3.2230 \(\int \frac{f+g x}{(d+e x)^3 (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=358 \[ \frac{256 c^3 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{63 e (2 c d-b e)^7 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{32 c^2 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{63 e (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{4 c (-3 b e g+2 c d g+4 c e f)}{21 e^2 (d+e x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (-3 b e g+2 c d g+4 c e f)}{21 e^2 (d+e x)^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{9 e^2 (d+e x)^3 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(32*c^2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x))/(63*e*(2*c*d - b*e)^5*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2)) - (2*(e*f - d*g))/(9*e^2*(2*c*d - b*e)*(d + e*x)^3*(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 - 3*b*e*g))/(21*e^2*(2*c*d - b*e)^2*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))
- (4*c*(4*c*e*f + 2*c*d*g - 3*b*e*g))/(21*e^2*(2*c*d - b*e)^3*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2)) + (256*c^3*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x))/(63*e*(2*c*d - b*e)^7*Sqrt[d*(c*d - b*e) - b*e^2
*x - c*e^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.511422, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {792, 658, 614, 613} \[ \frac{256 c^3 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{63 e (2 c d-b e)^7 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{32 c^2 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{63 e (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{4 c (-3 b e g+2 c d g+4 c e f)}{21 e^2 (d+e x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (-3 b e g+2 c d g+4 c e f)}{21 e^2 (d+e x)^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{9 e^2 (d+e x)^3 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)/((d + e*x)^3*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]

[Out]

(32*c^2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x))/(63*e*(2*c*d - b*e)^5*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2)) - (2*(e*f - d*g))/(9*e^2*(2*c*d - b*e)*(d + e*x)^3*(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 - 3*b*e*g))/(21*e^2*(2*c*d - b*e)^2*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))
- (4*c*(4*c*e*f + 2*c*d*g - 3*b*e*g))/(21*e^2*(2*c*d - b*e)^3*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2)) + (256*c^3*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x))/(63*e*(2*c*d - b*e)^7*Sqrt[d*(c*d - b*e) - b*e^2
*x - c*e^2*x^2])

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 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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{f+g x}{(d+e x)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=-\frac{2 (e f-d g)}{9 e^2 (2 c d-b e) (d+e x)^3 \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-3 b e g) \int \frac{1}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{3 e (2 c d-b e)}\\ &=-\frac{2 (e f-d g)}{9 e^2 (2 c d-b e) (d+e x)^3 \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-3 b e g)}{21 e^2 (2 c d-b e)^2 (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(10 c (4 c e f+2 c d g-3 b e g)) \int \frac{1}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{21 e (2 c d-b e)^2}\\ &=-\frac{2 (e f-d g)}{9 e^2 (2 c d-b e) (d+e x)^3 \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-3 b e g)}{21 e^2 (2 c d-b e)^2 (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{4 c (4 c e f+2 c d g-3 b e g)}{21 e^2 (2 c d-b e)^3 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{\left (16 c^2 (4 c e f+2 c d g-3 b e g)\right ) \int \frac{1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{21 e (2 c d-b e)^3}\\ &=\frac{32 c^2 (4 c e f+2 c d g-3 b e g) (b+2 c x)}{63 e (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{9 e^2 (2 c d-b e) (d+e x)^3 \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-3 b e g)}{21 e^2 (2 c d-b e)^2 (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{4 c (4 c e f+2 c d g-3 b e g)}{21 e^2 (2 c d-b e)^3 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{\left (128 c^3 (4 c e f+2 c d g-3 b e g)\right ) \int \frac{1}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{63 e (2 c d-b e)^5}\\ &=\frac{32 c^2 (4 c e f+2 c d g-3 b e g) (b+2 c x)}{63 e (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{9 e^2 (2 c d-b e) (d+e x)^3 \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-3 b e g)}{21 e^2 (2 c d-b e)^2 (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{4 c (4 c e f+2 c d g-3 b e g)}{21 e^2 (2 c d-b e)^3 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{256 c^3 (4 c e f+2 c d g-3 b e g) (b+2 c x)}{63 e (2 c d-b e)^7 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.403436, size = 596, normalized size = 1.66 \[ -\frac{2 (d+e x)^{5/2} \left (\frac{10 c e \left (\frac{8 c e \left (\frac{6 c e \left (\frac{4 c e \left (\frac{2}{\sqrt{d+e x} (-e (c d-b e)-c d e) \sqrt{-b e+c d-c e x}}-\frac{4 c e \sqrt{d+e x}}{(-e (c d-b e)-c d e) (e (c d-b e)+c d e) \sqrt{-b e+c d-c e x}}\right )}{3 (e (c d-b e)+c d e)}-\frac{2}{3 (d+e x)^{3/2} (e (c d-b e)+c d e) \sqrt{-b e+c d-c e x}}\right )}{5 (e (c d-b e)+c d e)}-\frac{2}{5 (d+e x)^{5/2} (e (c d-b e)+c d e) \sqrt{-b e+c d-c e x}}\right )}{7 (e (c d-b e)+c d e)}-\frac{2}{7 (d+e x)^{7/2} (e (c d-b e)+c d e) \sqrt{-b e+c d-c e x}}\right )}{9 (e (c d-b e)+c d e)}-\frac{2}{9 (d+e x)^{9/2} (e (c d-b e)+c d e) \sqrt{-b e+c d-c e x}}\right ) (-b e+c d-c e x)^{5/2} \left (6 c e^2 f-g \left (\frac{3 c d e}{2}-\frac{9}{2} e (c d-b e)\right )\right )}{3 c e (-e (c d-b e)-c d e) ((d+e x) (c (d-e x)-b e))^{5/2}}-\frac{2 (-b e+c d-c e x) (g (c d-b e)+c e f)}{3 c e (d+e x)^2 (-e (c d-b e)-c d e) ((d+e x) (c (d-e x)-b e))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)/((d + e*x)^3*(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 - b*e)*g)*(c*d - b*e - c*e*x))/(3*c*e*(-(c*d*e) - e*(c*d - b*e))*(d + e*x)^2*((d + e*x)*(-(b
*e) + c*(d - e*x)))^(5/2)) - (2*(6*c*e^2*f - ((3*c*d*e)/2 - (9*e*(c*d - b*e))/2)*g)*(d + e*x)^(5/2)*(c*d - b*e
- c*e*x)^(5/2)*(-2/(9*(c*d*e + e*(c*d - b*e))*(d + e*x)^(9/2)*Sqrt[c*d - b*e - c*e*x]) + (10*c*e*(-2/(7*(c*d*
e + e*(c*d - b*e))*(d + e*x)^(7/2)*Sqrt[c*d - b*e - c*e*x]) + (8*c*e*(-2/(5*(c*d*e + e*(c*d - b*e))*(d + e*x)^
(5/2)*Sqrt[c*d - b*e - c*e*x]) + (6*c*e*(-2/(3*(c*d*e + e*(c*d - b*e))*(d + e*x)^(3/2)*Sqrt[c*d - b*e - c*e*x]
) + (4*c*e*(2/((-(c*d*e) - e*(c*d - b*e))*Sqrt[d + e*x]*Sqrt[c*d - b*e - c*e*x]) - (4*c*e*Sqrt[d + e*x])/((-(c
*d*e) - e*(c*d - b*e))*(c*d*e + e*(c*d - b*e))*Sqrt[c*d - b*e - c*e*x])))/(3*(c*d*e + e*(c*d - b*e)))))/(5*(c*
d*e + e*(c*d - b*e)))))/(7*(c*d*e + e*(c*d - b*e)))))/(9*(c*d*e + e*(c*d - b*e)))))/(3*c*e*(-(c*d*e) - e*(c*d
- b*e))*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))

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Maple [B]  time = 0.013, size = 1036, normalized size = 2.9 \begin{align*} \text{result too large to display} \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)^(5/2),x)

[Out]

-2/63*(c*e*x+b*e-c*d)*(-768*b*c^5*e^7*g*x^6+512*c^6*d*e^6*g*x^6+1024*c^6*e^7*f*x^6-1152*b^2*c^4*e^7*g*x^5-1536
*b*c^5*d*e^6*g*x^5+1536*b*c^5*e^7*f*x^5+1536*c^6*d^2*e^5*g*x^5+3072*c^6*d*e^6*f*x^5-288*b^3*c^3*e^7*g*x^4-4416
*b^2*c^4*d*e^6*g*x^4+384*b^2*c^4*e^7*f*x^4+1920*b*c^5*d^2*e^5*g*x^4+6144*b*c^5*d*e^6*f*x^4+768*c^6*d^3*e^4*g*x
^4+1536*c^6*d^2*e^5*f*x^4+48*b^4*c^2*e^7*g*x^3-1472*b^3*c^3*d*e^6*g*x^3-64*b^3*c^3*e^7*f*x^3-5376*b^2*c^4*d^2*
e^5*g*x^3+1920*b^2*c^4*d*e^6*f*x^3+6912*b*c^5*d^3*e^4*g*x^3+8448*b*c^5*d^2*e^5*f*x^3-1792*c^6*d^4*e^3*g*x^3-35
84*c^6*d^3*e^4*f*x^3-18*b^5*c*e^7*g*x^2+300*b^4*c^2*d*e^6*g*x^2+24*b^4*c^2*e^7*f*x^2-3216*b^3*c^3*d^2*e^5*g*x^
2-384*b^3*c^3*d*e^6*f*x^2-288*b^2*c^4*d^3*e^4*g*x^2+4032*b^2*c^4*d^2*e^5*f*x^2+4704*b*c^5*d^4*e^3*g*x^2+3072*b
*c^5*d^3*e^4*f*x^2-2112*c^6*d^5*e^2*g*x^2-4224*c^6*d^4*e^3*f*x^2+9*b^6*e^7*g*x-132*b^5*c*d*e^6*g*x-12*b^5*c*e^
7*f*x+876*b^4*c^2*d^2*e^5*g*x+168*b^4*c^2*d*e^6*f*x-4128*b^3*c^3*d^3*e^4*g*x-1056*b^3*c^3*d^2*e^5*f*x+4848*b^2
*c^4*d^4*e^3*g*x+4800*b^2*c^4*d^3*e^4*f*x-1344*b*c^5*d^5*e^2*g*x-3264*b*c^5*d^4*e^3*f*x-192*c^6*d^6*e*g*x-384*
c^6*d^5*e^2*f*x+2*b^6*d*e^6*g+7*b^6*e^7*f-30*b^5*c*d^2*e^5*g-96*b^5*c*d*e^6*f+204*b^4*c^2*d^3*e^4*g+564*b^4*c^
2*d^2*e^5*f-976*b^3*c^3*d^4*e^3*g-1856*b^3*c^3*d^3*e^4*f+1344*b^2*c^4*d^5*e^2*g+3984*b^2*c^4*d^4*e^3*f-480*b*c
^5*d^6*e*g-3840*b*c^5*d^5*e^2*f-64*c^6*d^7*g+1216*c^6*d^6*e*f)/(e*x+d)^2/(b^7*e^7-14*b^6*c*d*e^6+84*b^5*c^2*d^
2*e^5-280*b^4*c^3*d^3*e^4+560*b^3*c^4*d^4*e^3-672*b^2*c^5*d^5*e^2+448*b*c^6*d^6*e-128*c^7*d^7)/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((g*x+f)/(e*x+d)^3/(-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 [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)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),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)/(e*x+d)**3/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\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)^(5/2),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, 1]