3.2255 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=118 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-2 b e g-5 c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{7/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2 (d+e x)^{5/2}} \]

[Out]

(-2*(9*c*e*f - 5*c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(63*c^2*e^2*(d + e*x)^(7/2)) -
(2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9*c*e^2*(d + e*x)^(5/2))

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Rubi [A]  time = 0.190536, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {794, 648} \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-2 b e g-5 c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{7/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(9*c*e*f - 5*c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(63*c^2*e^2*(d + e*x)^(7/2)) -
(2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9*c*e^2*(d + e*x)^(5/2))

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(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] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 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{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2 (d+e x)^{5/2}}-\frac{\left (2 \left (\frac{7}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac{5}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{9 c e^3}\\ &=-\frac{2 (9 c e f-5 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 c^2 e^2 (d+e x)^{7/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2 (d+e x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.136092, size = 78, normalized size = 0.66 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} (c (2 d g+9 e f+7 e g x)-2 b e g)}{63 c^2 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 79, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -7\,cegx+2\,beg-2\,cdg-9\,cef \right ) }{63\,{c}^{2}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/63*(c*e*x+b*e-c*d)*(-7*c*e*g*x+2*b*e*g-2*c*d*g-9*c*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c^2/e^2/(e*x
+d)^(5/2)

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Maxima [B]  time = 1.15934, size = 424, normalized size = 3.59 \begin{align*} \frac{2 \,{\left (c^{3} e^{3} x^{3} - c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + b^{3} e^{3} - 3 \,{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} + 3 \,{\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x\right )} \sqrt{-c e x + c d - b e} f}{7 \, c e} + \frac{2 \,{\left (7 \, c^{4} e^{4} x^{4} - 2 \, c^{4} d^{4} + 8 \, b c^{3} d^{3} e - 12 \, b^{2} c^{2} d^{2} e^{2} + 8 \, b^{3} c d e^{3} - 2 \, b^{4} e^{4} - 19 \,{\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + 15 \,{\left (c^{4} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{2} -{\left (c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{63 \, c^{2} e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*(c^3*e^3*x^3 - c^3*d^3 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + b^3*e^3 - 3*(c^3*d*e^2 - b*c^2*e^3)*x^2 + 3*(c^3*
d^2*e - 2*b*c^2*d*e^2 + b^2*c*e^3)*x)*sqrt(-c*e*x + c*d - b*e)*f/(c*e) + 2/63*(7*c^4*e^4*x^4 - 2*c^4*d^4 + 8*b
*c^3*d^3*e - 12*b^2*c^2*d^2*e^2 + 8*b^3*c*d*e^3 - 2*b^4*e^4 - 19*(c^4*d*e^3 - b*c^3*e^4)*x^3 + 15*(c^4*d^2*e^2
- 2*b*c^3*d*e^3 + b^2*c^2*e^4)*x^2 - (c^4*d^3*e - 3*b*c^3*d^2*e^2 + 3*b^2*c^2*d*e^3 - b^3*c*e^4)*x)*sqrt(-c*e
*x + c*d - b*e)*g/(c^2*e^2)

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Fricas [B]  time = 1.39592, size = 684, normalized size = 5.8 \begin{align*} \frac{2 \,{\left (7 \, c^{4} e^{4} g x^{4} +{\left (9 \, c^{4} e^{4} f - 19 \,{\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} g\right )} x^{3} - 3 \,{\left (9 \,{\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} f - 5 \,{\left (c^{4} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} g\right )} x^{2} - 9 \,{\left (c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f - 2 \,{\left (c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + b^{4} e^{4}\right )} g +{\left (27 \,{\left (c^{4} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} f -{\left (c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - 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}}{63 \,{\left (c^{2} e^{3} x + c^{2} d e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63*(7*c^4*e^4*g*x^4 + (9*c^4*e^4*f - 19*(c^4*d*e^3 - b*c^3*e^4)*g)*x^3 - 3*(9*(c^4*d*e^3 - b*c^3*e^4)*f - 5*
(c^4*d^2*e^2 - 2*b*c^3*d*e^3 + b^2*c^2*e^4)*g)*x^2 - 9*(c^4*d^3*e - 3*b*c^3*d^2*e^2 + 3*b^2*c^2*d*e^3 - b^3*c*
e^4)*f - 2*(c^4*d^4 - 4*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 4*b^3*c*d*e^3 + b^4*e^4)*g + (27*(c^4*d^2*e^2 - 2*b*
c^3*d*e^3 + b^2*c^2*e^4)*f - (c^4*d^3*e - 3*b*c^3*d^2*e^2 + 3*b^2*c^2*d*e^3 - 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^2*e^3*x + c^2*d*e^2)

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

[Out]

Timed out

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Giac [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)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

Timed out