3.2264 \(\int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=117 \[ -\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+c d g+3 c e f)}{3 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2} \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {794, 648} \[ -\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+c d g+3 c e f)}{3 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2} \]

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]

(-2*(3*c*e*f + c*d*g - 2*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*c^2*e^2*Sqrt[d + e*x]) - (2*g*Sq
rt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*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 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 {\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} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}-\frac {\left (2 \left (\frac {1}{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 {\sqrt {d+e x}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{3 c e^3}\\ &=-\frac {2 (3 c e f+c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c^2 e^2 \sqrt {d+e x}}-\frac {2 g \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 63, normalized size = 0.54 \[ -\frac {2 \sqrt {(d+e x) (c (d-e x)-b e)} (c (2 d g+3 e f+e g x)-2 b e g)}{3 c^2 e^2 \sqrt {d+e x}} \]

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

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fricas [A]  time = 1.06, size = 79, normalized size = 0.68 \[ -\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c e g x + 3 \, c e f + 2 \, {\left (c d - b e\right )} g\right )} \sqrt {e x + d}}{3 \, {\left (c^{2} e^{3} x + c^{2} d e^{2}\right )}} \]

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]

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

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giac [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((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 = 0.05, size = 79, normalized size = 0.68 \[ -\frac {2 \left (c e x +b e -c d \right ) \left (-c e g x +2 b e g -2 c d g -3 c e f \right ) \sqrt {e x +d}}{3 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{2} e^{2}} \]

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]

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

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maxima [A]  time = 0.74, size = 110, normalized size = 0.94 \[ \frac {2 \, {\left (c e x - c d + b e\right )} f}{\sqrt {-c e x + c d - b e} c e} + \frac {2 \, {\left (c^{2} e^{2} x^{2} - 2 \, c^{2} d^{2} + 4 \, b c d e - 2 \, b^{2} e^{2} + {\left (c^{2} d e - b c e^{2}\right )} x\right )} g}{3 \, \sqrt {-c e x + c d - b e} c^{2} e^{2}} \]

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*(c*e*x - c*d + b*e)*f/(sqrt(-c*e*x + c*d - b*e)*c*e) + 2/3*(c^2*e^2*x^2 - 2*c^2*d^2 + 4*b*c*d*e - 2*b^2*e^2
+ (c^2*d*e - b*c*e^2)*x)*g/(sqrt(-c*e*x + c*d - b*e)*c^2*e^2)

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mupad [B]  time = 2.52, size = 89, normalized size = 0.76 \[ -\frac {\left (\frac {\sqrt {d+e\,x}\,\left (4\,c\,d\,g-4\,b\,e\,g+6\,c\,e\,f\right )}{3\,c^2\,e^3}+\frac {2\,g\,x\,\sqrt {d+e\,x}}{3\,c\,e^2}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x+\frac {d}{e}} \]

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]

-((((d + e*x)^(1/2)*(4*c*d*g - 4*b*e*g + 6*c*e*f))/(3*c^2*e^3) + (2*g*x*(d + e*x)^(1/2))/(3*c*e^2))*(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 \[ \int \frac {\sqrt {d + e x} \left (f + g x\right )}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \]

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

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