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

Optimal. Leaf size=193 \[ -\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{693 c^3 e^2 (d+e x)^{7/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{99 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}} \]

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Rubi [A]  time = 0.34, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {794, 656, 648} \begin {gather*} -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{99 c^2 e^2 (d+e x)^{5/2}}-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{693 c^3 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}}{11 c e^2 (d+e x)^{3/2}} \end {gather*}

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)^(3/2),x]

[Out]

(-4*(2*c*d - b*e)*(11*c*e*f - 3*c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(693*c^3*e^2*(d
+ e*x)^(7/2)) - (2*(11*c*e*f - 3*c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(99*c^2*e^2*(d
+ e*x)^(5/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(11*c*e^2*(d + e*x)^(3/2))

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]

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 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])

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)^{3/2}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}-\frac {\left (2 \left (\frac {7}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac {3}{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)^{3/2}} \, dx}{11 c e^3}\\ &=-\frac {2 (11 c e f-3 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{99 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}+\frac {(2 (2 c d-b e) (11 c e f-3 c d g-4 b e g)) \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}{99 c^2 e}\\ &=-\frac {4 (2 c d-b e) (11 c e f-3 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{693 c^3 e^2 (d+e x)^{7/2}}-\frac {2 (11 c e f-3 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{99 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 121, normalized size = 0.63 \begin {gather*} \frac {2 (b e-c d+c e x)^3 \sqrt {(d+e x) (c (d-e x)-b e)} \left (8 b^2 e^2 g-2 b c e (19 d g+11 e f+14 e g x)+c^2 \left (30 d^2 g+d e (121 f+105 g x)+7 e^2 x (11 f+9 g x)\right )\right )}{693 c^3 e^2 \sqrt {d+e x}} \end {gather*}

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)^(3/2),x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(8*b^2*e^2*g - 2*b*c*e*(11*e*f + 19*d*g + 1
4*e*g*x) + c^2*(30*d^2*g + 7*e^2*x*(11*f + 9*g*x) + d*e*(121*f + 105*g*x))))/(693*c^3*e^2*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 2.16, size = 139, normalized size = 0.72 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{7/2} \left (8 b^2 e^2 g-28 b c e g (d+e x)-10 b c d e g-22 b c e^2 f-12 c^2 d^2 g+77 c^2 e f (d+e x)+44 c^2 d e f+63 c^2 g (d+e x)^2-21 c^2 d g (d+e x)\right )}{693 c^3 e^2 (d+e x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2)^(7/2)*(44*c^2*d*e*f - 22*b*c*e^2*f - 12*c^2*d^2*g - 10*b*c*d*e*g
+ 8*b^2*e^2*g + 77*c^2*e*f*(d + e*x) - 21*c^2*d*g*(d + e*x) - 28*b*c*e*g*(d + e*x) + 63*c^2*g*(d + e*x)^2))/(
693*c^3*e^2*(d + e*x)^(7/2))

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fricas [B]  time = 0.41, size = 500, normalized size = 2.59 \begin {gather*} \frac {2 \, {\left (63 \, c^{5} e^{5} g x^{5} + 7 \, {\left (11 \, c^{5} e^{5} f - {\left (12 \, c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} g\right )} x^{4} - {\left (11 \, {\left (10 \, c^{5} d e^{4} - 19 \, b c^{4} e^{5}\right )} f + {\left (96 \, c^{5} d^{2} e^{3} + 17 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} - 3 \, {\left (11 \, {\left (4 \, c^{5} d^{2} e^{3} + b c^{4} d e^{4} - 5 \, b^{2} c^{3} e^{5}\right )} f - {\left (54 \, c^{5} d^{3} e^{2} - 107 \, b c^{4} d^{2} e^{3} + 52 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} g\right )} x^{2} - 11 \, {\left (11 \, c^{5} d^{4} e - 35 \, b c^{4} d^{3} e^{2} + 39 \, b^{2} c^{3} d^{2} e^{3} - 17 \, b^{3} c^{2} d e^{4} + 2 \, b^{4} c e^{5}\right )} f - 2 \, {\left (15 \, c^{5} d^{5} - 64 \, b c^{4} d^{4} e + 106 \, b^{2} c^{3} d^{3} e^{2} - 84 \, b^{3} c^{2} d^{2} e^{3} + 31 \, b^{4} c d e^{4} - 4 \, b^{5} e^{5}\right )} g + {\left (11 \, {\left (26 \, c^{5} d^{3} e^{2} - 51 \, b c^{4} d^{2} e^{3} + 24 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} f - {\left (15 \, c^{5} d^{4} e - 49 \, b c^{4} d^{3} e^{2} + 57 \, b^{2} c^{3} d^{2} e^{3} - 27 \, b^{3} c^{2} d e^{4} + 4 \, b^{4} c e^{5}\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}}{693 \, {\left (c^{3} e^{3} x + c^{3} d e^{2}\right )}} \end {gather*}

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)^(3/2),x, algorithm="fricas")

[Out]

2/693*(63*c^5*e^5*g*x^5 + 7*(11*c^5*e^5*f - (12*c^5*d*e^4 - 23*b*c^4*e^5)*g)*x^4 - (11*(10*c^5*d*e^4 - 19*b*c^
4*e^5)*f + (96*c^5*d^2*e^3 + 17*b*c^4*d*e^4 - 113*b^2*c^3*e^5)*g)*x^3 - 3*(11*(4*c^5*d^2*e^3 + b*c^4*d*e^4 - 5
*b^2*c^3*e^5)*f - (54*c^5*d^3*e^2 - 107*b*c^4*d^2*e^3 + 52*b^2*c^3*d*e^4 + b^3*c^2*e^5)*g)*x^2 - 11*(11*c^5*d^
4*e - 35*b*c^4*d^3*e^2 + 39*b^2*c^3*d^2*e^3 - 17*b^3*c^2*d*e^4 + 2*b^4*c*e^5)*f - 2*(15*c^5*d^5 - 64*b*c^4*d^4
*e + 106*b^2*c^3*d^3*e^2 - 84*b^3*c^2*d^2*e^3 + 31*b^4*c*d*e^4 - 4*b^5*e^5)*g + (11*(26*c^5*d^3*e^2 - 51*b*c^4
*d^2*e^3 + 24*b^2*c^3*d*e^4 + b^3*c^2*e^5)*f - (15*c^5*d^4*e - 49*b*c^4*d^3*e^2 + 57*b^2*c^3*d^2*e^3 - 27*b^3*
c^2*d*e^4 + 4*b^4*c*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^3*e^3*x + c^3*d*e^2
)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

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)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 139, normalized size = 0.72 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (63 g \,x^{2} c^{2} e^{2}-28 b c \,e^{2} g x +105 c^{2} d e g x +77 c^{2} e^{2} f x +8 b^{2} e^{2} g -38 b c d e g -22 b c \,e^{2} f +30 c^{2} d^{2} g +121 c^{2} d e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{693 \left (e x +d \right )^{\frac {5}{2}} c^{3} e^{2}} \end {gather*}

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)^(3/2),x)

[Out]

2/693*(c*e*x+b*e-c*d)*(63*c^2*e^2*g*x^2-28*b*c*e^2*g*x+105*c^2*d*e*g*x+77*c^2*e^2*f*x+8*b^2*e^2*g-38*b*c*d*e*g
-22*b*c*e^2*f+30*c^2*d^2*g+121*c^2*d*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c^3/e^2/(e*x+d)^(5/2)

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maxima [B]  time = 0.89, size = 465, normalized size = 2.41 \begin {gather*} \frac {2 \, {\left (7 \, c^{4} e^{4} x^{4} - 11 \, c^{4} d^{4} + 35 \, b c^{3} d^{3} e - 39 \, b^{2} c^{2} d^{2} e^{2} + 17 \, b^{3} c d e^{3} - 2 \, b^{4} e^{4} - {\left (10 \, c^{4} d e^{3} - 19 \, b c^{3} e^{4}\right )} x^{3} - 3 \, {\left (4 \, c^{4} d^{2} e^{2} + b c^{3} d e^{3} - 5 \, b^{2} c^{2} e^{4}\right )} x^{2} + {\left (26 \, c^{4} d^{3} e - 51 \, b c^{3} d^{2} e^{2} + 24 \, b^{2} c^{2} d e^{3} + b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} f}{63 \, c^{2} e} + \frac {2 \, {\left (63 \, c^{5} e^{5} x^{5} - 30 \, c^{5} d^{5} + 128 \, b c^{4} d^{4} e - 212 \, b^{2} c^{3} d^{3} e^{2} + 168 \, b^{3} c^{2} d^{2} e^{3} - 62 \, b^{4} c d e^{4} + 8 \, b^{5} e^{5} - 7 \, {\left (12 \, c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} x^{4} - {\left (96 \, c^{5} d^{2} e^{3} + 17 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \, {\left (54 \, c^{5} d^{3} e^{2} - 107 \, b c^{4} d^{2} e^{3} + 52 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} x^{2} - {\left (15 \, c^{5} d^{4} e - 49 \, b c^{4} d^{3} e^{2} + 57 \, b^{2} c^{3} d^{2} e^{3} - 27 \, b^{3} c^{2} d e^{4} + 4 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{693 \, c^{3} e^{2}} \end {gather*}

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)^(3/2),x, algorithm="maxima")

[Out]

2/63*(7*c^4*e^4*x^4 - 11*c^4*d^4 + 35*b*c^3*d^3*e - 39*b^2*c^2*d^2*e^2 + 17*b^3*c*d*e^3 - 2*b^4*e^4 - (10*c^4*
d*e^3 - 19*b*c^3*e^4)*x^3 - 3*(4*c^4*d^2*e^2 + b*c^3*d*e^3 - 5*b^2*c^2*e^4)*x^2 + (26*c^4*d^3*e - 51*b*c^3*d^2
*e^2 + 24*b^2*c^2*d*e^3 + b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e)*f/(c^2*e) + 2/693*(63*c^5*e^5*x^5 - 30*c^5*d^
5 + 128*b*c^4*d^4*e - 212*b^2*c^3*d^3*e^2 + 168*b^3*c^2*d^2*e^3 - 62*b^4*c*d*e^4 + 8*b^5*e^5 - 7*(12*c^5*d*e^4
- 23*b*c^4*e^5)*x^4 - (96*c^5*d^2*e^3 + 17*b*c^4*d*e^4 - 113*b^2*c^3*e^5)*x^3 + 3*(54*c^5*d^3*e^2 - 107*b*c^4
*d^2*e^3 + 52*b^2*c^3*d*e^4 + b^3*c^2*e^5)*x^2 - (15*c^5*d^4*e - 49*b*c^4*d^3*e^2 + 57*b^2*c^3*d^2*e^3 - 27*b^
3*c^2*d*e^4 + 4*b^4*c*e^5)*x)*sqrt(-c*e*x + c*d - b*e)*g/(c^3*e^2)

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mupad [B]  time = 3.23, size = 320, normalized size = 1.66 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,c^2\,e^3\,g\,x^5}{11}+\frac {2\,c\,e^2\,x^4\,\left (23\,b\,e\,g-12\,c\,d\,g+11\,c\,e\,f\right )}{99}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\left (g\,b^2\,e^2+53\,g\,b\,c\,d\,e+55\,f\,b\,c\,e^2-54\,g\,c^2\,d^2+44\,f\,c^2\,d\,e\right )}{231\,c}-\frac {x^3\,\left (-226\,g\,b^2\,c^3\,e^5+34\,g\,b\,c^4\,d\,e^4-418\,f\,b\,c^4\,e^5+192\,g\,c^5\,d^2\,e^3+220\,f\,c^5\,d\,e^4\right )}{693\,c^3\,e^2}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\left (8\,g\,b^2\,e^2-38\,g\,b\,c\,d\,e-22\,f\,b\,c\,e^2+30\,g\,c^2\,d^2+121\,f\,c^2\,d\,e\right )}{693\,c^3\,e^2}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\left (-4\,g\,b^2\,e^2+19\,g\,b\,c\,d\,e+11\,f\,b\,c\,e^2-15\,g\,c^2\,d^2+286\,f\,c^2\,d\,e\right )}{693\,c^2\,e}\right )}{\sqrt {d+e\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*c^2*e^3*g*x^5)/11 + (2*c*e^2*x^4*(23*b*e*g - 12*c*d*g + 11*c*
e*f))/99 + (2*x^2*(b*e - c*d)*(b^2*e^2*g - 54*c^2*d^2*g + 55*b*c*e^2*f + 44*c^2*d*e*f + 53*b*c*d*e*g))/(231*c)
- (x^3*(192*c^5*d^2*e^3*g - 226*b^2*c^3*e^5*g - 418*b*c^4*e^5*f + 220*c^5*d*e^4*f + 34*b*c^4*d*e^4*g))/(693*c
^3*e^2) + (2*(b*e - c*d)^3*(8*b^2*e^2*g + 30*c^2*d^2*g - 22*b*c*e^2*f + 121*c^2*d*e*f - 38*b*c*d*e*g))/(693*c^
3*e^2) + (2*x*(b*e - c*d)^2*(11*b*c*e^2*f - 15*c^2*d^2*g - 4*b^2*e^2*g + 286*c^2*d*e*f + 19*b*c*d*e*g))/(693*c
^2*e)))/(d + e*x)^(1/2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)**(3/2),x)

[Out]

Timed out

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