^3*e^5*f*x^3+32*c^4*d^2*e^3*g*x^3+128*c^4*d*e^4*f*x^3-30*b^3*c*e^5*g*x^2-228*b^2*c^2*d*e^4
*g*x^2+48*b^2*c^2*e^5*f*x^2+216*b*c^3*d^2*e^3*g*x^2+384*b*c^3*d*e^4*f*x^2-48*c^4*d^3*e^2*g*x^2-192*c^4*d^2*e^3
*f*x^2+5*b^4*e^5*g*x-92*b^3*c*d*e^4*g*x-8*b^3*c*e^5*f*x-24*b^2*c^2*d^2*e^3*g*x+144*b^2*c^2*d*e^4*f*x+144*b*c^3
*d^3*e^2*g*x+96*b*c^3*d^2*e^3*f*x-48*c^4*d^4*e*g*x-192*c^4*d^3*e^2*f*x+2*b^4*d*e^4*g+3*b^4*e^5*f-38*b^3*c*d^2*
e^3*g-32*b^3*c*d*e^4*f+12*b^2*c^2*d^3*e^2*g+168*b^2*c^2*d^2*e^3*f+72*b*c^3*d^4*e*g-192*b*c^3*d^3*e^2*f-48*c^4*
d^5*g+48*c^4*d^4*e*f)/(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)^(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)/(-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)/(-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)/(-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*} \left [\mathit{undef}, \mathit{undef}, \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)^(5/2),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, 1]