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

Optimal. Leaf size=137 \[ -\frac{2 \sqrt{f+g x} (e f-d g) \left (a g^2-b f g+c f^2\right )}{g^4}+\frac{2 (f+g x)^{3/2} (c f (3 e f-2 d g)-g (-a e g-b d g+2 b e f))}{3 g^4}-\frac{2 (f+g x)^{5/2} (-b e g-c d g+3 c e f)}{5 g^4}+\frac{2 c e (f+g x)^{7/2}}{7 g^4} \]

[Out]

(-2*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*Sqrt[f + g*x])/g^4 + (2*(c*f*(3*e*f - 2*d*g) - g*(2*b*e*f - b*d*g - a*
e*g))*(f + g*x)^(3/2))/(3*g^4) - (2*(3*c*e*f - c*d*g - b*e*g)*(f + g*x)^(5/2))/(5*g^4) + (2*c*e*(f + g*x)^(7/2
))/(7*g^4)

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Rubi [A]  time = 0.109117, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {771} \[ -\frac{2 \sqrt{f+g x} (e f-d g) \left (a g^2-b f g+c f^2\right )}{g^4}+\frac{2 (f+g x)^{3/2} (c f (3 e f-2 d g)-g (-a e g-b d g+2 b e f))}{3 g^4}-\frac{2 (f+g x)^{5/2} (-b e g-c d g+3 c e f)}{5 g^4}+\frac{2 c e (f+g x)^{7/2}}{7 g^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2)*Sqrt[f + g*x])/g^4 + (2*(c*f*(3*e*f - 2*d*g) - g*(2*b*e*f - b*d*g - a*
e*g))*(f + g*x)^(3/2))/(3*g^4) - (2*(3*c*e*f - c*d*g - b*e*g)*(f + g*x)^(5/2))/(5*g^4) + (2*c*e*(f + g*x)^(7/2
))/(7*g^4)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(d+e x) \left (a+b x+c x^2\right )}{\sqrt{f+g x}} \, dx &=\int \left (\frac{(-e f+d g) \left (c f^2-b f g+a g^2\right )}{g^3 \sqrt{f+g x}}+\frac{(c f (3 e f-2 d g)-g (2 b e f-b d g-a e g)) \sqrt{f+g x}}{g^3}+\frac{(-3 c e f+c d g+b e g) (f+g x)^{3/2}}{g^3}+\frac{c e (f+g x)^{5/2}}{g^3}\right ) \, dx\\ &=-\frac{2 (e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt{f+g x}}{g^4}+\frac{2 (c f (3 e f-2 d g)-g (2 b e f-b d g-a e g)) (f+g x)^{3/2}}{3 g^4}-\frac{2 (3 c e f-c d g-b e g) (f+g x)^{5/2}}{5 g^4}+\frac{2 c e (f+g x)^{7/2}}{7 g^4}\\ \end{align*}

Mathematica [A]  time = 0.208884, size = 131, normalized size = 0.96 \[ \frac{2 \sqrt{f+g x} \left (7 g \left (5 a g (3 d g-2 e f+e g x)+5 b d g (g x-2 f)+b e \left (8 f^2-4 f g x+3 g^2 x^2\right )\right )+c \left (7 d g \left (8 f^2-4 f g x+3 g^2 x^2\right )-3 e \left (-8 f^2 g x+16 f^3+6 f g^2 x^2-5 g^3 x^3\right )\right )\right )}{105 g^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[f + g*x]*(7*g*(5*b*d*g*(-2*f + g*x) + 5*a*g*(-2*e*f + 3*d*g + e*g*x) + b*e*(8*f^2 - 4*f*g*x + 3*g^2*x^
2)) + c*(7*d*g*(8*f^2 - 4*f*g*x + 3*g^2*x^2) - 3*e*(16*f^3 - 8*f^2*g*x + 6*f*g^2*x^2 - 5*g^3*x^3))))/(105*g^4)

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Maple [A]  time = 0.05, size = 144, normalized size = 1.1 \begin{align*}{\frac{30\,ce{x}^{3}{g}^{3}+42\,be{g}^{3}{x}^{2}+42\,cd{g}^{3}{x}^{2}-36\,cef{g}^{2}{x}^{2}+70\,ae{g}^{3}x+70\,bd{g}^{3}x-56\,bef{g}^{2}x-56\,cdf{g}^{2}x+48\,ce{f}^{2}gx+210\,ad{g}^{3}-140\,aef{g}^{2}-140\,bdf{g}^{2}+112\,be{f}^{2}g+112\,cd{f}^{2}g-96\,ce{f}^{3}}{105\,{g}^{4}}\sqrt{gx+f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(g*x+f)^(1/2)*(15*c*e*g^3*x^3+21*b*e*g^3*x^2+21*c*d*g^3*x^2-18*c*e*f*g^2*x^2+35*a*e*g^3*x+35*b*d*g^3*x-2
8*b*e*f*g^2*x-28*c*d*f*g^2*x+24*c*e*f^2*g*x+105*a*d*g^3-70*a*e*f*g^2-70*b*d*f*g^2+56*b*e*f^2*g+56*c*d*f^2*g-48
*c*e*f^3)/g^4

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Maxima [A]  time = 0.967548, size = 174, normalized size = 1.27 \begin{align*} \frac{2 \,{\left (15 \,{\left (g x + f\right )}^{\frac{7}{2}} c e - 21 \,{\left (3 \, c e f -{\left (c d + b e\right )} g\right )}{\left (g x + f\right )}^{\frac{5}{2}} + 35 \,{\left (3 \, c e f^{2} - 2 \,{\left (c d + b e\right )} f g +{\left (b d + a e\right )} g^{2}\right )}{\left (g x + f\right )}^{\frac{3}{2}} - 105 \,{\left (c e f^{3} - a d g^{3} -{\left (c d + b e\right )} f^{2} g +{\left (b d + a e\right )} f g^{2}\right )} \sqrt{g x + f}\right )}}{105 \, g^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*(g*x + f)^(7/2)*c*e - 21*(3*c*e*f - (c*d + b*e)*g)*(g*x + f)^(5/2) + 35*(3*c*e*f^2 - 2*(c*d + b*e)*f
*g + (b*d + a*e)*g^2)*(g*x + f)^(3/2) - 105*(c*e*f^3 - a*d*g^3 - (c*d + b*e)*f^2*g + (b*d + a*e)*f*g^2)*sqrt(g
*x + f))/g^4

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Fricas [A]  time = 1.7422, size = 297, normalized size = 2.17 \begin{align*} \frac{2 \,{\left (15 \, c e g^{3} x^{3} - 48 \, c e f^{3} + 105 \, a d g^{3} + 56 \,{\left (c d + b e\right )} f^{2} g - 70 \,{\left (b d + a e\right )} f g^{2} - 3 \,{\left (6 \, c e f g^{2} - 7 \,{\left (c d + b e\right )} g^{3}\right )} x^{2} +{\left (24 \, c e f^{2} g - 28 \,{\left (c d + b e\right )} f g^{2} + 35 \,{\left (b d + a e\right )} g^{3}\right )} x\right )} \sqrt{g x + f}}{105 \, g^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*c*e*g^3*x^3 - 48*c*e*f^3 + 105*a*d*g^3 + 56*(c*d + b*e)*f^2*g - 70*(b*d + a*e)*f*g^2 - 3*(6*c*e*f*g^
2 - 7*(c*d + b*e)*g^3)*x^2 + (24*c*e*f^2*g - 28*(c*d + b*e)*f*g^2 + 35*(b*d + a*e)*g^3)*x)*sqrt(g*x + f)/g^4

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Sympy [A]  time = 67.7122, size = 549, normalized size = 4.01 \begin{align*} \begin{cases} - \frac{\frac{2 a d f}{\sqrt{f + g x}} + 2 a d \left (- \frac{f}{\sqrt{f + g x}} - \sqrt{f + g x}\right ) + \frac{2 a e f \left (- \frac{f}{\sqrt{f + g x}} - \sqrt{f + g x}\right )}{g} + \frac{2 a e \left (\frac{f^{2}}{\sqrt{f + g x}} + 2 f \sqrt{f + g x} - \frac{\left (f + g x\right )^{\frac{3}{2}}}{3}\right )}{g} + \frac{2 b d f \left (- \frac{f}{\sqrt{f + g x}} - \sqrt{f + g x}\right )}{g} + \frac{2 b d \left (\frac{f^{2}}{\sqrt{f + g x}} + 2 f \sqrt{f + g x} - \frac{\left (f + g x\right )^{\frac{3}{2}}}{3}\right )}{g} + \frac{2 b e f \left (\frac{f^{2}}{\sqrt{f + g x}} + 2 f \sqrt{f + g x} - \frac{\left (f + g x\right )^{\frac{3}{2}}}{3}\right )}{g^{2}} + \frac{2 b e \left (- \frac{f^{3}}{\sqrt{f + g x}} - 3 f^{2} \sqrt{f + g x} + f \left (f + g x\right )^{\frac{3}{2}} - \frac{\left (f + g x\right )^{\frac{5}{2}}}{5}\right )}{g^{2}} + \frac{2 c d f \left (\frac{f^{2}}{\sqrt{f + g x}} + 2 f \sqrt{f + g x} - \frac{\left (f + g x\right )^{\frac{3}{2}}}{3}\right )}{g^{2}} + \frac{2 c d \left (- \frac{f^{3}}{\sqrt{f + g x}} - 3 f^{2} \sqrt{f + g x} + f \left (f + g x\right )^{\frac{3}{2}} - \frac{\left (f + g x\right )^{\frac{5}{2}}}{5}\right )}{g^{2}} + \frac{2 c e f \left (- \frac{f^{3}}{\sqrt{f + g x}} - 3 f^{2} \sqrt{f + g x} + f \left (f + g x\right )^{\frac{3}{2}} - \frac{\left (f + g x\right )^{\frac{5}{2}}}{5}\right )}{g^{3}} + \frac{2 c e \left (\frac{f^{4}}{\sqrt{f + g x}} + 4 f^{3} \sqrt{f + g x} - 2 f^{2} \left (f + g x\right )^{\frac{3}{2}} + \frac{4 f \left (f + g x\right )^{\frac{5}{2}}}{5} - \frac{\left (f + g x\right )^{\frac{7}{2}}}{7}\right )}{g^{3}}}{g} & \text{for}\: g \neq 0 \\\frac{a d x + \frac{c e x^{4}}{4} + \frac{x^{3} \left (b e + c d\right )}{3} + \frac{x^{2} \left (a e + b d\right )}{2}}{\sqrt{f}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(2*a*d*f/sqrt(f + g*x) + 2*a*d*(-f/sqrt(f + g*x) - sqrt(f + g*x)) + 2*a*e*f*(-f/sqrt(f + g*x) - sq
rt(f + g*x))/g + 2*a*e*(f**2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g + 2*b*d*f*(-f/sqrt(f +
g*x) - sqrt(f + g*x))/g + 2*b*d*(f**2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g + 2*b*e*f*(f**
2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 + 2*b*e*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f +
g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**2 + 2*c*d*f*(f**2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) - (f +
g*x)**(3/2)/3)/g**2 + 2*c*d*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2
)/5)/g**2 + 2*c*e*f*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**
3 + 2*c*e*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**2*(f + g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f +
g*x)**(7/2)/7)/g**3)/g, Ne(g, 0)), ((a*d*x + c*e*x**4/4 + x**3*(b*e + c*d)/3 + x**2*(a*e + b*d)/2)/sqrt(f), T
rue))

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Giac [A]  time = 1.13976, size = 269, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (105 \, \sqrt{g x + f} a d + \frac{35 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} - 3 \, \sqrt{g x + f} f\right )} b d}{g} + \frac{35 \,{\left ({\left (g x + f\right )}^{\frac{3}{2}} - 3 \, \sqrt{g x + f} f\right )} a e}{g} + \frac{7 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f + 15 \, \sqrt{g x + f} f^{2}\right )} c d}{g^{2}} + \frac{7 \,{\left (3 \,{\left (g x + f\right )}^{\frac{5}{2}} - 10 \,{\left (g x + f\right )}^{\frac{3}{2}} f + 15 \, \sqrt{g x + f} f^{2}\right )} b e}{g^{2}} + \frac{3 \,{\left (5 \,{\left (g x + f\right )}^{\frac{7}{2}} - 21 \,{\left (g x + f\right )}^{\frac{5}{2}} f + 35 \,{\left (g x + f\right )}^{\frac{3}{2}} f^{2} - 35 \, \sqrt{g x + f} f^{3}\right )} c e}{g^{3}}\right )}}{105 \, g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(105*sqrt(g*x + f)*a*d + 35*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*b*d/g + 35*((g*x + f)^(3/2) - 3*sqrt(g
*x + f)*f)*a*e/g + 7*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*c*d/g^2 + 7*(3*(g*x + f
)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*b*e/g^2 + 3*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f +
35*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*c*e/g^3)/g