3.598 \(\int \frac{(d+e x) (a+c x^2)}{(f+g x)^{3/2}} \, dx\)

Optimal. Leaf size=111 \[ \frac{2 \left (a g^2+c f^2\right ) (e f-d g)}{g^4 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} \left (a e g^2+c f (3 e f-2 d g)\right )}{g^4}-\frac{2 c (f+g x)^{3/2} (3 e f-d g)}{3 g^4}+\frac{2 c e (f+g x)^{5/2}}{5 g^4} \]

[Out]

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

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Rubi [A]  time = 0.0661105, antiderivative size = 111, 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} \[ \frac{2 \left (a g^2+c f^2\right ) (e f-d g)}{g^4 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} \left (a e g^2+c f (3 e f-2 d g)\right )}{g^4}-\frac{2 c (f+g x)^{3/2} (3 e f-d g)}{3 g^4}+\frac{2 c e (f+g x)^{5/2}}{5 g^4} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(a + c*x^2))/(f + g*x)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0860907, size = 92, normalized size = 0.83 \[ \frac{30 a g^2 (-d g+2 e f+e g x)+10 c d g \left (-8 f^2-4 f g x+g^2 x^2\right )+6 c e \left (8 f^2 g x+16 f^3-2 f g^2 x^2+g^3 x^3\right )}{15 g^4 \sqrt{f+g x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(a + c*x^2))/(f + g*x)^(3/2),x]

[Out]

(30*a*g^2*(2*e*f - d*g + e*g*x) + 10*c*d*g*(-8*f^2 - 4*f*g*x + g^2*x^2) + 6*c*e*(16*f^3 + 8*f^2*g*x - 2*f*g^2*
x^2 + g^3*x^3))/(15*g^4*Sqrt[f + g*x])

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Maple [A]  time = 0.046, size = 101, normalized size = 0.9 \begin{align*} -{\frac{-6\,ce{x}^{3}{g}^{3}-10\,cd{g}^{3}{x}^{2}+12\,cef{g}^{2}{x}^{2}-30\,ae{g}^{3}x+40\,cdf{g}^{2}x-48\,ce{f}^{2}gx+30\,ad{g}^{3}-60\,aef{g}^{2}+80\,cd{f}^{2}g-96\,ce{f}^{3}}{15\,{g}^{4}}{\frac{1}{\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)^(3/2),x)

[Out]

-2/15/(g*x+f)^(1/2)*(-3*c*e*g^3*x^3-5*c*d*g^3*x^2+6*c*e*f*g^2*x^2-15*a*e*g^3*x+20*c*d*f*g^2*x-24*c*e*f^2*g*x+1
5*a*d*g^3-30*a*e*f*g^2+40*c*d*f^2*g-48*c*e*f^3)/g^4

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Maxima [A]  time = 0.998517, size = 151, normalized size = 1.36 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (g x + f\right )}^{\frac{5}{2}} c e - 5 \,{\left (3 \, c e f - c d g\right )}{\left (g x + f\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, c e f^{2} - 2 \, c d f g + a e g^{2}\right )} \sqrt{g x + f}}{g^{3}} + \frac{15 \,{\left (c e f^{3} - c d f^{2} g + a e f g^{2} - a d g^{3}\right )}}{\sqrt{g x + f} g^{3}}\right )}}{15 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/15*((3*(g*x + f)^(5/2)*c*e - 5*(3*c*e*f - c*d*g)*(g*x + f)^(3/2) + 15*(3*c*e*f^2 - 2*c*d*f*g + a*e*g^2)*sqrt
(g*x + f))/g^3 + 15*(c*e*f^3 - c*d*f^2*g + a*e*f*g^2 - a*d*g^3)/(sqrt(g*x + f)*g^3))/g

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Fricas [A]  time = 1.72792, size = 252, normalized size = 2.27 \begin{align*} \frac{2 \,{\left (3 \, c e g^{3} x^{3} + 48 \, c e f^{3} - 40 \, c d f^{2} g + 30 \, a e f g^{2} - 15 \, a d g^{3} -{\left (6 \, c e f g^{2} - 5 \, c d g^{3}\right )} x^{2} +{\left (24 \, c e f^{2} g - 20 \, c d f g^{2} + 15 \, a e g^{3}\right )} x\right )} \sqrt{g x + f}}{15 \,{\left (g^{5} x + f g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x^2+a)/(g*x+f)^(3/2),x, algorithm="fricas")

[Out]

2/15*(3*c*e*g^3*x^3 + 48*c*e*f^3 - 40*c*d*f^2*g + 30*a*e*f*g^2 - 15*a*d*g^3 - (6*c*e*f*g^2 - 5*c*d*g^3)*x^2 +
(24*c*e*f^2*g - 20*c*d*f*g^2 + 15*a*e*g^3)*x)*sqrt(g*x + f)/(g^5*x + f*g^4)

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Sympy [A]  time = 14.7237, size = 112, normalized size = 1.01 \begin{align*} \frac{2 c e \left (f + g x\right )^{\frac{5}{2}}}{5 g^{4}} + \frac{\left (f + g x\right )^{\frac{3}{2}} \left (2 c d g - 6 c e f\right )}{3 g^{4}} + \frac{\sqrt{f + g x} \left (2 a e g^{2} - 4 c d f g + 6 c e f^{2}\right )}{g^{4}} - \frac{2 \left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )}{g^{4} \sqrt{f + g x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x**2+a)/(g*x+f)**(3/2),x)

[Out]

2*c*e*(f + g*x)**(5/2)/(5*g**4) + (f + g*x)**(3/2)*(2*c*d*g - 6*c*e*f)/(3*g**4) + sqrt(f + g*x)*(2*a*e*g**2 -
4*c*d*f*g + 6*c*e*f**2)/g**4 - 2*(a*g**2 + c*f**2)*(d*g - e*f)/(g**4*sqrt(f + g*x))

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Giac [A]  time = 1.12359, size = 193, normalized size = 1.74 \begin{align*} -\frac{2 \,{\left (c d f^{2} g + a d g^{3} - c f^{3} e - a f g^{2} e\right )}}{\sqrt{g x + f} g^{4}} + \frac{2 \,{\left (5 \,{\left (g x + f\right )}^{\frac{3}{2}} c d g^{17} - 30 \, \sqrt{g x + f} c d f g^{17} + 3 \,{\left (g x + f\right )}^{\frac{5}{2}} c g^{16} e - 15 \,{\left (g x + f\right )}^{\frac{3}{2}} c f g^{16} e + 45 \, \sqrt{g x + f} c f^{2} g^{16} e + 15 \, \sqrt{g x + f} a g^{18} e\right )}}{15 \, g^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(c*x^2+a)/(g*x+f)^(3/2),x, algorithm="giac")

[Out]

-2*(c*d*f^2*g + a*d*g^3 - c*f^3*e - a*f*g^2*e)/(sqrt(g*x + f)*g^4) + 2/15*(5*(g*x + f)^(3/2)*c*d*g^17 - 30*sqr
t(g*x + f)*c*d*f*g^17 + 3*(g*x + f)^(5/2)*c*g^16*e - 15*(g*x + f)^(3/2)*c*f*g^16*e + 45*sqrt(g*x + f)*c*f^2*g^
16*e + 15*sqrt(g*x + f)*a*g^18*e)/g^20