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

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

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Rubi [A]  time = 0.18, antiderivative size = 165, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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]

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]

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+e x)}{3 c e^2 (2 c d-b e) \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+e x)}{3 c e^2 (2 c d-b e) \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*}

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Mathematica [A]  time = 0.09, size = 151, normalized size = 0.92 \begin {gather*} \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^3 g+d^2 e (f-g x)+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)}} \end {gather*}

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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IntegrateAlgebraic [B]  time = 43.04, size = 7201, normalized size = 43.64 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((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]

Result too large to show

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fricas [B]  time = 9.47, size = 431, normalized size = 2.61 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 0.67, size = 521, normalized size = 3.16 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.06, size = 227, normalized size = 1.38 \begin {gather*} \frac {2 \left (e x +d \right )^{2} \left (c e x +b e -c d \right ) \left (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 \right )}{3 \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}} e^{2}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?

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mupad [B]  time = 3.44, size = 795, normalized size = 4.82 \begin {gather*} -\frac {8\,c^2\,d^3\,g\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-6\,b^2\,e^3\,f\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-16\,c^2\,e^3\,f\,x^2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+12\,b^2\,d\,e^2\,g\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+8\,c^2\,d^2\,e\,f\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+6\,b^2\,e^3\,g\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+4\,b\,c\,e^3\,g\,x^2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+16\,c^2\,d\,e^2\,f\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-8\,c^2\,d^2\,e\,g\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+8\,c^2\,d\,e^2\,g\,x^2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-20\,b\,c\,d^2\,e\,g\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}-24\,b\,c\,e^3\,f\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}+8\,b\,c\,d\,e^2\,g\,x\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{3\,b^5\,d\,e^7+3\,b^5\,e^8\,x-24\,b^4\,c\,d^2\,e^6-18\,b^4\,c\,d\,e^7\,x+6\,b^4\,c\,e^8\,x^2+75\,b^3\,c^2\,d^3\,e^5+33\,b^3\,c^2\,d^2\,e^6\,x-39\,b^3\,c^2\,d\,e^7\,x^2+3\,b^3\,c^2\,e^8\,x^3-114\,b^2\,c^3\,d^4\,e^4-6\,b^2\,c^3\,d^3\,e^5\,x+90\,b^2\,c^3\,d^2\,e^6\,x^2-18\,b^2\,c^3\,d\,e^7\,x^3+84\,b\,c^4\,d^5\,e^3-36\,b\,c^4\,d^4\,e^4\,x-84\,b\,c^4\,d^3\,e^5\,x^2+36\,b\,c^4\,d^2\,e^6\,x^3-24\,c^5\,d^6\,e^2+24\,c^5\,d^5\,e^3\,x+24\,c^5\,d^4\,e^4\,x^2-24\,c^5\,d^3\,e^5\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(8*c^2*d^3*g*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - 6*b^2*e^3*f*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^
(1/2) - 16*c^2*e^3*f*x^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + 12*b^2*d*e^2*g*(c*d^2 - c*e^2*x^2 - b*d
*e - b*e^2*x)^(1/2) + 8*c^2*d^2*e*f*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + 6*b^2*e^3*g*x*(c*d^2 - c*e^2
*x^2 - b*d*e - b*e^2*x)^(1/2) + 4*b*c*e^3*g*x^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + 16*c^2*d*e^2*f*x
*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - 8*c^2*d^2*e*g*x*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + 8
*c^2*d*e^2*g*x^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) - 20*b*c*d^2*e*g*(c*d^2 - c*e^2*x^2 - b*d*e - b*e
^2*x)^(1/2) - 24*b*c*e^3*f*x*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + 8*b*c*d*e^2*g*x*(c*d^2 - c*e^2*x^2
- b*d*e - b*e^2*x)^(1/2))/(3*b^5*d*e^7 + 3*b^5*e^8*x - 24*c^5*d^6*e^2 + 84*b*c^4*d^5*e^3 - 24*b^4*c*d^2*e^6 +
6*b^4*c*e^8*x^2 + 24*c^5*d^5*e^3*x - 114*b^2*c^3*d^4*e^4 + 75*b^3*c^2*d^3*e^5 + 3*b^3*c^2*e^8*x^3 + 24*c^5*d^4
*e^4*x^2 - 24*c^5*d^3*e^5*x^3 - 18*b^4*c*d*e^7*x + 90*b^2*c^3*d^2*e^6*x^2 - 36*b*c^4*d^4*e^4*x - 6*b^2*c^3*d^3
*e^5*x + 33*b^3*c^2*d^2*e^6*x - 84*b*c^4*d^3*e^5*x^2 - 39*b^3*c^2*d*e^7*x^2 + 36*b*c^4*d^2*e^6*x^3 - 18*b^2*c^
3*d*e^7*x^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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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