3.2278 \(\int \frac{(d+e x)^{9/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=291 \[ \frac{2 (d+e x)^{5/2} (-2 b e g+3 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{8 (d+e x)^{3/2} (-2 b e g+3 c d g+c e f)}{3 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{16 \sqrt{d+e x} (2 c d-b e) (-2 b e g+3 c d g+c e f)}{3 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.412777, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {788, 656, 648} \[ \frac{2 (d+e x)^{5/2} (-2 b e g+3 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{8 (d+e x)^{3/2} (-2 b e g+3 c d g+c e f)}{3 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{16 \sqrt{d+e x} (2 c d-b e) (-2 b e g+3 c d g+c e f)}{3 c^4 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

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)^(5/2),x]

[Out]

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

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

Mathematica [A]  time = 0.16805, size = 180, normalized size = 0.62 \[ \frac{2 \sqrt{d+e x} \left (8 b^2 c e^2 (8 d g+e (f-3 g x))-16 b^3 e^3 g-2 b c^2 e \left (41 d^2 g+2 d e (5 f-18 g x)+3 e^2 x (g x-2 f)\right )+c^3 \left (d^2 e (11 f-51 g x)+34 d^3 g+6 d e^2 x (2 g x-3 f)+e^3 x^2 (3 f+g x)\right )\right )}{3 c^4 e^2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(9/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[d + e*x]*(-16*b^3*e^3*g + 8*b^2*c*e^2*(8*d*g + e*(f - 3*g*x)) - 2*b*c^2*e*(41*d^2*g + 2*d*e*(5*f - 18*
g*x) + 3*e^2*x*(-2*f + g*x)) + c^3*(34*d^3*g + d^2*e*(11*f - 51*g*x) + e^3*x^2*(3*f + g*x) + 6*d*e^2*x*(-3*f +
2*g*x))))/(3*c^4*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 235, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -g{e}^{3}{x}^{3}{c}^{3}+6\,b{c}^{2}{e}^{3}g{x}^{2}-12\,{c}^{3}d{e}^{2}g{x}^{2}-3\,{c}^{3}{e}^{3}f{x}^{2}+24\,{b}^{2}c{e}^{3}gx-72\,b{c}^{2}d{e}^{2}gx-12\,b{c}^{2}{e}^{3}fx+51\,{c}^{3}{d}^{2}egx+18\,{c}^{3}d{e}^{2}fx+16\,{b}^{3}{e}^{3}g-64\,{b}^{2}cd{e}^{2}g-8\,{b}^{2}c{e}^{3}f+82\,b{c}^{2}{d}^{2}eg+20\,b{c}^{2}d{e}^{2}f-34\,{c}^{3}{d}^{3}g-11\,{c}^{3}{d}^{2}ef \right ) }{3\,{c}^{4}{e}^{2}} \left ( ex+d \right ) ^{{\frac{5}{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)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

-2/3*(c*e*x+b*e-c*d)*(-c^3*e^3*g*x^3+6*b*c^2*e^3*g*x^2-12*c^3*d*e^2*g*x^2-3*c^3*e^3*f*x^2+24*b^2*c*e^3*g*x-72*
b*c^2*d*e^2*g*x-12*b*c^2*e^3*f*x+51*c^3*d^2*e*g*x+18*c^3*d*e^2*f*x+16*b^3*e^3*g-64*b^2*c*d*e^2*g-8*b^2*c*e^3*f
+82*b*c^2*d^2*e*g+20*b*c^2*d*e^2*f-34*c^3*d^3*g-11*c^3*d^2*e*f)*(e*x+d)^(5/2)/c^4/e^2/(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 1.38911, size = 332, normalized size = 1.14 \begin{align*} \frac{2 \,{\left (3 \, c^{2} e^{2} x^{2} + 11 \, c^{2} d^{2} - 20 \, b c d e + 8 \, b^{2} e^{2} - 6 \,{\left (3 \, c^{2} d e - 2 \, b c e^{2}\right )} x\right )} f}{3 \,{\left (c^{4} e^{2} x - c^{4} d e + b c^{3} e^{2}\right )} \sqrt{-c e x + c d - b e}} + \frac{2 \,{\left (c^{3} e^{3} x^{3} + 34 \, c^{3} d^{3} - 82 \, b c^{2} d^{2} e + 64 \, b^{2} c d e^{2} - 16 \, b^{3} e^{3} + 6 \,{\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} - 3 \,{\left (17 \, c^{3} d^{2} e - 24 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} g}{3 \,{\left (c^{5} e^{3} x - c^{5} d e^{2} + b c^{4} e^{3}\right )} \sqrt{-c e x + c d - b e}} \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)^(5/2),x, algorithm="maxima")

[Out]

2/3*(3*c^2*e^2*x^2 + 11*c^2*d^2 - 20*b*c*d*e + 8*b^2*e^2 - 6*(3*c^2*d*e - 2*b*c*e^2)*x)*f/((c^4*e^2*x - c^4*d*
e + b*c^3*e^2)*sqrt(-c*e*x + c*d - b*e)) + 2/3*(c^3*e^3*x^3 + 34*c^3*d^3 - 82*b*c^2*d^2*e + 64*b^2*c*d*e^2 - 1
6*b^3*e^3 + 6*(2*c^3*d*e^2 - b*c^2*e^3)*x^2 - 3*(17*c^3*d^2*e - 24*b*c^2*d*e^2 + 8*b^2*c*e^3)*x)*g/((c^5*e^3*x
- c^5*d*e^2 + b*c^4*e^3)*sqrt(-c*e*x + c*d - b*e))

________________________________________________________________________________________

Fricas [A]  time = 1.47718, size = 624, normalized size = 2.14 \begin{align*} -\frac{2 \,{\left (c^{3} e^{3} g x^{3} + 3 \,{\left (c^{3} e^{3} f + 2 \,{\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} g\right )} x^{2} +{\left (11 \, c^{3} d^{2} e - 20 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f + 2 \,{\left (17 \, c^{3} d^{3} - 41 \, b c^{2} d^{2} e + 32 \, b^{2} c d e^{2} - 8 \, b^{3} e^{3}\right )} g - 3 \,{\left (2 \,{\left (3 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f +{\left (17 \, c^{3} d^{2} e - 24 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\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}}{3 \,{\left (c^{6} e^{5} x^{3} + c^{6} d^{3} e^{2} - 2 \, b c^{5} d^{2} e^{3} + b^{2} c^{4} d e^{4} -{\left (c^{6} d e^{4} - 2 \, b c^{5} e^{5}\right )} x^{2} -{\left (c^{6} d^{2} e^{3} - b^{2} c^{4} e^{5}\right )} x\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)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(c^3*e^3*g*x^3 + 3*(c^3*e^3*f + 2*(2*c^3*d*e^2 - b*c^2*e^3)*g)*x^2 + (11*c^3*d^2*e - 20*b*c^2*d*e^2 + 8*b
^2*c*e^3)*f + 2*(17*c^3*d^3 - 41*b*c^2*d^2*e + 32*b^2*c*d*e^2 - 8*b^3*e^3)*g - 3*(2*(3*c^3*d*e^2 - 2*b*c^2*e^3
)*f + (17*c^3*d^2*e - 24*b*c^2*d*e^2 + 8*b^2*c*e^3)*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^5*x^3 + c^6*d^3*e^2 - 2*b*c^5*d^2*e^3 + b^2*c^4*d*e^4 - (c^6*d*e^4 - 2*b*c^5*e^5)*x^2 - (c^6*d^2*e
^3 - b^2*c^4*e^5)*x)

________________________________________________________________________________________

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)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

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)^(5/2),x, algorithm="giac")

[Out]

sage0*x