3.2215 \(\int \frac{f+g x}{(d+e x)^4 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=285 \[ -\frac{16 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x) (2 c d-b e)^4}-\frac{8 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x)^2 (2 c d-b e)^3}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{35 e^2 (d+e x)^3 (2 c d-b e)^2}-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (d+e x)^4 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(7*e^2*(2*c*d - b*e)*(d + e*x)^4) - (2*(6*c*e*f + 8
*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^3) - (8*c*(6*c*
e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^3*(d + e*x)^2) - (1
6*c^2*(6*c*e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^4*(d + e
*x))

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Rubi [A]  time = 0.441794, antiderivative size = 285, 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, 650} \[ -\frac{16 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x) (2 c d-b e)^4}-\frac{8 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x)^2 (2 c d-b e)^3}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{35 e^2 (d+e x)^3 (2 c d-b e)^2}-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (d+e x)^4 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(7*e^2*(2*c*d - b*e)*(d + e*x)^4) - (2*(6*c*e*f + 8
*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^3) - (8*c*(6*c*
e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^3*(d + e*x)^2) - (1
6*c^2*(6*c*e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^4*(d + e
*x))

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 650

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))/((p + 1)*(2*c*d - b*e)), 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] && EqQ[m + 2*p + 2, 0]

Rubi steps

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

Mathematica [A]  time = 0.186849, size = 247, normalized size = 0.87 \[ \frac{2 (b e-c d+c e x) \left (2 b^2 c e^2 \left (23 d^2 g+d e (54 f+82 g x)+e^2 x (9 f+14 g x)\right )-3 b^3 e^3 (2 d g+5 e f+7 e g x)-4 b c^2 e \left (d^2 e (69 f+131 g x)+36 d^3 g+2 d e^2 x (15 f+32 g x)+2 e^3 x^2 (3 f+7 g x)\right )+8 c^3 \left (d^2 e^2 x (39 f+32 g x)+4 d^3 e (9 f+13 g x)+13 d^4 g+8 d e^3 x^2 (3 f+g x)+6 e^4 f x^3\right )\right )}{105 e^2 (d+e x)^3 (b e-2 c d)^4 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-(c*d) + b*e + c*e*x)*(-3*b^3*e^3*(5*e*f + 2*d*g + 7*e*g*x) + 8*c^3*(13*d^4*g + 6*e^4*f*x^3 + 8*d*e^3*x^2*
(3*f + g*x) + 4*d^3*e*(9*f + 13*g*x) + d^2*e^2*x*(39*f + 32*g*x)) + 2*b^2*c*e^2*(23*d^2*g + e^2*x*(9*f + 14*g*
x) + d*e*(54*f + 82*g*x)) - 4*b*c^2*e*(36*d^3*g + 2*e^3*x^2*(3*f + 7*g*x) + 2*d*e^2*x*(15*f + 32*g*x) + d^2*e*
(69*f + 131*g*x))))/(105*e^2*(-2*c*d + b*e)^4*(d + e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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Maple [A]  time = 0.01, size = 382, normalized size = 1.3 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 56\,b{c}^{2}{e}^{4}g{x}^{3}-64\,{c}^{3}d{e}^{3}g{x}^{3}-48\,{c}^{3}{e}^{4}f{x}^{3}-28\,{b}^{2}c{e}^{4}g{x}^{2}+256\,b{c}^{2}d{e}^{3}g{x}^{2}+24\,b{c}^{2}{e}^{4}f{x}^{2}-256\,{c}^{3}{d}^{2}{e}^{2}g{x}^{2}-192\,{c}^{3}d{e}^{3}f{x}^{2}+21\,{b}^{3}{e}^{4}gx-164\,{b}^{2}cd{e}^{3}gx-18\,{b}^{2}c{e}^{4}fx+524\,b{c}^{2}{d}^{2}{e}^{2}gx+120\,b{c}^{2}d{e}^{3}fx-416\,{c}^{3}{d}^{3}egx-312\,{c}^{3}{d}^{2}{e}^{2}fx+6\,{b}^{3}d{e}^{3}g+15\,{b}^{3}{e}^{4}f-46\,{b}^{2}c{d}^{2}{e}^{2}g-108\,{b}^{2}cd{e}^{3}f+144\,b{c}^{2}{d}^{3}eg+276\,b{c}^{2}{d}^{2}{e}^{2}f-104\,{c}^{3}{d}^{4}g-288\,{c}^{3}{d}^{3}ef \right ) }{105\, \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 ){e}^{2} \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)/(e*x+d)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

-2/105*(c*e*x+b*e-c*d)*(56*b*c^2*e^4*g*x^3-64*c^3*d*e^3*g*x^3-48*c^3*e^4*f*x^3-28*b^2*c*e^4*g*x^2+256*b*c^2*d*
e^3*g*x^2+24*b*c^2*e^4*f*x^2-256*c^3*d^2*e^2*g*x^2-192*c^3*d*e^3*f*x^2+21*b^3*e^4*g*x-164*b^2*c*d*e^3*g*x-18*b
^2*c*e^4*f*x+524*b*c^2*d^2*e^2*g*x+120*b*c^2*d*e^3*f*x-416*c^3*d^3*e*g*x-312*c^3*d^2*e^2*f*x+6*b^3*d*e^3*g+15*
b^3*e^4*f-46*b^2*c*d^2*e^2*g-108*b^2*c*d*e^3*f+144*b*c^2*d^3*e*g+276*b*c^2*d^2*e^2*f-104*c^3*d^4*g-288*c^3*d^3
*e*f)/(e*x+d)^3/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)^(1/2)

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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)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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)**4/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError