3.2242 \(\int \sqrt{d+e x} (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=267 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{99 c^2 e^2 \sqrt{d+e x}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{693 c^3 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{3465 c^4 e^2 (d+e x)^{5/2}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2} \]

[Out]

(-16*(2*c*d - b*e)^2*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3465*c^4*e^2*(
d + e*x)^(5/2)) - (8*(2*c*d - b*e)*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(
693*c^3*e^2*(d + e*x)^(3/2)) - (2*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9
9*c^2*e^2*Sqrt[d + e*x]) - (2*g*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(11*c*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.427803, antiderivative size = 267, 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 = {794, 656, 648} \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{99 c^2 e^2 \sqrt{d+e x}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{693 c^3 e^2 (d+e x)^{3/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{3465 c^4 e^2 (d+e x)^{5/2}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*(2*c*d - b*e)^2*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3465*c^4*e^2*(
d + e*x)^(5/2)) - (8*(2*c*d - b*e)*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(
693*c^3*e^2*(d + e*x)^(3/2)) - (2*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9
9*c^2*e^2*Sqrt[d + e*x]) - (2*g*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(11*c*e^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 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 \sqrt{d+e x} (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 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2}-\frac{\left (2 \left (\frac{5}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac{1}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \sqrt{d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{11 c e^3}\\ &=-\frac{2 (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2}+\frac{(4 (2 c d-b e) (11 c e f+c d g-6 b e g)) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{99 c^2 e}\\ &=-\frac{8 (2 c d-b e) (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{693 c^3 e^2 (d+e x)^{3/2}}-\frac{2 (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2}+\frac{\left (8 (2 c d-b e)^2 (11 c e f+c d g-6 b e g)\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{693 c^3 e}\\ &=-\frac{16 (2 c d-b e)^2 (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3465 c^4 e^2 (d+e x)^{5/2}}-\frac{8 (2 c d-b e) (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{693 c^3 e^2 (d+e x)^{3/2}}-\frac{2 (11 c e f+c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{99 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2}\\ \end{align*}

Mathematica [A]  time = 0.187301, size = 183, normalized size = 0.69 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (8 b^2 c e^2 (40 d g+11 e f+15 e g x)-48 b^3 e^3 g-2 b c^2 e \left (347 d^2 g+d e (286 f+340 g x)+5 e^2 x (22 f+21 g x)\right )+c^3 \left (d^2 e (1177 f+1055 g x)+422 d^3 g+10 d e^2 x (121 f+98 g x)+35 e^3 x^2 (11 f+9 g x)\right )\right )}{3465 c^4 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-48*b^3*e^3*g + 8*b^2*c*e^2*(11*e*f + 40*
d*g + 15*e*g*x) - 2*b*c^2*e*(347*d^2*g + 5*e^2*x*(22*f + 21*g*x) + d*e*(286*f + 340*g*x)) + c^3*(422*d^3*g + 3
5*e^3*x^2*(11*f + 9*g*x) + 10*d*e^2*x*(121*f + 98*g*x) + d^2*e*(1177*f + 1055*g*x))))/(3465*c^4*e^2*Sqrt[d + e
*x])

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 235, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -315\,g{e}^{3}{x}^{3}{c}^{3}+210\,b{c}^{2}{e}^{3}g{x}^{2}-980\,{c}^{3}d{e}^{2}g{x}^{2}-385\,{c}^{3}{e}^{3}f{x}^{2}-120\,{b}^{2}c{e}^{3}gx+680\,b{c}^{2}d{e}^{2}gx+220\,b{c}^{2}{e}^{3}fx-1055\,{c}^{3}{d}^{2}egx-1210\,{c}^{3}d{e}^{2}fx+48\,{b}^{3}{e}^{3}g-320\,{b}^{2}cd{e}^{2}g-88\,{b}^{2}c{e}^{3}f+694\,b{c}^{2}{d}^{2}eg+572\,b{c}^{2}d{e}^{2}f-422\,{c}^{3}{d}^{3}g-1177\,f{d}^{2}{c}^{3}e \right ) }{3465\,{c}^{4}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

-2/3465*(c*e*x+b*e-c*d)*(-315*c^3*e^3*g*x^3+210*b*c^2*e^3*g*x^2-980*c^3*d*e^2*g*x^2-385*c^3*e^3*f*x^2-120*b^2*
c*e^3*g*x+680*b*c^2*d*e^2*g*x+220*b*c^2*e^3*f*x-1055*c^3*d^2*e*g*x-1210*c^3*d*e^2*f*x+48*b^3*e^3*g-320*b^2*c*d
*e^2*g-88*b^2*c*e^3*f+694*b*c^2*d^2*e*g+572*b*c^2*d*e^2*f-422*c^3*d^3*g-1177*c^3*d^2*e*f)*(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(3/2)/c^4/e^2/(e*x+d)^(3/2)

________________________________________________________________________________________

Maxima [B]  time = 1.38542, size = 678, normalized size = 2.54 \begin{align*} -\frac{2 \,{\left (35 \, c^{4} e^{4} x^{4} + 107 \, c^{4} d^{4} - 266 \, b c^{3} d^{3} e + 219 \, b^{2} c^{2} d^{2} e^{2} - 68 \, b^{3} c d e^{3} + 8 \, b^{4} e^{4} + 10 \,{\left (4 \, c^{4} d e^{3} + 5 \, b c^{3} e^{4}\right )} x^{3} - 3 \,{\left (26 \, c^{4} d^{2} e^{2} - 46 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} x^{2} - 2 \,{\left (52 \, c^{4} d^{3} e - 39 \, b c^{3} d^{2} e^{2} - 15 \, b^{2} c^{2} d e^{3} + 2 \, b^{3} c e^{4}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} f}{315 \,{\left (c^{3} e^{2} x + c^{3} d e\right )}} - \frac{2 \,{\left (315 \, c^{5} e^{5} x^{5} + 422 \, c^{5} d^{5} - 1538 \, b c^{4} d^{4} e + 2130 \, b^{2} c^{3} d^{3} e^{2} - 1382 \, b^{3} c^{2} d^{2} e^{3} + 416 \, b^{4} c d e^{4} - 48 \, b^{5} e^{5} + 70 \,{\left (5 \, c^{5} d e^{4} + 6 \, b c^{4} e^{5}\right )} x^{4} - 5 \,{\left (118 \, c^{5} d^{2} e^{3} - 214 \, b c^{4} d e^{4} - 3 \, b^{2} c^{3} e^{5}\right )} x^{3} - 6 \,{\left (118 \, c^{5} d^{3} e^{2} - 101 \, b c^{4} d^{2} e^{3} - 20 \, b^{2} c^{3} d e^{4} + 3 \, b^{3} c^{2} e^{5}\right )} x^{2} +{\left (211 \, c^{5} d^{4} e - 558 \, b c^{4} d^{3} e^{2} + 507 \, b^{2} c^{3} d^{2} e^{3} - 184 \, b^{3} c^{2} d e^{4} + 24 \, b^{4} c e^{5}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} g}{3465 \,{\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/315*(35*c^4*e^4*x^4 + 107*c^4*d^4 - 266*b*c^3*d^3*e + 219*b^2*c^2*d^2*e^2 - 68*b^3*c*d*e^3 + 8*b^4*e^4 + 10
*(4*c^4*d*e^3 + 5*b*c^3*e^4)*x^3 - 3*(26*c^4*d^2*e^2 - 46*b*c^3*d*e^3 - b^2*c^2*e^4)*x^2 - 2*(52*c^4*d^3*e - 3
9*b*c^3*d^2*e^2 - 15*b^2*c^2*d*e^3 + 2*b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^3*e^2*x + c^3*d*e
) - 2/3465*(315*c^5*e^5*x^5 + 422*c^5*d^5 - 1538*b*c^4*d^4*e + 2130*b^2*c^3*d^3*e^2 - 1382*b^3*c^2*d^2*e^3 + 4
16*b^4*c*d*e^4 - 48*b^5*e^5 + 70*(5*c^5*d*e^4 + 6*b*c^4*e^5)*x^4 - 5*(118*c^5*d^2*e^3 - 214*b*c^4*d*e^4 - 3*b^
2*c^3*e^5)*x^3 - 6*(118*c^5*d^3*e^2 - 101*b*c^4*d^2*e^3 - 20*b^2*c^3*d*e^4 + 3*b^3*c^2*e^5)*x^2 + (211*c^5*d^4
*e - 558*b*c^4*d^3*e^2 + 507*b^2*c^3*d^2*e^3 - 184*b^3*c^2*d*e^4 + 24*b^4*c*e^5)*x)*sqrt(-c*e*x + c*d - b*e)*(
e*x + d)*g/(c^4*e^3*x + c^4*d*e^2)

________________________________________________________________________________________

Fricas [B]  time = 1.41612, size = 1080, normalized size = 4.04 \begin{align*} -\frac{2 \,{\left (315 \, c^{5} e^{5} g x^{5} + 35 \,{\left (11 \, c^{5} e^{5} f + 2 \,{\left (5 \, c^{5} d e^{4} + 6 \, b c^{4} e^{5}\right )} g\right )} x^{4} + 5 \,{\left (22 \,{\left (4 \, c^{5} d e^{4} + 5 \, b c^{4} e^{5}\right )} f -{\left (118 \, c^{5} d^{2} e^{3} - 214 \, b c^{4} d e^{4} - 3 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} - 3 \,{\left (11 \,{\left (26 \, c^{5} d^{2} e^{3} - 46 \, b c^{4} d e^{4} - b^{2} c^{3} e^{5}\right )} f + 2 \,{\left (118 \, c^{5} d^{3} e^{2} - 101 \, b c^{4} d^{2} e^{3} - 20 \, b^{2} c^{3} d e^{4} + 3 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} + 11 \,{\left (107 \, c^{5} d^{4} e - 266 \, b c^{4} d^{3} e^{2} + 219 \, b^{2} c^{3} d^{2} e^{3} - 68 \, b^{3} c^{2} d e^{4} + 8 \, b^{4} c e^{5}\right )} f + 2 \,{\left (211 \, c^{5} d^{5} - 769 \, b c^{4} d^{4} e + 1065 \, b^{2} c^{3} d^{3} e^{2} - 691 \, b^{3} c^{2} d^{2} e^{3} + 208 \, b^{4} c d e^{4} - 24 \, b^{5} e^{5}\right )} g -{\left (22 \,{\left (52 \, c^{5} d^{3} e^{2} - 39 \, b c^{4} d^{2} e^{3} - 15 \, b^{2} c^{3} d e^{4} + 2 \, b^{3} c^{2} e^{5}\right )} f -{\left (211 \, c^{5} d^{4} e - 558 \, b c^{4} d^{3} e^{2} + 507 \, b^{2} c^{3} d^{2} e^{3} - 184 \, b^{3} c^{2} d e^{4} + 24 \, 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}}{3465 \,{\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-2/3465*(315*c^5*e^5*g*x^5 + 35*(11*c^5*e^5*f + 2*(5*c^5*d*e^4 + 6*b*c^4*e^5)*g)*x^4 + 5*(22*(4*c^5*d*e^4 + 5*
b*c^4*e^5)*f - (118*c^5*d^2*e^3 - 214*b*c^4*d*e^4 - 3*b^2*c^3*e^5)*g)*x^3 - 3*(11*(26*c^5*d^2*e^3 - 46*b*c^4*d
*e^4 - b^2*c^3*e^5)*f + 2*(118*c^5*d^3*e^2 - 101*b*c^4*d^2*e^3 - 20*b^2*c^3*d*e^4 + 3*b^3*c^2*e^5)*g)*x^2 + 11
*(107*c^5*d^4*e - 266*b*c^4*d^3*e^2 + 219*b^2*c^3*d^2*e^3 - 68*b^3*c^2*d*e^4 + 8*b^4*c*e^5)*f + 2*(211*c^5*d^5
- 769*b*c^4*d^4*e + 1065*b^2*c^3*d^3*e^2 - 691*b^3*c^2*d^2*e^3 + 208*b^4*c*d*e^4 - 24*b^5*e^5)*g - (22*(52*c^
5*d^3*e^2 - 39*b*c^4*d^2*e^3 - 15*b^2*c^3*d*e^4 + 2*b^3*c^2*e^5)*f - (211*c^5*d^4*e - 558*b*c^4*d^3*e^2 + 507*
b^2*c^3*d^2*e^3 - 184*b^3*c^2*d*e^4 + 24*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^4*e^3*x + c^4*d*e^2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}} \sqrt{d + e x} \left (f + g x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

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

________________________________________________________________________________________

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((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")

[Out]

Timed out