3.2180 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=285 \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{315 e^2 (d+e x)^3 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{105 e^2 (d+e x)^4 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{21 e^2 (d+e x)^5 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 e^2 (d+e x)^6 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(9*e^2*(2*c*d - b*e)*(d + e*x)^6) - (2*(2*c*e*f +
4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(21*e^2*(2*c*d - b*e)^2*(d + e*x)^5) - (8*c*(
2*c*e*f + 4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(105*e^2*(2*c*d - b*e)^3*(d + e*x)^4
) - (16*c^2*(2*c*e*f + 4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(315*e^2*(2*c*d - b*e)^
4*(d + e*x)^3)

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Rubi [A]  time = 0.456655, 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} \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{315 e^2 (d+e x)^3 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{105 e^2 (d+e x)^4 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{21 e^2 (d+e x)^5 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 e^2 (d+e x)^6 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^6,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(9*e^2*(2*c*d - b*e)*(d + e*x)^6) - (2*(2*c*e*f +
4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(21*e^2*(2*c*d - b*e)^2*(d + e*x)^5) - (8*c*(
2*c*e*f + 4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(105*e^2*(2*c*d - b*e)^3*(d + e*x)^4
) - (16*c^2*(2*c*e*f + 4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(315*e^2*(2*c*d - b*e)^
4*(d + e*x)^3)

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) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 e^2 (2 c d-b e) (d+e x)^6}+\frac{(2 c e f+4 c d g-3 b e g) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^5} \, dx}{3 e (2 c d-b e)}\\ &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 e^2 (2 c d-b e) (d+e x)^6}-\frac{2 (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{21 e^2 (2 c d-b e)^2 (d+e x)^5}+\frac{(4 c (2 c e f+4 c d g-3 b e g)) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx}{21 e (2 c d-b e)^2}\\ &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 e^2 (2 c d-b e) (d+e x)^6}-\frac{2 (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{21 e^2 (2 c d-b e)^2 (d+e x)^5}-\frac{8 c (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{105 e^2 (2 c d-b e)^3 (d+e x)^4}+\frac{\left (8 c^2 (2 c e f+4 c d g-3 b e g)\right ) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx}{105 e (2 c d-b e)^3}\\ &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 e^2 (2 c d-b e) (d+e x)^6}-\frac{2 (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{21 e^2 (2 c d-b e)^2 (d+e x)^5}-\frac{8 c (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{105 e^2 (2 c d-b e)^3 (d+e x)^4}-\frac{16 c^2 (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{315 e^2 (2 c d-b e)^4 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.202175, size = 232, normalized size = 0.81 \[ -\frac{2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (6 b^2 c e^2 \left (11 d^2 g+d e (40 f+52 g x)+e^2 x (5 f+6 g x)\right )-5 b^3 e^3 (2 d g+7 e f+9 e g x)-12 b c^2 e \left (d^2 e (47 f+61 g x)+12 d^3 g+2 d e^2 x (7 f+8 g x)+2 e^3 x^2 (f+g x)\right )+8 c^3 \left (3 d^2 e^2 x (11 f+8 g x)+d^3 e (58 f+66 g x)+11 d^4 g+4 d e^3 x^2 (3 f+g x)+2 e^4 f x^3\right )\right )}{315 e^2 (d+e x)^6 (b e-2 c d)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^6,x]

[Out]

(-2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-5*b^3*e^3*(7*e*f + 2*d*g + 9*e*g*x) + 6*b^2*c*e^2*(11*d^2*g + e
^2*x*(5*f + 6*g*x) + d*e*(40*f + 52*g*x)) - 12*b*c^2*e*(12*d^3*g + 2*e^3*x^2*(f + g*x) + 2*d*e^2*x*(7*f + 8*g*
x) + d^2*e*(47*f + 61*g*x)) + 8*c^3*(11*d^4*g + 2*e^4*f*x^3 + 4*d*e^3*x^2*(3*f + g*x) + 3*d^2*e^2*x*(11*f + 8*
g*x) + d^3*e*(58*f + 66*g*x))))/(315*e^2*(-2*c*d + b*e)^4*(d + e*x)^6)

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Maple [A]  time = 0.013, size = 382, normalized size = 1.3 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 24\,b{c}^{2}{e}^{4}g{x}^{3}-32\,{c}^{3}d{e}^{3}g{x}^{3}-16\,{c}^{3}{e}^{4}f{x}^{3}-36\,{b}^{2}c{e}^{4}g{x}^{2}+192\,b{c}^{2}d{e}^{3}g{x}^{2}+24\,b{c}^{2}{e}^{4}f{x}^{2}-192\,{c}^{3}{d}^{2}{e}^{2}g{x}^{2}-96\,{c}^{3}d{e}^{3}f{x}^{2}+45\,{b}^{3}{e}^{4}gx-312\,{b}^{2}cd{e}^{3}gx-30\,{b}^{2}c{e}^{4}fx+732\,b{c}^{2}{d}^{2}{e}^{2}gx+168\,b{c}^{2}d{e}^{3}fx-528\,{c}^{3}{d}^{3}egx-264\,{c}^{3}{d}^{2}{e}^{2}fx+10\,{b}^{3}d{e}^{3}g+35\,{b}^{3}{e}^{4}f-66\,{b}^{2}c{d}^{2}{e}^{2}g-240\,{b}^{2}cd{e}^{3}f+144\,b{c}^{2}{d}^{3}eg+564\,b{c}^{2}{d}^{2}{e}^{2}f-88\,{c}^{3}{d}^{4}g-464\,{c}^{3}{d}^{3}ef \right ) }{315\, \left ( ex+d \right ) ^{5}{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 ) }\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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^6,x)

[Out]

-2/315*(c*e*x+b*e-c*d)*(24*b*c^2*e^4*g*x^3-32*c^3*d*e^3*g*x^3-16*c^3*e^4*f*x^3-36*b^2*c*e^4*g*x^2+192*b*c^2*d*
e^3*g*x^2+24*b*c^2*e^4*f*x^2-192*c^3*d^2*e^2*g*x^2-96*c^3*d*e^3*f*x^2+45*b^3*e^4*g*x-312*b^2*c*d*e^3*g*x-30*b^
2*c*e^4*f*x+732*b*c^2*d^2*e^2*g*x+168*b*c^2*d*e^3*f*x-528*c^3*d^3*e*g*x-264*c^3*d^2*e^2*f*x+10*b^3*d*e^3*g+35*
b^3*e^4*f-66*b^2*c*d^2*e^2*g-240*b^2*c*d*e^3*f+144*b*c^2*d^3*e*g+564*b*c^2*d^2*e^2*f-88*c^3*d^4*g-464*c^3*d^3*
e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^5/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)

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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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^6,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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^6,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)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**6,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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError