3.2178 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=137 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-5 b e g+8 c d g+2 c e f)}{15 e^2 (d+e x)^3 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 e^2 (d+e x)^4 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(5*e^2*(2*c*d - b*e)*(d + e*x)^4) - (2*(2*c*e*f +
8*c*d*g - 5*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(15*e^2*(2*c*d - b*e)^2*(d + e*x)^3)

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Rubi [A]  time = 0.206598, antiderivative size = 137, 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, 650} \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-5 b e g+8 c d g+2 c e f)}{15 e^2 (d+e x)^3 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 e^2 (d+e x)^4 (2 c d-b e)} \]

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)^4,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(5*e^2*(2*c*d - b*e)*(d + e*x)^4) - (2*(2*c*e*f +
8*c*d*g - 5*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(15*e^2*(2*c*d - b*e)^2*(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 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)^4} \, 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}}{5 e^2 (2 c d-b e) (d+e x)^4}+\frac{(2 c e f+8 c d g-5 b e g) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx}{5 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}}{5 e^2 (2 c d-b e) (d+e x)^4}-\frac{2 (2 c e f+8 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15 e^2 (2 c d-b e)^2 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0666737, size = 102, normalized size = 0.74 \[ \frac{2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)} \left (2 c \left (d^2 g+4 d e (f+g x)+e^2 f x\right )-b e (2 d g+3 e f+5 e g x)\right )}{15 e^2 (d+e x)^3 (b e-2 c d)^2} \]

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)^4,x]

[Out]

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

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Maple [A]  time = 0.007, size = 128, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 5\,b{e}^{2}gx-8\,cdegx-2\,c{e}^{2}fx+2\,bdeg+3\,b{e}^{2}f-2\,c{d}^{2}g-8\,cdef \right ) }{15\, \left ( ex+d \right ) ^{3}{e}^{2} \left ({b}^{2}{e}^{2}-4\,bcde+4\,{c}^{2}{d}^{2} \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)^4,x)

[Out]

-2/15*(c*e*x+b*e-c*d)*(5*b*e^2*g*x-8*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g+3*b*e^2*f-2*c*d^2*g-8*c*d*e*f)*(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^3/e^2/(b^2*e^2-4*b*c*d*e+4*c^2*d^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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 58.2361, size = 620, normalized size = 4.53 \begin{align*} \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left ({\left (2 \, c^{2} e^{3} f +{\left (8 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} g\right )} x^{2} -{\left (8 \, c^{2} d^{2} e - 11 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f - 2 \,{\left (c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2}\right )} g +{\left ({\left (6 \, c^{2} d e^{2} - b c e^{3}\right )} f -{\left (6 \, c^{2} d^{2} e - 11 \, b c d e^{2} + 5 \, b^{2} e^{3}\right )} g\right )} x\right )}}{15 \,{\left (4 \, c^{2} d^{5} e^{2} - 4 \, b c d^{4} e^{3} + b^{2} d^{3} e^{4} +{\left (4 \, c^{2} d^{2} e^{5} - 4 \, b c d e^{6} + b^{2} e^{7}\right )} x^{3} + 3 \,{\left (4 \, c^{2} d^{3} e^{4} - 4 \, b c d^{2} e^{5} + b^{2} d e^{6}\right )} x^{2} + 3 \,{\left (4 \, c^{2} d^{4} e^{3} - 4 \, b c d^{3} e^{4} + b^{2} d^{2} e^{5}\right )} x\right )}} \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)^4,x, algorithm="fricas")

[Out]

2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((2*c^2*e^3*f + (8*c^2*d*e^2 - 5*b*c*e^3)*g)*x^2 - (8*c^2*d^2*
e - 11*b*c*d*e^2 + 3*b^2*e^3)*f - 2*(c^2*d^3 - 2*b*c*d^2*e + b^2*d*e^2)*g + ((6*c^2*d*e^2 - b*c*e^3)*f - (6*c^
2*d^2*e - 11*b*c*d*e^2 + 5*b^2*e^3)*g)*x)/(4*c^2*d^5*e^2 - 4*b*c*d^4*e^3 + b^2*d^3*e^4 + (4*c^2*d^2*e^5 - 4*b*
c*d*e^6 + b^2*e^7)*x^3 + 3*(4*c^2*d^3*e^4 - 4*b*c*d^2*e^5 + b^2*d*e^6)*x^2 + 3*(4*c^2*d^4*e^3 - 4*b*c*d^3*e^4
+ b^2*d^2*e^5)*x)

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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)**4,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)^4,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError