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

Optimal. Leaf size=113 \[ -\frac{2 \sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)}{g^4}+\frac{2 (f+g x)^{3/2} \left (a e g^2+c f (3 e f-2 d g)\right )}{3 g^4}-\frac{2 c (f+g x)^{5/2} (3 e f-d g)}{5 g^4}+\frac{2 c e (f+g x)^{7/2}}{7 g^4} \]

[Out]

(-2*(e*f - d*g)*(c*f^2 + a*g^2)*Sqrt[f + g*x])/g^4 + (2*(a*e*g^2 + c*f*(3*e*f - 2*d*g))*(f + g*x)^(3/2))/(3*g^
4) - (2*c*(3*e*f - d*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 = 0.0742448, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ -\frac{2 \sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)}{g^4}+\frac{2 (f+g x)^{3/2} \left (a e g^2+c f (3 e f-2 d g)\right )}{3 g^4}-\frac{2 c (f+g x)^{5/2} (3 e f-d g)}{5 g^4}+\frac{2 c e (f+g x)^{7/2}}{7 g^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{(d+e x) \left (a+c x^2\right )}{\sqrt{f+g x}} \, dx &=\int \left (\frac{(-e f+d g) \left (c f^2+a g^2\right )}{g^3 \sqrt{f+g x}}+\frac{\left (a e g^2+c f (3 e f-2 d g)\right ) \sqrt{f+g x}}{g^3}+\frac{c (-3 e f+d 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+a g^2\right ) \sqrt{f+g x}}{g^4}+\frac{2 \left (a e g^2+c f (3 e f-2 d g)\right ) (f+g x)^{3/2}}{3 g^4}-\frac{2 c (3 e f-d 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 = 0.0912615, size = 94, normalized size = 0.83 \[ \frac{2 \sqrt{f+g x} \left (35 a g^2 (3 d g-2 e f+e g x)+7 c d g \left (8 f^2-4 f g x+3 g^2 x^2\right )-3 c e \left (-8 f^2 g x+16 f^3+6 f g^2 x^2-5 g^3 x^3\right )\right )}{105 g^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[f + g*x]*(35*a*g^2*(-2*e*f + 3*d*g + e*g*x) + 7*c*d*g*(8*f^2 - 4*f*g*x + 3*g^2*x^2) - 3*c*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 = 0.046, size = 101, normalized size = 0.9 \begin{align*}{\frac{30\,ce{x}^{3}{g}^{3}+42\,cd{g}^{3}{x}^{2}-36\,cef{g}^{2}{x}^{2}+70\,ae{g}^{3}x-56\,cdf{g}^{2}x+48\,ce{f}^{2}gx+210\,ad{g}^{3}-140\,aef{g}^{2}+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+a)/(g*x+f)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.00632, size = 140, normalized size = 1.24 \begin{align*} \frac{2 \,{\left (15 \,{\left (g x + f\right )}^{\frac{7}{2}} c e - 21 \,{\left (3 \, c e f - c d g\right )}{\left (g x + f\right )}^{\frac{5}{2}} + 35 \,{\left (3 \, c e f^{2} - 2 \, c d f g + a e g^{2}\right )}{\left (g x + f\right )}^{\frac{3}{2}} - 105 \,{\left (c e f^{3} - c d f^{2} g + a e f g^{2} - a d g^{3}\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+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*g)*(g*x + f)^(5/2) + 35*(3*c*e*f^2 - 2*c*d*f*g + a*e*g^2)*(g
*x + f)^(3/2) - 105*(c*e*f^3 - c*d*f^2*g + a*e*f*g^2 - a*d*g^3)*sqrt(g*x + f))/g^4

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Fricas [A]  time = 1.65952, size = 243, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (15 \, c e g^{3} x^{3} - 48 \, c e f^{3} + 56 \, c d f^{2} g - 70 \, a e f g^{2} + 105 \, a d g^{3} - 3 \,{\left (6 \, c e f g^{2} - 7 \, c d g^{3}\right )} x^{2} +{\left (24 \, c e f^{2} g - 28 \, c d f g^{2} + 35 \, a e 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+a)/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 35.1896, size = 374, normalized size = 3.31 \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 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{a e x^{2}}{2} + \frac{c d x^{3}}{3} + \frac{c e x^{4}}{4}}{\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+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*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 + a*e*x**2/2 + c*d*x**3/3 + c*e*
x**4/4)/sqrt(f), True))

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Giac [A]  time = 1.1359, size = 181, normalized size = 1.6 \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 )} 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{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+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)*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 + 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