3.2268 \(\int \frac{(d+e x)^{9/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=369 \[ \frac{2 (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{7 c^2 e^2 (2 c d-b e)}+\frac{12 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^3 e^2}+\frac{16 \sqrt{d+e x} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^4 e^2}+\frac{32 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^5 e^2 \sqrt{d+e x}}+\frac{2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(9/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) +
(32*(2*c*d - b*e)^2*(7*c*e*f + 9*c*d*g - 8*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(35*c^5*e^2*Sqrt[
d + e*x]) + (16*(2*c*d - b*e)*(7*c*e*f + 9*c*d*g - 8*b*e*g)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2])/(35*c^4*e^2) + (12*(7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2])/(35*c^3*e^2) + (2*(7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]
)/(7*c^2*e^2*(2*c*d - b*e))

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Rubi [A]  time = 0.554529, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {788, 656, 648} \[ \frac{2 (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{7 c^2 e^2 (2 c d-b e)}+\frac{12 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^3 e^2}+\frac{16 \sqrt{d+e x} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^4 e^2}+\frac{32 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+9 c d g+7 c e f)}{35 c^5 e^2 \sqrt{d+e x}}+\frac{2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^(9/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(9/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) +
(32*(2*c*d - b*e)^2*(7*c*e*f + 9*c*d*g - 8*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(35*c^5*e^2*Sqrt[
d + e*x]) + (16*(2*c*d - b*e)*(7*c*e*f + 9*c*d*g - 8*b*e*g)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2])/(35*c^4*e^2) + (12*(7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2])/(35*c^3*e^2) + (2*(7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]
)/(7*c^2*e^2*(2*c*d - b*e))

Rule 788

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

Rule 656

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(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] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), 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 + p, 0]

Rubi steps

\begin{align*} \int \frac{(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=\frac{2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(7 c e f+9 c d g-8 b e g) \int \frac{(d+e x)^{7/2}}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c e (2 c d-b e)}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}-\frac{(6 (7 c e f+9 c d g-8 b e g)) \int \frac{(d+e x)^{5/2}}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{7 c^2 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{12 (7 c e f+9 c d g-8 b e g) (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^3 e^2}+\frac{2 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}-\frac{(24 (2 c d-b e) (7 c e f+9 c d g-8 b e g)) \int \frac{(d+e x)^{3/2}}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{35 c^3 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (2 c d-b e) (7 c e f+9 c d g-8 b e g) \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^4 e^2}+\frac{12 (7 c e f+9 c d g-8 b e g) (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^3 e^2}+\frac{2 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}-\frac{\left (16 (2 c d-b e)^2 (7 c e f+9 c d g-8 b e g)\right ) \int \frac{\sqrt{d+e x}}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{35 c^4 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{9/2}}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{32 (2 c d-b e)^2 (7 c e f+9 c d g-8 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^5 e^2 \sqrt{d+e x}}+\frac{16 (2 c d-b e) (7 c e f+9 c d g-8 b e g) \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^4 e^2}+\frac{12 (7 c e f+9 c d g-8 b e g) (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{35 c^3 e^2}+\frac{2 (7 c e f+9 c d g-8 b e g) (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c^2 e^2 (2 c d-b e)}\\ \end{align*}

Mathematica [A]  time = 0.249249, size = 245, normalized size = 0.66 \[ -\frac{2 \sqrt{d+e x} \left (-8 b^2 c^2 e^2 \left (257 d^2 g+d e (77 f-45 g x)-e^2 x (7 f+2 g x)\right )+16 b^3 c e^3 (53 d g+7 e f-4 e g x)-128 b^4 e^4 g-2 b c^3 e \left (d^2 e (334 g x-553 f)-1075 d^3 g+d e^2 x (126 f+37 g x)+e^3 x^2 (7 f+4 g x)\right )+c^4 \left (d^2 e^2 x (301 f+93 g x)+d^3 e (407 g x-637 f)-814 d^4 g+d e^3 x^2 (49 f+29 g x)+e^4 x^3 (7 f+5 g x)\right )\right )}{35 c^5 e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(9/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(-128*b^4*e^4*g + 16*b^3*c*e^3*(7*e*f + 53*d*g - 4*e*g*x) - 8*b^2*c^2*e^2*(257*d^2*g + d*e*(
77*f - 45*g*x) - e^2*x*(7*f + 2*g*x)) - 2*b*c^3*e*(-1075*d^3*g + e^3*x^2*(7*f + 4*g*x) + d*e^2*x*(126*f + 37*g
*x) + d^2*e*(-553*f + 334*g*x)) + c^4*(-814*d^4*g + e^4*x^3*(7*f + 5*g*x) + d*e^3*x^2*(49*f + 29*g*x) + d^2*e^
2*x*(301*f + 93*g*x) + d^3*e*(-637*f + 407*g*x))))/(35*c^5*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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Maple [A]  time = 0.009, size = 367, normalized size = 1. \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -5\,g{e}^{4}{x}^{4}{c}^{4}+8\,b{c}^{3}{e}^{4}g{x}^{3}-29\,{c}^{4}d{e}^{3}g{x}^{3}-7\,{c}^{4}{e}^{4}f{x}^{3}-16\,{b}^{2}{c}^{2}{e}^{4}g{x}^{2}+74\,b{c}^{3}d{e}^{3}g{x}^{2}+14\,b{c}^{3}{e}^{4}f{x}^{2}-93\,{c}^{4}{d}^{2}{e}^{2}g{x}^{2}-49\,{c}^{4}d{e}^{3}f{x}^{2}+64\,{b}^{3}c{e}^{4}gx-360\,{b}^{2}{c}^{2}d{e}^{3}gx-56\,{b}^{2}{c}^{2}{e}^{4}fx+668\,b{c}^{3}{d}^{2}{e}^{2}gx+252\,b{c}^{3}d{e}^{3}fx-407\,{c}^{4}{d}^{3}egx-301\,{c}^{4}{d}^{2}{e}^{2}fx+128\,{b}^{4}{e}^{4}g-848\,{b}^{3}cd{e}^{3}g-112\,{b}^{3}c{e}^{4}f+2056\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}g+616\,{b}^{2}{c}^{2}d{e}^{3}f-2150\,b{c}^{3}{d}^{3}eg-1106\,b{c}^{3}{d}^{2}{e}^{2}f+814\,{c}^{4}{d}^{4}g+637\,f{d}^{3}{c}^{4}e \right ) }{35\,{c}^{5}{e}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

-2/35*(c*e*x+b*e-c*d)*(-5*c^4*e^4*g*x^4+8*b*c^3*e^4*g*x^3-29*c^4*d*e^3*g*x^3-7*c^4*e^4*f*x^3-16*b^2*c^2*e^4*g*
x^2+74*b*c^3*d*e^3*g*x^2+14*b*c^3*e^4*f*x^2-93*c^4*d^2*e^2*g*x^2-49*c^4*d*e^3*f*x^2+64*b^3*c*e^4*g*x-360*b^2*c
^2*d*e^3*g*x-56*b^2*c^2*e^4*f*x+668*b*c^3*d^2*e^2*g*x+252*b*c^3*d*e^3*f*x-407*c^4*d^3*e*g*x-301*c^4*d^2*e^2*f*
x+128*b^4*e^4*g-848*b^3*c*d*e^3*g-112*b^3*c*e^4*f+2056*b^2*c^2*d^2*e^2*g+616*b^2*c^2*d*e^3*f-2150*b*c^3*d^3*e*
g-1106*b*c^3*d^2*e^2*f+814*c^4*d^4*g+637*c^4*d^3*e*f)*(e*x+d)^(3/2)/c^5/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
3/2)

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Maxima [A]  time = 1.18426, size = 428, normalized size = 1.16 \begin{align*} -\frac{2 \,{\left (c^{3} e^{3} x^{3} - 91 \, c^{3} d^{3} + 158 \, b c^{2} d^{2} e - 88 \, b^{2} c d e^{2} + 16 \, b^{3} e^{3} +{\left (7 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} x^{2} +{\left (43 \, c^{3} d^{2} e - 36 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} f}{5 \, \sqrt{-c e x + c d - b e} c^{4} e} - \frac{2 \,{\left (5 \, c^{4} e^{4} x^{4} - 814 \, c^{4} d^{4} + 2150 \, b c^{3} d^{3} e - 2056 \, b^{2} c^{2} d^{2} e^{2} + 848 \, b^{3} c d e^{3} - 128 \, b^{4} e^{4} +{\left (29 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} x^{3} +{\left (93 \, c^{4} d^{2} e^{2} - 74 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} x^{2} +{\left (407 \, c^{4} d^{3} e - 668 \, b c^{3} d^{2} e^{2} + 360 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} x\right )} g}{35 \, \sqrt{-c e x + c d - b e} c^{5} e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/5*(c^3*e^3*x^3 - 91*c^3*d^3 + 158*b*c^2*d^2*e - 88*b^2*c*d*e^2 + 16*b^3*e^3 + (7*c^3*d*e^2 - 2*b*c^2*e^3)*x
^2 + (43*c^3*d^2*e - 36*b*c^2*d*e^2 + 8*b^2*c*e^3)*x)*f/(sqrt(-c*e*x + c*d - b*e)*c^4*e) - 2/35*(5*c^4*e^4*x^4
- 814*c^4*d^4 + 2150*b*c^3*d^3*e - 2056*b^2*c^2*d^2*e^2 + 848*b^3*c*d*e^3 - 128*b^4*e^4 + (29*c^4*d*e^3 - 8*b
*c^3*e^4)*x^3 + (93*c^4*d^2*e^2 - 74*b*c^3*d*e^3 + 16*b^2*c^2*e^4)*x^2 + (407*c^4*d^3*e - 668*b*c^3*d^2*e^2 +
360*b^2*c^2*d*e^3 - 64*b^3*c*e^4)*x)*g/(sqrt(-c*e*x + c*d - b*e)*c^5*e^2)

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Fricas [A]  time = 1.41679, size = 794, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (5 \, c^{4} e^{4} g x^{4} +{\left (7 \, c^{4} e^{4} f +{\left (29 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} g\right )} x^{3} +{\left (7 \,{\left (7 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} f +{\left (93 \, c^{4} d^{2} e^{2} - 74 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} g\right )} x^{2} - 7 \,{\left (91 \, c^{4} d^{3} e - 158 \, b c^{3} d^{2} e^{2} + 88 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} f - 2 \,{\left (407 \, c^{4} d^{4} - 1075 \, b c^{3} d^{3} e + 1028 \, b^{2} c^{2} d^{2} e^{2} - 424 \, b^{3} c d e^{3} + 64 \, b^{4} e^{4}\right )} g +{\left (7 \,{\left (43 \, c^{4} d^{2} e^{2} - 36 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} f +{\left (407 \, c^{4} d^{3} e - 668 \, b c^{3} d^{2} e^{2} + 360 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d}}{35 \,{\left (c^{6} e^{4} x^{2} + b c^{5} e^{4} x - c^{6} d^{2} e^{2} + b c^{5} d e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*c^4*e^4*g*x^4 + (7*c^4*e^4*f + (29*c^4*d*e^3 - 8*b*c^3*e^4)*g)*x^3 + (7*(7*c^4*d*e^3 - 2*b*c^3*e^4)*f
+ (93*c^4*d^2*e^2 - 74*b*c^3*d*e^3 + 16*b^2*c^2*e^4)*g)*x^2 - 7*(91*c^4*d^3*e - 158*b*c^3*d^2*e^2 + 88*b^2*c^2
*d*e^3 - 16*b^3*c*e^4)*f - 2*(407*c^4*d^4 - 1075*b*c^3*d^3*e + 1028*b^2*c^2*d^2*e^2 - 424*b^3*c*d*e^3 + 64*b^4
*e^4)*g + (7*(43*c^4*d^2*e^2 - 36*b*c^3*d*e^3 + 8*b^2*c^2*e^4)*f + (407*c^4*d^3*e - 668*b*c^3*d^2*e^2 + 360*b^
2*c^2*d*e^3 - 64*b^3*c*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^6*e^4*x^2 + b*c^
5*e^4*x - c^6*d^2*e^2 + b*c^5*d*e^3)

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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((e*x+d)**(9/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x