3.2214 \(\int \frac{f+g x}{(d+e x)^3 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=210 \[ -\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (d+e x)^3 (2 c d-b e)}-\frac{4 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x) (2 c d-b e)^3}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x)^2 (2 c d-b e)^2} \]

[Out]

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

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Rubi [A]  time = 0.32978, antiderivative size = 210, 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, 650} \[ -\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (d+e x)^3 (2 c d-b e)}-\frac{4 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x) (2 c d-b e)^3}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{15 e^2 (d+e x)^2 (2 c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.114256, size = 166, normalized size = 0.79 \[ -\frac{2 (b e-c d+c e x) \left (b^2 e^2 (2 d g+3 e f+5 e g x)-2 b c e \left (7 d^2 g+2 d e (4 f+9 g x)+e^2 x (2 f+5 g x)\right )+4 c^2 \left (d^2 e (7 f+9 g x)+3 d^3 g+3 d e^2 x (2 f+g x)+2 e^3 f x^2\right )\right )}{15 e^2 (d+e x)^2 (b e-2 c d)^3 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.009, size = 236, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -10\,bc{e}^{3}g{x}^{2}+12\,{c}^{2}d{e}^{2}g{x}^{2}+8\,{c}^{2}{e}^{3}f{x}^{2}+5\,{b}^{2}{e}^{3}gx-36\,bcd{e}^{2}gx-4\,bc{e}^{3}fx+36\,{c}^{2}{d}^{2}egx+24\,{c}^{2}d{e}^{2}fx+2\,{b}^{2}d{e}^{2}g+3\,{b}^{2}{e}^{3}f-14\,bc{d}^{2}eg-16\,bcd{e}^{2}f+12\,{c}^{2}{d}^{3}g+28\,{c}^{2}{d}^{2}ef \right ) }{15\,{e}^{2} \left ({b}^{3}{e}^{3}-6\,{b}^{2}cd{e}^{2}+12\,b{c}^{2}{d}^{2}e-8\,{c}^{3}{d}^{3} \right ) \left ( ex+d \right ) ^{2}}{\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)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

-2/15*(c*e*x+b*e-c*d)*(-10*b*c*e^3*g*x^2+12*c^2*d*e^2*g*x^2+8*c^2*e^3*f*x^2+5*b^2*e^3*g*x-36*b*c*d*e^2*g*x-4*b
*c*e^3*f*x+36*c^2*d^2*e*g*x+24*c^2*d*e^2*f*x+2*b^2*d*e^2*g+3*b^2*e^3*f-14*b*c*d^2*e*g-16*b*c*d*e^2*f+12*c^2*d^
3*g+28*c^2*d^2*e*f)/(e*x+d)^2/(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)^(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)^3/(-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 [A]  time = 60.4129, size = 745, normalized size = 3.55 \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 (6 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} g\right )} x^{2} +{\left (28 \, c^{2} d^{2} e - 16 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + 2 \,{\left (6 \, c^{2} d^{3} - 7 \, b c d^{2} e + b^{2} d e^{2}\right )} g +{\left (4 \,{\left (6 \, c^{2} d e^{2} - b c e^{3}\right )} f +{\left (36 \, c^{2} d^{2} e - 36 \, b c d e^{2} + 5 \, b^{2} e^{3}\right )} g\right )} x\right )}}{15 \,{\left (8 \, c^{3} d^{6} e^{2} - 12 \, b c^{2} d^{5} e^{3} + 6 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5} +{\left (8 \, c^{3} d^{3} e^{5} - 12 \, b c^{2} d^{2} e^{6} + 6 \, b^{2} c d e^{7} - b^{3} e^{8}\right )} x^{3} + 3 \,{\left (8 \, c^{3} d^{4} e^{4} - 12 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} - b^{3} d e^{7}\right )} x^{2} + 3 \,{\left (8 \, c^{3} d^{5} e^{3} - 12 \, b c^{2} d^{4} e^{4} + 6 \, b^{2} c d^{3} e^{5} - b^{3} d^{2} e^{6}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

-2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^2*e^3*f + (6*c^2*d*e^2 - 5*b*c*e^3)*g)*x^2 + (28*c^2*
d^2*e - 16*b*c*d*e^2 + 3*b^2*e^3)*f + 2*(6*c^2*d^3 - 7*b*c*d^2*e + b^2*d*e^2)*g + (4*(6*c^2*d*e^2 - b*c*e^3)*f
+ (36*c^2*d^2*e - 36*b*c*d*e^2 + 5*b^2*e^3)*g)*x)/(8*c^3*d^6*e^2 - 12*b*c^2*d^5*e^3 + 6*b^2*c*d^4*e^4 - b^3*d
^3*e^5 + (8*c^3*d^3*e^5 - 12*b*c^2*d^2*e^6 + 6*b^2*c*d*e^7 - b^3*e^8)*x^3 + 3*(8*c^3*d^4*e^4 - 12*b*c^2*d^3*e^
5 + 6*b^2*c*d^2*e^6 - b^3*d*e^7)*x^2 + 3*(8*c^3*d^5*e^3 - 12*b*c^2*d^4*e^4 + 6*b^2*c*d^3*e^5 - b^3*d^2*e^6)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)**3/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

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

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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)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError