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*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)

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Rubi [A]  time = 0.179782, antiderivative size = 117, normalized size of antiderivative = 1., 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 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 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 \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*}

Mathematica [A]  time = 0.0725006, 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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Maple [A]  time = 0.004, size = 79, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -cegx+2\,beg-2\,cdg-3\,cef \right ) }{3\,{c}^{2}{e}^{2}}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{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)^(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 = 1.72485, size = 149, normalized size = 1.27 \begin{align*} \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}} \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)^(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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Fricas [A]  time = 1.44703, size = 169, normalized size = 1.44 \begin{align*} -\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 )}} \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)^(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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \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)**(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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \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)^(1/2),x, algorithm="giac")

[Out]

sage0*x