3.2227 \(\int \frac{(d+e x) (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=165 \[ \frac{2 (b+2 c x) (-b e g-2 c d g+4 c e f)}{3 c e (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (e x (2 c d-b e)+d (2 c d-b e)) (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.179259, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {777, 613} \[ \frac{2 (b+2 c x) (-b e g-2 c d g+4 c e f)}{3 c e (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (e x (2 c d-b e)+d (2 c d-b e)) (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

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

Mathematica [A]  time = 0.107284, size = 151, normalized size = 0.92 \[ \frac{6 b^2 e^2 (2 d g-e f+e g x)-4 b c e \left (5 d^2 g-2 d e g x+e^2 x (6 f-g x)\right )+8 c^2 \left (d^2 e (f-g x)+d^3 g+d e^2 x (2 f+g x)-2 e^3 f x^2\right )}{3 e^2 (b e-2 c d)^3 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b^2*e^2*(-(e*f) + 2*d*g + e*g*x) - 4*b*c*e*(5*d^2*g - 2*d*e*g*x + e^2*x*(6*f - g*x)) + 8*c^2*(d^3*g - 2*e^3
*f*x^2 + d^2*e*(f - g*x) + d*e^2*x*(2*f + g*x)))/(3*e^2*(-2*c*d + b*e)^3*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)
*(-(b*e) + c*(d - e*x))])

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Maple [A]  time = 0.01, size = 227, normalized size = 1.4 \begin{align*}{\frac{2\, \left ( ex+d \right ) ^{2} \left ( cex+be-cd \right ) \left ( 2\,bc{e}^{3}g{x}^{2}+4\,{c}^{2}d{e}^{2}g{x}^{2}-8\,{c}^{2}{e}^{3}f{x}^{2}+3\,{b}^{2}{e}^{3}gx+4\,bcd{e}^{2}gx-12\,bc{e}^{3}fx-4\,{c}^{2}{d}^{2}egx+8\,{c}^{2}d{e}^{2}fx+6\,{b}^{2}d{e}^{2}g-3\,{b}^{2}{e}^{3}f-10\,bc{d}^{2}eg+4\,{c}^{2}{d}^{3}g+4\,{c}^{2}{d}^{2}ef \right ) }{ \left ( 3\,{b}^{3}{e}^{3}-18\,{b}^{2}cd{e}^{2}+36\,b{c}^{2}{d}^{2}e-24\,{c}^{3}{d}^{3} \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((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

2/3*(e*x+d)^2*(c*e*x+b*e-c*d)*(2*b*c*e^3*g*x^2+4*c^2*d*e^2*g*x^2-8*c^2*e^3*f*x^2+3*b^2*e^3*g*x+4*b*c*d*e^2*g*x
-12*b*c*e^3*f*x-4*c^2*d^2*e*g*x+8*c^2*d*e^2*f*x+6*b^2*d*e^2*g-3*b^2*e^3*f-10*b*c*d^2*e*g+4*c^2*d^3*g+4*c^2*d^2
*e*f)/(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)^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 81.3107, size = 859, normalized size = 5.21 \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 (2 \, c^{2} d e^{2} + b c e^{3}\right )} g\right )} x^{2} -{\left (4 \, c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} f - 2 \,{\left (2 \, c^{2} d^{3} - 5 \, b c d^{2} e + 3 \, b^{2} d e^{2}\right )} g -{\left (4 \,{\left (2 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} f -{\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} - 3 \, b^{2} e^{3}\right )} g\right )} x\right )}}{3 \,{\left (8 \, c^{5} d^{6} e^{2} - 28 \, b c^{4} d^{5} e^{3} + 38 \, b^{2} c^{3} d^{4} e^{4} - 25 \, b^{3} c^{2} d^{3} e^{5} + 8 \, b^{4} c d^{2} e^{6} - b^{5} d e^{7} +{\left (8 \, c^{5} d^{3} e^{5} - 12 \, b c^{4} d^{2} e^{6} + 6 \, b^{2} c^{3} d e^{7} - b^{3} c^{2} e^{8}\right )} x^{3} -{\left (8 \, c^{5} d^{4} e^{4} - 28 \, b c^{4} d^{3} e^{5} + 30 \, b^{2} c^{3} d^{2} e^{6} - 13 \, b^{3} c^{2} d e^{7} + 2 \, b^{4} c e^{8}\right )} x^{2} -{\left (8 \, c^{5} d^{5} e^{3} - 12 \, b c^{4} d^{4} e^{4} - 2 \, b^{2} c^{3} d^{3} e^{5} + 11 \, b^{3} c^{2} d^{2} e^{6} - 6 \, b^{4} c d e^{7} + b^{5} e^{8}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^2*e^3*f - (2*c^2*d*e^2 + b*c*e^3)*g)*x^2 - (4*c^2*d^2*
e - 3*b^2*e^3)*f - 2*(2*c^2*d^3 - 5*b*c*d^2*e + 3*b^2*d*e^2)*g - (4*(2*c^2*d*e^2 - 3*b*c*e^3)*f - (4*c^2*d^2*e
- 4*b*c*d*e^2 - 3*b^2*e^3)*g)*x)/(8*c^5*d^6*e^2 - 28*b*c^4*d^5*e^3 + 38*b^2*c^3*d^4*e^4 - 25*b^3*c^2*d^3*e^5
+ 8*b^4*c*d^2*e^6 - b^5*d*e^7 + (8*c^5*d^3*e^5 - 12*b*c^4*d^2*e^6 + 6*b^2*c^3*d*e^7 - b^3*c^2*e^8)*x^3 - (8*c^
5*d^4*e^4 - 28*b*c^4*d^3*e^5 + 30*b^2*c^3*d^2*e^6 - 13*b^3*c^2*d*e^7 + 2*b^4*c*e^8)*x^2 - (8*c^5*d^5*e^3 - 12*
b*c^4*d^4*e^4 - 2*b^2*c^3*d^3*e^5 + 11*b^3*c^2*d^2*e^6 - 6*b^4*c*d*e^7 + b^5*e^8)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right ) \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Integral((d + e*x)*(f + g*x)/(-(d + e*x)*(b*e - c*d + c*e*x))**(5/2), x)

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Giac [B]  time = 1.26342, size = 703, normalized size = 4.26 \begin{align*} \frac{2 \, \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left ({\left ({\left (\frac{2 \,{\left (4 \, c^{3} d^{2} g e^{3} - 8 \, c^{3} d f e^{4} + 4 \, b c^{2} f e^{5} - b^{2} c g e^{5}\right )} x}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}} + \frac{3 \,{\left (4 \, b c^{2} d^{2} g e^{3} - 8 \, b c^{2} d f e^{4} + 4 \, b^{2} c f e^{5} - b^{3} g e^{5}\right )}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )} x + \frac{3 \,{\left (8 \, c^{3} d^{3} f e^{2} - 4 \, b c^{2} d^{3} g e^{2} - 12 \, b c^{2} d^{2} f e^{3} + 8 \, b^{2} c d^{2} g e^{3} + 2 \, b^{2} c d f e^{4} - 3 \, b^{3} d g e^{4} + b^{3} f e^{5}\right )}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )} x + \frac{8 \, c^{3} d^{5} g + 8 \, c^{3} d^{4} f e - 24 \, b c^{2} d^{4} g e - 4 \, b c^{2} d^{3} f e^{2} + 22 \, b^{2} c d^{3} g e^{2} - 6 \, b^{2} c d^{2} f e^{3} - 6 \, b^{3} d^{2} g e^{3} + 3 \, b^{3} d f e^{4}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )}}{3 \,{\left (c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

2/3*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(((2*(4*c^3*d^2*g*e^3 - 8*c^3*d*f*e^4 + 4*b*c^2*f*e^5 - b^2*c*g
*e^5)*x/(16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d^2*e^4 - 8*b^3*c*d*e^5 + b^4*e^6) + 3*(4*b*c^2*d^2*g*
e^3 - 8*b*c^2*d*f*e^4 + 4*b^2*c*f*e^5 - b^3*g*e^5)/(16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d^2*e^4 - 8
*b^3*c*d*e^5 + b^4*e^6))*x + 3*(8*c^3*d^3*f*e^2 - 4*b*c^2*d^3*g*e^2 - 12*b*c^2*d^2*f*e^3 + 8*b^2*c*d^2*g*e^3 +
2*b^2*c*d*f*e^4 - 3*b^3*d*g*e^4 + b^3*f*e^5)/(16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d^2*e^4 - 8*b^3*
c*d*e^5 + b^4*e^6))*x + (8*c^3*d^5*g + 8*c^3*d^4*f*e - 24*b*c^2*d^4*g*e - 4*b*c^2*d^3*f*e^2 + 22*b^2*c*d^3*g*e
^2 - 6*b^2*c*d^2*f*e^3 - 6*b^3*d^2*g*e^3 + 3*b^3*d*f*e^4)/(16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d^2*
e^4 - 8*b^3*c*d*e^5 + b^4*e^6))/(c*x^2*e^2 - c*d^2 + b*x*e^2 + b*d*e)^2