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

Optimal. Leaf size=191 \[ -\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-4 b e g+c d g+7 c e f)}{105 c^3 e^2 (d+e x)^{3/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-4 b e g+c d g+7 c e f)}{35 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2} \]

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Rubi [A]  time = 0.30, antiderivative size = 191, 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 )^{3/2} (-4 b e g+c d g+7 c e f)}{35 c^2 e^2 \sqrt {d+e x}}-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-4 b e g+c d g+7 c e f)}{105 c^3 e^2 (d+e x)^{3/2}}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(2*c*d - b*e)*(7*c*e*f + c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(105*c^3*e^2*(d + e
*x)^(3/2)) - (2*(7*c*e*f + c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(35*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)^(3/2))/(7*c*e^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 \sqrt {d+e x} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^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 )^{3/2}}{7 c e^2}-\frac {\left (2 \left (\frac {3}{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} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{7 c e^3}\\ &=-\frac {2 (7 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 )^{3/2}}{35 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 )^{3/2}}{7 c e^2}+\frac {(2 (2 c d-b e) (7 c e f+c d g-4 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}} \, dx}{35 c^2 e}\\ &=-\frac {4 (2 c d-b e) (7 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 )^{3/2}}{105 c^3 e^2 (d+e x)^{3/2}}-\frac {2 (7 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 )^{3/2}}{35 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 )^{3/2}}{7 c e^2}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 119, normalized size = 0.62 \begin {gather*} \frac {2 (b e-c d+c e x) \sqrt {(d+e x) (c (d-e x)-b e)} \left (8 b^2 e^2 g-2 b c e (15 d g+7 e f+6 e g x)+c^2 \left (22 d^2 g+d e (49 f+33 g x)+3 e^2 x (7 f+5 g x)\right )\right )}{105 c^3 e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]*(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))]*(8*b^2*e^2*g - 2*b*c*e*(7*e*f + 15*d*g + 6*e*
g*x) + c^2*(22*d^2*g + 3*e^2*x*(7*f + 5*g*x) + d*e*(49*f + 33*g*x))))/(105*c^3*e^2*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.30, size = 139, normalized size = 0.73 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{3/2} \left (8 b^2 e^2 g-12 b c e g (d+e x)-18 b c d e g-14 b c e^2 f+4 c^2 d^2 g+21 c^2 e f (d+e x)+28 c^2 d e f+15 c^2 g (d+e x)^2+3 c^2 d g (d+e x)\right )}{105 c^3 e^2 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2)^(3/2)*(28*c^2*d*e*f - 14*b*c*e^2*f + 4*c^2*d^2*g - 18*b*c*d*e*g
+ 8*b^2*e^2*g + 21*c^2*e*f*(d + e*x) + 3*c^2*d*g*(d + e*x) - 12*b*c*e*g*(d + e*x) + 15*c^2*g*(d + e*x)^2))/(10
5*c^3*e^2*(d + e*x)^(3/2))

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fricas [A]  time = 0.67, size = 233, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f + {\left (6 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} g\right )} x^{2} - 7 \, {\left (7 \, c^{3} d^{2} e - 9 \, b c^{2} d e^{2} + 2 \, b^{2} c e^{3}\right )} f - 2 \, {\left (11 \, c^{3} d^{3} - 26 \, b c^{2} d^{2} e + 19 \, b^{2} c d e^{2} - 4 \, b^{3} e^{3}\right )} g + {\left (7 \, {\left (4 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} f - {\left (11 \, c^{3} d^{2} e - 15 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\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}}{105 \, {\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((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

2/105*(15*c^3*e^3*g*x^3 + 3*(7*c^3*e^3*f + (6*c^3*d*e^2 + b*c^2*e^3)*g)*x^2 - 7*(7*c^3*d^2*e - 9*b*c^2*d*e^2 +
2*b^2*c*e^3)*f - 2*(11*c^3*d^3 - 26*b*c^2*d^2*e + 19*b^2*c*d*e^2 - 4*b^3*e^3)*g + (7*(4*c^3*d*e^2 + b*c^2*e^3
)*f - (11*c^3*d^2*e - 15*b*c^2*d*e^2 + 4*b^2*c*e^3)*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(-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((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.04, size = 139, normalized size = 0.73 \begin {gather*} \frac {2 \left (c e x +b e -c d \right ) \left (15 g \,x^{2} c^{2} e^{2}-12 b c \,e^{2} g x +33 c^{2} d e g x +21 c^{2} e^{2} f x +8 b^{2} e^{2} g -30 b c d e g -14 b c \,e^{2} f +22 c^{2} d^{2} g +49 c^{2} d e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{105 \sqrt {e x +d}\, c^{3} e^{2}} \end {gather*}

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)^(1/2),x)

[Out]

2/105*(c*e*x+b*e-c*d)*(15*c^2*e^2*g*x^2-12*b*c*e^2*g*x+33*c^2*d*e*g*x+21*c^2*e^2*f*x+8*b^2*e^2*g-30*b*c*d*e*g-
14*b*c*e^2*f+22*c^2*d^2*g+49*c^2*d*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/c^3/e^2/(e*x+d)^(1/2)

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maxima [A]  time = 0.79, size = 236, normalized size = 1.24 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} e^{2} x^{2} - 7 \, c^{2} d^{2} + 9 \, b c d e - 2 \, b^{2} e^{2} + {\left (4 \, c^{2} d e + b c e^{2}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{15 \, {\left (c^{2} e^{2} x + c^{2} d e\right )}} + \frac {2 \, {\left (15 \, c^{3} e^{3} x^{3} - 22 \, c^{3} d^{3} + 52 \, b c^{2} d^{2} e - 38 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} + 3 \, {\left (6 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{2} - {\left (11 \, c^{3} d^{2} e - 15 \, 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 )} g}{105 \, {\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((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

2/15*(3*c^2*e^2*x^2 - 7*c^2*d^2 + 9*b*c*d*e - 2*b^2*e^2 + (4*c^2*d*e + b*c*e^2)*x)*sqrt(-c*e*x + c*d - b*e)*(e
*x + d)*f/(c^2*e^2*x + c^2*d*e) + 2/105*(15*c^3*e^3*x^3 - 22*c^3*d^3 + 52*b*c^2*d^2*e - 38*b^2*c*d*e^2 + 8*b^3
*e^3 + 3*(6*c^3*d*e^2 + b*c^2*e^3)*x^2 - (11*c^3*d^2*e - 15*b*c^2*d*e^2 + 4*b^2*c*e^3)*x)*sqrt(-c*e*x + c*d -
b*e)*(e*x + d)*g/(c^3*e^3*x + c^3*d*e^2)

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mupad [B]  time = 2.65, size = 219, normalized size = 1.15 \begin {gather*} \frac {\left (\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{7}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (b\,e\,g+6\,c\,d\,g+7\,c\,e\,f\right )}{35\,c\,e}+\frac {2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (8\,g\,b^2\,e^2-30\,g\,b\,c\,d\,e-14\,f\,b\,c\,e^2+22\,g\,c^2\,d^2+49\,f\,c^2\,d\,e\right )}{105\,c^3\,e^3}+\frac {x\,\sqrt {d+e\,x}\,\left (-8\,g\,b^2\,c\,e^3+30\,g\,b\,c^2\,d\,e^2+14\,f\,b\,c^2\,e^3-22\,g\,c^3\,d^2\,e+56\,f\,c^3\,d\,e^2\right )}{105\,c^3\,e^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((2*g*x^3*(d + e*x)^(1/2))/7 + (2*x^2*(d + e*x)^(1/2)*(b*e*g + 6*c*d*g + 7*c*e*f))/(35*c*e) + (2*(b*e - c*d)*
(d + e*x)^(1/2)*(8*b^2*e^2*g + 22*c^2*d^2*g - 14*b*c*e^2*f + 49*c^2*d*e*f - 30*b*c*d*e*g))/(105*c^3*e^3) + (x*
(d + e*x)^(1/2)*(14*b*c^2*e^3*f - 8*b^2*c*e^3*g + 56*c^3*d*e^2*f - 22*c^3*d^2*e*g + 30*b*c^2*d*e^2*g))/(105*c^
3*e^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(x + d/e)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \sqrt {d + e x} \left (f + g x\right )\, dx \end {gather*}

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)**(1/2),x)

[Out]

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

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