3.2243 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=193 \[ -\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{315 c^3 e^2 (d+e x)^{5/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt {d+e x}} \]

[Out]

-4/315*(-b*e+2*c*d)*(-4*b*e*g-c*d*g+9*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^3/e^2/(e*x+d)^(5/2)-2/63
*(-4*b*e*g-c*d*g+9*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^2/e^2/(e*x+d)^(3/2)-2/9*g*(d*(-b*e+c*d)-b*e
^2*x-c*e^2*x^2)^(5/2)/c/e^2/(e*x+d)^(1/2)

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Rubi [A]  time = 0.31, 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} \[ -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{3/2}}-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{315 c^3 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(2*c*d - b*e)*(9*c*e*f - c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(315*c^3*e^2*(d + e
*x)^(5/2)) - (2*(9*c*e*f - c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(63*c^2*e^2*(d + e*x)
^(3/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9*c*e^2*Sqrt[d + e*x])

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 )^{3/2}}{\sqrt {d+e x}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt {d+e x}}-\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 \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}{9 c e^3}\\ &=-\frac {2 (9 c e f-c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{63 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt {d+e x}}+\frac {(2 (2 c d-b e) (9 c e f-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 )^{3/2}}{(d+e x)^{3/2}} \, dx}{63 c^2 e}\\ &=-\frac {4 (2 c d-b e) (9 c e f-c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{315 c^3 e^2 (d+e x)^{5/2}}-\frac {2 (9 c e f-c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{63 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 121, normalized size = 0.63 \[ -\frac {2 (b e-c d+c e x)^2 \sqrt {(d+e x) (c (d-e x)-b e)} \left (8 b^2 e^2 g-2 b c e (17 d g+9 e f+10 e g x)+c^2 \left (26 d^2 g+d e (81 f+65 g x)+5 e^2 x (9 f+7 g x)\right )\right )}{315 c^3 e^2 \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(8*b^2*e^2*g - 2*b*c*e*(9*e*f + 17*d*g + 1
0*e*g*x) + c^2*(26*d^2*g + 5*e^2*x*(9*f + 7*g*x) + d*e*(81*f + 65*g*x))))/(315*c^3*e^2*Sqrt[d + e*x])

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fricas [B]  time = 1.24, size = 354, normalized size = 1.83 \[ -\frac {2 \, {\left (35 \, c^{4} e^{4} g x^{4} + 5 \, {\left (9 \, c^{4} e^{4} f - {\left (c^{4} d e^{3} - 10 \, b c^{3} e^{4}\right )} g\right )} x^{3} - 3 \, {\left (3 \, {\left (c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} f + {\left (23 \, c^{4} d^{2} e^{2} - 22 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} g\right )} x^{2} + 9 \, {\left (9 \, c^{4} d^{3} e - 20 \, b c^{3} d^{2} e^{2} + 13 \, b^{2} c^{2} d e^{3} - 2 \, b^{3} c e^{4}\right )} f + 2 \, {\left (13 \, c^{4} d^{4} - 43 \, b c^{3} d^{3} e + 51 \, b^{2} c^{2} d^{2} e^{2} - 25 \, b^{3} c d e^{3} + 4 \, b^{4} e^{4}\right )} g - {\left (9 \, {\left (13 \, c^{4} d^{2} e^{2} - 12 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} f - {\left (13 \, c^{4} d^{3} e - 30 \, b c^{3} d^{2} e^{2} + 21 \, b^{2} c^{2} d e^{3} - 4 \, b^{3} c e^{4}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{315 \, {\left (c^{3} e^{3} x + c^{3} d e^{2}\right )}} \]

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)^(3/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

-2/315*(35*c^4*e^4*g*x^4 + 5*(9*c^4*e^4*f - (c^4*d*e^3 - 10*b*c^3*e^4)*g)*x^3 - 3*(3*(c^4*d*e^3 - 8*b*c^3*e^4)
*f + (23*c^4*d^2*e^2 - 22*b*c^3*d*e^3 - b^2*c^2*e^4)*g)*x^2 + 9*(9*c^4*d^3*e - 20*b*c^3*d^2*e^2 + 13*b^2*c^2*d
*e^3 - 2*b^3*c*e^4)*f + 2*(13*c^4*d^4 - 43*b*c^3*d^3*e + 51*b^2*c^2*d^2*e^2 - 25*b^3*c*d*e^3 + 4*b^4*e^4)*g -
(9*(13*c^4*d^2*e^2 - 12*b*c^3*d*e^3 - b^2*c^2*e^4)*f - (13*c^4*d^3*e - 30*b*c^3*d^2*e^2 + 21*b^2*c^2*d*e^3 - 4
*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^3*e^3*x + c^3*d*e^2)

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

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)^(3/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 139, normalized size = 0.72 \[ \frac {2 \left (c e x +b e -c d \right ) \left (35 g \,x^{2} c^{2} e^{2}-20 b c \,e^{2} g x +65 c^{2} d e g x +45 c^{2} e^{2} f x +8 b^{2} e^{2} g -34 b c d e g -18 b c \,e^{2} f +26 c^{2} d^{2} g +81 c^{2} d e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{315 \left (e x +d \right )^{\frac {3}{2}} c^{3} e^{2}} \]

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)^(3/2)/(e*x+d)^(1/2),x)

[Out]

2/315*(c*e*x+b*e-c*d)*(35*c^2*e^2*g*x^2-20*b*c*e^2*g*x+65*c^2*d*e*g*x+45*c^2*e^2*f*x+8*b^2*e^2*g-34*b*c*d*e*g-
18*b*c*e^2*f+26*c^2*d^2*g+81*c^2*d*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c^3/e^2/(e*x+d)^(3/2)

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maxima [A]  time = 0.84, size = 320, normalized size = 1.66 \[ -\frac {2 \, {\left (5 \, c^{3} e^{3} x^{3} + 9 \, c^{3} d^{3} - 20 \, b c^{2} d^{2} e + 13 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3} - {\left (c^{3} d e^{2} - 8 \, b c^{2} e^{3}\right )} x^{2} - {\left (13 \, c^{3} d^{2} e - 12 \, b c^{2} d e^{2} - b^{2} c e^{3}\right )} x\right )} \sqrt {-c e x + c d - b e} f}{35 \, c^{2} e} - \frac {2 \, {\left (35 \, c^{4} e^{4} x^{4} + 26 \, c^{4} d^{4} - 86 \, b c^{3} d^{3} e + 102 \, b^{2} c^{2} d^{2} e^{2} - 50 \, b^{3} c d e^{3} + 8 \, b^{4} e^{4} - 5 \, {\left (c^{4} d e^{3} - 10 \, b c^{3} e^{4}\right )} x^{3} - 3 \, {\left (23 \, c^{4} d^{2} e^{2} - 22 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} x^{2} + {\left (13 \, c^{4} d^{3} e - 30 \, b c^{3} d^{2} e^{2} + 21 \, b^{2} c^{2} d e^{3} - 4 \, b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{315 \, c^{3} e^{2}} \]

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)^(3/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

-2/35*(5*c^3*e^3*x^3 + 9*c^3*d^3 - 20*b*c^2*d^2*e + 13*b^2*c*d*e^2 - 2*b^3*e^3 - (c^3*d*e^2 - 8*b*c^2*e^3)*x^2
- (13*c^3*d^2*e - 12*b*c^2*d*e^2 - b^2*c*e^3)*x)*sqrt(-c*e*x + c*d - b*e)*f/(c^2*e) - 2/315*(35*c^4*e^4*x^4 +
26*c^4*d^4 - 86*b*c^3*d^3*e + 102*b^2*c^2*d^2*e^2 - 50*b^3*c*d*e^3 + 8*b^4*e^4 - 5*(c^4*d*e^3 - 10*b*c^3*e^4)
*x^3 - 3*(23*c^4*d^2*e^2 - 22*b*c^3*d*e^3 - b^2*c^2*e^4)*x^2 + (13*c^4*d^3*e - 30*b*c^3*d^2*e^2 + 21*b^2*c^2*d
*e^3 - 4*b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e)*g/(c^3*e^2)

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mupad [B]  time = 3.05, size = 239, normalized size = 1.24 \[ -\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e\,x^3\,\left (10\,b\,e\,g-c\,d\,g+9\,c\,e\,f\right )}{63}+\frac {2\,x^2\,\left (g\,b^2\,e^2+22\,g\,b\,c\,d\,e+24\,f\,b\,c\,e^2-23\,g\,c^2\,d^2-3\,f\,c^2\,d\,e\right )}{105\,c}+\frac {2\,c\,e^2\,g\,x^4}{9}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\left (8\,g\,b^2\,e^2-34\,g\,b\,c\,d\,e-18\,f\,b\,c\,e^2+26\,g\,c^2\,d^2+81\,f\,c^2\,d\,e\right )}{315\,c^3\,e^2}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\left (-4\,g\,b^2\,e^2+17\,g\,b\,c\,d\,e+9\,f\,b\,c\,e^2-13\,g\,c^2\,d^2+117\,f\,c^2\,d\,e\right )}{315\,c^2\,e}\right )}{\sqrt {d+e\,x}} \]

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)^(3/2))/(d + e*x)^(1/2),x)

[Out]

-((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*e*x^3*(10*b*e*g - c*d*g + 9*c*e*f))/63 + (2*x^2*(b^2*e^2*g -
23*c^2*d^2*g + 24*b*c*e^2*f - 3*c^2*d*e*f + 22*b*c*d*e*g))/(105*c) + (2*c*e^2*g*x^4)/9 + (2*(b*e - c*d)^2*(8*
b^2*e^2*g + 26*c^2*d^2*g - 18*b*c*e^2*f + 81*c^2*d*e*f - 34*b*c*d*e*g))/(315*c^3*e^2) + (2*x*(b*e - c*d)*(9*b*
c*e^2*f - 13*c^2*d^2*g - 4*b^2*e^2*g + 117*c^2*d*e*f + 17*b*c*d*e*g))/(315*c^2*e)))/(d + e*x)^(1/2)

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

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)**(3/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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