3.2232 \(\int (d+e x)^{3/2} (f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx\)

Optimal. Leaf size=267 \[ -\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{315 c^4 e^2 (d+e x)^{3/2}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{105 c^3 e^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{21 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2} \]

[Out]

(-16*(2*c*d - b*e)^2*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(315*c^4*e^2*(d
+ e*x)^(3/2)) - (8*(2*c*d - b*e)*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(105
*c^3*e^2*Sqrt[d + e*x]) - (2*(3*c*e*f + c*d*g - 2*b*e*g)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(
3/2))/(21*c^2*e^2) - (2*g*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(9*c*e^2)

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Rubi [A]  time = 0.43535, 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} \[ -\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{315 c^4 e^2 (d+e x)^{3/2}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{105 c^3 e^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{21 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*(2*c*d - b*e)^2*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(315*c^4*e^2*(d
+ e*x)^(3/2)) - (8*(2*c*d - b*e)*(3*c*e*f + c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(105
*c^3*e^2*Sqrt[d + e*x]) - (2*(3*c*e*f + c*d*g - 2*b*e*g)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(
3/2))/(21*c^2*e^2) - (2*g*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(9*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 (d+e x)^{3/2} (f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx &=-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2}-\frac{\left (2 \left (\frac{3}{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 (d+e x)^{3/2} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{9 c e^3}\\ &=-\frac{2 (3 c e f+c d g-2 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{21 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2}+\frac{(4 (2 c d-b e) (3 c e f+c d g-2 b e g)) \int \sqrt{d+e x} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{21 c^2 e}\\ &=-\frac{8 (2 c d-b e) (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{105 c^3 e^2 \sqrt{d+e x}}-\frac{2 (3 c e f+c d g-2 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{21 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2}+\frac{\left (8 (2 c d-b e)^2 (3 c e f+c d g-2 b e g)\right ) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt{d+e x}} \, dx}{105 c^3 e}\\ &=-\frac{16 (2 c d-b e)^2 (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{315 c^4 e^2 (d+e x)^{3/2}}-\frac{8 (2 c d-b e) (3 c e f+c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{105 c^3 e^2 \sqrt{d+e x}}-\frac{2 (3 c e f+c d g-2 b e g) \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{21 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2}\\ \end{align*}

Mathematica [A]  time = 0.170961, size = 179, normalized size = 0.67 \[ \frac{2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)} \left (24 b^2 c e^2 (4 d g+e (f+g x))-16 b^3 e^3 g-6 b c^2 e \left (31 d^2 g+d e (22 f+20 g x)+e^2 x (6 f+5 g x)\right )+c^3 \left (3 d^2 e (71 f+53 g x)+106 d^3 g+6 d e^2 x (27 f+20 g x)+5 e^3 x^2 (9 f+7 g x)\right )\right )}{315 c^4 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-16*b^3*e^3*g + 24*b^2*c*e^2*(4*d*g + e*(f +
g*x)) - 6*b*c^2*e*(31*d^2*g + e^2*x*(6*f + 5*g*x) + d*e*(22*f + 20*g*x)) + c^3*(106*d^3*g + 5*e^3*x^2*(9*f +
7*g*x) + 6*d*e^2*x*(27*f + 20*g*x) + 3*d^2*e*(71*f + 53*g*x))))/(315*c^4*e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.008, size = 235, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -35\,g{e}^{3}{x}^{3}{c}^{3}+30\,b{c}^{2}{e}^{3}g{x}^{2}-120\,{c}^{3}d{e}^{2}g{x}^{2}-45\,{c}^{3}{e}^{3}f{x}^{2}-24\,{b}^{2}c{e}^{3}gx+120\,b{c}^{2}d{e}^{2}gx+36\,b{c}^{2}{e}^{3}fx-159\,{c}^{3}{d}^{2}egx-162\,{c}^{3}d{e}^{2}fx+16\,{b}^{3}{e}^{3}g-96\,{b}^{2}cd{e}^{2}g-24\,{b}^{2}c{e}^{3}f+186\,b{c}^{2}{d}^{2}eg+132\,b{c}^{2}d{e}^{2}f-106\,{c}^{3}{d}^{3}g-213\,f{d}^{2}{c}^{3}e \right ) }{315\,{c}^{4}{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(c*e*x+b*e-c*d)*(-35*c^3*e^3*g*x^3+30*b*c^2*e^3*g*x^2-120*c^3*d*e^2*g*x^2-45*c^3*e^3*f*x^2-24*b^2*c*e^3
*g*x+120*b*c^2*d*e^2*g*x+36*b*c^2*e^3*f*x-159*c^3*d^2*e*g*x-162*c^3*d*e^2*f*x+16*b^3*e^3*g-96*b^2*c*d*e^2*g-24
*b^2*c*e^3*f+186*b*c^2*d^2*e*g+132*b*c^2*d*e^2*f-106*c^3*d^3*g-213*c^3*d^2*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(1/2)/c^4/e^2/(e*x+d)^(1/2)

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Maxima [A]  time = 1.25903, size = 478, normalized size = 1.79 \begin{align*} \frac{2 \,{\left (15 \, c^{3} e^{3} x^{3} - 71 \, c^{3} d^{3} + 115 \, b c^{2} d^{2} e - 52 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} + 3 \,{\left (13 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{2} +{\left (17 \, c^{3} d^{2} e + 22 \, b c^{2} d e^{2} - 4 \, b^{2} c e^{3}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} f}{105 \,{\left (c^{3} e^{2} x + c^{3} d e\right )}} + \frac{2 \,{\left (35 \, c^{4} e^{4} x^{4} - 106 \, c^{4} d^{4} + 292 \, b c^{3} d^{3} e - 282 \, b^{2} c^{2} d^{2} e^{2} + 112 \, b^{3} c d e^{3} - 16 \, b^{4} e^{4} + 5 \,{\left (17 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} x^{3} + 3 \,{\left (13 \, c^{4} d^{2} e^{2} + 10 \, b c^{3} d e^{3} - 2 \, b^{2} c^{2} e^{4}\right )} x^{2} -{\left (53 \, c^{4} d^{3} e - 93 \, b c^{3} d^{2} e^{2} + 48 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} g}{315 \,{\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)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

2/105*(15*c^3*e^3*x^3 - 71*c^3*d^3 + 115*b*c^2*d^2*e - 52*b^2*c*d*e^2 + 8*b^3*e^3 + 3*(13*c^3*d*e^2 + b*c^2*e^
3)*x^2 + (17*c^3*d^2*e + 22*b*c^2*d*e^2 - 4*b^2*c*e^3)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^3*e^2*x + c^
3*d*e) + 2/315*(35*c^4*e^4*x^4 - 106*c^4*d^4 + 292*b*c^3*d^3*e - 282*b^2*c^2*d^2*e^2 + 112*b^3*c*d*e^3 - 16*b^
4*e^4 + 5*(17*c^4*d*e^3 + b*c^3*e^4)*x^3 + 3*(13*c^4*d^2*e^2 + 10*b*c^3*d*e^3 - 2*b^2*c^2*e^4)*x^2 - (53*c^4*d
^3*e - 93*b*c^3*d^2*e^2 + 48*b^2*c^2*d*e^3 - 8*b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^4*e^3*x +
c^4*d*e^2)

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Fricas [A]  time = 1.89747, size = 740, normalized size = 2.77 \begin{align*} \frac{2 \,{\left (35 \, c^{4} e^{4} g x^{4} + 5 \,{\left (9 \, c^{4} e^{4} f +{\left (17 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} g\right )} x^{3} + 3 \,{\left (3 \,{\left (13 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} f +{\left (13 \, c^{4} d^{2} e^{2} + 10 \, b c^{3} d e^{3} - 2 \, b^{2} c^{2} e^{4}\right )} g\right )} x^{2} - 3 \,{\left (71 \, c^{4} d^{3} e - 115 \, b c^{3} d^{2} e^{2} + 52 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} f - 2 \,{\left (53 \, c^{4} d^{4} - 146 \, b c^{3} d^{3} e + 141 \, b^{2} c^{2} d^{2} e^{2} - 56 \, b^{3} c d e^{3} + 8 \, b^{4} e^{4}\right )} g +{\left (3 \,{\left (17 \, c^{4} d^{2} e^{2} + 22 \, b c^{3} d e^{3} - 4 \, b^{2} c^{2} e^{4}\right )} f -{\left (53 \, c^{4} d^{3} e - 93 \, b c^{3} d^{2} e^{2} + 48 \, b^{2} c^{2} d e^{3} - 8 \, 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}}{315 \,{\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)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*c^4*e^4*g*x^4 + 5*(9*c^4*e^4*f + (17*c^4*d*e^3 + b*c^3*e^4)*g)*x^3 + 3*(3*(13*c^4*d*e^3 + b*c^3*e^4)
*f + (13*c^4*d^2*e^2 + 10*b*c^3*d*e^3 - 2*b^2*c^2*e^4)*g)*x^2 - 3*(71*c^4*d^3*e - 115*b*c^3*d^2*e^2 + 52*b^2*c
^2*d*e^3 - 8*b^3*c*e^4)*f - 2*(53*c^4*d^4 - 146*b*c^3*d^3*e + 141*b^2*c^2*d^2*e^2 - 56*b^3*c*d*e^3 + 8*b^4*e^4
)*g + (3*(17*c^4*d^2*e^2 + 22*b*c^3*d*e^3 - 4*b^2*c^2*e^4)*f - (53*c^4*d^3*e - 93*b*c^3*d^2*e^2 + 48*b^2*c^2*d
*e^3 - 8*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^4*e^3*x + c^4*d*e^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError